"""
This file contains preprocessing tools based on polynomials.
"""
import collections
import numbers
from itertools import chain, combinations
from itertools import combinations_with_replacement as combinations_w_r
import numpy as np
from scipy import sparse
from scipy.interpolate import BSpline
from scipy.special import comb
from ..base import BaseEstimator, TransformerMixin
from ..utils import check_array
from ..utils.deprecation import deprecated
from ..utils.fixes import linspace
from ..utils.validation import check_is_fitted, FLOAT_DTYPES, _check_sample_weight
from ..utils.validation import _check_feature_names_in
from ..utils.stats import _weighted_percentile
from ._csr_polynomial_expansion import _csr_polynomial_expansion
__all__ = [
"PolynomialFeatures",
"SplineTransformer",
]
[docs]class PolynomialFeatures(TransformerMixin, BaseEstimator):
"""Generate polynomial and interaction features.
Generate a new feature matrix consisting of all polynomial combinations
of the features with degree less than or equal to the specified degree.
For example, if an input sample is two dimensional and of the form
[a, b], the degree-2 polynomial features are [1, a, b, a^2, ab, b^2].
Read more in the :ref:`User Guide <polynomial_features>`.
Parameters
----------
degree : int or tuple (min_degree, max_degree), default=2
If a single int is given, it specifies the maximal degree of the
polynomial features. If a tuple `(min_degree, max_degree)` is passed,
then `min_degree` is the minimum and `max_degree` is the maximum
polynomial degree of the generated features. Note that `min_degree=0`
and `min_degree=1` are equivalent as outputting the degree zero term is
determined by `include_bias`.
interaction_only : bool, default=False
If `True`, only interaction features are produced: features that are
products of at most `degree` *distinct* input features, i.e. terms with
power of 2 or higher of the same input feature are excluded:
- included: `x[0]`, `x[1]`, `x[0] * x[1]`, etc.
- excluded: `x[0] ** 2`, `x[0] ** 2 * x[1]`, etc.
include_bias : bool, default=True
If `True` (default), then include a bias column, the feature in which
all polynomial powers are zero (i.e. a column of ones - acts as an
intercept term in a linear model).
order : {'C', 'F'}, default='C'
Order of output array in the dense case. `'F'` order is faster to
compute, but may slow down subsequent estimators.
.. versionadded:: 0.21
Attributes
----------
powers_ : ndarray of shape (`n_output_features_`, `n_features_in_`)
`powers_[i, j]` is the exponent of the jth input in the ith output.
n_input_features_ : int
The total number of input features.
.. deprecated:: 1.0
This attribute is deprecated in 1.0 and will be removed in 1.2.
Refer to `n_features_in_` instead.
n_features_in_ : int
Number of features seen during :term:`fit`.
.. versionadded:: 0.24
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
n_output_features_ : int
The total number of polynomial output features. The number of output
features is computed by iterating over all suitably sized combinations
of input features.
See Also
--------
SplineTransformer : Transformer that generates univariate B-spline bases
for features.
Notes
-----
Be aware that the number of features in the output array scales
polynomially in the number of features of the input array, and
exponentially in the degree. High degrees can cause overfitting.
See :ref:`examples/linear_model/plot_polynomial_interpolation.py
<sphx_glr_auto_examples_linear_model_plot_polynomial_interpolation.py>`
Examples
--------
>>> import numpy as np
>>> from sklearn.preprocessing import PolynomialFeatures
>>> X = np.arange(6).reshape(3, 2)
>>> X
array([[0, 1],
[2, 3],
[4, 5]])
>>> poly = PolynomialFeatures(2)
>>> poly.fit_transform(X)
array([[ 1., 0., 1., 0., 0., 1.],
[ 1., 2., 3., 4., 6., 9.],
[ 1., 4., 5., 16., 20., 25.]])
>>> poly = PolynomialFeatures(interaction_only=True)
>>> poly.fit_transform(X)
array([[ 1., 0., 1., 0.],
[ 1., 2., 3., 6.],
[ 1., 4., 5., 20.]])
