patsy  Describing statistical models in Python¶
Contents:
Overview¶
patsy
is a Python package for describing statistical models
(especially linear models, or models that have a linear component)
and building design matrices. It is closely inspired by and compatible
with the formula minilanguage used in R and S.
For instance, if we have some variable y, and we want to regress it against some other variables x, a, b, and the interaction of a and b, then we simply write:
patsy.dmatrices("y ~ x + a + b + a:b", data)
and Patsy takes care of building appropriate matrices. Furthermore, it:
 Allows data transformations to be specified using arbitrary Python
code: instead of
x
, we could have writtenlog(x)
,(x > 0)
, or evenlog(x) if x > 1e5 else log(1e5)
,  Provides a range of convenient options for coding categorical variables, including automatic detection and removal of redundancies,
 Knows how to apply ‘the same’ transformation used on original data to new data, even for tricky transformations like centering or standardization (critical if you want to use your model to make predictions),
 Has an incremental mode to handle data sets which are too large to fit into memory at one time,
 Provides a language for symbolic, humanreadable specification of linear constraint matrices,
 Has a thorough test suite (>97% statement coverage) and solid underlying theory, allowing it to correctly handle corner cases that even R gets wrong, and
 Features a simple API for integration into statistical packages.
What Patsy won’t do is, well, statistics — it just lets you describe models in general terms. It doesn’t know or care whether you ultimately want to do linear regression, timeseries analysis, or fit a forest of decision trees, and it certainly won’t do any of those things for you — it just gives a highlevel language for describing which factors you want your underlying model to take into account. It’s not suitable for implementing arbitrary nonlinear models from scratch; for that, you’ll be better off with something like Theano, SymPy, or just plain Python. But if you’re using a statistical package that requires you to provide a raw model matrix, then you can use Patsy to painlessly construct that model matrix; and if you’re the author of a statistics package, then I hope you’ll consider integrating Patsy as part of your frontend.
Patsy’s goal is to become the standard highlevel interface to describing statistical models in Python, regardless of what particular model or library is being used underneath.
Download¶
The current release may be downloaded from the Python Package index at
Or the latest development version may be found in our Git repository:
git clone git://github.com/pydata/patsy.git
Installation¶
If you have pip
installed, then a simple
pip install upgrade patsy
should get you the latest version. Otherwise, download and unpack the source distribution, and then run
python setup.py install
Contact¶
Post your suggestions and questions directly to the pydata mailing list (pydata@googlegroups.com, gmane archive), or to our bug tracker. You could also contact Nathaniel J. Smith directly, but really the mailing list is almost always a better bet, because more people will see your query and others will be able to benefit from any answers you get.
License¶
2clause BSD. See the file LICENSE.txt for details.
Users¶
We currently know of the following projects using Patsy to provide a highlevel interface to their statistical code:
If you’d like your project to appear here, see our documentation for library developers!
Quickstart¶
If you prefer to learn by diving in and getting your feet wet, then here are some cutandpasteable examples to play with.
First, let’s import stuff and get some data to work with:
In [1]: import numpy as np
In [2]: from patsy import dmatrices, dmatrix, demo_data
In [3]: data = demo_data("a", "b", "x1", "x2", "y", "z column")
demo_data()
gives us a mix of categorical and numerical
variables:
In [4]: data
Out[4]:
{'a': ['a1', 'a1', 'a2', 'a2', 'a1', 'a1', 'a2', 'a2'],
'b': ['b1', 'b2', 'b1', 'b2', 'b1', 'b2', 'b1', 'b2'],
'x1': array([ 1.76405235, 0.40015721, 0.97873798, 2.2408932 , 1.86755799,
0.97727788, 0.95008842, 0.15135721]),
'x2': array([0.10321885, 0.4105985 , 0.14404357, 1.45427351, 0.76103773,
0.12167502, 0.44386323, 0.33367433]),
'y': array([ 1.49407907, 0.20515826, 0.3130677 , 0.85409574, 2.55298982,
0.6536186 , 0.8644362 , 0.74216502]),
'z column': array([ 2.26975462, 1.45436567, 0.04575852, 0.18718385, 1.53277921,
1.46935877, 0.15494743, 0.37816252])}
Of course Patsy doesn’t much care what sort of object you store
your data in, so long as it can be indexed like a Python dictionary,
data[varname]
. You may prefer to store your data in a pandas DataFrame, or a numpy
record array... whatever makes you happy.
Now, let’s generate design matrices suitable for regressing y
onto
x1
and x2
.
In [5]: dmatrices("y ~ x1 + x2", data)
Out[5]:
(DesignMatrix with shape (8, 1)
y
1.49408
0.20516
0.31307
0.85410
2.55299
0.65362
0.86444
0.74217
Terms:
'y' (column 0),
DesignMatrix with shape (8, 3)
Intercept x1 x2
1 1.76405 0.10322
1 0.40016 0.41060
1 0.97874 0.14404
1 2.24089 1.45427
1 1.86756 0.76104
1 0.97728 0.12168
1 0.95009 0.44386
1 0.15136 0.33367
Terms:
'Intercept' (column 0)
'x1' (column 1)
'x2' (column 2))
The return value is a Python tuple containing two DesignMatrix
objects, the first representing the lefthand side of our formula, and
the second representing the righthand side. Notice that an intercept
term was automatically added to the righthand side. These are just
ordinary numpy arrays with some extra metadata and a fancy __repr__
method attached, so we can pass them directly to a regression function
like np.linalg.lstsq()
:
In [6]: outcome, predictors = dmatrices("y ~ x1 + x2", data)
In [7]: betas = np.linalg.lstsq(predictors, outcome)[0].ravel()
In [8]: for name, beta in zip(predictors.design_info.column_names, betas):
...: print("%s: %s" % (name, beta))
...:
Intercept: 0.579662344123
x1: 0.0885991903554
x2: 1.76479205551
Of course the resulting numbers aren’t very interesting, since this is just random data.
If you just want the design matrix alone, without the y
values,
use dmatrix()
and leave off the y ~
part at the beginning:
In [9]: dmatrix("x1 + x2", data)
Out[9]:
DesignMatrix with shape (8, 3)
Intercept x1 x2
1 1.76405 0.10322
1 0.40016 0.41060
1 0.97874 0.14404
1 2.24089 1.45427
1 1.86756 0.76104
1 0.97728 0.12168
1 0.95009 0.44386
1 0.15136 0.33367
Terms:
'Intercept' (column 0)
'x1' (column 1)
'x2' (column 2)
We’ll use dmatrix for the rest of the examples, since seeing the
outcome matrix over and over would get boring. This matrix’s metadata
is stored in an extra attribute called .design_info
, which is a
DesignInfo
object you can explore at your leisure:
In [10]: d = dmatrix("x1 + x2", data)
In [11]: d.design_info.<TAB>
d.design_info.builder d.design_info.slice
d.design_info.column_name_indexes d.design_info.term_name_slices
d.design_info.column_names d.design_info.term_names
d.design_info.describe d.design_info.term_slices
d.design_info.linear_constraint d.design_info.terms
Usually the intercept is useful, but if we don’t want it we can get rid of it:
In [12]: dmatrix("x1 + x2  1", data)
Out[12]:
DesignMatrix with shape (8, 2)
x1 x2
1.76405 0.10322
0.40016 0.41060
0.97874 0.14404
2.24089 1.45427
1.86756 0.76104
0.97728 0.12168
0.95009 0.44386
0.15136 0.33367
Terms:
'x1' (column 0)
'x2' (column 1)
We can transform variables using arbitrary Python code:
In [13]: dmatrix("x1 + np.log(x2 + 10)", data)
Out[13]:
DesignMatrix with shape (8, 3)
Intercept x1 np.log(x2 + 10)
1 1.76405 2.29221
1 0.40016 2.34282
1 0.97874 2.31689
1 2.24089 2.43836
1 1.86756 2.37593
1 0.97728 2.31468
1 0.95009 2.34601
1 0.15136 2.33541
Terms:
'Intercept' (column 0)
'x1' (column 1)
'np.log(x2 + 10)' (column 2)
Notice that np.log
is being pulled out of the environment where
dmatrix()
was called – np.log
is accessible because we did
import numpy as np
up above. Any functions or variables that you
could reference when calling dmatrix()
can also be used inside
the formula passed to dmatrix()
. For example:
In [14]: new_x2 = data["x2"] * 100
In [15]: dmatrix("new_x2")
Out[15]:
DesignMatrix with shape (8, 2)
Intercept new_x2
1 10.32189
1 41.05985
1 14.40436
1 145.42735
1 76.10377
1 12.16750
1 44.38632
1 33.36743
Terms:
'Intercept' (column 0)
'new_x2' (column 1)
Patsy has some transformation functions “built in”, that are automatically accessible to your code:
In [16]: dmatrix("center(x1) + standardize(x2)", data)
Out[16]:
DesignMatrix with shape (8, 3)
Intercept center(x1) standardize(x2)
1 0.87995 1.21701
1 0.48395 0.07791
1 0.09463 0.66885
1 1.35679 2.23584
1 0.98345 0.69899
1 1.86138 0.71844
1 0.06598 0.00417
1 1.03546 0.24845
Terms:
'Intercept' (column 0)
'center(x1)' (column 1)
'standardize(x2)' (column 2)
See patsy.builtins
for a complete list of functions made
available to formulas. You can also define your own transformation
functions in the ordinary Python way:
In [17]: def double(x):
....: return 2 * x
....:
In [18]: dmatrix("x1 + double(x1)", data)
Out[18]:
DesignMatrix with shape (8, 3)
Intercept x1 double(x1)
1 1.76405 3.52810
1 0.40016 0.80031
1 0.97874 1.95748
1 2.24089 4.48179
1 1.86756 3.73512
1 0.97728 1.95456
1 0.95009 1.90018
1 0.15136 0.30271
Terms:
'Intercept' (column 0)
'x1' (column 1)
'double(x1)' (column 2)
This flexibility does create problems in one case, though – because
we interpret whatever you write inbetween the +
signs as Python
code, you do in fact have to write valid Python code. And this can be
tricky if your variable names have funny characters in them, like
whitespace or punctuation. Fortunately, patsy has a builtin
“transformation” called Q()
that lets you “quote” such
variables:
In [19]: weird_data = demo_data("weird column!", "x1")
# This is an error...
In [20]: dmatrix("weird column! + x1", weird_data)
[...]
PatsyError: error tokenizing input (maybe an unclosed string?)
weird column! + x1
^
# ...but this works:
In [21]: dmatrix("Q('weird column!') + x1", weird_data)
Out[21]:
DesignMatrix with shape (5, 3)
Intercept Q('weird column!') x1
1 1.76405 0.97728
1 0.40016 0.95009
1 0.97874 0.15136
1 2.24089 0.10322
1 1.86756 0.41060
Terms:
'Intercept' (column 0)
"Q('weird column!')" (column 1)
'x1' (column 2)
Q()
even plays well with other transformations:
In [22]: dmatrix("double(Q('weird column!')) + x1", weird_data)
Out[22]:
DesignMatrix with shape (5, 3)
Intercept double(Q('weird column!')) x1
1 3.52810 0.97728
1 0.80031 0.95009
1 1.95748 0.15136
1 4.48179 0.10322
1 3.73512 0.41060
Terms:
'Intercept' (column 0)
"double(Q('weird column!'))" (column 1)
'x1' (column 2)
Arithmetic transformations are also possible, but you’ll need to
“protect” them by wrapping them in I()
, so that Patsy knows
that you really do want +
to mean addition:
In [23]: dmatrix("I(x1 + x2)", data) # compare to "x1 + x2"
Out[23]:
DesignMatrix with shape (8, 2)
Intercept I(x1 + x2)
1 1.66083
1 0.81076
1 1.12278
1 3.69517
1 2.62860
1 0.85560
1 1.39395
1 0.18232
Terms:
'Intercept' (column 0)
'I(x1 + x2)' (column 1)
Note that while Patsy goes to considerable efforts to take in data represented using different Python data types and convert them into a standard representation, all this work happens after any transformations you perform as part of your formula. So, for example, if your data is in the form of numpy arrays, “+” will perform elementwise addition, but if it is in standard Python lists, it will perform concatentation:
In [24]: dmatrix("I(x1 + x2)", {"x1": np.array([1, 2, 3]), "x2": np.array([4, 5, 6])})
Out[24]:
DesignMatrix with shape (3, 2)
Intercept I(x1 + x2)
1 5
1 7
1 9
Terms:
'Intercept' (column 0)
'I(x1 + x2)' (column 1)
In [25]: dmatrix("I(x1 + x2)", {"x1": [1, 2, 3], "x2": [4, 5, 6]})
Out[25]:
DesignMatrix with shape (6, 2)
Intercept I(x1 + x2)
1 1
1 2
1 3
1 4
1 5
1 6
Terms:
'Intercept' (column 0)
'I(x1 + x2)' (column 1)
Patsy becomes particularly useful when you have categorical data. If you use a predictor that has a categorical type (e.g. strings or bools), it will be automatically coded. Patsy automatically chooses an appropriate way to code categorical data to avoid producing a redundant, overdetermined model.
If there is just one categorical variable alone, the default is to dummy code it:
In [26]: dmatrix("0 + a", data)
Out[26]:
DesignMatrix with shape (8, 2)
a[a1] a[a2]
1 0
1 0
0 1
0 1
1 0
1 0
0 1
0 1
Terms:
'a' (columns 0:2)
But if you did that and put the intercept back in, you’d get a redundant model. So if the intercept is present, Patsy uses a reducedrank contrast code (treatment coding by default):
In [27]: dmatrix("a", data)
Out[27]:
DesignMatrix with shape (8, 2)
Intercept a[T.a2]
1 0
1 0
1 1
1 1
1 0
1 0
1 1
1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
The T.
notation is there to remind you that these columns are
treatment coded.
Interactions are also easy – they represent the cartesian product of
all the factors involved. Here’s a dummy coding of each combination
of values taken by a
and b
:
In [28]: dmatrix("0 + a:b", data)
Out[28]:
DesignMatrix with shape (8, 4)
a[a1]:b[b1] a[a2]:b[b1] a[a1]:b[b2] a[a2]:b[b2]
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
Terms:
'a:b' (columns 0:4)
But interactions also know how to use contrast coding to avoid redundancy. If you have both main effects and interactions in a model, then Patsy goes from lowerorder effects to higherorder effects, adding in just enough columns to produce a welldefined model. The result is that each set of columns measures the additional contribution of this effect – just what you want for a traditional ANOVA:
In [29]: dmatrix("a + b + a:b", data)
Out[29]:
DesignMatrix with shape (8, 4)
Intercept a[T.a2] b[T.b2] a[T.a2]:b[T.b2]
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
'b' (column 2)
'a:b' (column 3)
Since this is so common, there’s a convenient shorthand:
In [30]: dmatrix("a*b", data)
Out[30]:
DesignMatrix with shape (8, 4)
Intercept a[T.a2] b[T.b2] a[T.a2]:b[T.b2]
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
'b' (column 2)
'a:b' (column 3)
Of course you can use other coding schemes too (or even define your own). Here’s orthogonal polynomial coding
:
In [31]: dmatrix("C(c, Poly)", {"c": ["c1", "c1", "c2", "c2", "c3", "c3"]})
Out[31]:
DesignMatrix with shape (6, 3)
Intercept C(c, Poly).Linear C(c, Poly).Quadratic
1 0.70711 0.40825
1 0.70711 0.40825
1 0.00000 0.81650
1 0.00000 0.81650
1 0.70711 0.40825
1 0.70711 0.40825
Terms:
'Intercept' (column 0)
'C(c, Poly)' (columns 1:3)
You can even write interactions between categorical and numerical
variables. Here we fit two different slope coefficients for x1
;
one for the a1
group, and one for the a2
group:
In [32]: dmatrix("a:x1", data)
Out[32]:
DesignMatrix with shape (8, 3)
Intercept a[a1]:x1 a[a2]:x1
1 1.76405 0.00000
1 0.40016 0.00000
1 0.00000 0.97874
1 0.00000 2.24089
1 1.86756 0.00000
1 0.97728 0.00000
1 0.00000 0.95009
1 0.00000 0.15136
Terms:
'Intercept' (column 0)
'a:x1' (columns 1:3)
The same redundancy avoidance code works here, so if you’d rather have
treatmentcoded slopes (one slope for the a1
group, and a second
for the difference between the a1
and a2
group slopes), then
you can request it like this:
# compare to the difference between "0 + a" and "1 + a"
In [33]: dmatrix("x1 + a:x1", data)
Out[33]:
DesignMatrix with shape (8, 3)
Intercept x1 a[T.a2]:x1
1 1.76405 0.00000
1 0.40016 0.00000
1 0.97874 0.97874
1 2.24089 2.24089
1 1.86756 0.00000
1 0.97728 0.00000
1 0.95009 0.95009
1 0.15136 0.15136
Terms:
'Intercept' (column 0)
'x1' (column 1)
'a:x1' (column 2)
And more complex expressions work too:
In [34]: dmatrix("C(a, Poly):center(x1)", data)
Out[34]:
DesignMatrix with shape (8, 3)
Intercept C(a, Poly).Constant:center(x1) C(a, Poly).Linear:center(x1)
1 0.87995 0.62222
1 0.48395 0.34220
1 0.09463 0.06691
1 1.35679 0.95939
1 0.98345 0.69541
1 1.86138 1.31620
1 0.06598 0.04666
1 1.03546 0.73218
Terms:
'Intercept' (column 0)
'C(a, Poly):center(x1)' (columns 1:3)
How formulas work¶
Now we’ll describe the fully nittygritty of how formulas are parsed and interpreted. Here’s the picture you’ll want to keep in mind:
Say we have a formula like:
y ~ a + a:b + np.log(x)
This overall thing is a formula, and it’s divided into a lefthand
side, y
, and a righthand side, a + a:b +
np.log(x)
. (Sometimes you want a formula that has no lefthand
side, and you can write that as ~ x1 + x2
or even x1 + x2
.)
Each side contains a list of terms separated by +
; on the left
there is one term, y
, and on the right, there are four terms:
a
and a:b
and np.log(x)
, plus an invisible intercept
term. And finally, each term is the interaction of zero or more
factors. A factor is the minimal, indivisible unit that each
formula is built up out of; the factors here are y
, a
, b
,
and np.log(x)
. Most of these terms have only one factor – for
example, the term y
is a kind of trivial interaction between the
factor y
and, well... and nothing. There’s only one factor in that
“interaction”. The term a:b
is an interaction between two factors,
a
and b
. And the intercept term is an interaction between
zero factors. (This may seem odd, but it turns out that defining the
zeroorder interaction to produce a column of all ones is very
convenient, just like it turns out to be convenient to define the
product of an empty list to be np.prod([]) ==
1
.)
Note
In the context of Patsy, the word factor does not refer specifically to categorical data. What we call a “factor” can represent either categorical or numerical data. Think of factors like in multiplying factors together, not like in factorial design. When we want to refer to categorical data, this manual and the Patsy API use the word “categorical”.
To make this more concrete, here’s how you could manually construct the same objects that Patsy will construct if given the above formula:
from patsy import ModelDesc
ModelDesc([Term([EvalFactor("y")])],
[Term([]),
Term([EvalFactor("a")]),
Term([EvalFactor("a"), EvalFactor("b")]),
Term([EvalFactor("np.log(x)")])])
Compare to what you get from parsing the above formula:
ModelDesc.from_formula("y ~ a + a:b + np.log(x)")
ModelDesc
represents an overall formula; it just takes two
lists of Term
objects, representing the lefthand side and
the righthand side. And each Term
object just takes a list of
factor objects. In this case our factors are of type
EvalFactor
, which evaluates arbitrary Python code, but in
general any object that implements the factor protocol will do – for
details see Model specification for experts and computers.
Of course as a user you never have to actually touch
ModelDesc
, Term
, or EvalFactor
objects by
hand – but it’s useful to know that this lower layer exists in case
you ever want to generate a formula programmatically, and to have an
image in your mind of what a formula really is.
The formula language¶
Now let’s talk about exactly how those magic formula strings are processed.
Since a term is nothing but a set of factors, and a model is nothing
but two sets of terms, you can write any Patsy model just using
:
to create interactions, +
to join terms together into a set,
and ~
to separate the lefthand side from the righthand side.
But for convenience, Patsy also understands a number of other
shorthand operators, and evaluates them all using a fullfledged
parser
complete with robust error reporting, etc.
Operators¶
The builtin binary operators, ordered by precedence, are:
~ 
lowest precedence (binds most loosely) 
+ ,  

