Polynomial features

Polynomial features#

The modelling tools included in ISLP allow for construction of orthogonal polynomials of features.

Force rebuild

import numpy as np
from ISLP import load_data
from ISLP.models import ModelSpec, poly

Carseats = load_data('Carseats')
Carseats.columns

Index(['Sales', 'CompPrice', 'Income', 'Advertising', 'Population', 'Price',
       'ShelveLoc', 'Age', 'Education', 'Urban', 'US'],
      dtype='object')

Let’s make a term representing a quartic effect for Population.

quartic = poly('Population', 4)

The object quartic does not refer to any data yet, it must be included in a ModelSpec object and fit using the fit method.

design = ModelSpec([quartic], intercept=False)
ISLP_features = design.fit_transform(Carseats)
ISLP_features.columns

Index(['poly(Population, degree=4)[0]', 'poly(Population, degree=4)[1]',
       'poly(Population, degree=4)[2]', 'poly(Population, degree=4)[3]'],
      dtype='object')

Compare to R#

We can compare our polynomials to a similar function in R

%load_ext rpy2.ipython

We’ll recompute these features using poly in R.

%%R -i Carseats -o R_features
R_features = poly(Carseats$Population, 4)

In addition: Warning message:
In (function (package, help, pos = 2, lib.loc = NULL, character.only = FALSE,  :
  libraries ‘/usr/local/lib/R/site-library’, ‘/usr/lib/R/site-library’ contain no packages

np.linalg.norm(ISLP_features - R_features)

6.611410814977955e-15

Underlying model#

If we look at quartic, we see it is a Feature, i.e. it can be used to produce a set of columns in a design matrix when it is a term used in creating the ModelSpec.

Its encoder is Poly(degree=4). This is a special sklearn transform that expects a single column in its fit() method and constructs a matrix of corresponding orthogonal polynomials.

The spline helpers ns and bs as well as pca follow a similar structure.

quartic

Feature(variables=('Population',), name='poly(Population, degree=4)', encoder=Poly(degree=4), use_transform=True, pure_columns=False, override_encoder_colnames=True)

Raw polynomials#

One can compute raw polynomials (which results in a less well-conditioned design matrix) of course.

quartic_raw = poly('Population', degree=4, raw=True)

Let’s compare the features again.

design = ModelSpec([quartic_raw], intercept=False)
raw_features = design.fit_transform(Carseats)

%%R -i Carseats -o R_features
R_features = poly(Carseats$Population, 4, raw=TRUE)

np.linalg.norm(raw_features - R_features)

0.0

Intercept#

Looking at py_features we see it contains columns: [Population**i for i in range(1, 4)]. That is, it doesn’t contain an intercept, the order 0 term. This can be include with intercept=True

quartic_int = poly('Population', degree=4, raw=True, intercept=True)
design = ModelSpec([quartic_int], intercept=False)
intercept_features = design.fit_transform(Carseats)

np.linalg.norm(intercept_features.iloc[:,1:] - R_features)

0.0
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