"""
def __init__(
self, degree=2, *, interaction_only=False, include_bias=True, order="C"
):
self.degree = degree
self.interaction_only = interaction_only
self.include_bias = include_bias
self.order = order
@staticmethod
def _combinations(
n_features, min_degree, max_degree, interaction_only, include_bias
):
comb = combinations if interaction_only else combinations_w_r
start = max(1, min_degree)
iter = chain.from_iterable(
comb(range(n_features), i) for i in range(start, max_degree + 1)
)
if include_bias:
iter = chain(comb(range(n_features), 0), iter)
return iter
@staticmethod
def _num_combinations(
n_features, min_degree, max_degree, interaction_only, include_bias
):
"""Calculate number of terms in polynomial expansion
This should be equivalent to counting the number of terms returned by
_combinations(...) but much faster.
"""
if interaction_only:
combinations = sum(
[
comb(n_features, i, exact=True)
for i in range(max(1, min_degree), min(max_degree, n_features) + 1)
]
)
else:
combinations = comb(n_features + max_degree, max_degree, exact=True) - 1
if min_degree > 0:
d = min_degree - 1
combinations -= comb(n_features + d, d, exact=True) - 1
if include_bias:
combinations += 1
return combinations
@property
def powers_(self):
"""Exponent for each of the inputs in the output."""
check_is_fitted(self)
combinations = self._combinations(
n_features=self.n_features_in_,
min_degree=self._min_degree,
max_degree=self._max_degree,
interaction_only=self.interaction_only,
include_bias=self.include_bias,
)
return np.vstack(
[np.bincount(c, minlength=self.n_features_in_) for c in combinations]
)
[docs] @deprecated(
"get_feature_names is deprecated in 1.0 and will be removed "
"in 1.2. Please use get_feature_names_out instead."
)
def get_feature_names(self, input_features=None):
"""Return feature names for output features.
Parameters
----------
input_features : list of str of shape (n_features,), default=None
String names for input features if available. By default,
"x0", "x1", ... "xn_features" is used.
Returns
-------
output_feature_names : list of str of shape (n_output_features,)
Transformed feature names.
"""
powers = self.powers_
if input_features is None:
input_features = ["x%d" % i for i in range(powers.shape[1])]
feature_names = []
for row in powers:
inds = np.where(row)[0]
if len(inds):
name = " ".join(
"%s^%d" % (input_features[ind], exp)
if exp != 1
else input_features[ind]
for ind, exp in zip(inds, row[inds])
)
else:
name = "1"
feature_names.append(name)
return feature_names
[docs] def get_feature_names_out(self, input_features=None):
"""Get output feature names for transformation.
Parameters
----------
input_features : array-like of str or None, default=None
Input features.
- If `input_features is None`, then `feature_names_in_` is
used as feature names in. If `feature_names_in_` is not defined,
then names are generated: `[x0, x1, ..., x(n_features_in_)]`.
- If `input_features` is an array-like, then `input_features` must
match `feature_names_in_` if `feature_names_in_` is defined.
Returns
-------
feature_names_out : ndarray of str objects
Transformed feature names.
"""
powers = self.powers_
input_features = _check_feature_names_in(self, input_features)
feature_names = []
for row in powers:
inds = np.where(row)[0]
if len(inds):
name = " ".join(
"%s^%d" % (input_features[ind], exp)
if exp != 1
else input_features[ind]
for ind, exp in zip(inds, row[inds])
)
else:
name = "1"
feature_names.append(name)
return np.asarray(feature_names, dtype=object)
[docs] def fit(self, X, y=None):
"""
Compute number of output features.
Parameters
----------
X : {array-like, sparse matrix} of shape (n_samples, n_features)
The data.
y : Ignored
Not used, present here for API consistency by convention.
Returns
-------
self : object
Fitted transformer.
"""
_, n_features = self._validate_data(X, accept_sparse=True).shape
if isinstance(self.degree, numbers.Integral):
if self.degree < 0:
raise ValueError(
f"degree must be a non-negative integer, got {self.degree}."
)
self._min_degree = 0
self._max_degree = self.degree
elif (
isinstance(self.degree, collections.abc.Iterable) and len(self.degree) == 2
):
self._min_degree, self._max_degree = self.degree
if not (
isinstance(self._min_degree, numbers.Integral)
and isinstance(self._max_degree, numbers.Integral)
and self._min_degree >= 0
and self._min_degree <= self._max_degree
):
raise ValueError(
"degree=(min_degree, max_degree) must "
"be non-negative integers that fulfil "
"min_degree <= max_degree, got "
f"{self.degree}."