* , / 

: 

** 
highest precedence (binds most tightly) 
Of course, you can override the order of operations using
parentheses. All operations are leftassociative (so a  b  c
means
the same as (a  b)  c
, not a  (b  c)
). Their meanings are as
follows:
~
 Separates the lefthand side and righthand side of a formula. Optional. If not present, then the formula is considered to contain a righthand side only.
+
 Takes the set of terms given on the left and the set of terms given
on the right, and returns a set of terms that combines both (i.e.,
it computes a set union). Note that this means that
a + a
is justa
. 
 Takes the set of terms given on the left and removes any terms which are given on the right (i.e., it computes a set difference).
*
a * b
is shorthand fora + b + a:b
, and is useful for the common case of wanting to include all interactions between a set of variables while partitioning their variance between lower and higherorder interactions. Standard ANOVA models are of the forma * b * c * ...
./
 This one is a bit quirky.
a / b
is shorthand fora + a:b
, and is intended to be useful in cases where you want to fit a standard sort of ANOVA model, butb
is nested withina
, soa*b
doesn’t make sense. So far so good. Also, if you have multiple terms on the right, then the obvious thing happens:a / (b + c)
is equivalent toa + a:b + a:c
(/
is rightward distributive over+
). But, if you have multiple terms on the left, then there is a surprising special case:(a + b)/c
is equivalent toa + b + a:b:c
(and note that this is different from what you’d get out ofa/c + b/c
–/
is not leftward distributive over+
). Again, this is motivated by the idea of using this for nested variables. It doesn’t make sense forc
to be nested within botha
andb
separately, unlessb
is itself nested ina
– but if that were true, then you’d writea/b/c
instead. So if we see(a + b)/c
, we decide thata
andb
must be independent factors, but thatc
is nested within each combination of levels ofa
andb
, which is whata:b:c
gives us. If this is confusing, then my apologies... S has been working this way for >20 years, so it’s a bit late to change it now. :
 This takes two sets of terms, and computes the interaction between
each term on the left and each term on the right. So, for example,
(a + b):(c + d)
is the same asa:c + a:d + b:c + b:d
. Calculating the interaction between two terms is also a kind of set union operation, but:
takes the union of factors within two terms, while+
takes the union of two sets of terms. Note that this means thata:a
is justa
, and(a:b):(a:c)
is the same asa:b:c
. **
This takes a set of terms on the left, and an integer n on the right, and computes the
*
of that set of terms with itself n times. This is useful if you want to compute all interactions up to order n, but no further. Example:(a + b + c + d) ** 3
is expanded to:
(a + b + c + d) * (a + b + c + d) * (a + b + c + d)
Note that an equivalent way to write this particular expression would be:
a*b*c*d  a:b:c:d
(Exercise: why?)
The parser also understands unary +
and 
, though they aren’t very
useful. +
is a noop, and 
can only be used in the forms 1
(which means the same as 0
) and 0
(which means the same as
1
). See below for more on 0
and
1
.
Factors and terms¶
So that explains how the operators work – the verbs in the formula
language – but what about the nouns, the terms like y
and
np.log(x)
that are actually picking out bits of your data?
Individual factors are allowed to be arbitrary Python code. Scanning
arbitrary Python code can be quite complicated, but Patsy uses the
official Python tokenizer that’s built into the standard library, so
it’s able to do it robustly. There is still a bit of a problem,
though, since Patsy operators like +
are also valid Python
operators. When we see a +
, how do we know which interpretation to
use?
The answer is that a Python factor begins whenever we see a token which
 is not a Patsy operator listed in that table up above, and
 is not a parentheses
And then the factor ends whenever we see a token which
 is a Patsy operator listed in that table up above, and
 it not enclosed in any kind of parentheses (where “any kind” includes regular, square, and curly bracket varieties)
This will be clearer with an example:
f(x1 + x2) + x3
First, we see f
, which is not an operator or a parentheses, so we
know this string begins with a Pythondefined factor. Then we keep
reading from there. The next Patsy operator we see is the +
in
x1 + x2
... but since at this point we have seen the opening (
but not the closing )
, we know that we’re inside parentheses and
ignore it. Eventually we come to the second +
, and by this time we
have seen the closing parentheses, so we know that this is the end of
the first factor and we interpret the +
as a Patsy operator.
One sideeffect of this is that if you do want to perform some
arithmetic inside your formula object, you can hide it from the
Patsy parser by putting it inside a function call. To make this
more convenient, Patsy provides a builtin function I()
that simply returns its input. (Hence the name: it’s the Identity
function.) This means you can use I(x1 + x2)
inside a formula to
represent the sum of x1
and x2
.
Note
The above plays a bit fastandloose with the distinction
between factors and terms. If you want to get more technical, then
given something like a:b
, what’s happening is first that we
create a factor a
and then we package it up into a
singlefactor term. And then we create a factor b
, and we
package it up into a singlefactor term. And then we evaluate the
:
, and compute the interaction between these two terms. When
we encounter embedded Python code, it’s always converted straight
to a singlefactor term before doing anything else.
Intercept handling¶
There are two special things about how intercept terms are handled inside the formula parser.
First, since an intercept term is an interaction of zero factors, we
have no way to write it down using the parts of the language described
so far. Therefore, as a special case, the string 1
is taken to
represent the intercept term.
Second, since intercept terms are almost always wanted and remembering
to include them by hand all the time is quite tedious, they are always
included by default in the righthand side of any formula. The way
this is implemented is exactly as if there is an invisible 1 +
inserted at the beginning of every righthand side.
Of course, if you don’t want an intercept, you can remove it again
just like any other unwanted term, using the 
operator. The only
thing that’s special about the 1 +
is that it’s invisible;
otherwise it acts just like any other term. This formula has an
intercept:
y ~ x
because it is processed like y ~ 1 + x
.
This formula does not have an intercept:
y ~ x  1
because it is processed like y ~ 1 + x  1
.
Of course if you want to be really explicit you can mention the intercept explicitly:
y ~ 1 + x
Once the invisible 1 +
is added, this formula is processed like
y ~ 1 + 1 + x
, and as you’ll recall from the definition of +
above, adding the same term twice produces the same result as adding
it just once.
For compatibility with S and R, we also allow the magic terms 0
and
1
which represent the “antiintercept”. Adding one of these terms
has exactly the same effect as subtracting the intercept term, and
subtracting one of these terms has exactly the same effect as adding
the intercept term. That means that all of these formulas are
equivalent:
y ~ x  1
y ~ x + 1
y ~ 1 + x
y ~ 0 + x
y ~ x  (0)
Explore!¶
The formula language is actually fairly simple once you get the hang of it, but if you’re ever in doubt as to what some construction means, you can always ask Patsy how it expands.
Here’s some code to try out at the Python prompt to get started:
from patsy import ModelDesc
ModelDesc.from_formula("y ~ x")
ModelDesc.from_formula("y ~ x + x + x")
ModelDesc.from_formula("y ~ 1 + x")
ModelDesc.from_formula("~ 1")
ModelDesc.from_formula("y ~ a:b")
ModelDesc.from_formula("y ~ a*b")
ModelDesc.from_formula("y ~ (a + b + c + d) ** 2")
ModelDesc.from_formula("y ~ (a + b)/(c + d)")
ModelDesc.from_formula("np.log(x1 + x2) "
"+ (x + {6: x3, 8 + 1: x4}[3 * i])")
Sometimes it might be easier to read if you put the processed formula
back into formula notation using ModelDesc.describe()
:
desc = ModelDesc.from_formula("y ~ (a + b + c + d) ** 2")
desc.describe()
From terms to matrices¶
So at this point, you hopefully understand how a string is parsed into
the ModelDesc
structure shown in the figure at the top of
this page. And if you like you can also produce such structures
directly without going through the formula parser (see
Model specification for experts and computers). But these terms and factors
objects are still a fairly highlevel, symbolic representation of a
model. Now we’ll talk about how they get converted into actual
matrices with numbers in.
There are two core operations here. The first takes a list of
Term
objects (a termlist) and some data, and produces a
DesignMatrixBuilder
. The second takes a
DesignMatrixBuilder
and some data, and produces a design
matrix. In practice, these operations are implemented by
design_matrix_builders()
and build_design_matrices()
,
respectively, and each of these functions is “vectorized” to process
an arbitrary number of matrices together in a single operation. But
we’ll ignore that for now, and just focus on what happens to a single
termlist.
First, each individual factor is given a chance to set up any Stateful transforms it may have, and then is evaluated on the data, to determine:
 Whether it is categorical or numerical
 If it is categorical, what levels it has
 If it is numerical, how many columns it has.
Next, we sort terms based on the factors they contain. This is done by dividing terms into groups based on what combination of numerical factors each one contains. The group of terms that have no numerical factors comes first, then the rest of the groups in the order they are first mentioned within the term list. Then within each group, lowerorder interactions are ordered to come before higherorder interactions. (Interactions of the same order are left alone.)
Example:
In [3]: data = demo_data("a", "b", "x1", "x2")
In [4]: mat = dmatrix("x1:x2 + a:b + b + x1:a:b + a + x2:a:x1", data)
In [5]: mat.design_info.term_names
Out[5]: ['Intercept', 'b', 'a', 'a:b', 'x1:x2', 'x2:a:x1', 'x1:a:b']
The nonnumerical terms are Intercept, b, a, a:b and they come first, sorted from lowerorder to higherorder. b comes before a because it did in the original formula. Next come the terms that involved x1 and x2 together, and x1:x2 comes before x2:a:x1 because it is a lowerorder term. Finally comes the sole term involving x1 without x2.
Note
These ordering rules may seem a bit arbitrary, but will make more sense after our discussion of redundancy below. Basically the motivation is that terms like b and a represent overlapping vector spaces, which means that the presence of one will affect how the other is coded. So, we group to them together, to make these relationships easier to see in the final analysis. And, a term like b represents a subspace of a term like a:b, so if you’re including both terms in your model you presumably want the variance represented by b to be partitioned out separately from the overall a:b term, and for that to happen, b should come first in the final model.
After sorting the terms, we determine appropriate coding schemes for
categorical factors, as described in the next section. And that’s it
– we now know exactly how to produce this design matrix, and
design_matrix_builders()
packages this knowledge up into a
DesignMatrixBuilder
and returns it. To get the design matrix
itself, we then use build_design_matrices()
.
Redundancy and categorical factors¶
Here’s the basic idea about how Patsy codes categorical factors: each
term that’s included means that we want our outcome variable to be
able to vary in a certain way – for example, the a:b in y ~ a:b
means that we want our model to be flexible enough to assign y a
different value for every possible combination of a and b
values. So what Patsy does is build up a design matrix incrementally
by working from left to right in the sorted term list, and for each
term it adds just the right columns needed to make sure that the model
will be flexible enough to include the kind of variation this term
represents, while keeping the overall design matrix full rank. The
result is that the columns associated with each term always represent
the additional flexibility that the models gains by adding that
term, on top of the terms to its left. Numerical factors are assumed
not to be redundant with each other, and are always included “as is”;
categorical factors and interactions might be redundant, so Patsy
chooses either fullrank or reducedrank contrast coding for each one
to keep the overall design matrix at full rank.
Note
We’re only worried here about “structural redundancies”, those which occur inevitably no matter what the particular values occur in your data set. If you enter two different factors x1 and x2, but set them to be numerically equal, then Patsy will indeed produce a design matrix that isn’t full rank. Avoiding that is your problem.
Okay, now for the more the more detailed explanation. Each term represents a certain space of linear combinations of column vectors:
 A numerical factor represents the vector space spanned by its columns.
 A categorical factor represents the vector space spanned by the columns you get if you apply “dummy coding”.
 An interaction between two factors represents the vector space
spanned by the elementwise products between vectors in the first
factor’s space with vectors in the second factor’s space. For
example, if and are two columns that
form a basis for the vector space represented by factor ,
and likewise and are a basis for the
vector space represented by , then , , ,
is a basis for the vector space represented
by . Here the operator represents
elementwise multiplication, like numpy
*
. (Exercise: show that the choice of basis does not matter.)  The empty interaction represents the space spanned by the identity element for elementwise multiplication, i.e., the allones “intercept” term.
So suppose that a is a categorical factor with two levels a1 and a2, and b is a categorical factor with two levels b1 and b1. Then:
a represents the space spanned by two vectors: one that has a 1 everywhere that
a == "a1"
, and a zero everywhere else, and another that’s similar but fora == "a2"
. (dummy coding)b works similarly
and a:b represents the space spanned by four vectors: one that has a 1 everywhere that has
a == "a1"
andb == "b1"
, another that has a 1 everywhere that hasa1 == "a2"
andb == "b1"
, etc. So if you are familiar with ANOVA terminology, then these are not the kinds of interactions you are expecting! They represent a more fundamental idea, that when we write:y ~ a:b
we mean that the value of y can vary depending on every possible combination of a and b.
Notice that this means that the space spanned by the intercept term is always a vector subspace of the spaces spanned by a and b, and these subspaces in turn are always subspaces of the space spanned by a:b. (Another way to say this is that a and b are “marginal to” a:b.) The diagram on the right shows these relationships graphically. This reflects the intuition that allowing y to depend on every combination of a and b gives you a more flexible model than allowing it to vary based on just a or just b.
So what this means is that once you have a:b in your model, adding a or b or the intercept term won’t actually give you any additional flexibility; the most they can do is to create redundancies that your linear algebra package will have to somehow detect and remove later. These two models are identical in terms of how flexible they are:
y ~ 0 + a:b
y ~ 1 + a + b + a:b
And, indeed, we can check that the matrices that Patsy generates for these two formulas have identical column spans:
In [6]: data = demo_data("a", "b", "y")
In [7]: mat1 = dmatrices("y ~ 0 + a:b", data)[1]
In [8]: mat2 = dmatrices("y ~ 1 + a + b + a:b", data)[1]
In [9]: np.linalg.matrix_rank(mat1)
Out[9]: 4
In [10]: np.linalg.matrix_rank(mat2)
Out[10]: 4
In [11]: np.linalg.matrix_rank(np.column_stack((mat1, mat2)))
Out[11]: 4
But, of course, their actual contents is different:
In [12]: mat1
Out[12]:
DesignMatrix with shape (8, 4)
a[a1]:b[b1] a[a2]:b[b1] a[a1]:b[b2] a[a2]:b[b2]
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
Terms:
'a:b' (columns 0:4)
In [13]: mat2
Out[13]:
DesignMatrix with shape (8, 4)
Intercept a[T.a2] b[T.b2] a[T.a2]:b[T.b2]
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
'b' (column 2)
'a:b' (column 3)
This happens because Patsy is finding ways to avoid creating redundancy while coding each term. To understand how this works, it’s useful to draw some pictures. Patsy has two general strategies for coding a categorical factor with levels. The first is to use a fullrank encoding with columns. Here are some pictures of this style of coding:
Obviously if we lay these images on top of each other, they’ll overlap, which corresponds to their overlap when considered as vector spaces. If we try just putting them all into the same model, we get mud:
Patsy avoids this by using its second strategy: coding an level factor in columns which, critically, do not span the intercept. We’ll call this style of coding reducedrank, and use notation like a to refer to factors coded this way.
Note
Each of the categorical coding schemes included in patsy
come in both fullrank and reducedrank flavours. If you ask for,
say, Poly
coding, then this is the mechanism used to
decide whether you get full or reducedrank Poly
coding.
For coding a there are two options:
And likewise for b:
When it comes to a:b, things get more interesting: it can choose whether to use a full or reducedrank encoding separately for each factor, leading to four choices overall:
So when interpreting a formula like 1 + a + b + a:b
, Patsy’s
job is to pick and choose from the above pieces and then assemble them
together like a jigsaw puzzle.
Let’s walk through the formula 1 + a + b + a:b
to see how this
works. First it encodes the intercept:
In [14]: dmatrices("y ~ 1", data)[1]
Out[14]:
DesignMatrix with shape (8, 1)
Intercept
1
1
1
1
1
1
1
1