)
else:
raise ValueError(
"degree must be a non-negative int or tuple "
"(min_degree, max_degree), got "
f"{self.degree}."
)
self.n_output_features_ = self._num_combinations(
n_features=n_features,
min_degree=self._min_degree,
max_degree=self._max_degree,
interaction_only=self.interaction_only,
include_bias=self.include_bias,
)
# We also record the number of output features for
# _max_degree = 0
self._n_out_full = self._num_combinations(
n_features=n_features,
min_degree=0,
max_degree=self._max_degree,
interaction_only=self.interaction_only,
include_bias=self.include_bias,
)
return self
# TODO: Remove in 1.2
# mypy error: Decorated property not supported
@deprecated( # type: ignore
"The attribute `n_input_features_` was "
"deprecated in version 1.0 and will be removed in 1.2."
)
@property
def n_input_features_(self):
return self.n_features_in_
# TODO:
# - sparse support (either scipy or own cython solution)?
class SplineTransformer(TransformerMixin, BaseEstimator):
"""Generate univariate B-spline bases for features.
Generate a new feature matrix consisting of
`n_splines=n_knots + degree - 1` (`n_knots - 1` for
`extrapolation="periodic"`) spline basis functions
(B-splines) of polynomial order=`degree` for each feature.
Read more in the :ref:`User Guide <spline_transformer>`.
.. versionadded:: 1.0
Parameters
----------
n_knots : int, default=5
Number of knots of the splines if `knots` equals one of
{'uniform', 'quantile'}. Must be larger or equal 2. Ignored if `knots`
is array-like.
degree : int, default=3
The polynomial degree of the spline basis. Must be a non-negative
integer.
knots : {'uniform', 'quantile'} or array-like of shape \
(n_knots, n_features), default='uniform'
Set knot positions such that first knot <= features <= last knot.
- If 'uniform', `n_knots` number of knots are distributed uniformly
from min to max values of the features.
- If 'quantile', they are distributed uniformly along the quantiles of
the features.
- If an array-like is given, it directly specifies the sorted knot
positions including the boundary knots. Note that, internally,
`degree` number of knots are added before the first knot, the same
after the last knot.
extrapolation : {'error', 'constant', 'linear', 'continue', 'periodic'}, \
default='constant'
If 'error', values outside the min and max values of the training
features raises a `ValueError`. If 'constant', the value of the
splines at minimum and maximum value of the features is used as
constant extrapolation. If 'linear', a linear extrapolation is used.
If 'continue', the splines are extrapolated as is, i.e. option
`extrapolate=True` in :class:`scipy.interpolate.BSpline`. If
'periodic', periodic splines with a periodicity equal to the distance
between the first and last knot are used. Periodic splines enforce
equal function values and derivatives at the first and last knot.
For example, this makes it possible to avoid introducing an arbitrary
jump between Dec 31st and Jan 1st in spline features derived from a
naturally periodic "day-of-year" input feature. In this case it is
recommended to manually set the knot values to control the period.
include_bias : bool, default=True
If True (default), then the last spline element inside the data range
of a feature is dropped. As B-splines sum to one over the spline basis
functions for each data point, they implicitly include a bias term,
i.e. a column of ones. It acts as an intercept term in a linear models.
order : {'C', 'F'}, default='C'
Order of output array. 'F' order is faster to compute, but may slow
down subsequent estimators.
Attributes
----------
bsplines_ : list of shape (n_features,)
List of BSplines objects, one for each feature.
n_features_in_ : int
The total number of input features.
feature_names_in_ : ndarray of shape (`n_features_in_`,)
Names of features seen during :term:`fit`. Defined only when `X`
has feature names that are all strings.
.. versionadded:: 1.0
n_features_out_ : int
The total number of output features, which is computed as
`n_features * n_splines`, where `n_splines` is
the number of bases elements of the B-splines,
`n_knots + degree - 1` for non-periodic splines and
`n_knots - 1` for periodic ones.
If `include_bias=False`, then it is only
`n_features * (n_splines - 1)`.
See Also
--------
KBinsDiscretizer : Transformer that bins continuous data into intervals.
PolynomialFeatures : Transformer that generates polynomial and interaction
features.