Terms:
'Intercept' (column 0)
Then it adds the a term. It has two choices, either the fullrank coding or the reduced rank a coding. Using the fullrank coding would overlap with the alreadyexisting intercept term, though, so it chooses the reduced rank coding:
In [15]: dmatrices("y ~ 1 + a", data)[1]
Out[15]:
DesignMatrix with shape (8, 2)
Intercept a[T.a2]
1 0
1 0
1 1
1 1
1 0
1 0
1 1
1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
The b term is treated similarly:
In [16]: dmatrices("y ~ 1 + a + b", data)[1]
Out[16]:
DesignMatrix with shape (8, 3)
Intercept a[T.a2] b[T.b2]
1 0 0
1 0 1
1 1 0
1 1 1
1 0 0
1 0 1
1 1 0
1 1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
'b' (column 2)
And finally, there are four options for the a:b term, but only one of them will fit without creating overlap:
In [17]: dmatrices("y ~ 1 + a + b + a:b", data)[1]
Out[17]:
DesignMatrix with shape (8, 4)
Intercept a[T.a2] b[T.b2] a[T.a2]:b[T.b2]
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
1 0 0 0
1 0 1 0
1 1 0 0
1 1 1 1
Terms:
'Intercept' (column 0)
'a' (column 1)
'b' (column 2)
'a:b' (column 3)
Patsy tries to use the fewest pieces possible to cover the space. For instance, in this formula, the a:b term is able to fill the remaining space by using a single piece:
In [18]: dmatrices("y ~ 1 + b + a:b", data)[1]
Out[18]:
DesignMatrix with shape (8, 4)
Intercept b[T.b2] a[T.a2]:b[b1] a[T.a2]:b[b2]
1 0 0 0
1 1 0 0
1 0 1 0
1 1 0 1
1 0 0 0
1 1 0 0
1 0 1 0
1 1 0 1
Terms:
'Intercept' (column 0)
'b' (column 1)
'a:b' (columns 2:4)
However, this is not always possible. In such cases, Patsy will assemble multiple pieces to code a single term [1], e.g.:
In [19]: dmatrices("y ~ 1 + a:b", data)[1]
Out[19]:
DesignMatrix with shape (8, 4)
Intercept b[T.b2] a[T.a2]:b[b1] a[T.a2]:b[b2]
1 0 0 0
1 1 0 0
1 0 1 0
1 1 0 1
1 0 0 0
1 1 0 0
1 0 1 0
1 1 0 1
Terms:
'Intercept' (column 0)
'a:b' (columns 1:4)
Notice that the matrix entries and column names here are identical to those produced by the previous example, but the association between terms and columns shown at the bottom is different.
In all of these cases, the final model spans the same space; a:b is included in the formula, and therefore the final matrix must fill in the full a:b square. By including different combinations of lowerorder interactions, we can control how this overall variance is partitioned into distinct terms.
Exercise: create the similar diagram for a formula that includes a threeway interaction, like1 + a + a:b + a:b:c
or1 + a:b:c
. Hint: it’s a cube. Then, send us your diagram for inclusion in this documentation [2].
Finally, we’ve so far only discussed purely categorical interactions. Bringing numerical interactions into the mix doesn’t make things much more complicated. Each combination of numerical factors is considered to be distinct from all other combinations, so we divide all of our terms into groups based on which numerical factors they contain (just like we do when sorting terms, as described above), and then within each group we separately apply the algorithm described here to the categorical parts of each term.
Technical details¶
The actual algorithm Patsy uses to produce the above coding is very simple. Within the group of terms associated with each combination of numerical factors, it works from left to right. For each term it encounters, it breaks the categorical part of the interaction down into minimal pieces, e.g. a:b is replaced by 1 + (a) + (b) + (a):(b):
(Formally speaking, these “minimal pieces” consist of the set of all subsets of the original interaction.) Then, any of the minimal pieces which were used by a previous term within this group are deleted, since they are redundant:
and then we greedily recombine the pieces that are left by repeatedly merging adjacent pieces according to the rule ANYTHING + ANYTHING : FACTOR = ANYTHING : FACTOR:
Exercise: Prove formally that the space spanned by ANYTHING + ANYTHING : FACTOR is identical to the space spanned by ANYTHING : FACTOR.
Exercise: Either show that the greedy algorithm here is produces optimal encodings in some sense (e.g., smallest number of pieces used), or else find a better algorithm. (Extra credit: implement your algorithm and submit a pull request [3].)
Is this algorithm correct? A full formal proof would be too tedious for this reference manual, but here’s a sketch of the analysis.
Recall that our goal is to maintain two invariants: the design matrix column space should include the space associated with each term, and should avoid “structural redundancy”, i.e. it should be full rank on at least some data sets. It’s easy to see the above algorithm will never “lose” columns, since the only time it eliminates a subspace is when it has previously processed that exact subspace within the same design. But will it always detect all the redundancies that are present?
That is guaranteed by the following theorem:
Theorem: Let two sets of factors, and be given, and let be the numerical and categorical factors, respectively (and similarly for . Then the space represented by the interaction has a nontrivial intersection with the space represented by the interaction whenever:
 , and
And, furthermore, whenever this condition does not hold, then there exists some assignment of values to the factors for which the associated vector spaces have only a trivial intersection.
Exercise: Prove it.
Exercise: Show that given a sufficient number of rows, the set of factor assignments on which represents a subspace of without the above conditions being satisfied is actually a zero set.
Corollary: Patsy’s strategy of dividing into groups by numerical factors, and then comparing all subsets of the remaining categorical factors, allows it to precisely identify and avoid structural redundancies.
Footnotes¶
[1]  This is one of the places where Patsy improves on R, which produces incorrect output in this case (see Differences between R and Patsy formulas). 
[2]  Yes, I’m lazy. And shameless. 
[3]  Yes, still shameless. 
Coding categorical data¶
Patsy allows great flexibility in how categorical data is coded,
via the function C()
. C()
marks some data as being
categorical (including data which would not automatically be treated
as categorical, such as a column of integers), while also optionally
setting the preferred coding scheme and level ordering.
Let’s get some categorical data to work with:
In [1]: from patsy import dmatrix, demo_data, ContrastMatrix, Poly
In [2]: data = demo_data("a", nlevels=3)
In [3]: data
Out[3]: {'a': ['a1', 'a2', 'a3', 'a1', 'a2', 'a3']}
As you know, simply giving Patsy a categorical variable causes it
to be coded using the default Treatment
coding
scheme. (Strings and booleans are treated as categorical by default.)
In [4]: dmatrix("a", data)
Out[4]:
DesignMatrix with shape (6, 3)
Intercept a[T.a2] a[T.a3]
1 0 0
1 1 0
1 0 1
1 0 0
1 1 0
1 0 1
Terms:
'Intercept' (column 0)
'a' (columns 1:3)
We can also alter the level ordering, which is useful for, e.g.,
Diff
coding:
In [5]: l = ["a3", "a2", "a1"]
In [6]: dmatrix("C(a, levels=l)", data)
Out[6]:
DesignMatrix with shape (6, 3)
Intercept C(a, levels=l)[T.a2] C(a, levels=l)[T.a1]
1 0 1
1 1 0
1 0 0
1 0 1
1 1 0
1 0 0
Terms:
'Intercept' (column 0)
'C(a, levels=l)' (columns 1:3)
But the default coding is just that – a default. The easiest alternative is to use one of the other builtin coding schemes, like orthogonal polynomial coding:
In [7]: dmatrix("C(a, Poly)", data)
Out[7]:
DesignMatrix with shape (6, 3)
Intercept C(a, Poly).Linear C(a, Poly).Quadratic
1 0.70711 0.40825
1 0.00000 0.81650
1 0.70711 0.40825
1 0.70711 0.40825
1 0.00000 0.81650
1 0.70711 0.40825
Terms:
'Intercept' (column 0)
'C(a, Poly)' (columns 1:3)
There are a number of builtin coding schemes; for details you can check the API reference. But we aren’t restricted to those. We can also provide a custom contrast matrix, which allows us to produce all kinds of strange designs:
In [8]: contrast = [[1, 2], [3, 4], [5, 6]]
In [9]: dmatrix("C(a, contrast)", data)
Out[9]:
DesignMatrix with shape (6, 3)
Intercept C(a, contrast)[custom0] C(a, contrast)[custom1]
1 1 2
1 3 4
1 5 6
1 1 2
1 3 4
1 5 6
Terms:
'Intercept' (column 0)
'C(a, contrast)' (columns 1:3)
In [10]: dmatrix("C(a, [[1], [2], [4]])", data)
Out[10]:
DesignMatrix with shape (6, 2)
Intercept C(a, [[1], [2], [4]])[custom0]
1 1
1 2
1 4
1 1
1 2
1 4
Terms:
'Intercept' (column 0)
'C(a, [[1], [2], [4]])' (column 1)
Hmm, those [custom0]
, [custom1]
names that Patsy
autogenerated for us are a bit ugly looking. We can attach names to
our contrast matrix by creating a ContrastMatrix
object, and
make things prettier:
In [11]: contrast_mat = ContrastMatrix(contrast, ["[pretty0]", "[pretty1]"])
In [12]: dmatrix("C(a, contrast_mat)", data)
Out[12]:
DesignMatrix with shape (6, 3)
Intercept C(a, contrast_mat)[pretty0] C(a, contrast_mat)[pretty1]
1 1 2
1 3 4
1 5 6
1 1 2
1 3 4
1 5 6
Terms:
'Intercept' (column 0)
'C(a, contrast_mat)' (columns 1:3)
And, finally, if we want to get really fancy, we can also define our
own “smart” coding schemes like Poly
. Just define a class
that has two methods, code_with_intercept()
and
code_without_intercept()
. They have identical signatures, taking
a list of levels as their argument and returning a
ContrastMatrix
. Patsy will automatically choose the
appropriate method to call to produce a fullrank design matrix
without redundancy; see Redundancy and categorical factors for the full details on how
Patsy makes this decision.
As an example, here’s a simplified version of the builtin
Treatment
coding object:
import numpy as np
class MyTreat(object):
def __init__(self, reference=0):
self.reference = reference
def code_with_intercept(self, levels):
return ContrastMatrix(np.eye(len(levels)),
["[My.%s]" % (level,) for level in levels])
def code_without_intercept(self, levels):
eye = np.eye(len(levels)  1)
contrasts = np.vstack((eye[:self.reference, :],
np.zeros((1, len(levels)  1)),
eye[self.reference:, :]))
suffixes = ["[MyT.%s]" % (level,) for level in
levels[:self.reference] + levels[self.reference + 1:]]
return ContrastMatrix(contrasts, suffixes)
And it can now be used just like the builtin methods:
# Full rank:
In [14]: dmatrix("0 + C(a, MyTreat)", data)
Out[14]:
DesignMatrix with shape (6, 3)
C(a, MyTreat)[My.a1] C(a, MyTreat)[My.a2] C(a, MyTreat)[My.a3]
1 0 0
0 1 0
0 0 1
1 0 0
0 1 0
0 0 1
Terms:
'C(a, MyTreat)' (columns 0:3)
# Reduced rank:
In [15]: dmatrix("C(a, MyTreat)", data)
Out[15]:
DesignMatrix with shape (6, 3)
Intercept C(a, MyTreat)[MyT.a2] C(a, MyTreat)[MyT.a3]
1 0 0
1 1 0
1 0 1
1 0 0
1 1 0
1 0 1
Terms:
'Intercept' (column 0)
'C(a, MyTreat)' (columns 1:3)
# With argument:
In [16]: dmatrix("C(a, MyTreat(2))", data)
Out[16]:
DesignMatrix with shape (6, 3)
Intercept C(a, MyTreat(2))[MyT.a1] C(a, MyTreat(2))[MyT.a2]
1 1 0
1 0 1
1 0 0
1 1 0
1 0 1
1 0 0
Terms:
'Intercept' (column 0)
'C(a, MyTreat(2))' (columns 1:3)
Stateful transforms¶
There’s a subtle problem that sometimes bites people when working with
formulas. Suppose that I have some numerical data called x
, and I
would like to center it before fitting. The obvious way would be to
write:
y ~ I(x  np.mean(x)) # BROKEN! Don't do this!
or, even better we could package it up into a function:
In [1]: def naive_center(x): # BROKEN! don't use!
...: x = np.asarray(x)
...: return x  np.mean(x)
...:
and then write our formula like:
y ~ naive_center(x)
Why is this a bad idea? Let’s set up an example.
In [2]: import numpy as np
In [3]: from patsy import dmatrix, build_design_matrices, incr_dbuilder
In [4]: data = {"x": [1, 2, 3, 4]}
Now we can build a design matrix and see what we get:
In [5]: mat = dmatrix("naive_center(x)", data)
In [6]: mat
Out[6]:
DesignMatrix with shape (4, 2)
Intercept naive_center(x)
1 1.5
1 0.5
1 0.5
1 1.5
Terms:
'Intercept' (column 0)
'naive_center(x)' (column 1)
Those numbers look correct, and in fact they are correct. If all we’re
going to do with this model is call dmatrix()
once, then
everything is fine – which is what makes this problem so insidious.
Often we want to do more with a model than this. For instance, we might find some new data, and want to feed it into our model to make predictions. To do this, though, we first need to reapply the same transformation, like so:
In [7]: new_data = {"x": [5, 6, 7, 8]}
# Broken!
In [8]: build_design_matrices([mat.design_info.builder], new_data)[0]
Out[8]:
DesignMatrix with shape (4, 2)
Intercept naive_center(x)
1 1.5
1 0.5
1 0.5
1 1.5
Terms:
'Intercept' (column 0)
'naive_center(x)' (column 1)
So it’s clear what’s happened here – Patsy has centered the new data, just like it centered the old data. But if you think about what this means statistically, it makes no sense. According to this, the new data point where x is 5 will behave exactly like the old data point where x is 1, because they both produce the same input to the actual model.
The problem is what it means to apply “the same transformation”. Here, what we really want to do is to subtract the mean of the original data from the new data.
Patsy’s solution is called a stateful transform. These look like
ordinary functions, but they perform a bit of magic to remember the
state of the original data, and use it in transforming new data.
Several useful stateful transforms are included out of the box,
including one called center()
.
Using center()
instead of naive_center()
produces the same
correct result for our original matrix. It’s used in exactly the same
way:
In [9]: fixed_mat = dmatrix("center(x)", data)
In [10]: fixed_mat
Out[10]:
DesignMatrix with shape (4, 2)
Intercept center(x)
1 1.5
1 0.5
1 0.5
1 1.5
Terms:
'Intercept' (column 0)
'center(x)' (column 1)
But if we then feed in our new data, we also get out the correct result:
# Correct!
In [11]: build_design_matrices([fixed_mat.design_info.builder], new_data)[0]
Out[11]:
DesignMatrix with shape (4, 2)
Intercept center(x)
1 2.5
1 3.5
1 4.5
1 5.5
Terms:
'Intercept' (column 0)
'center(x)' (column 1)
Another situation where we need some stateful transform magic is when
we are working with data that is too large to fit into memory at
once. To handle such cases, Patsy allows you to set up a design
matrix while working our way incrementally through the data. But if we
use naive_center()
when building a matrix incrementally, then it
centers each chunk of data, not the data as a whole. (Of course,
depending on how your data is distributed, this might end up being
just similar enough for you to miss the problem until it’s too late.)
In [12]: data_chunked = [{"x": data["x"][:2]},
....: {"x": data["x"][2:]}]
....:
In [13]: builder = incr_dbuilder("naive_center(x)", lambda: iter(data_chunked))
# Broken!
In [14]: np.row_stack([build_design_matrices([builder], chunk)[0]
....: for chunk in data_chunked])
....:
Out[14]:
array([[ 1. , 0.5],
[ 1. , 0.5],
[ 1. , 0.5],
[ 1. , 0.5]])
But if we use the proper stateful transform, this just works:
In [15]: builder = incr_dbuilder("center(x)", lambda: iter(data_chunked))
# Correct!
In [16]: np.row_stack([build_design_matrices([builder], chunk)[0]
....: for chunk in data_chunked])
....:
Out[16]:
array([[ 1. , 1.5],
[ 1. , 0.5],
[ 1. , 0.5],
[ 1. , 1.5]])
Note
Under the hood, the way this works is that incr_dbuilder()
iterates through the data once to calculate the mean, and then we
use build_design_matrices()
to iterate through it a second
time creating our design matrix. While taking two passes through a
large data set may be slow, there’s really no other way to
accomplish what the user asked for. The good news is that Patsy is
smart enough to make only the minimum number of passes
necessary. For example, in our example with naive_center()
above, incr_dbuilder()
would not have done a full pass
through the data at all. And if you have multiple stateful
transforms in the same formula, then Patsy will process them in
parallel in a single pass.
And, of course, we can use the resulting builder for prediction as well:
# Correct!
In [17]: build_design_matrices([builder], new_data)[0]
Out[17]:
DesignMatrix with shape (4, 2)
Intercept center(x)
1 2.5
1 3.5
1 4.5
1 5.5
Terms:
'Intercept' (column 0)
'center(x)' (column 1)
In fact, Patsy’s stateful transform handling is clever enough that
it can support arbitrary mixing of stateful transforms with other
Python code. E.g., if center()
and spline()
were both
stateful transforms, then even a silly a formula like this will be
handled 100% correctly:
y ~ I(spline(center(x1)) + center(x2))
However, it isn’t perfect – there are two things you have to be careful of. Let’s put them in red:
Warning
If you are unwise enough to ignore this section, write a function
like naive_center above, and use it in a formula, then Patsy will
not notice. If you use that formula with incr_dbuilders()
or
for predictions, then you will just silently get the wrong
results. We have a plan to detect such cases, but it isn’t
implemented yet (and in any case can never be 100% reliable). So be
careful!
Warning
Even if you do use a “real” stateful transform like center()
or standardize()
, still have to make sure that Patsy can
“see” that you are using such a transform. Currently the rule is
that you must access the stateful transform function using a
simple, bare variable reference, without any dots or other
lookups:
dmatrix("y ~ center(x)", data) # okay
asdf = patsy.center
dmatrix("y ~ asdf(x)", data) # okay
dmatrix("y ~ patsy.center(x)", data) # BROKEN! DON'T DO THIS!
funcs = {"center": patsy.center}
dmatrix("y ~ funcs['center'](x)", data) # BROKEN! DON'T DO THIS!
Builtin stateful transforms¶
There are a number of builtin stateful transforms beyond
center()
; see stateful transforms in the API reference for a complete list.
Defining a stateful transform¶
You can also easily define your own stateful transforms. The first step is to define a class which fulfills the stateful transform protocol. The lifecycle of a stateful transform object is as follows:
 An instance of your type will be constructed.
memorize_chunk()
will be called one or more times.memorize_finish()
will be called once.transform()
will be called one or more times, on either the same or different data to what was initially passed tomemorize_chunk()
. You can trust that any nondata arguments will be identical between calls tomemorize_chunk()
andtransform()
.
And here are the methods and call signatures you need to define:

class
patsy.
stateful_transform_protocol
¶ 
__init__
() It must be possible to create an instance of the class by calling the constructor with no arguments.

memorize_chunk
(*args, **kwargs)¶ Update any internal state, based on the data passed into memorize_chunk.

memorize_finish
()¶ Do any housekeeping you want to do between the last call to
memorize_chunk()
and the first call totransform()
. For example, if you are computing some summary statistic that cannot be done incrementally, then yourmemorize_chunk()
method might just store the data that’s passed in, and thenmemorize_finish()
could compute the summary statistic and delete the stored data to free up the associated memory.

transform
(*args, **kwargs)¶ This method should transform the input data passed to it. It should be deterministic, and it should be “pointwise”, in the sense that when passed an array it performs an independent transformation on each data point that is not affected by any other data points passed to
transform()
.

Then once you have created your class, pass it to
stateful_transform()
to create a callable stateful transform
object suitable for use inside or outside formulas.
Here’s a simple example of how you might implement a working version
of center()
(though it’s less robust and featureful than the
real builtin):
class MyExampleCenter(object):
def __init__(self):
self._total = 0
self._count = 0
self._mean = None
def memorize_chunk(self, x):
self._total += np.sum(x)
self._count += len(x)
def memorize_finish(self):
self._mean = self.total * 1. / self._count
def transform(self, x):
return x  self._mean
my_example_center = patsy.stateful_transform(MyExampleCenter)
print(my_example_center(np.array([1, 2, 3])))
But of course, if you come up with any useful ones, please let us know so we can incorporate them into patsy itself!
Spline regression¶
Patsy offers a set of specific stateful transforms (for more details about stateful transforms see Stateful transforms) that you can use in formulas to generate splines bases and express nonlinear fits.
General Bsplines¶
Bspline bases can be generated with the bs()
stateful
transform. The spline bases returned by bs()
are designed to be
compatible with those produced by the R bs
function.
The following code illustrates a typical basis and the resulting spline:
In [3]: import matplotlib.pyplot as plt
In [4]: plt.title("Bspline basis example (degree=3)");
In [5]: x = np.linspace(0., 1., 100)
In [6]: y = dmatrix("bs(x, df=6, degree=3, include_intercept=True)  1", {"x": x})
# Define some coefficients
In [7]: b = np.array([1.3, 0.6, 0.9, 0.4, 1.6, 0.7])
# Plot Bspline basis functions (colored curves) each multiplied by its coeff
In [8]: plt.plot(x, y*b);
# Plot the spline itself (sum of the basis functions, thick black curve)
In [9]: plt.plot(x, np.dot(y, b), color='k', linewidth=3);
In the following example we first set up our Bspline basis using some data and then make predictions on a new set of data:
In [10]: data = {"x": np.linspace(0., 1., 100)}
In [11]: design_matrix = dmatrix("bs(x, df=4)", data)
In [12]: new_data = {"x": [0.1, 0.25, 0.9]}
In [13]: build_design_matrices([design_matrix.design_info.builder], new_data)[0]
Out[13]:
DesignMatrix with shape (3, 5)
Intercept bs(x, df=4)[0] bs(x, df=4)[1] bs(x, df=4)[2] bs(x, df=4)[3]
1 0.43400 0.052 0.00200 0.000
1 0.59375 0.250 0.03125 0.000
1 0.00200 0.052 0.43400 0.512
Terms:
'Intercept' (column 0)
'bs(x, df=4)' (columns 1:5)
bs()
can produce Bspline bases of arbitrary degrees – e.g.,
degree=0
will give produce piecewiseconstant functions,
degree=1
will produce piecewiselinear functions, and the default
degree=3
produces cubic splines. The next section describes more
specialized functions for producing different types of cubic splines.
Natural and cyclic cubic regression splines¶
Natural and cyclic cubic regression splines are provided through the stateful
transforms cr()
and cc()
respectively. Here the spline is
parameterized directly using its values at the knots. These splines were designed
to be compatible with those found in the R package
mgcv
(these are called cr, cs and cc in the context of mgcv), but
can be used with any model.
Warning
Note that the compatibility with mgcv applies only to the generation of spline bases: we do not implement any kind of mgcvcompatible penalized fitting process. Thus these spline bases can be used to precisely reproduce predictions from a model previously fitted with mgcv, or to serve as building blocks for other regression models (like OLS).
Here are some illustrations of typical natural and cyclic spline bases:
In [14]: plt.title("Natural cubic regression spline basis example");
In [15]: y = dmatrix("cr(x, df=6)  1", {"x": x})
# Plot natural cubic regression spline basis functions (colored curves) each multiplied by its coeff
In [16]: plt.plot(x, y*b);
# Plot the spline itself (sum of the basis functions, thick black curve)
In [17]: plt.plot(x, np.dot(y, b), color='k', linewidth=3);
In [18]: plt.title("Cyclic cubic regression spline basis example");
In [19]: y = dmatrix("cc(x, df=6)  1", {"x": x})
# Plot cyclic cubic regression spline basis functions (colored curves) each multiplied by its coeff
In [20]: plt.plot(x, y*b);
# Plot the spline itself (sum of the basis functions, thick black curve)
In [21]: plt.plot(x, np.dot(y, b), color='k', linewidth=3);
In the following example we first set up our spline basis using same data as for the Bspline example above and then make predictions on a new set of data:
In [22]: design_matrix = dmatrix("cr(x, df=4, constraints='center')", data)
In [23]: new_design_matrix = build_design_matrices([design_matrix.design_info.builder], new_data)[0]
In [24]: new_design_matrix
Out[24]:
DesignMatrix with shape (3, 5)
Columns:
['Intercept',
"cr(x, df=4, constraints='center')[0]",
"cr(x, df=4, constraints='center')[1]",
"cr(x, df=4, constraints='center')[2]",
"cr(x, df=4, constraints='center')[3]"]
Terms:
'Intercept' (column 0)
"cr(x, df=4, constraints='center')" (columns 1:5)
(to view full data, use np.asarray(this_obj))
In [25]: np.asarray(new_design_matrix)
Out[25]:
array([[ 1. , 0.15855682, 0.5060419 , 0.40944318, 0.16709613],
[ 1. , 0.71754625, 0.22956933, 0.28245375, 0.10215042],
[ 1. , 0.1602992 , 0.30354568, 0.4077008 , 0.43900769]])
Note that in the above example 5 knots are actually used to achieve 4 degrees of freedom since a centering constraint is requested.
Note that the API is different from mgcv:
 In patsy one can specify the number of degrees of freedom directly (actual number of columns of the resulting design matrix) whereas in mgcv one has to specify the number of knots to use. For instance, in the case of cyclic regression splines (with no additional constraints) the actual degrees of freedom is the number of knots minus one.
 In patsy one can specify inner knots as well as lower and upper exterior knots which can be useful for cyclic spline for instance.
 In mgcv a centering/identifiability constraint is automatically computed and
absorbed in the resulting design matrix.
The purpose of this is to ensure that if
b
is the array of initial parameters (corresponding to the initial unconstrained design matrixdm
), our model is centered, ienp.mean(np.dot(dm, b))
is zero. We can rewrite this asnp.dot(c, b)
being zero withc
a 1row constraint matrix containing the mean of each column ofdm
. Absorbing this constraint in the final design matrix means that we rewrite the model in terms of unconstrained parameters (this is done through a QRdecomposition of the constraint matrix). Those unconstrained parameters have the property that when projected back into the initial parameters space (let’s callb_back
the result of this projection), the constraintnp.dot(c, b_back)
being zero is automatically verified. In patsy one can choose between no constraint, a centering constraint like mgcv ('center'
) or a user provided constraint matrix.
Tensor product smooths¶
Smooths of several covariates can be generated through a tensor product of
the bases of marginal univariate smooths. For these marginal smooths one can
use the above defined splines as well as user defined smooths provided they
actually transform input univariate data into some kind of smooth functions
basis producing a 2d array output with the (i, j)
element corresponding
to the value of the j
th basis function at the i
th data point.
The tensor product stateful transform is called te()
.
Note
The implementation of this tensor product is compatible with mgcv when considering only cubic regression spline marginal smooths, which means that generated bases will match those produced by mgcv. Recall that we do not implement any kind of mgcvcompatible penalized fitting process.
In the following code we show an example of tensor product basis functions
used to represent a smooth of two variables x1
and x2
. Note how
marginal spline bases patterns can be observed on the x and y contour projections:
In [26]: from matplotlib import cm
In [27]: from mpl_toolkits.mplot3d.axes3d import Axes3D
In [28]: x1 = np.linspace(0., 1., 100)
In [29]: x2 = np.linspace(0., 1., 100)
In [30]: x1, x2 = np.meshgrid(x1, x2)
In [31]: df = 3
In [32]: y = dmatrix("te(cr(x1, df), cc(x2, df))  1",
....: {"x1": x1.ravel(), "x2": x2.ravel(), "df": df})
....:
In [33]: print y.shape
(10000, 9)
In [34]: fig = plt.figure()
In [35]: fig.suptitle("Tensor product basis example (2 covariates)");
In [36]: for i in xrange(df * df):
....: ax = fig.add_subplot(df, df, i + 1, projection='3d')
....: yi = y[:, i].reshape(x1.shape)
....: ax.plot_surface(x1, x2, yi, rstride=4, cstride=4, alpha=0.15)
....: ax.contour(x1, x2, yi, zdir='z', cmap=cm.coolwarm, offset=0.5)
....: ax.contour(x1, x2, yi, zdir='y', cmap=cm.coolwarm, offset=1.2)
....: ax.contour(x1, x2, yi, zdir='x', cmap=cm.coolwarm, offset=0.2)
....: ax.set_xlim3d(0.2, 1.0)
....: ax.set_ylim3d(0, 1.2)
....: ax.set_zlim3d(0.5, 1)
....: ax.set_xticks([0, 1])
....: ax.set_yticks([0, 1])
....: ax.set_zticks([0.5, 0, 1])
....:
In [37]: fig.tight_layout()
Following what we did for univariate splines in the preceding sections, we will now set up a 3d smooth basis using some data and then make predictions on a new set of data:
In [38]: data = {"x1": np.linspace(0., 1., 100),
....: "x2": np.linspace(0., 1., 100),
....: "x3": np.linspace(0., 1., 100)}
....:
In [39]: design_matrix = dmatrix("te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')",
....: data)
....:
In [40]: new_data = {"x1": [0.1, 0.2],
....: "x2": [0.2, 0.3],
....: "x3": [0.3, 0.4]}
....:
In [41]: new_design_matrix = build_design_matrices([design_matrix.design_info.builder], new_data)[0]
In [42]: new_design_matrix
Out[42]:
DesignMatrix with shape (2, 27)
Columns:
['Intercept',
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[0]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[1]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[2]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[3]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[4]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[5]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[6]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[7]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[8]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[9]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[10]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[11]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[12]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[13]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[14]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[15]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[16]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[17]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[18]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[19]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[20]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[21]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[22]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[23]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[24]",
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')[25]"]
Terms:
'Intercept' (column 0)
"te(cr(x1, df=3), cr(x2, df=3), cc(x3, df=3), constraints='center')" (columns 1:27)
(to view full data, use np.asarray(this_obj))
In [43]: np.asarray(new_design_matrix)
Out[43]:
array([[ 1.00000000e+00, 3.50936277e01, 2.42388845e02,
2.79787994e02, 3.82596528e01, 2.22904932e02,
4.12367917e04, 5.83969029e02, 9.25602506e03,
1.93080061e03, 1.13410128e01, 2.40550675e03,
1.44263739e02, 3.13527293e02, 1.48712359e01,
2.12892006e02, 8.23983725e03, 3.41861279e02,
3.77323208e03, 2.07265029e02, 5.80002506e03,
2.15508006e02, 1.05942372e02, 3.39701279e02,
6.27553548e02, 1.46393875e02, 3.00859954e02],
[ 1.00000000e+00, 1.09386577e01, 4.93985311e02,
8.80045587e02, 3.34907265e01, 1.14474114e01,
1.58055508e02, 4.16223930e02, 7.14533704e03,
5.02445587e02, 1.15899265e01, 5.97221139e02,
7.67240826e02, 2.54852719e01, 5.82470575e02,
1.38925587e02, 3.09397741e02, 5.69908474e02,
1.19655508e02, 1.93503930e02, 1.57733704e03,
1.19725587e02, 4.20757741e02, 5.97748474e02,
7.82200162e02, 2.04894751e02, 3.43123674e02]])
Model specification for experts and computers¶
While the formula language is great for interactive modelfitting and exploratory data analysis, there are times when we want a different or more systematic interface for creating design matrices. If you ever find yourself writing code that pastes together bits of strings to create a formula, then stop! And read this chapter.
Our first option, of course, is that we can go ahead and write some
code to construct our design matrices directly, just like we did in
the old days. Since this is supported directly by dmatrix()
and
dmatrices()
, it also works with any thirdparty library
functions that use Patsy internally. Just pass in an array_like or
a tuple (y_array_like, X_array_like)
in place of the formula.
In [1]: from patsy import dmatrix
In [2]: X = [[1, 10], [1, 20], [1, 2]]
In [3]: dmatrix(X)
Out[3]:
DesignMatrix with shape (3, 2)
x0 x1
1 10
1 20
1 2
Terms:
'x0' (column 0)
'x1' (column 1)
By using a DesignMatrix
with DesignInfo
attached, we
can also specify custom names for our custom matrix (or even term
slices and so forth), so that we still get the nice output and such
that Patsy would otherwise provide:
In [4]: from patsy import DesignMatrix, DesignInfo
In [5]: design_info = DesignInfo(["Intercept!", "Not intercept!"])
In [6]: X_dm = DesignMatrix(X, design_info)
In [7]: dmatrix(X_dm)
Out[7]:
DesignMatrix with shape (3, 2)
Intercept! Not intercept!
1 10
1 20
1 2
Terms:
'Intercept!' (column 0)
'Not intercept!' (column 1)
Or if all we want to do is to specify column names, we could also just
use a pandas.DataFrame
:
In [8]: import pandas
In [9]: df = pandas.DataFrame([[1, 10], [1, 20], [1, 2]],
...: columns=["Intercept!", "Not intercept!"])
...:
In [10]: dmatrix(df)
Out[10]:
DesignMatrix with shape (3, 2)
Intercept! Not intercept!
1 10
1 20
1 2
Terms:
'Intercept!' (column 0)
'Not intercept!' (column 1)
However, there is also a middle ground between pasting together strings and going back to putting together design matrices out of string and baling wire. Patsy has a straightforward Python interface for representing the result of parsing formulas, and you can use it directly. This lets you keep Patsy’s normal advantages – handling of categorical data and interactions, predictions, term tracking, etc. – while using a nice highlevel Python API. An example of somewhere this might be useful is if, say, you had a GUI with a tick box next to each variable in your data set, and wanted to construct a formula containing all the variables that had been checked, and letting Patsy deal with categorical data handling. Or this would be the approach you’d take for doing stepwise regression, where you need to programatically add and remove terms.
Whatever your particular situation, the strategy is this:
 Construct some factor objects (probably using
LookupFactor
orEvalFactor
 Put them into some
Term
objects,  Put the
Term
objects into two lists, representing the left and righthand side of your formula,  And then wrap the whole thing up in a
ModelDesc
.
(See How formulas work if you need a refresher on what each of these things are.)
In [11]: import numpy as np
In [12]: from patsy import (ModelDesc, EvalEnvironment, Term, EvalFactor,
....: LookupFactor, demo_data, dmatrix)
....:
In [13]: data = demo_data("a", "x")
# LookupFactor takes a dictionary key:
In [14]: a_lookup = LookupFactor("a")
# EvalFactor takes arbitrary Python code:
In [15]: x_transform = EvalFactor("np.log(x ** 2)")
# First argument is empty list for dmatrix; we would need to put
# something there if we were calling dmatrices.
In [16]: desc = ModelDesc([],
....: [Term([a_lookup]),
....: Term([x_transform]),
....: Term([a_lookup, x_transform])])
....:
# Create the matrix (or pass 'desc' to any statistical library
# function that uses patsy.dmatrix internally):
In [17]: dmatrix(desc, data)
Out[17]:
DesignMatrix with shape (6, 4)
a[a1] a[a2] np.log(x ** 2) a[T.a2]:np.log(x ** 2)
1 0 1.13523 0.00000
0 1 1.83180 1.83180
1 0 0.04298 0.00000
0 1 1.61375 1.61375
1 0 1.24926 0.00000
0 1 0.04597 0.04597
Terms:
'a' (columns 0:2)
'np.log(x ** 2)' (column 2)
'a:np.log(x ** 2)' (column 3)
Notice that no intercept term is included. Implicit intercepts are a
feature of the formula parser, not the underlying represention. If you
want an intercept, include the constant INTERCEPT
in your
list of terms (which is just sugar for Term([])
).
Note
Another option is to just pass your term lists directly to
design_matrix_builders()
, and skip the ModelDesc
entirely – all of the highlevel API functions like dmatrix()
accept DesignMatrixBuilder
objects as well as
ModelDesc
objects.
Example: say our data has 100 different numerical columns that we want to include in our design – and we also have a few categorical variables with a more complex interaction structure. Here’s one solution:
def add_predictors(base_formula, extra_predictors):
desc = ModelDesc.from_formula(base_formula)
# Using LookupFactor here ensures that everything will work correctly even
# if one of the column names in extra_columns is named like "weight.in.kg"
# or "sys.exit()" or "LittleBobbyTables()".
desc.rhs_termlist += [Term([LookupFactor(p)]) for p in extra_predictors]
return desc
In [19]: extra_predictors = ["x%s" % (i,) for i in range(10)]
In [20]: desc = add_predictors("np.log(y) ~ a*b + c:d", extra_predictors)
In [21]: desc.describe()
Out[21]: 'np.log(y) ~ a + b + a:b + c:d + x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9'
The factor protocol¶
If LookupFactor
and EvalFactor
aren’t enough for
you, then you can define your own factor class.
The full interface looks like this:

class
patsy.
factor_protocol
¶ 
name
()¶ This must return a short string describing this factor. It will be used to create column names, among other things.

origin
¶ A
patsy.Origin
if this factor has one; otherwise, just set it to None.

__eq__
(obj)¶ 
__ne__
(obj)¶ 
__hash__
()¶ If your factor will ever contain categorical data or participate in interactions, then it’s important to make sure you’ve defined
__eq__()
and__ne__()
and that your type is hashable. These methods will determine which factors Patsy considers equal for purposes of redundancy elimination.

memorize_passes_needed
(state, eval_env)¶ Return the number of passes through the data that this factor will need in order to set up any Stateful transforms.
If you don’t want to support stateful transforms, just return 0. In this case,
memorize_chunk()
andmemorize_finish()
will never be called.state is an (initially) empty dict which is maintained by the builder machinery, and that we can do whatever we like with. It will be passed back in to all memorization and evaluation methods.
eval_env is an
EvalEnvironment
object, describing the Python environment where the factor is being evaluated.

memorize_chunk
(state, which_pass, data)¶ Called repeatedly with each ‘chunk’ of data produced by the data_iter_maker passed to
design_matrix_builders()
.state is the state dictionary. which_pass will be zero on the first pass through the data, and eventually reach the value you returned from
memorize_passes_needed()
, minus one.Return value is ignored.

memorize_finish
(state, which_pass)¶ Called once after each pass through the data.
Return value is ignored.

eval
(state, data)¶ Evaluate this factor on the given data. Return value should ideally be a 1d or 2d array or
Categorical()
object, but this will be checked and converted as needed.