Notes
-----
High degrees and a high number of knots can cause overfitting.
See :ref:`examples/linear_model/plot_polynomial_interpolation.py
<sphx_glr_auto_examples_linear_model_plot_polynomial_interpolation.py>`.
Examples
--------
>>> import numpy as np
>>> from sklearn.preprocessing import SplineTransformer
>>> X = np.arange(6).reshape(6, 1)
>>> spline = SplineTransformer(degree=2, n_knots=3)
>>> spline.fit_transform(X)
array([[0.5 , 0.5 , 0. , 0. ],
[0.18, 0.74, 0.08, 0. ],
[0.02, 0.66, 0.32, 0. ],
[0. , 0.32, 0.66, 0.02],
[0. , 0.08, 0.74, 0.18],
[0. , 0. , 0.5 , 0.5 ]])
"""
def __init__(
self,
n_knots=5,
degree=3,
*,
knots="uniform",
extrapolation="constant",
include_bias=True,
order="C",
):
self.n_knots = n_knots
self.degree = degree
self.knots = knots
self.extrapolation = extrapolation
self.include_bias = include_bias
self.order = order
@staticmethod
def _get_base_knot_positions(X, n_knots=10, knots="uniform", sample_weight=None):
"""Calculate base knot positions.
Base knots such that first knot <= feature <= last knot. For the
B-spline construction with scipy.interpolate.BSpline, 2*degree knots
beyond the base interval are added.
Returns
-------
knots : ndarray of shape (n_knots, n_features), dtype=np.float64
Knot positions (points) of base interval.
"""
if knots == "quantile":
percentiles = 100 * np.linspace(
start=0, stop=1, num=n_knots, dtype=np.float64
)
if sample_weight is None:
knots = np.percentile(X, percentiles, axis=0)
else:
knots = np.array(
[
_weighted_percentile(X, sample_weight, percentile)
for percentile in percentiles
]
)
else:
# knots == 'uniform':
# Note that the variable `knots` has already been validated and
# `else` is therefore safe.
# Disregard observations with zero weight.
mask = slice(None, None, 1) if sample_weight is None else sample_weight > 0
x_min = np.amin(X[mask], axis=0)
x_max = np.amax(X[mask], axis=0)
knots = linspace(
start=x_min,
stop=x_max,
num=n_knots,
endpoint=True,
dtype=np.float64,
)
return knots
@deprecated(
"get_feature_names is deprecated in 1.0 and will be removed "
"in 1.2. Please use get_feature_names_out instead."
)
def get_feature_names(self, input_features=None):
"""Return feature names for output features.
Parameters
----------
input_features : list of str of shape (n_features,), default=None
String names for input features if available. By default,
"x0", "x1", ... "xn_features" is used.
Returns
-------
output_feature_names : list of str of shape (n_output_features,)
Transformed feature names.
"""
n_splines = self.bsplines_[0].c.shape[0]
if input_features is None:
input_features = ["x%d" % i for i in range(self.n_features_in_)]
feature_names = []
for i in range(self.n_features_in_):
for j in range(n_splines - 1 + self.include_bias):
feature_names.append(f"{input_features[i]}_sp_{j}")
return feature_names
def get_feature_names_out(self, input_features=None):
"""Get output feature names for transformation.
Parameters
----------
input_features : array-like of str or None, default=None
Input features.
- If `input_features` is `None`, then `feature_names_in_` is
used as feature names in. If `feature_names_in_` is not defined,
then names are generated: `[x0, x1, ..., x(n_features_in_)]`.
- If `input_features` is an array-like, then `input_features` must
match `feature_names_in_` if `feature_names_in_` is defined.
Returns
-------
feature_names_out : ndarray of str objects
Transformed feature names.
"""
n_splines = self.bsplines_[0].c.shape[0]
input_features = _check_feature_names_in(self, input_features)
feature_names = []
for i in range(self.n_features_in_):
for j in range(n_splines - 1 + self.include_bias):
feature_names.append(f"{input_features[i]}_sp_{j}")
return np.asarray(feature_names, dtype=object)
def fit(self, X, y=None, sample_weight=None):
"""Compute knot positions of splines.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The data.
y : None
Ignored.
sample_weight : array-like of shape (n_samples,), default = None
Individual weights for each sample. Used to calculate quantiles if
`knots="quantile"`. For `knots="uniform"`, zero weighted
observations are ignored for finding the min and max of `X`.