In addition, factor objects should be pickleable/unpickleable, so as to allow models containing them to be pickled/unpickled. (Or, if for some reason your factor objects are not safely pickleable, you should consider giving them a __getstate__ method which raises an error, so that any users which attempt to pickle a model containing your factors will get a clear failure immediately, instead of only later when they try to unpickle.)
Warning
Do not store evaluationrelated state in
attributes of your factor object! The same factor object may
appear in two totally different formulas, or if you have two
factor objects which compare equally, then only one may be
executed, and which one this is may vary randomly depending
on how build_design_matrices()
is called! Use only the
state dictionary for storing state.
The lifecycle of a factor object therefore looks like:
 Initialized.
memorize_passes_needed()
is called.for i in range(passes_needed):
memorize_chunk()
is called one or more timesmemorize_finish()
is called
eval()
is called zero or more times.
Alternative formula implementations¶
Even if you hate Patsy’s formulas all together, to the extent that
you’re going to go and implement your own competing mechanism for
defining formulas, you can still Patsybased
interfaces. Unfortunately, this isn’t quite as clean as we’d like,
because for now there’s no way to define a custom
DesignMatrixBuilder
. So you do still have to go through
Patsy’s formulabuilding machinery. But, this machinery simply
passes numerical data through unchanged, so in extremis you can:
 Define a special factor object that simply defers to your existing machinery
 Define the magic
__patsy_get_model_desc__
method on your formula object.dmatrix()
and friends check for the presence of this method on any object that is passed in, and if found, it is called (passing in theEvalEnvironment
), and expected to return aModelDesc
. And yourModelDesc
can, of course, include your special factor object(s).
Put together, it looks something like this:
class MyAlternativeFactor(object):
# A factor object that simply returns the design
def __init__(self, alternative_formula, side):
self.alternative_formula = alternative_formula
self.side = side
def name(self):
return self.side
def memorize_passes_needed(self, state):
return 0
def eval(self, state, data):
return self.alternative_formula.get_matrix(self.side, data)
class MyAlternativeFormula(object):
...
def __patsy_get_model_desc__(self, eval_env):
return ModelDesc([Term([MyAlternativeFactor(self, side="left")])],
[Term([MyAlternativeFactor(self, side="right")])],
my_formula = MyAlternativeFormula(...)
dmatrix(my_formula, data)
The only downside to this approach is that you can’t control the names of individual columns. (A workaround would be to create multiple terms each with its own factor that returns a different pieces of your overall matrix.) If this is a problem for you, though, then let’s talk – we can probably work something out.
Using Patsy in your library¶
Our goal is to make Patsy the de facto standard for describing models in Python, regardless of the underlying package in use – just as formulas are the standard interface to all R packages. Therefore we’ve tried to make it as easy as possible for you to build Patsy support into your libraries.
Patsy is a good houseguest:
 Pure Python, no compilation necessary.
 Exhaustive tests (>98% statement coverage at time of writing) and documentation (you’re looking at it).
 No dependencies besides numpy.
 Tested and supported on every version of Python since 2.5. (And 2.4 probably still works too if you really want it, it’s just become too hard to keep a working 2.4 environment on the test server.)
So you can be pretty confident that adding a dependency on Patsy won’t create much hassle for your users.
And, of course, the fundamental design is very conservative – the formula minilanguage in S was first described in Chambers and Hastie (1992), more than two decades ago. It’s still in heavy use today in R, which is one of the most popular environments for statistical programming. Many of your users may already be familiar with it. So we can be pretty certain that it will hold up to realworld usage.
Using the highlevel interface¶
If you have a function whose signature currently looks like this:
def mymodel2(X, y, ...):
...
or this:
def mymodel1(X, ...):
...
then adding Patsy support is extremely easy (though of course like any other API change, you may have to deprecate the old interface, or provide two interfaces in parallel, depending on your situation). Just write something like:
def mymodel2_patsy(formula_like, data={}, ...):
y, X = patsy.dmatrices(formula_like, data, 1)
...
or:
def mymodel1_patsy(formula_like, data={}, ...):
X = patsy.dmatrix(formula_like, data, 1)
...
(See dmatrices()
and dmatrix()
for details.) This won’t
force your users to switch to formulas immediately; they can replace
code that looks like this:
X, y = build_matrices_laboriously()
result = mymodel2(X, y, ...)
other_result = mymodel1(X, ...)
with code like this:
X, y = build_matrices_laboriously()
result = mymodel2((y, X), data=None, ...)
other_result = mymodel1(X, data=None, ...)
Of course in the long run they might want to throw away that
build_matrices_laboriously()
function and start using formulas,
but they aren’t forced to just to start using your new interface.
Working with metadata¶
Once you’ve started using Patsy to handle formulas, you’ll probably
want to take advantage of the metadata that Patsy provides, so that
you can display regression coefficients by name and so forth. Design
matrices processed by Patsy always have a .design_info
attribute which contains lots of information about the design: see
DesignInfo
for details.
Predictions¶
Another nice feature is making predictions on new data. But this requires that we can take in new data, and transform it to create a new X matrix. Or if we want to compute the likelihood of our model on new data, we need both new X and y matrices.
This is also easily done with Patsy – first fetch the relevant
DesignMatrixBuilder
objects by doing
input_data.design_info.builder
, and then pass them to
build_design_matrices()
along with the new data.
Example¶
Here’s a simplified class for doing ordinary leastsquares regression, demonstrating the above techniques:
Warning
This code has not been validated for numerical correctness.
import numpy as np
from patsy import dmatrices, build_design_matrices
class LM(object):
"""An example ordinary least squares linear model class, analogous to R's
lm() function. Don't use this in real life, it isn't properly tested."""
def __init__(self, formula_like, data={}):
y, x = dmatrices(formula_like, data, 1)
self.nobs = x.shape[0]
self.betas, self.rss, _, _ = np.linalg.lstsq(x, y)
self._y_design_info = y.design_info
self._x_design_info = x.design_info
def __repr__(self):
summary = ("Ordinary leastsquares regression\n"
" Model: %s ~ %s\n"
" Regression (beta) coefficients:\n"
% (self._y_design_info.describe(),
self._x_design_info.describe()))
for name, value in zip(self._x_design_info.column_names, self.betas):
summary += " %s: %0.3g\n" % (name, value[0])
return summary
def predict(self, new_data):
(new_x,) = build_design_matrices([self._x_design_info.builder],
new_data)
return np.dot(new_x, self.betas)
def loglik(self, new_data):
(new_y, new_x) = build_design_matrices([self._y_design_info.builder,
self._x_design_info.builder],
new_data)
new_pred = np.dot(new_x, self.betas)
sigma2 = self.rss / self.nobs
# It'd be more elegant to use scipy.stats.norm.logpdf here, but adding
# a dependency on scipy makes the docs build more complicated:
Z = 0.5 * np.log(2 * np.pi * sigma2)
return Z + 0.5 * (new_y  new_x) ** 2/sigma2
And here’s how it can be used:
In [2]: from patsy import demo_data
In [3]: data = demo_data("x", "y", "a")
# Old and boring approach (but it still works):
In [4]: X = np.column_stack(([1] * len(data["y"]), data["x"]))
In [5]: LM((data["y"], X))
Out[5]:
Ordinary leastsquares regression
Model: y0 ~ x0 + x1
Regression (beta) coefficients:
x0: 0.677
x1: 0.217
# Fancy new way:
In [6]: m = LM("y ~ x", data)
In [7]: m
Out[7]:
Ordinary leastsquares regression
Model: y ~ 1 + x
Regression (beta) coefficients:
Intercept: 0.677
x: 0.217
In [8]: m.predict({"x": [10, 20, 30]})
Out[8]:
array([[1.48944498],
[3.65620297],
[5.82296096]])
In [9]: m.loglik(data)
Out[9]:
array([[ 0.28884193, 1.46289596],
[ 2.64235743, 0.8254485 ],
[ 2.44930737, 2.36666465],
[ 0.90233651, 6.24317017],
[ 1.58762894, 5.56817766],
[ 0.65148056, 10.80114045]])
In [10]: m.loglik({"x": [10, 20, 30], "y": [1, 2, 3]})
Out[10]:
array([[ 7.39939265, 215.51261221],
[ 16.29311998, 861.19721649],
[ 28.74433824, 1937.33822362]])
# Your users get support for categorical predictors for free:
In [11]: LM("y ~ a", data)
Out[11]:
Ordinary leastsquares regression
Model: y ~ 1 + a
Regression (beta) coefficients:
Intercept: 0.33
a[T.a2]: 0.241
# And variable transformations too:
In [12]: LM("y ~ np.log(x ** 2)", data)
Out[12]:
Ordinary leastsquares regression
Model: y ~ 1 + np.log(x ** 2)
Regression (beta) coefficients:
Intercept: 0.399
np.log(x ** 2): 0.148
Other cool tricks¶
If you want to compute ANOVAs, then check out
DesignInfo.term_name_slices
, DesignInfo.slice()
.
If you support linear hypothesis tests or otherwise allow your users
to specify linear constraints on model parameters, consider taking
advantage of DesignInfo.linear_constraint()
.
Extending the formula syntax¶
The above documentation assumes that you have a relatively simple model that can be described by one or two matrices (plus whatever other arguments you take). This covers many of the most popular models, but it’s definitely not sufficient for every model out there.
Internally, Patsy is designed to be very flexible – for example,
it’s quite straightforward to add custom operators to the formula
parser, or otherwise extend the formula evaluation machinery. (Heck,
it only took an hour or two to repurpose it for a totally different
purpose, parsing linear constraints.) But extending Patsy in a
more fundamental way then this will require just a wee bit more complicated
API than just calling dmatrices()
, and for this initial release,
we’ve been busy enough getting the basics working that we haven’t yet
taken the time to pin down a public extension API we can support.
So, if you want something fancier – please give us a nudge, it’s entirely likely we can work something out.
Differences between R and Patsy formulas¶
Patsy has a very high degree of compatibility with R. Almost any formula you would use in R will also work in Patsy – with a few caveats.
Note
All R quirks described herein were last verified with R 2.15.0.
Differences from R:
Most obviously, we both support using arbitrary code to perform variable transformations, but in Patsy this code is written in Python, not R.
Patsy has no
%in%
. In R,a %in% b
is identical tob:a
. Patsy only supports theb:a
version of this syntax.In Patsy, only
**
can be used for exponentiation. In R, both^
and**
can be used for exponentiation, i.e., you can write either(a + b)^2
or(a + b)**2
. In Patsy (as in Python generally), only**
indicates exponentiation;^
is ignored by the parser (and if present, will be interpreted as a call to the Python binary XOR operator).In Patsy, the lefthand side of a formula uses the same evaluation rules as the righthand side. In R, the left hand side is treated as R code, so a formula like
y1 + y2 ~ x1 + x2
actually regresses the sum ofy1
andy2
onto the set of predictorsx1
andx2
. In Patsy, the only difference between the lefthand side and the righthand side is that there is no automatic intercept added to the lefthand side. (In this regard Patsy is similar to the R enhanced formula package Formula.)Patsy produces a different column ordering for formulas involving numeric predictors. In R, there are two rules for term ordering: first, lowerorder interactions are sorted before higherorder interactions, and second, interactions of the same order are listed in whatever order they appeared in the formula. In Patsy, we add another rule: terms are first grouped together based on which numeric factors they include. Then within each group, we use the same ordering as R.
Patsy has more rigorous handling of the presence or absence of the intercept term. In R, the rules for when deciding whether to include an intercept are somewhat idiosyncratic and can ignore things like parentheses. To understand the difference, first consider the formula
a + (b  a)
. In both Patsy and R, we first evaluate the(b  a)
part; since there is noa
term to remove, this simplifies to justb
. We then evaluatea + b
: the end result is a model which contains ana
term in it.Now consider the formula
1 + (b  1)
. In Patsy, this is analogous to the case above: first(b  1)
is reduced to justb
, and then1 + b
produces a model with intercept included. In R, the parentheses are ignored, and1 + (b  1)
gives a model that does not include the intercept.This can be slightly more confusing when it comes to the implicit intercept term. In Patsy, this is handled exactly as if the righthand side of each formula has an invisible
"1 +"
inserted at the beginning. Therefore in Patsy, these formulas are different:# Python: dmatrices("y ~ b  1") # equivalent to 1 + b  1: no intercept dmatrices("y ~ (b  1)") # equivalent to 1 + (b  1): has intercept
In R, these two formulas are equivalent.
Patsy has a more accurate algorithm for deciding whether to use a full or reducedrank coding scheme for categorical factors. There are two situations in which R’s coding algorithm for categorical variables can become confused and produce over or underspecified model matrices. Patsy, so far as we are aware, produces correctly specified matrices in all cases. It’s unlikely that you’ll run into these in actual usage, but they’re worth mentioning. To illustrate, let’s define
a
andb
as categorical predictors, each with 2 levels:# R: > a < factor(c("a1", "a1", "a2", "a2")) > b < factor(c("b1", "b2", "b1", "b2"))
The first problem occurs for formulas like
1 + a:b
. This produces a model matrix with rank 4, just like many other formulas that includea:b
, such as0 + a:b
,1 + a + a:b
, anda*b
:# R: > qr(model.matrix(~ 1 + a:b))$rank [1] 4
However, the matrix produced for this formula has 5 columns, meaning that it contains redundant overspecification:
# R: > mat < model.matrix(~ 1 + a:b) > ncol(mat) [1] 5
The underlying problem is that R’s algorithm does not pay attention to ‘nonlocal’ redundancies – it will adjust its coding to avoid a redundancy between two terms of degreen, or a term of degreen and one of degree(n+1), but it is blind to a redundancy between a term of degreen and one of degree(n+2), as we have here.
Patsy’s algorithm has no such limitation:
# Python: In [4]: a = ["a1", "a1", "a2", "a2"] In [5]: b = ["b1", "b2", "b1", "b2"] In [6]: mat = dmatrix("1 + a:b") In [7]: mat.shape[1] Out[7]: 4
To produce this result, it codes
a:b
uses the same columns that would be used to codeb + a:b
in the formula"1 + b + a:b"
.The second problem occurs for formulas involving numeric predictors. Effectively, when determining coding schemes, R assumes that all factors are categorical. So for the formula
0 + a:c + a:b
, R will notice that if it used a fullrank coding for thec
andb
factors, then both terms would be collinear witha
, and thus each other. Therefore, it encodesc
with a fullrank encoding, and uses a reducedrank encoding forb
. (And the0 +
lets it avoid the previous bug.) So far, so good.But now consider the formula
0 + a:x + a:b
, wherex
is numeric. Here,a:x
anda:b
will not be collinear, even if we do use a fullrank encoding forb
. Therefore, we should use a fullrank encoding forb
, and produce a model matrix with 6 columns. But in fact, R gives us only 4:# R: > x < c(1, 2, 3, 4) > mat < model.matrix(~ 0 + a:x + a:b) > ncol(mat) [1] 4
The problem is that it cannot tell the difference between
0 + a:x + a:b
and0 + a:c + a:b
: it uses the same coding for both, whether it’s appropriate or not.(The alert reader might wonder whether this bug could be triggered by a simpler formula, like
0 + x + b
. It turns out that R’s codedo_modelmatrix
function has a specialcase where for firstorder interactions only, it will peek at the type of the data before deciding on a coding scheme.)Patsy always checks whether each factor is categorical or numeric before it makes coding decisions, and thus handles this case correctly:
# Python: In [8]: x = [1, 2, 3, 4] In [9]: mat = dmatrix("0 + a:x + a:b") In [10]: mat.shape[1] Out[10]: 6
Python 2 versus Python 3¶
The biggest difference between Python 2 and Python 3 is in their
string handling, and this is particularly relevant to Patsy since
it parses user input. We follow a simple rule: input to Patsy
should always be of type str. That means that on Python 2, you
should pass bytestrings (not unicode), and on Python 3, you should
pass unicode strings (not bytestrings). Similarly, when Patsy
passes text back (e.g. DesignInfo.column_names
), it’s always
in the form of a str.
In addition to this being the most convenient for users (you never
need to use any b”weird” u”prefixes” when writing a formula string),
it’s actually a necessary consequence of a deeper change in the Python
language: in Python 2, Python code itself is represented as
bytestrings, and that’s the only form of input accepted by the
tokenize
module. On the other hand, Python 3’s tokenizer and
parser use unicode, and since Patsy processes Python code, it has
to follow suit.
patsy
API reference¶
This is a complete reference for everything you get when you import patsy.
Basic API¶

patsy.
dmatrix
(formula_like, data={}, eval_env=0, NA_action='drop', return_type='matrix')¶ Construct a single design matrix given a formula_like and data.
Parameters:  formula_like – An object that can be used to construct a design matrix. See below.
 data – A dictlike object that can be used to look up variables referenced in formula_like.
 eval_env – Either a
EvalEnvironment
which will be used to look up any variables referenced in formula_like that cannot be found in data, or else a depth represented as an integer which will be passed toEvalEnvironment.capture()
.eval_env=0
means to use the context of the function callingdmatrix()
for lookups. If calling this function from a library, you probably wanteval_env=1
, which means that variables should be resolved in your caller’s namespace.  NA_action – What to do with rows that contain missing values. You can
"drop"
them,"raise"
an error, or for customization, pass anNAAction
object. SeeNAAction
for details on what values count as ‘missing’ (and how to alter this).  return_type – Either
"matrix"
or"dataframe"
. See below.
The formula_like can take a variety of forms. You can use any of the following:
 (The most common option) A formula string like
"x1 + x2"
(fordmatrix()
) or"y ~ x1 + x2"
(fordmatrices()
). For details see How formulas work.  A
ModelDesc
, which is a Python object representation of a formula. See How formulas work and Model specification for experts and computers for details.  A
DesignInfo
.  An object that has a method called
__patsy_get_model_desc__()
. For details see Model specification for experts and computers.  A numpy array_like (for
dmatrix()
) or a tuple (array_like, array_like) (fordmatrices()
). These will have metadata added, representation normalized, and then be returned directly. In this case data and eval_env are ignored. There is special handling for two cases:DesignMatrix
objects will have theirDesignInfo
preserved. This allows you to set up custom column names and term information even if you aren’t using the rest of the patsy machinery.pandas.DataFrame
orpandas.Series
objects will have their (row) indexes checked. If two are passed in, their indexes must be aligned. Ifreturn_type="dataframe"
, then their indexes will be preserved on the output.
Regardless of the input, the return type is always either:
 A
DesignMatrix
, ifreturn_type="matrix"
(the default)  A
pandas.DataFrame
, ifreturn_type="dataframe"
.
The actual contents of the design matrix is identical in both cases, and in both cases a
DesignInfo
object will be available in a.design_info
attribute on the return value. However, forreturn_type="dataframe"
, any pandas indexes on the input (either in data or directly passed through formula_like) will be preserved, which may be useful for e.g. timeseries models.New in version 0.2.0: The
NA_action
argument.

patsy.
dmatrices
(formula_like, data={}, eval_env=0, NA_action='drop', return_type='matrix')¶ Construct two design matrices given a formula_like and data.
This function is identical to
dmatrix()
, except that it requires (and returns) two matrices instead of one. By convention, the first matrix is the “outcome” or “y” data, and the second is the “predictor” or “x” data.See
dmatrix()
for details.

patsy.
incr_dbuilders
(formula_like, data_iter_maker, eval_env=0, NA_action='drop')¶ Construct two design matrix builders incrementally from a large data set.
incr_dbuilders()
is toincr_dbuilder()
asdmatrices()
is todmatrix()
. Seeincr_dbuilder()
for details.

patsy.
incr_dbuilder
(formula_like, data_iter_maker, eval_env=0, NA_action='drop')¶ Construct a design matrix builder incrementally from a large data set.
Parameters:  formula_like – Similar to
dmatrix()
, except that explicit matrices are not allowed. Must be a formula string, aModelDesc
, aDesignInfo
, or an object with a__patsy_get_model_desc__
method.  data_iter_maker – A zeroargument callable which returns an iterator over dictlike data objects. This must be a callable rather than a simple iterator because sufficiently complex formulas may require multiple passes over the data (e.g. if there are nested stateful transforms).
 eval_env – Either a
EvalEnvironment
which will be used to look up any variables referenced in formula_like that cannot be found in data, or else a depth represented as an integer which will be passed toEvalEnvironment.capture()
.eval_env=0
means to use the context of the function callingincr_dbuilder()
for lookups. If calling this function from a library, you probably wanteval_env=1
, which means that variables should be resolved in your caller’s namespace.  NA_action – An
NAAction
object or string, used to determine what values count as ‘missing’ for purposes of determining the levels of categorical factors.
Returns: Tip: for data_iter_maker, write a generator like:
def iter_maker(): for data_chunk in my_data_store: yield data_chunk
and pass iter_maker (not iter_maker()).
New in version 0.2.0: The
NA_action
argument. formula_like – Similar to

exception
patsy.
PatsyError
(message, origin=None)¶ This is the main error type raised by Patsy functions.
In addition to the usual Python exception features, you can pass a second argument to this function specifying the origin of the error; this is included in any error message, and used to help the user locate errors arising from malformed formulas. This second argument should be an
Origin
object, or else an arbitrary object with a.origin
attribute. (If it is neither of these things, then it will simply be ignored.)For ordinary display to the user with default formatting, use
str(exc)
. If you want to do something cleverer, you can use the.message
and.origin
attributes directly. (The latter may be None.)
Convenience utilities¶

patsy.
balanced
(factor_name=num_levels[, factor_name=num_levels, ..., repeat=1])¶ Create simple balanced factorial designs for testing.
Given some factor names and the number of desired levels for each, generates a balanced factorial design in the form of a data dictionary. For example:
In [1]: balanced(a=2, b=3) Out[1]: {'a': ['a1', 'a1', 'a1', 'a2', 'a2', 'a2'], 'b': ['b1', 'b2', 'b3', 'b1', 'b2', 'b3']}
By default it produces exactly one instance of each combination of levels, but if you want multiple replicates this can be accomplished via the repeat argument:
In [2]: balanced(a=2, b=2, repeat=2) Out[2]: {'a': ['a1', 'a1', 'a2', 'a2', 'a1', 'a1', 'a2', 'a2'], 'b': ['b1', 'b2', 'b1', 'b2', 'b1', 'b2', 'b1', 'b2']}

patsy.
demo_data
(*names, nlevels=2, min_rows=5)¶ Create simple categorical/numerical demo data.
Pass in a set of variable names, and this function will return a simple data set using those variable names.
Names whose first letter falls in the range “a” through “m” will be made categorical (with nlevels levels). Those that start with a “p” through “z” are numerical.
We attempt to produce a balanced design on the categorical variables, repeating as necessary to generate at least min_rows data points. Categorical variables are returned as a list of strings.
Numerical data is generated by sampling from a normal distribution. A fixed random seed is used, so that identical calls to demo_data() will produce identical results. Numerical data is returned in a numpy array.
Example:
Design metadata¶

class
patsy.
DesignInfo
(column_names, factor_infos=None, term_codings=None)¶ A DesignInfo object holds metadata about a design matrix.
This is the main object that Patsy uses to pass metadata about a design matrix to statistical libraries, in order to allow further downstream processing like intelligent tests, prediction on new data, etc. Usually encountered as the .design_info attribute on design matrices.
Here’s an example of the most common way to get a
DesignInfo
:In [3]: mat = dmatrix("a + x", demo_data("a", "x", nlevels=3)) In [4]: di = mat.design_info

column_names
¶ The names of each column, represented as a list of strings in the proper order. Guaranteed to exist.
In [5]: di.column_names Out[5]: ['Intercept', 'a[T.a2]', 'a[T.a3]', 'x']

column_name_indexes
¶ An
OrderedDict
mapping column names (as strings) to column indexes (as integers). Guaranteed to exist and to be sorted from low to high.In [6]: di.column_name_indexes Out[6]: OrderedDict([('Intercept', 0), ('a[T.a2]', 1), ('a[T.a3]', 2), ('x', 3)])

term_names
¶ The names of each term, represented as a list of strings in the proper order. Guaranteed to exist. There is a onetomany relationship between columns and terms – each term generates one or more columns.
In [7]: di.term_names Out[7]: ['Intercept', 'a', 'x']

term_name_slices
¶ An
OrderedDict
mapping term names (as strings) to Pythonslice()
objects indicating which columns correspond to each term. Guaranteed to exist. The slices are guaranteed to be sorted from left to right and to cover the whole range of columns with no overlaps or gaps.In [8]: di.term_name_slices Out[8]: OrderedDict([('Intercept', slice(0, 1, None)), ('a', slice(1, 3, None)), ('x', slice(3, 4, None))])

terms
¶ A list of
Term
objects representing each term. May be None, for example if a user passed in a plain preassembled design matrix rather than using the Patsy machinery.In [9]: di.terms Out[9]: [Term([]), Term([EvalFactor('a')]), Term([EvalFactor('x')])] In [10]: [term.name() for term in di.terms] Out[10]: ['Intercept', 'a', 'x']

term_slices
¶ An
OrderedDict
mappingTerm
objects to Pythonslice()
objects indicating which columns correspond to which terms. Liketerms
, this may be None.In [11]: di.term_slices Out[11]: OrderedDict([(Term([]), slice(0, 1, None)), (Term([EvalFactor('a')]), slice(1, 3, None)), (Term([EvalFactor('x')]), slice(3, 4, None))])

factor_infos
¶ A dict mapping factor objects to
FactorInfo
objects providing information about each factor. Liketerms
, this may be None.In [12]: di.term_slices Out[12]: OrderedDict([(Term([]), slice(0, 1, None)), (Term([EvalFactor('a')]), slice(1, 3, None)), (Term([EvalFactor('x')]), slice(3, 4, None))])

term_codings
¶ An
OrderedDict
mapping eachTerm
object to a list ofSubtermInfo
objects which together describe how this term is encoded in the final design matrix. Liketerms
, this may be None.In [13]: di.term_codings Out[13]: OrderedDict([(Term([]), [SubtermInfo(factors=(), contrast_matrices={}, num_columns=1)]), (Term([EvalFactor('a')]), [SubtermInfo(factors=(EvalFactor('a'),), contrast_matrices={EvalFactor('a'): ContrastMatrix(array([[ 0., 0.], [ 1., 0.], [ 0., 1.]]), ['[T.a2]', '[T.a3]'])}, num_columns=2)]), (Term([EvalFactor('x')]), [SubtermInfo(factors=(EvalFactor('x'),), contrast_matrices={}, num_columns=1)])])

builder
¶ In versions of patsy before 0.4.0, this returned a
DesignMatrixBuilder
object which could be passed tobuild_design_matrices()
. Starting in 0.4.0,build_design_matrices()
now acceptsDesignInfo
objects directly, and writingf(design_info.builder)
is now a deprecated alias for simply writingf(design_info)
.
A number of convenience methods are also provided that take advantage of the above metadata:

describe
()¶ Returns a humanreadable string describing this design info.
Example:
In [1]: y, X = dmatrices("y ~ x1 + x2", demo_data("y", "x1", "x2")) In [2]: y.design_info.describe() Out[2]: 'y' In [3]: X.design_info.describe() Out[3]: '1 + x1 + x2'
Warning
There is no guarantee that the strings returned by this function can be parsed as formulas, or that if they can be parsed as a formula that they will produce a model equivalent to the one you started with. This function produces a besteffort description intended for humans to read.

linear_constraint
(constraint_likes)¶ Construct a linear constraint in matrix form from a (possibly symbolic) description.
Possible inputs:
 A dictionary which is taken as a set of equality constraint. Keys can be either string column names, or integer column indexes.
 A string giving a arithmetic expression referring to the matrix columns by name.
 A list of such strings which are ANDed together.
 A tuple (A, b) where A and b are array_likes, and the constraint is Ax = b. If necessary, these will be coerced to the proper dimensionality by appending dimensions with size 1.
The stringbased language has the standard arithmetic operators, / * +  and parentheses, plus “=” is used for equality and ”,” is used to AND together multiple constraint equations within a string. You can If no = appears in some expression, then that expression is assumed to be equal to zero. Division is always floatbased, even if
__future__.true_division
isn’t in effect.Returns a
LinearConstraint
object.Examples:
di = DesignInfo(["x1", "x2", "x3"]) # Equivalent ways to write x1 == 0: di.linear_constraint({"x1": 0}) # by name di.linear_constraint({0: 0}) # by index di.linear_constraint("x1 = 0") # string based di.linear_constraint("x1") # can leave out "= 0" di.linear_constraint("2 * x1 = (x1 + 2 * x1) / 3") di.linear_constraint(([1, 0, 0], 0)) # constraint matrices # Equivalent ways to write x1 == 0 and x3 == 10 di.linear_constraint({"x1": 0, "x3": 10}) di.linear_constraint({0: 0, 2: 10}) di.linear_constraint({0: 0, "x3": 10}) di.linear_constraint("x1 = 0, x3 = 10") di.linear_constraint("x1, x3 = 10") di.linear_constraint(["x1", "x3 = 0"]) # list of strings di.linear_constraint("x1 = 0, x3  10 = x1") di.linear_constraint([[1, 0, 0], [0, 0, 1]], [0, 10]) # You can also chain together equalities, just like Python: di.linear_constraint("x1 = x2 = 3")

slice
(columns_specifier)¶ Locate a subset of design matrix columns, specified symbolically.
A patsy design matrix has two levels of structure: the individual columns (which are named), and the terms in the formula that generated those columns. This is a onetomany relationship: a single term may span several columns. This method provides a userfriendly API for locating those columns.
(While we talk about columns here, this is probably most useful for indexing into other arrays that are derived from the design matrix, such as regression coefficients or covariance matrices.)
The columns_specifier argument can take a number of forms:
 A term name
 A column name
 A
Term
object  An integer giving a raw index
 A raw slice object
In all cases, a Python
slice()
object is returned, which can be used directly for indexing.Example:
y, X = dmatrices("y ~ a", demo_data("y", "a", nlevels=3)) betas = np.linalg.lstsq(X, y)[0] a_betas = betas[X.design_info.slice("a")]
(If you want to look up a single individual column by name, use
design_info.column_name_indexes[name]
.)

subset
(which_terms)¶ Create a new
DesignInfo
for design matrices that contain a subset of the terms that the currentDesignInfo
does.For example, if
design_info
has termsx
,y
, andz
, then:design_info2 = design_info.subset(["x", "z"])
will return a new DesignInfo that can be used to construct design matrices with only the columns corresponding to the terms
x
andz
. After we do this, then in general these two expressions will return the same thing (here we assume thatx
,y
, andz
each generate a single column of the output):build_design_matrix([design_info], data)[0][:, [0, 2]] build_design_matrix([design_info2], data)[0]
However, a critical difference is that in the second case,
data
need not contain any values fory
. This is very useful when doing prediction using a subset of a model, in which situation R usually forces you to specify dummy values fory
.If using a formula to specify the terms to include, remember that like any formula, the intercept term will be included by default, so use
0
or1
in your formula if you want to avoid this.This method can also be used to reorder the terms in your design matrix, in case you want to do that for some reason. I can’t think of any.
Note that this method will generally not produce the same result as creating a new model directly. Consider these DesignInfo objects:
design1 = dmatrix("1 + C(a)", data) design2 = design1.subset("0 + C(a)") design3 = dmatrix("0 + C(a)", data)
Here
design2
anddesign3
will both produce design matrices that contain an encoding ofC(a)
without any intercept term. Butdesign3
uses a fullrank encoding for the categorical termC(a)
, whiledesign2
uses the same reducedrank encoding asdesign1
.Parameters: which_terms – The terms which should be kept in the new DesignMatrixBuilder
. If this is a string, then it is parsed as a formula, and then the names of the resulting terms are taken as the terms to keep. If it is a list, then it can contain a mixture of term names (as strings) andTerm
objects.

classmethod
from_array
(array_like, default_column_prefix='column')¶ Find or construct a DesignInfo appropriate for a given array_like.
If the input array_like already has a
.design_info
attribute, then it will be returned. Otherwise, a new DesignInfo object will be constructed, using names either taken from the array_like (e.g., for a pandas DataFrame with named columns), or constructed using default_column_prefix.This is how
dmatrix()
(for example) creates a DesignInfo object if an arbitrary matrix is passed in.Parameters:  array_like – An ndarray or pandas container.
 default_column_prefix – If it’s necessary to invent column names, then this will be used to construct them.
Returns: a DesignInfo object


class
patsy.
FactorInfo
(factor, type, state, num_columns=None, categories=None)¶ A FactorInfo object is a simple class that provides some metadata about the role of a factor within a model.
DesignInfo.factor_infos
is a dictionary which maps factor objects to FactorInfo objects for each factor in the model.New in version 0.4.0.
Attributes:

factor
¶ The factor object being described.

type
¶ The type of the factor – either the string
"numerical"
or the string"categorical"
.

state
¶ An opaque object which holds the state needed to evaluate this factor on new data (e.g., for prediction). See
factor_protocol.eval()
.

num_columns
¶ For numerical factors, the number of columns this factor produces. For categorical factors, this attribute will always be
None
.

categories
¶ For categorical factors, a tuple of the possible categories this factor takes on, in order. For numerical factors, this attribute will always be
None
.


class
patsy.
SubtermInfo
(factors, contrast_matrices, num_columns)¶ A SubtermInfo object is a simple metadata container describing a single primitive interaction and how it is coded in our design matrix. Our final design matrix is produced by coding each primitive interaction in order from left to right, and then stacking the resulting columns. For each
Term
, we have one or more of these objects which describe how that term is encoded.DesignInfo.term_codings
is a dictionary which maps term objects to lists of SubtermInfo objects.To code a primitive interaction, the following steps are performed:
 Evaluate each factor on the provided data.
 Encode each factor into one or more protocolumns. For numerical factors, these protocolumns are identical to whatever the factor evaluates to; for categorical factors, they are encoded using a specified contrast matrix.
 Form all pairwise, elementwise products between protocolumns generated
by different factors. (For example, if factor 1 generated protocolumns
A and B, and factor 2 generated protocolumns C and D, then our final
columns are
A * C
,B * C
,A * D
,B * D
.)  The resulting columns are stored directly into the final design matrix.
Sometimes multiple primitive interactions are needed to encode a single term; this occurs, for example, in the formula
"1 + a:b"
whena
andb
are categorical. See From terms to matrices for full details.New in version 0.4.0.
Attributes:

factors
¶ The factors which appear in this subterm’s interaction.

contrast_matrices
¶ A dict mapping factor objects to
ContrastMatrix
objects, describing how each categorical factor in this interaction is coded.

num_columns
¶ The number of design matrix columns which this interaction generates.

class
patsy.
DesignMatrix
¶ A simple numpy array subclass that carries design matrix metadata.

design_info
¶ A
DesignInfo
object containing metadata about this design matrix.
This class also defines a fancy __repr__ method with labeled columns. Otherwise it is identical to a regular numpy ndarray.
Warning
You should never check for this class using
isinstance()
. Limitations of the numpy API mean that it is impossible to prevent the creation of numpy arrays that have type DesignMatrix, but that are not actually design matrices (and such objects will behave like regular ndarrays in every way). Instead, check for the presence of a.design_info
attribute – this will be present only on “real” DesignMatrix objects.
static
__new__
(input_array, design_info=None, default_column_prefix='column')¶ Create a DesignMatrix, or cast an existing matrix to a DesignMatrix.
A call like:
DesignMatrix(my_array)
will convert an arbitrary array_like object into a DesignMatrix.
The return from this function is guaranteed to be a twodimensional ndarray with a realvalued floating point dtype, and a
.design_info
attribute which matches its shape. If the design_info argument is not given, then one is created viaDesignInfo.from_array()
using the given default_column_prefix.Depending on the input array, it is possible this will pass through its input unchanged, or create a view.

Stateful transforms¶
Patsy comes with a number of stateful transforms built in:

patsy.
center
(x)¶ A stateful transform that centers input data, i.e., subtracts the mean.
If input has multiple columns, centers each column separately.
Equivalent to
standardize(x, rescale=False)

patsy.
standardize
(x, center=True, rescale=True, ddof=0)¶ A stateful transform that standardizes input data, i.e. it subtracts the mean and divides by the sample standard deviation.
Either centering or rescaling or both can be disabled by use of keyword arguments. The ddof argument controls the delta degrees of freedom when computing the standard deviation (cf.
numpy.std()
). The default ofddof=0
produces the maximum likelihood estimate; useddof=1
if you prefer the square root of the unbiased estimate of the variance.If input has multiple columns, standardizes each column separately.
Note
This function computes the mean and standard deviation using a memoryefficient online algorithm, making it suitable for use with large incrementally processed datasets.

patsy.
scale
(x, center=True, rescale=True, ddof=0)¶ An alias for
standardize()
, for R compatibility.
Finally, this is not itself a stateful transform, but it’s useful if you want to define your own:

patsy.
stateful_transform
(class_)¶ Create a stateful transform callable object from a class that fulfills the stateful transform protocol.
Handling categorical data¶

class
patsy.
Treatment
(reference=None)¶ Treatment coding (also known as dummy coding).
This is the default coding.
For reducedrank coding, one level is chosen as the “reference”, and its mean behaviour is represented by the intercept. Each column of the resulting matrix represents the difference between the mean of one level and this reference level.
For fullrank coding, classic “dummy” coding is used, and each column of the resulting matrix represents the mean of the corresponding level.
The reference level defaults to the first level, or can be specified explicitly.
# reduced rank In [1]: dmatrix("C(a, Treatment)", balanced(a=3)) Out[1]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment)[T.a2] C(a, Treatment)[T.a3] 1 0 0 1 1 0 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Treatment)' (columns 1:3) # full rank In [2]: dmatrix("0 + C(a, Treatment)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Treatment)[a1] C(a, Treatment)[a2] C(a, Treatment)[a3] 1 0 0 0 1 0 0 0 1 Terms: 'C(a, Treatment)' (columns 0:3) # Setting a reference level In [3]: dmatrix("C(a, Treatment(1))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment(1))[T.a1] C(a, Treatment(1))[T.a3] 1 1 0 1 0 0 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Treatment(1))' (columns 1:3) In [4]: dmatrix("C(a, Treatment('a2'))", balanced(a=3)) Out[4]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment('a2'))[T.a1] C(a, Treatment('a2'))[T.a3] 1 1 0 1 0 0 1 0 1 Terms: 'Intercept' (column 0) "C(a, Treatment('a2'))" (columns 1:3)
Equivalent to R
contr.treatment
. The R documentation suggests that usingTreatment(reference=1)
will produce contrasts that are “equivalent to those produced by many (but not all) SAS procedures”.

class
patsy.
Diff
¶ Backward difference coding.
This coding scheme is useful for ordered factors, and compares the mean of each level with the preceding level. So you get the second level minus the first, the third level minus the second, etc.
For fullrank coding, a standard intercept term is added (which gives the mean value for the first level).
Examples:
# Reduced rank In [1]: dmatrix("C(a, Diff)", balanced(a=3)) Out[1]: DesignMatrix with shape (3, 3) Intercept C(a, Diff)[D.a1] C(a, Diff)[D.a2] 1 0.66667 0.33333 1 0.33333 0.33333 1 0.33333 0.66667 Terms: 'Intercept' (column 0) 'C(a, Diff)' (columns 1:3) # Full rank In [2]: dmatrix("0 + C(a, Diff)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Diff)[D.a1] C(a, Diff)[D.a2] C(a, Diff)[D.a3] 1 0.66667 0.33333 1 0.33333 0.33333 1 0.33333 0.66667 Terms: 'C(a, Diff)' (columns 0:3)

class
patsy.
Poly
(scores=None)¶ Orthogonal polynomial contrast coding.
This coding scheme treats the levels as ordered samples from an underlying continuous scale, whose effect takes an unknown functional form which is Taylordecomposed into the sum of a linear, quadratic, etc. components.
For reducedrank coding, you get a linear column, a quadratic column, etc., up to the number of levels provided.
For fullrank coding, the same scheme is used, except that the zeroorder constant polynomial is also included. I.e., you get an intercept column included as part of your categorical term.
By default the levels are treated as equally spaced, but you can override this by providing a value for the scores argument.
Examples:
# Reduced rank In [1]: dmatrix("C(a, Poly)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Poly).Linear C(a, Poly).Quadratic C(a, Poly).Cubic 1 0.67082 0.5 0.22361 1 0.22361 0.5 0.67082 1 0.22361 0.5 0.67082 1 0.67082 0.5 0.22361 Terms: 'Intercept' (column 0) 'C(a, Poly)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Poly)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Poly).Constant C(a, Poly).Linear C(a, Poly).Quadratic 1 0.70711 0.40825 1 0.00000 0.81650 1 0.70711 0.40825 Terms: 'C(a, Poly)' (columns 0:3) # Explicit scores In [3]: dmatrix("C(a, Poly([1, 2, 10]))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Poly([1, 2, 10])).Linear C(a, Poly([1, 2, 10])).Quadratic 1 0.47782 0.66208 1 0.33447 0.74485 1 0.81229 0.08276 Terms: 'Intercept' (column 0) 'C(a, Poly([1, 2, 10]))' (columns 1:3)
This is equivalent to R’s
contr.poly
. (But note that in R, reduced rank encodings are always dummycoded, regardless of what contrast you have set.)

class
patsy.
Sum
(omit=None)¶ Deviation coding (also known as sumtozero coding).
Compares the mean of each level to the meanofmeans. (In a balanced design, compares the mean of each level to the overall mean.)
For fullrank coding, a standard intercept term is added.
One level must be omitted to avoid redundancy; by default this is the last level, but this can be adjusted via the omit argument.
Warning
There are multiple definitions of ‘deviation coding’ in use. Make sure this is the one you expect before trying to interpret your results!
Examples:
# Reduced rank In [1]: dmatrix("C(a, Sum)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Sum)[S.a1] C(a, Sum)[S.a2] C(a, Sum)[S.a3] 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 Terms: 'Intercept' (column 0) 'C(a, Sum)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Sum)", balanced(a=4)) Out[2]: DesignMatrix with shape (4, 4) C(a, Sum)[mean] C(a, Sum)[S.a1] C(a, Sum)[S.a2] C(a, Sum)[S.a3] 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 Terms: 'C(a, Sum)' (columns 0:4) # Omit a different level In [3]: dmatrix("C(a, Sum(1))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Sum(1))[S.a1] C(a, Sum(1))[S.a3] 1 1 0 1 1 1 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Sum(1))' (columns 1:3) In [4]: dmatrix("C(a, Sum('a1'))", balanced(a=3)) Out[4]: DesignMatrix with shape (3, 3) Intercept C(a, Sum('a1'))[S.a2] C(a, Sum('a1'))[S.a3] 1 1 1 1 1 0 1 0 1 Terms: 'Intercept' (column 0) "C(a, Sum('a1'))" (columns 1:3)
This is equivalent to R’s contr.sum.

class
patsy.
Helmert
¶ Helmert contrasts.
Compares the second level with the first, the third with the average of the first two, and so on.
For fullrank coding, a standard intercept term is added.
Warning
There are multiple definitions of ‘Helmert coding’ in use. Make sure this is the one you expect before trying to interpret your results!
Examples:
# Reduced rank In [1]: dmatrix("C(a, Helmert)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Helmert)[H.a2] C(a, Helmert)[H.a3] C(a, Helmert)[H.a4] 1 1 1 1 1 1 1 1 1 0 2 1 1 0 0 3 Terms: 'Intercept' (column 0) 'C(a, Helmert)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Helmert)", balanced(a=4)) Out[2]: DesignMatrix with shape (4, 4) Columns: ['C(a, Helmert)[H.intercept]', 'C(a, Helmert)[H.a2]', 'C(a, Helmert)[H.a3]', 'C(a, Helmert)[H.a4]'] Terms: 'C(a, Helmert)' (columns 0:4) (to view full data, use np.asarray(this_obj))
This is equivalent to R’s contr.helmert.

class
patsy.
ContrastMatrix
(matrix, column_suffixes)¶ A simple container for a matrix used for coding categorical factors.
Attributes:

matrix
¶ A 2d ndarray, where each column corresponds to one column of the resulting design matrix, and each row contains the entries for a single categorical variable level. Usually nbyn for a full rank coding or nby(n1) for a reduced rank coding, though other options are possible.

column_suffixes
¶ A list of strings to be appended to the factor name, to produce the final column names. E.g. for treatment coding the entries will look like
"[T.level1]"
.