Returns
-------
self : object
Fitted transformer.
"""
X = self._validate_data(
X,
reset=True,
accept_sparse=False,
ensure_min_samples=2,
ensure_2d=True,
)
if sample_weight is not None:
sample_weight = _check_sample_weight(sample_weight, X, dtype=X.dtype)
_, n_features = X.shape
if not (isinstance(self.degree, numbers.Integral) and self.degree >= 0):
raise ValueError(
f"degree must be a non-negative integer, got {self.degree}."
)
if isinstance(self.knots, str) and self.knots in [
"uniform",
"quantile",
]:
if not (isinstance(self.n_knots, numbers.Integral) and self.n_knots >= 2):
raise ValueError(
f"n_knots must be a positive integer >= 2, got: {self.n_knots}"
)
base_knots = self._get_base_knot_positions(
X, n_knots=self.n_knots, knots=self.knots, sample_weight=sample_weight
)
else:
base_knots = check_array(self.knots, dtype=np.float64)
if base_knots.shape[0] < 2:
raise ValueError("Number of knots, knots.shape[0], must be >= 2.")
elif base_knots.shape[1] != n_features:
raise ValueError("knots.shape[1] == n_features is violated.")
elif not np.all(np.diff(base_knots, axis=0) > 0):
raise ValueError("knots must be sorted without duplicates.")
if self.extrapolation not in (
"error",
"constant",
"linear",
"continue",
"periodic",
):
raise ValueError(
"extrapolation must be one of 'error', "
"'constant', 'linear', 'continue' or 'periodic'."
)
if not isinstance(self.include_bias, (bool, np.bool_)):
raise ValueError("include_bias must be bool.")
# number of knots for base interval
n_knots = base_knots.shape[0]
if self.extrapolation == "periodic" and n_knots <= self.degree:
raise ValueError(
"Periodic splines require degree < n_knots. Got n_knots="
f"{n_knots} and degree={self.degree}."
)
# number of splines basis functions
if self.extrapolation != "periodic":
n_splines = n_knots + self.degree - 1
else:
# periodic splines have self.degree less degrees of freedom
n_splines = n_knots - 1
degree = self.degree
n_out = n_features * n_splines
# We have to add degree number of knots below, and degree number knots
# above the base knots in order to make the spline basis complete.
if self.extrapolation == "periodic":
# For periodic splines the spacing of the first / last degree knots
# needs to be a continuation of the spacing of the last / first
# base knots.
period = base_knots[-1] - base_knots[0]
knots = np.r_[
base_knots[-(degree + 1) : -1] - period,
base_knots,
base_knots[1 : (degree + 1)] + period,
]
else:
# Eilers & Marx in "Flexible smoothing with B-splines and
# penalties" https://doi.org/10.1214/ss/1038425655 advice
# against repeating first and last knot several times, which
# would have inferior behaviour at boundaries if combined with
# a penalty (hence P-Spline). We follow this advice even if our
# splines are unpenalized. Meaning we do not:
# knots = np.r_[
# np.tile(base_knots.min(axis=0), reps=[degree, 1]),
# base_knots,
# np.tile(base_knots.max(axis=0), reps=[degree, 1])
# ]
# Instead, we reuse the distance of the 2 fist/last knots.
dist_min = base_knots[1] - base_knots[0]
dist_max = base_knots[-1] - base_knots[-2]
knots = np.r_[
linspace(
base_knots[0] - degree * dist_min,
base_knots[0] - dist_min,
num=degree,
),
base_knots,
linspace(
base_knots[-1] + dist_max,
base_knots[-1] + degree * dist_max,
num=degree,
),
]
# With a diagonal coefficient matrix, we get back the spline basis
# elements, i.e. the design matrix of the spline.
# Note, BSpline appreciates C-contiguous float64 arrays as c=coef.
coef = np.eye(n_splines, dtype=np.float64)
if self.extrapolation == "periodic":
coef = np.concatenate((coef, coef[:degree, :]))
extrapolate = self.extrapolation in ["periodic", "continue"]
bsplines = [
BSpline.construct_fast(
knots[:, i], coef, self.degree, extrapolate=extrapolate
)
for i in range(n_features)
]
self.bsplines_ = bsplines
self.n_features_out_ = n_out - n_features * (1 - self.include_bias)
return self
def transform(self, X):
"""Transform each feature data to B-splines.