Spline regression¶

patsy.
bs
(x, df=None, knots=None, degree=3, include_intercept=False, lower_bound=None, upper_bound=None)¶ Generates a Bspline basis for
x
, allowing nonlinear fits. The usual usage is something like:y ~ 1 + bs(x, 4)
to fit
y
as a smooth function ofx
, with 4 degrees of freedom given to the smooth.Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  degree – The degree of the spline to use.
 include_intercept – If
True
, then the resulting spline basis will span the intercept term (i.e., the constant function). IfFalse
(the default) then this will not be the case, which is useful for avoiding overspecification in models that include multiple spline terms and/or an intercept term.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
A spline with
degree=0
is piecewise constant with breakpoints at each knot, and the default knot positions are quantiles of the input. So if you find yourself in the situation of wanting to quantize a continuous variable intonum_bins
equalsized bins with a constant effect across each bin, you can usebs(x, num_bins  1, degree=0)
. (The 1
is because one degree of freedom will be taken by the intercept; alternatively, you could leave the intercept term out of your model and usebs(x, num_bins, degree=0, include_intercept=True)
.A spline with
degree=1
is piecewise linear with breakpoints at each knot.The default is
degree=3
, which gives a cubic bspline.This is a stateful transform (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
Note
This function is very similar to the R function of the same name. In cases where both return output at all (e.g., R’s
bs
will raise an error ifdegree=0
, while patsy’s will not), they should produce identical output given identical input and parameter settings.Warning
I’m not sure on what the proper handling of points outside the lower/upper bounds is, so for now attempting to evaluate a spline basis at such points produces an error. Patches gratefully accepted.
New in version 0.2.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.
cr
(x, df=None, knots=None, lower_bound=None, upper_bound=None, constraints=None)¶ Generates a natural cubic spline basis for
x
(with the option of absorbing centering or more general parameters constraints), allowing nonlinear fits. The usual usage is something like:y ~ 1 + cr(x, df=5, constraints='center')
to fit
y
as a smooth function ofx
, with 5 degrees of freedom given to the smooth, and centering constraint absorbed in the resulting design matrix. Note that in this example, due to the centering constraint, 6 knots will get computed from the input datax
to achieve 5 degrees of freedom.Note
This function reproduce the cubic regression splines ‘cr’ and ‘cs’ as implemented in the R package ‘mgcv’ (GAM modelling).
Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
 constraints – Either a 2d array defining general linear constraints
(that is
np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.
This is a stateful transforms (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
New in version 0.3.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.
cc
(x, df=None, knots=None, lower_bound=None, upper_bound=None, constraints=None)¶ Generates a cyclic cubic spline basis for
x
(with the option of absorbing centering or more general parameters constraints), allowing nonlinear fits. The usual usage is something like:y ~ 1 + cc(x, df=7, constraints='center')
to fit
y
as a smooth function ofx
, with 7 degrees of freedom given to the smooth, and centering constraint absorbed in the resulting design matrix. Note that in this example, due to the centering and cyclic constraints, 9 knots will get computed from the input datax
to achieve 7 degrees of freedom.Note
This function reproduce the cubic regression splines ‘cc’ as implemented in the R package ‘mgcv’ (GAM modelling).
Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
 constraints – Either a 2d array defining general linear constraints
(that is
np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.
This is a stateful transforms (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
New in version 0.3.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.
te
(s1, .., sn, constraints=None)¶ Generates smooth of several covariates as a tensor product of the bases of marginal univariate smooths
s1, .., sn
. The marginal smooths are required to transform input univariate data into some kind of smooth functions basis producing a 2d array output with the(i, j)
element corresponding to the value of thej
th basis function at thei
th data point. The resulting basis dimension is the product of the basis dimensions of the marginal smooths. The usual usage is something like:y ~ 1 + te(cr(x1, df=5), cc(x2, df=6), constraints='center')
to fit
y
as a smooth function of bothx1
andx2
, with a natural cubic spline forx1
marginal smooth and a cyclic cubic spline forx2
(and centering constraint absorbed in the resulting design matrix).Parameters: constraints – Either a 2d array defining general linear constraints (that is np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.Using this function requires scipy be installed.
Note
This function reproduce the tensor product smooth ‘te’ as implemented in the R package ‘mgcv’ (GAM modelling). See also ‘Generalized Additive Models’, Simon N. Wood, 2006, pp 158163
New in version 0.3.0.
Working with formulas programmatically¶

class
patsy.
Term
(factors)¶ The interaction between a collection of factor objects.
This is one of the basic types used in representing formulas, and corresponds to an expression like
"a:b:c"
in a formula string. For details, see How formulas work and Model specification for experts and computers.Terms are hashable and compare by value.
Attributes:

factors
¶ A tuple of factor objects.


patsy.
INTERCEPT
¶ This is a preinstantiated zerofactors
Term
object representing the intercept, useful for making your code clearer. Do remember though that this is not a singleton object, i.e., you should compare against it using==
, notis
.

class
patsy.
LookupFactor
(varname, force_categorical=False, contrast=None, levels=None, origin=None)¶ A simple factor class that simply looks up a named entry in the given data.
Useful for programatically constructing formulas, and as a simple example of the factor protocol. For details see Model specification for experts and computers.
Example:
dmatrix(ModelDesc([], [Term([LookupFactor("x")])]), {"x": [1, 2, 3]})
Parameters:  varname – The name of this variable; used as a lookup key in the passed in data dictionary/DataFrame/whatever.
 force_categorical – If True, then treat this factor as
categorical. (Equivalent to using
C()
in a regular formula, but of course you can’t do that with aLookupFactor
.  contrast – If given, the contrast to use; see
C()
. (Requiresforce_categorical=True
.)  levels – If given, the categorical levels; see
C()
. (Requiresforce_categorical=True
.)  origin – Either
None
, or theOrigin
of this factor for use in error reporting.
New in version 0.2.0: The
force_categorical
and related arguments.

class
patsy.
EvalFactor
(code, origin=None)¶ A factor class that executes arbitrary Python code and supports stateful transforms.
Parameters: code – A string containing a Python expression, that will be evaluated to produce this factor’s value. This is the standard factor class that is used when parsing formula strings and implements the standard stateful transform processing. See Stateful transforms and Model specification for experts and computers.
Two EvalFactor’s are considered equal (e.g., for purposes of redundancy detection) if they contain the same token stream. Basically this means that the source code must be identical except for whitespace:
assert EvalFactor("a + b") == EvalFactor("a+b") assert EvalFactor("a + b") != EvalFactor("b + a")

class
patsy.
ModelDesc
(lhs_termlist, rhs_termlist)¶ A simple container representing the termlists parsed from a formula.
This is a simple container object which has exactly the same representational power as a formula string, but is a Python object instead. You can construct one by hand, and pass it to functions like
dmatrix()
orincr_dbuilder()
that are expecting a formula string, but without having to do any messy string manipulation. For details see Model specification for experts and computers.Attributes:

lhs_termlist
¶ 
rhs_termlist
¶ Two termlists representing the left and righthand sides of a formula, suitable for passing to
design_matrix_builders()
.

Working with the Python execution environment¶

class
patsy.
EvalEnvironment
(namespaces, flags=0)¶ Represents a Python execution environment.
Encapsulates a namespace for variable lookup and set of __future__ flags.

classmethod
capture
(eval_env=0, reference=0)¶ Capture an execution environment from the stack.
If eval_env is already an
EvalEnvironment
, it is returned unchanged. Otherwise, we walk up the stack byeval_env + reference
steps and capture that function’s evaluation environment.For
eval_env=0
andreference=0
, the default, this captures the stack frame of the function that callscapture()
. Ifeval_env + reference
is 1, then we capture that function’s caller, etc.This somewhat complicated calling convention is designed to be convenient for functions which want to capture their caller’s environment by default, but also allow explicit environments to be specified. See the second example.
Example:
x = 1 this_env = EvalEnvironment.capture() assert this_env["x"] == 1 def child_func(): return EvalEnvironment.capture(1) this_env_from_child = child_func() assert this_env_from_child["x"] == 1
Example:
# This function can be used like: # my_model(formula_like, data) # > evaluates formula_like in caller's environment # my_model(formula_like, data, eval_env=1) # > evaluates formula_like in caller's caller's environment # my_model(formula_like, data, eval_env=my_env) # > evaluates formula_like in environment 'my_env' def my_model(formula_like, data, eval_env=0): eval_env = EvalEnvironment.capture(eval_env, reference=1) return model_setup_helper(formula_like, data, eval_env)
This is how
dmatrix()
works.

eval
(expr, source_name='<string>', inner_namespace={})¶ Evaluate some Python code in the encapsulated environment.
Parameters:  expr – A string containing a Python expression.
 source_name – A name for this string, for use in tracebacks.
 inner_namespace – A dictlike object that will be checked first when expr attempts to access any variables.
Returns: The value of expr.

namespace
¶ A dictlike object that can be used to look up variables accessible from the encapsulated environment.

subset
(names)¶ Creates a new, flat EvalEnvironment that contains only the variables specified.

with_outer_namespace
(outer_namespace)¶ Return a new EvalEnvironment with an extra namespace added.
This namespace will be used only for variables that are not found in any existing namespace, i.e., it is “outside” them all.

classmethod
Building design matrices¶

patsy.
design_matrix_builders
(termlists, data_iter_maker, eval_env, NA_action='drop')¶ Construct several
DesignInfo
objects from termlists.This is one of Patsy’s fundamental functions. This function and
build_design_matrices()
together form the API to the core formula interpretation machinery.Parameters:  termlists – A list of termlists, where each termlist is a list of
Term
objects which together specify a design matrix.  data_iter_maker – A zeroargument callable which returns an iterator over dictlike data objects. This must be a callable rather than a simple iterator because sufficiently complex formulas may require multiple passes over the data (e.g. if there are nested stateful transforms).
 eval_env – Either a
EvalEnvironment
which will be used to look up any variables referenced in termlists that cannot be found in data_iter_maker, or else a depth represented as an integer which will be passed toEvalEnvironment.capture()
.eval_env=0
means to use the context of the function callingdesign_matrix_builders()
for lookups. If calling this function from a library, you probably wanteval_env=1
, which means that variables should be resolved in your caller’s namespace.  NA_action – An
NAAction
object or string, used to determine what values count as ‘missing’ for purposes of determining the levels of categorical factors.
Returns: A list of
DesignInfo
objects, one for each termlist passed in.This function performs zero or more iterations over the data in order to sniff out any necessary information about factor types, set up stateful transforms, pick column names, etc.
See How formulas work for details.
New in version 0.2.0: The
NA_action
argument.New in version 0.4.0: The
eval_env
argument. termlists – A list of termlists, where each termlist is a list of

patsy.
build_design_matrices
(design_infos, data, NA_action='drop', return_type='matrix', dtype=dtype('float64'))¶ Construct several design matrices from
DesignMatrixBuilder
objects.This is one of Patsy’s fundamental functions. This function and
design_matrix_builders()
together form the API to the core formula interpretation machinery.Parameters:  design_infos – A list of
DesignInfo
objects describing the design matrices to be built.  data – A dictlike object which will be used to look up data.
 NA_action – What to do with rows that contain missing values. You can
"drop"
them,"raise"
an error, or for customization, pass anNAAction
object. SeeNAAction
for details on what values count as ‘missing’ (and how to alter this).  return_type – Either
"matrix"
or"dataframe"
. See below.  dtype – The dtype of the returned matrix. Useful if you want to use singleprecision or extendedprecision.
This function returns either a list of
DesignMatrix
objects (forreturn_type="matrix"
) or a list ofpandas.DataFrame
objects (forreturn_type="dataframe"
). In both cases, all returned design matrices will have.design_info
attributes containing the appropriateDesignInfo
objects.Note that unlike
design_matrix_builders()
, this function takes only a simple data argument, not any kind of iterator. That’s because this function doesn’t need a global view of the data – everything that depends on the whole data set is already encapsulated in thedesign_infos
. If you are incrementally processing a large data set, simply call this function for each chunk.Index handling: This function always checks for indexes in the following places:
 If
data
is apandas.DataFrame
, its.index
attribute.  If any factors evaluate to a
pandas.Series
orpandas.DataFrame
, then their.index
attributes.
If multiple indexes are found, they must be identical (same values in the same order). If no indexes are found, then a default index is generated using
np.arange(num_rows)
. One way or another, we end up with a single index for all the data. Ifreturn_type="dataframe"
, then this index is used as the index of the returned DataFrame objects. Examining this index makes it possible to determine which rows were removed due to NAs.Determining the number of rows in design matrices: This is not as obvious as it might seem, because it’s possible to have a formula like “~ 1” that doesn’t depend on the data (it has no factors). For this formula, it’s obvious what every row in the design matrix should look like (just the value
1
); but, how many rows like this should there be? To determine the number of rows in a design matrix, this function always checks in the following places: If
data
is apandas.DataFrame
, then its number of rows.  The number of entries in any factors present in any of the design
 matrices being built.
All these values much match. In particular, if this function is called to generate multiple design matrices at once, then they must all have the same number of rows.
New in version 0.2.0: The
NA_action
argument. design_infos – A list of
Missing values¶

class
patsy.
NAAction
(on_NA='drop', NA_types=['None', 'NaN'])¶ An
NAAction
object defines a strategy for handling missing data.“NA” is short for “Not Available”, and is used to refer to any value which is somehow unmeasured or unavailable. In the long run, it is devoutly hoped that numpy will gain firstclass missing value support. Until then, we work around this lack as best we’re able.
There are two parts to this: First, we have to determine what counts as missing data. For numerical data, the default is to treat NaN values (e.g.,
numpy.nan
) as missing. For categorical data, the default is to treat NaN values, and also the Python object None, as missing. (This is consistent with how pandas does things, so if you’re already using None/NaN to mark missing data in your pandas DataFrames, you’re good to go.)Second, we have to decide what to do with any missing data when we encounter it. One option is to simply discard any rows which contain missing data from our design matrices (
drop
). Another option is to raise an error (raise
). A third option would be to simply let the missing values pass through into the returned design matrices. However, this last option is not yet implemented, because of the lack of any standard way to represent missing values in arbitrary numpy matrices; we’re hoping numpy will get this sorted out before we standardize on anything ourselves.You can control how patsy handles missing data through the
NA_action=
argument to functions likebuild_design_matrices()
anddmatrix()
. If all you want to do is to choose betweendrop
andraise
behaviour, you can pass one of those strings as theNA_action=
argument directly. If you want more finegrained control over how missing values are detected and handled, then you can create an instance of this class, or your own object that implements the same interface, and pass that as theNA_action=
argument instead.The
NAAction
constructor takes the following arguments:Parameters:  on_NA – How to handle missing values. The default is
"drop"
, which removes all rows from all matrices which contain any missing values. Also available is"raise"
, which raises an exception when any missing values are encountered.  NA_types –
Which rules are used to identify missing values, as a list of strings. Allowed values are:
"None"
: treat theNone
object as missing in categorical data."NaN"
: treat floating point NaN values as missing in categorical and numerical data.
New in version 0.2.0.

handle_NA
(values, is_NAs, origins)¶ Takes a set of factor values that may have NAs, and handles them appropriately.
Parameters:  values – A list of ndarray objects representing the data. These may be 1 or 2dimensional, and may be of varying dtype. All will have the same number of rows (or entries, for 1d arrays).
 is_NAs – A list with the same number of entries as values, containing boolean ndarray objects that indicate which rows contain NAs in the corresponding entry in values.
 origins – A list with the same number of entries as
values, containing information on the origin of each
value. If we encounter a problem with some particular value, we use
the corresponding entry in origins as the origin argument when
raising a
PatsyError
.
Returns: A list of new values (which may have a differing number of rows.)

is_categorical_NA
(obj)¶ Return True if obj is a categorical NA value.
Note that here obj is a single scalar value.

is_numerical_NA
(arr)¶ Returns a 1d mask array indicating which rows in an array of numerical values contain at least one NA value.
Note that here arr is a numpy array or pandas DataFrame.
 on_NA – How to handle missing values. The default is
Linear constraints¶

class
patsy.
LinearConstraint
(variable_names, coefs, constants=None)¶ A linear constraint in matrix form.
This object represents a linear constraint of the form Ax = b.
Usually you won’t be constructing these by hand, but instead get them as the return value from
DesignInfo.linear_constraint()
.
coefs
¶ A 2dimensional ndarray with float dtype, representing A.

constants
¶ A 2dimensional singlecolumn ndarray with float dtype, representing b.

variable_names
¶ A list of strings giving the names of the variables being constrained. (Used only for consistency checking.)

Origin tracking¶

class
patsy.
Origin
(code, start, end)¶ This represents the origin of some object in some string.
For example, if we have an object
x1_obj
that was produced by parsing thex1
in the formula"y ~ x1:x2"
, then we conventionally keep track of that relationship by doing:x1_obj.origin = Origin("y ~ x1:x2", 4, 6)
Then later if we run into a problem, we can do:
raise PatsyError("invalid factor", x1_obj)
and we’ll produce a nice error message like:
PatsyError: invalid factor y ~ x1:x2 ^^
Origins are compared by value, and hashable.

caretize
(indent=0)¶ Produces a userreadable two line string indicating the origin of some code. Example:
y ~ x1:x2 ^^
If optional argument ‘indent’ is given, then both lines will be indented by this much. The returned string does not have a trailing newline.

classmethod
combine
(origin_objs)¶ Class method for combining a set of Origins into one large Origin that spans them.
Example usage: if we wanted to represent the origin of the “x1:x2” term, we could do
Origin.combine([x1_obj, x2_obj])
.Single argument is an iterable, and each element in the iterable should be either:
 An Origin object
None
 An object that has a
.origin
attribute which fulfills the above criteria.
Returns either an Origin object, or None.

relevant_code
()¶ Extracts and returns the span of the original code represented by this Origin. Example:
x1
.

patsy.builtins
API reference¶
This module defines some tools that are automatically made available
to code evaluated in formulas. You can also access it directly; use
from patsy.builtins import *
to import the same variables that
formula code receives automatically.

patsy.builtins.
I
(x)¶ The identity function. Simply returns its input unchanged.
Since Patsy’s formula parser ignores anything inside a function call syntax, this is useful to ‘hide’ arithmetic operations from it. For instance:
y ~ x1 + x2
has
x1
andx2
as two separate predictors. But in:y ~ I(x1 + x2)
we instead have a single predictor, defined to be the sum of
x1
andx2
.

patsy.builtins.
Q
(name)¶ A way to ‘quote’ variable names, especially ones that do not otherwise meet Python’s variable name rules.
If
x
is a variable,Q("x")
returns the value ofx
. (Note thatQ
takes the string"x"
, not the value ofx
itself.) This works even if instead ofx
, we have a variable name that would not otherwise be legal in Python.For example, if you have a column of data named
weight.in.kg
, then you can’t write:y ~ weight.in.kg
because Python will try to find a variable named
weight
, that has an attribute namedin
, that has an attribute namedkg
. (And worse yet,in
is a reserved word, which makes this example doubly broken.) Instead, write:y ~ Q("weight.in.kg")
and all will be well. Note, though, that this requires embedding a Python string inside your formula, which may require some care with your quote marks. Some standard options include:
my_fit_function("y ~ Q('weight.in.kg')", ...) my_fit_function('y ~ Q("weight.in.kg")', ...) my_fit_function("y ~ Q(\"weight.in.kg\")", ...)
Note also that
Q
is an ordinary Python function, which means that you can use it in more complex expressions. For example, this is a legal formula:y ~ np.sqrt(Q("weight.in.kg"))

class
patsy.builtins.
ContrastMatrix
(matrix, column_suffixes)¶ A simple container for a matrix used for coding categorical factors.
Attributes:

matrix
¶ A 2d ndarray, where each column corresponds to one column of the resulting design matrix, and each row contains the entries for a single categorical variable level. Usually nbyn for a full rank coding or nby(n1) for a reduced rank coding, though other options are possible.

column_suffixes
¶ A list of strings to be appended to the factor name, to produce the final column names. E.g. for treatment coding the entries will look like
"[T.level1]"
.