Parameters
----------
X : array-like of shape (n_samples, n_features)
The data to transform.
Returns
-------
XBS : ndarray of shape (n_samples, n_features * n_splines)
The matrix of features, where n_splines is the number of bases
elements of the B-splines, n_knots + degree - 1.
"""
check_is_fitted(self)
X = self._validate_data(X, reset=False, accept_sparse=False, ensure_2d=True)
n_samples, n_features = X.shape
n_splines = self.bsplines_[0].c.shape[1]
degree = self.degree
# Note that scipy BSpline returns float64 arrays and converts input
# x=X[:, i] to c-contiguous float64.
n_out = self.n_features_out_ + n_features * (1 - self.include_bias)
if X.dtype in FLOAT_DTYPES:
dtype = X.dtype
else:
dtype = np.float64
XBS = np.zeros((n_samples, n_out), dtype=dtype, order=self.order)
for i in range(n_features):
spl = self.bsplines_[i]
if self.extrapolation in ("continue", "error", "periodic"):
if self.extrapolation == "periodic":
# With periodic extrapolation we map x to the segment
# [spl.t[k], spl.t[n]].
# This is equivalent to BSpline(.., extrapolate="periodic")
# for scipy>=1.0.0.
n = spl.t.size - spl.k - 1
# Assign to new array to avoid inplace operation
x = spl.t[spl.k] + (X[:, i] - spl.t[spl.k]) % (
spl.t[n] - spl.t[spl.k]
)
else:
x = X[:, i]
XBS[:, (i * n_splines) : ((i + 1) * n_splines)] = spl(x)
else:
xmin = spl.t[degree]
xmax = spl.t[-degree - 1]
mask = (xmin <= X[:, i]) & (X[:, i] <= xmax)
XBS[mask, (i * n_splines) : ((i + 1) * n_splines)] = spl(X[mask, i])
# Note for extrapolation:
# 'continue' is already returned as is by scipy BSplines
if self.extrapolation == "error":
# BSpline with extrapolate=False does not raise an error, but
# output np.nan.
if np.any(np.isnan(XBS[:, (i * n_splines) : ((i + 1) * n_splines)])):
raise ValueError(
"X contains values beyond the limits of the knots."
)
elif self.extrapolation == "constant":
# Set all values beyond xmin and xmax to the value of the
# spline basis functions at those two positions.
# Only the first degree and last degree number of splines
# have non-zero values at the boundaries.
# spline values at boundaries
f_min = spl(xmin)
f_max = spl(xmax)
mask = X[:, i] < xmin
if np.any(mask):
XBS[mask, (i * n_splines) : (i * n_splines + degree)] = f_min[
:degree
]
mask = X[:, i] > xmax
if np.any(mask):
XBS[
mask,
((i + 1) * n_splines - degree) : ((i + 1) * n_splines),
] = f_max[-degree:]
elif self.extrapolation == "linear":
# Continue the degree first and degree last spline bases
# linearly beyond the boundaries, with slope = derivative at
# the boundary.
# Note that all others have derivative = value = 0 at the
# boundaries.
# spline values at boundaries
f_min, f_max = spl(xmin), spl(xmax)
# spline derivatives = slopes at boundaries
fp_min, fp_max = spl(xmin, nu=1), spl(xmax, nu=1)
# Compute the linear continuation.
if degree <= 1:
# For degree=1, the derivative of 2nd spline is not zero at
# boundary. For degree=0 it is the same as 'constant'.
degree += 1
for j in range(degree):
mask = X[:, i] < xmin
if np.any(mask):
XBS[mask, i * n_splines + j] = (
f_min[j] + (X[mask, i] - xmin) * fp_min[j]
)
mask = X[:, i] > xmax
if np.any(mask):
k = n_splines - 1 - j
XBS[mask, i * n_splines + k] = (
f_max[k] + (X[mask, i] - xmax) * fp_max[k]
)
if self.include_bias:
return XBS
else:
# We throw away one spline basis per feature.
# We chose the last one.
indices = [j for j in range(XBS.shape[1]) if (j + 1) % n_splines != 0]
return XBS[:, indices]