class
patsy.builtins.
Treatment
(reference=None)¶ Treatment coding (also known as dummy coding).
This is the default coding.
For reducedrank coding, one level is chosen as the “reference”, and its mean behaviour is represented by the intercept. Each column of the resulting matrix represents the difference between the mean of one level and this reference level.
For fullrank coding, classic “dummy” coding is used, and each column of the resulting matrix represents the mean of the corresponding level.
The reference level defaults to the first level, or can be specified explicitly.
# reduced rank In [1]: dmatrix("C(a, Treatment)", balanced(a=3)) Out[1]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment)[T.a2] C(a, Treatment)[T.a3] 1 0 0 1 1 0 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Treatment)' (columns 1:3) # full rank In [2]: dmatrix("0 + C(a, Treatment)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Treatment)[a1] C(a, Treatment)[a2] C(a, Treatment)[a3] 1 0 0 0 1 0 0 0 1 Terms: 'C(a, Treatment)' (columns 0:3) # Setting a reference level In [3]: dmatrix("C(a, Treatment(1))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment(1))[T.a1] C(a, Treatment(1))[T.a3] 1 1 0 1 0 0 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Treatment(1))' (columns 1:3) In [4]: dmatrix("C(a, Treatment('a2'))", balanced(a=3)) Out[4]: DesignMatrix with shape (3, 3) Intercept C(a, Treatment('a2'))[T.a1] C(a, Treatment('a2'))[T.a3] 1 1 0 1 0 0 1 0 1 Terms: 'Intercept' (column 0) "C(a, Treatment('a2'))" (columns 1:3)
Equivalent to R
contr.treatment
. The R documentation suggests that usingTreatment(reference=1)
will produce contrasts that are “equivalent to those produced by many (but not all) SAS procedures”.
code_with_intercept
(levels)¶

code_without_intercept
(levels)¶


class
patsy.builtins.
Poly
(scores=None)¶ Orthogonal polynomial contrast coding.
This coding scheme treats the levels as ordered samples from an underlying continuous scale, whose effect takes an unknown functional form which is Taylordecomposed into the sum of a linear, quadratic, etc. components.
For reducedrank coding, you get a linear column, a quadratic column, etc., up to the number of levels provided.
For fullrank coding, the same scheme is used, except that the zeroorder constant polynomial is also included. I.e., you get an intercept column included as part of your categorical term.
By default the levels are treated as equally spaced, but you can override this by providing a value for the scores argument.
Examples:
# Reduced rank In [1]: dmatrix("C(a, Poly)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Poly).Linear C(a, Poly).Quadratic C(a, Poly).Cubic 1 0.67082 0.5 0.22361 1 0.22361 0.5 0.67082 1 0.22361 0.5 0.67082 1 0.67082 0.5 0.22361 Terms: 'Intercept' (column 0) 'C(a, Poly)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Poly)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Poly).Constant C(a, Poly).Linear C(a, Poly).Quadratic 1 0.70711 0.40825 1 0.00000 0.81650 1 0.70711 0.40825 Terms: 'C(a, Poly)' (columns 0:3) # Explicit scores In [3]: dmatrix("C(a, Poly([1, 2, 10]))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Poly([1, 2, 10])).Linear C(a, Poly([1, 2, 10])).Quadratic 1 0.47782 0.66208 1 0.33447 0.74485 1 0.81229 0.08276 Terms: 'Intercept' (column 0) 'C(a, Poly([1, 2, 10]))' (columns 1:3)
This is equivalent to R’s
contr.poly
. (But note that in R, reduced rank encodings are always dummycoded, regardless of what contrast you have set.)
code_with_intercept
(levels)¶

code_without_intercept
(levels)¶


class
patsy.builtins.
Sum
(omit=None)¶ Deviation coding (also known as sumtozero coding).
Compares the mean of each level to the meanofmeans. (In a balanced design, compares the mean of each level to the overall mean.)
For fullrank coding, a standard intercept term is added.
One level must be omitted to avoid redundancy; by default this is the last level, but this can be adjusted via the omit argument.
Warning
There are multiple definitions of ‘deviation coding’ in use. Make sure this is the one you expect before trying to interpret your results!
Examples:
# Reduced rank In [1]: dmatrix("C(a, Sum)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Sum)[S.a1] C(a, Sum)[S.a2] C(a, Sum)[S.a3] 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 Terms: 'Intercept' (column 0) 'C(a, Sum)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Sum)", balanced(a=4)) Out[2]: DesignMatrix with shape (4, 4) C(a, Sum)[mean] C(a, Sum)[S.a1] C(a, Sum)[S.a2] C(a, Sum)[S.a3] 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 1 Terms: 'C(a, Sum)' (columns 0:4) # Omit a different level In [3]: dmatrix("C(a, Sum(1))", balanced(a=3)) Out[3]: DesignMatrix with shape (3, 3) Intercept C(a, Sum(1))[S.a1] C(a, Sum(1))[S.a3] 1 1 0 1 1 1 1 0 1 Terms: 'Intercept' (column 0) 'C(a, Sum(1))' (columns 1:3) In [4]: dmatrix("C(a, Sum('a1'))", balanced(a=3)) Out[4]: DesignMatrix with shape (3, 3) Intercept C(a, Sum('a1'))[S.a2] C(a, Sum('a1'))[S.a3] 1 1 1 1 1 0 1 0 1 Terms: 'Intercept' (column 0) "C(a, Sum('a1'))" (columns 1:3)
This is equivalent to R’s contr.sum.

code_with_intercept
(levels)¶

code_without_intercept
(levels)¶


class
patsy.builtins.
Helmert
¶ Helmert contrasts.
Compares the second level with the first, the third with the average of the first two, and so on.
For fullrank coding, a standard intercept term is added.
Warning
There are multiple definitions of ‘Helmert coding’ in use. Make sure this is the one you expect before trying to interpret your results!
Examples:
# Reduced rank In [1]: dmatrix("C(a, Helmert)", balanced(a=4)) Out[1]: DesignMatrix with shape (4, 4) Intercept C(a, Helmert)[H.a2] C(a, Helmert)[H.a3] C(a, Helmert)[H.a4] 1 1 1 1 1 1 1 1 1 0 2 1 1 0 0 3 Terms: 'Intercept' (column 0) 'C(a, Helmert)' (columns 1:4) # Full rank In [2]: dmatrix("0 + C(a, Helmert)", balanced(a=4)) Out[2]: DesignMatrix with shape (4, 4) Columns: ['C(a, Helmert)[H.intercept]', 'C(a, Helmert)[H.a2]', 'C(a, Helmert)[H.a3]', 'C(a, Helmert)[H.a4]'] Terms: 'C(a, Helmert)' (columns 0:4) (to view full data, use np.asarray(this_obj))
This is equivalent to R’s contr.helmert.

code_with_intercept
(levels)¶

code_without_intercept
(levels)¶


class
patsy.builtins.
Diff
¶ Backward difference coding.
This coding scheme is useful for ordered factors, and compares the mean of each level with the preceding level. So you get the second level minus the first, the third level minus the second, etc.
For fullrank coding, a standard intercept term is added (which gives the mean value for the first level).
Examples:
# Reduced rank In [1]: dmatrix("C(a, Diff)", balanced(a=3)) Out[1]: DesignMatrix with shape (3, 3) Intercept C(a, Diff)[D.a1] C(a, Diff)[D.a2] 1 0.66667 0.33333 1 0.33333 0.33333 1 0.33333 0.66667 Terms: 'Intercept' (column 0) 'C(a, Diff)' (columns 1:3) # Full rank In [2]: dmatrix("0 + C(a, Diff)", balanced(a=3)) Out[2]: DesignMatrix with shape (3, 3) C(a, Diff)[D.a1] C(a, Diff)[D.a2] C(a, Diff)[D.a3] 1 0.66667 0.33333 1 0.33333 0.33333 1 0.33333 0.66667 Terms: 'C(a, Diff)' (columns 0:3)

code_with_intercept
(levels)¶

code_without_intercept
(levels)¶


patsy.builtins.
C
(data, contrast=None, levels=None)¶ Marks some data as being categorical, and specifies how to interpret it.
This is used for three reasons:
To explicitly mark some data as categorical. For instance, integer data is by default treated as numerical. If you have data that is stored using an integer type, but where you want patsy to treat each different value as a different level of a categorical factor, you can wrap it in a call to C to accomplish this. E.g., compare:
dmatrix("a", {"a": [1, 2, 3]}) dmatrix("C(a)", {"a": [1, 2, 3]})
To explicitly set the levels or override the default level ordering for categorical data, e.g.:
dmatrix("C(a, levels=["a2", "a1"])", balanced(a=2))
To override the default coding scheme for categorical data. The contrast argument can be any of:
 A
ContrastMatrix
object  A simple 2d ndarray (which is treated the same as a ContrastMatrix object except that you can’t specify column names)
 An object with methods called code_with_intercept and
code_without_intercept, like the builtin contrasts
(
Treatment
,Diff
,Poly
, etc.). See Coding categorical data for more details.  A callable that returns one of the above.
 A

patsy.builtins.
center
(x)¶ A stateful transform that centers input data, i.e., subtracts the mean.
If input has multiple columns, centers each column separately.
Equivalent to
standardize(x, rescale=False)

patsy.builtins.
standardize
(x, center=True, rescale=True, ddof=0)¶ A stateful transform that standardizes input data, i.e. it subtracts the mean and divides by the sample standard deviation.
Either centering or rescaling or both can be disabled by use of keyword arguments. The ddof argument controls the delta degrees of freedom when computing the standard deviation (cf.
numpy.std()
). The default ofddof=0
produces the maximum likelihood estimate; useddof=1
if you prefer the square root of the unbiased estimate of the variance.If input has multiple columns, standardizes each column separately.
Note
This function computes the mean and standard deviation using a memoryefficient online algorithm, making it suitable for use with large incrementally processed datasets.

patsy.builtins.
scale
(*args, **kwargs)¶ standardize(x, center=True, rescale=True, ddof=0)
A stateful transform that standardizes input data, i.e. it subtracts the mean and divides by the sample standard deviation.
Either centering or rescaling or both can be disabled by use of keyword arguments. The ddof argument controls the delta degrees of freedom when computing the standard deviation (cf.
numpy.std()
). The default ofddof=0
produces the maximum likelihood estimate; useddof=1
if you prefer the square root of the unbiased estimate of the variance.If input has multiple columns, standardizes each column separately.
Note
This function computes the mean and standard deviation using a memoryefficient online algorithm, making it suitable for use with large incrementally processed datasets.

patsy.builtins.
bs
(x, df=None, knots=None, degree=3, include_intercept=False, lower_bound=None, upper_bound=None)¶ Generates a Bspline basis for
x
, allowing nonlinear fits. The usual usage is something like:y ~ 1 + bs(x, 4)
to fit
y
as a smooth function ofx
, with 4 degrees of freedom given to the smooth.Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  degree – The degree of the spline to use.
 include_intercept – If
True
, then the resulting spline basis will span the intercept term (i.e., the constant function). IfFalse
(the default) then this will not be the case, which is useful for avoiding overspecification in models that include multiple spline terms and/or an intercept term.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
A spline with
degree=0
is piecewise constant with breakpoints at each knot, and the default knot positions are quantiles of the input. So if you find yourself in the situation of wanting to quantize a continuous variable intonum_bins
equalsized bins with a constant effect across each bin, you can usebs(x, num_bins  1, degree=0)
. (The 1
is because one degree of freedom will be taken by the intercept; alternatively, you could leave the intercept term out of your model and usebs(x, num_bins, degree=0, include_intercept=True)
.A spline with
degree=1
is piecewise linear with breakpoints at each knot.The default is
degree=3
, which gives a cubic bspline.This is a stateful transform (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
Note
This function is very similar to the R function of the same name. In cases where both return output at all (e.g., R’s
bs
will raise an error ifdegree=0
, while patsy’s will not), they should produce identical output given identical input and parameter settings.Warning
I’m not sure on what the proper handling of points outside the lower/upper bounds is, so for now attempting to evaluate a spline basis at such points produces an error. Patches gratefully accepted.
New in version 0.2.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.builtins.
cr
(x, df=None, knots=None, lower_bound=None, upper_bound=None, constraints=None)¶ Generates a natural cubic spline basis for
x
(with the option of absorbing centering or more general parameters constraints), allowing nonlinear fits. The usual usage is something like:y ~ 1 + cr(x, df=5, constraints='center')
to fit
y
as a smooth function ofx
, with 5 degrees of freedom given to the smooth, and centering constraint absorbed in the resulting design matrix. Note that in this example, due to the centering constraint, 6 knots will get computed from the input datax
to achieve 5 degrees of freedom.Note
This function reproduce the cubic regression splines ‘cr’ and ‘cs’ as implemented in the R package ‘mgcv’ (GAM modelling).
Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
 constraints – Either a 2d array defining general linear constraints
(that is
np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.
This is a stateful transforms (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
New in version 0.3.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.builtins.
cc
(x, df=None, knots=None, lower_bound=None, upper_bound=None, constraints=None)¶ Generates a cyclic cubic spline basis for
x
(with the option of absorbing centering or more general parameters constraints), allowing nonlinear fits. The usual usage is something like:y ~ 1 + cc(x, df=7, constraints='center')
to fit
y
as a smooth function ofx
, with 7 degrees of freedom given to the smooth, and centering constraint absorbed in the resulting design matrix. Note that in this example, due to the centering and cyclic constraints, 9 knots will get computed from the input datax
to achieve 7 degrees of freedom.Note
This function reproduce the cubic regression splines ‘cc’ as implemented in the R package ‘mgcv’ (GAM modelling).
Parameters:  df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of
df
andknots
.  knots – The interior knots to use for the spline. If unspecified, then
equally spaced quantiles of the input data are used. You must specify at
least one of
df
andknots
.  lower_bound – The lower exterior knot location.
 upper_bound – The upper exterior knot location.
 constraints – Either a 2d array defining general linear constraints
(that is
np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.
This is a stateful transforms (for details see Stateful transforms). If
knots
,lower_bound
, orupper_bound
are not specified, they will be calculated from the data and then the chosen values will be remembered and reused for prediction from the fitted model.Using this function requires scipy be installed.
New in version 0.3.0.
 df – The number of degrees of freedom to use for this spline. The
return value will have this many columns. You must specify at least one
of

patsy.builtins.
te
(s1, .., sn, constraints=None)¶ Generates smooth of several covariates as a tensor product of the bases of marginal univariate smooths
s1, .., sn
. The marginal smooths are required to transform input univariate data into some kind of smooth functions basis producing a 2d array output with the(i, j)
element corresponding to the value of thej
th basis function at thei
th data point. The resulting basis dimension is the product of the basis dimensions of the marginal smooths. The usual usage is something like:y ~ 1 + te(cr(x1, df=5), cc(x2, df=6), constraints='center')
to fit
y
as a smooth function of bothx1
andx2
, with a natural cubic spline forx1
marginal smooth and a cyclic cubic spline forx2
(and centering constraint absorbed in the resulting design matrix).Parameters: constraints – Either a 2d array defining general linear constraints (that is np.dot(constraints, betas)
is zero, wherebetas
denotes the array of initial parameters, corresponding to the initial unconstrained design matrix), or the string'center'
indicating that we should apply a centering constraint (this constraint will be computed from the input data, remembered and reused for prediction from the fitted model). The constraints are absorbed in the resulting design matrix which means that the model is actually rewritten in terms of unconstrained parameters. For more details see Spline regression.Using this function requires scipy be installed.
Note
This function reproduce the tensor product smooth ‘te’ as implemented in the R package ‘mgcv’ (GAM modelling). See also ‘Generalized Additive Models’, Simon N. Wood, 2006, pp 158163
New in version 0.3.0.
Changes¶
v0.4.0¶
Incompatible changes:
EvalFactor
andModelDesc.from_formula()
no longer take aneval_env
argument. The
design_matrix_builders()
function and thefactor_protocol.memorize_passes_needed()
method now require aneval_env
as an additional argument.  The
DesignInfo
constructor’s arguments have totally changed. In addition to the changes needed to support the new features below, we no longer support “shim” DesignInfo objects that have nontrivial term specifications. This was only included in the first place to provide a compatibility hook for competing formula libraries; four years later, no such libraries have shown up. If one does, we can readd it, but I’m not going to bother maintaining it in the mean time...  Dropped support for Python 3.2.
Other changes:
 Patsy now supports Pandas’s new (version 0.15 or later) categorical objects.
 Formulas (or more precisely,
EvalFactor
objects) now only keep a reference to the variables required from their environment instead of the whole environment where the formula was defined. (Thanks to Christian Hudon.) DesignInfo
has new attributesDesignInfo.factor_infos
andDesignInfo.term_codings
which provide detailed metadata about how each factor and term is encoded. As a result of the above changes, the split between
DesignInfo
andDesignMatrixBuilder
is no longer necessary;DesignMatrixBuiler
has been eliminated. So for example,design_matrix_builders()
now returns a list ofDesignInfo
objects, and you can now passDesignInfo
objects directly to any function for building design matrices. For compatibility,DesignInfo
continues to provide.builder
and.design_info
attributes, so that old code should continue to work; however, these attributes are deprecated.  Ensured that attempting to pickle most Patsy objects raises an error. This has never been supported, and the interesting cases failed in any case, but now we’re taking a more systematic approach. (Soon we will add real, supported pickling support.)
 Fixed a bug when running under
python OO
.
v0.3.0¶
New stateful transforms for computing natural and cylic cubic splines with constraints, and tensor spline bases with constraints. (Thanks to @broessli and GDF Suez for contributing this code.)
Dropped support for Python 2.5 and earlier.
Switched to using a single source tree for both Python 2 and Python 3.
Added a fastpath to skip NA detection for inputs with boolean dtypes (thanks to Matt Davis for patch).
Incompatible change: Sometimes when building a design matrix for a formula that does not depend on the data in any way, like
"1 ~ 1"
, we have no way to determine how many rows the resulting matrix should have. In previous versions of patsy, when this occurred we simply returned a matrix with 1 row. In 0.3.0+, we instead refuse to guess, and raise an error.Note that because of the next change listed, this situation occurs less frequently in 0.3.0 than in previous versions.
If the
data
argument tobuild_design_matrices()
(or derived functions likedmatrix()
,dmatrices()
) is apandas.DataFrame
, then we now check its number of rows and index, and insist that the output design matrices match. This also means that ifdata
is a DataFrame, then the error described in the first bullet above cannot occur – we will simply return a column of 1s that is the same size as the input dataframe.Worked around some more limitations in py2exe/py2app and friends.
v0.2.1¶
 Fixed a nasty bug in missing value handling where, if missing values
were present,
dmatrix(..., result_type="dataframe")
would always crash, anddmatrices("y ~ 1")
would produce left and righthand side matrices that had different numbers of rows. (As far as I can tell, this bug could not possibly cause incorrect results, only crashes, since it always involved the creation of matrices with incommensurate shapes. Therefore there is no need to worry about the accuracy of any analyses that were successfully performed with v0.2.0.)  Modified
patsy/__init__.py
to work around limitations in py2exe/py2app/etc.
v0.2.0¶
Warnings:
 The lowest officially supported Python version is now 2.5. So far as I know everything still works with Python 2.4, but as everyone else has continued to drop support for 2.4, testing on 2.4 has become so much trouble that I’ve given up.
New features:
 New support for automatically detecting and (optionally) removing
missing values (see
NAAction
).  New stateful transform for Bspline regression:
bs()
. (Requires scipy.)  Added a core API to make it possible to run predictions on only a
subset of model terms. (This is particularly useful for
e.g. plotting the isolated effect of a single fitted spline term.)
See
DesignMatrixBuilder.subset()
. LookupFactor
now allows users to mark variables as categorical directly.pandas.Categorical
objects are now recognized as representing categorical data and handled appropriately. Better error reporting for exceptions raised by user code inside formulas. We now, whenever possible, tag the generated exception with information about which factor’s code raised it, and use this information to give better error reporting.
EvalEnvironment.capture()
now takes a reference argument, to make it easier to implement newdmatrix()
like functions.
Other: miscellaneous doc improvements and bug fixes.