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Polynomial and Spline interpolation¶

This example demonstrates how to approximate a function with polynomials up to degree degree by using ridge regression. We show two different ways given n_samples of 1d points x_i:

PolynomialFeatures generates all monomials up to degree. This gives us the so called Vandermonde matrix with n_samples rows and degree + 1 columns:
```
[[1, x_0, x_0 ** 2, x_0 ** 3, ..., x_0 ** degree],
 [1, x_1, x_1 ** 2, x_1 ** 3, ..., x_1 ** degree],
 ...]
```
Intuitively, this matrix can be interpreted as a matrix of pseudo features (the points raised to some power). The matrix is akin to (but different from) the matrix induced by a polynomial kernel.
SplineTransformer generates B-spline basis functions. A basis function of a B-spline is a piece-wise polynomial function of degree degree that is non-zero only between degree+1 consecutive knots. Given n_knots number of knots, this results in matrix of n_samples rows and n_knots + degree - 1 columns:
```
[[basis_1(x_0), basis_2(x_0), ...],
 [basis_1(x_1), basis_2(x_1), ...],
 ...]
```

This example shows that these two transformers are well suited to model non-linear effects with a linear model, using a pipeline to add non-linear features. Kernel methods extend this idea and can induce very high (even infinite) dimensional feature spaces.

# Author: Mathieu Blondel
#         Jake Vanderplas
#         Christian Lorentzen
#         Malte Londschien
# License: BSD 3 clause

import matplotlib.pyplot as plt
import numpy as np

from sklearn.linear_model import Ridge
from sklearn.pipeline import make_pipeline
from sklearn.preprocessing import PolynomialFeatures, SplineTransformer

We start by defining a function that we intend to approximate and prepare plotting it.

def f(x):
    """Function to be approximated by polynomial interpolation."""
    return x * np.sin(x)


# whole range we want to plot
x_plot = np.linspace(-1, 11, 100)

To make it interesting, we only give a small subset of points to train on.

x_train = np.linspace(0, 10, 100)
rng = np.random.RandomState(0)
x_train = np.sort(rng.choice(x_train, size=20, replace=False))
y_train = f(x_train)

# create 2D-array versions of these arrays to feed to transformers
X_train = x_train[:, np.newaxis]
X_plot = x_plot[:, np.newaxis]

Now we are ready to create polynomial features and splines, fit on the training points and show how well they interpolate.

# plot function
lw = 2
fig, ax = plt.subplots()
ax.set_prop_cycle(
    color=["black", "teal", "yellowgreen", "gold", "darkorange", "tomato"]
)
ax.plot(x_plot, f(x_plot), linewidth=lw, label="ground truth")

# plot training points
ax.scatter(x_train, y_train, label="training points")

# polynomial features
for degree in [3, 4, 5]:
    model = make_pipeline(PolynomialFeatures(degree), Ridge(alpha=1e-3))
    model.fit(X_train, y_train)
    y_plot = model.predict(X_plot)
    ax.plot(x_plot, y_plot, label=f"degree {degree}")

# B-spline with 4 + 3 - 1 = 6 basis functions
model = make_pipeline(SplineTransformer(n_knots=4, degree=3), Ridge(alpha=1e-3))
model.fit(X_train, y_train)

y_plot = model.predict(X_plot)
ax.plot(x_plot, y_plot, label="B-spline")
ax.legend(loc="lower center")
ax.set_ylim(-20, 10)
plt.show()

This shows nicely that higher degree polynomials can fit the data better. But at the same time, too high powers can show unwanted oscillatory behaviour and are particularly dangerous for extrapolation beyond the range of fitted data. This is an advantage of B-splines. They usually fit the data as well as polynomials and show very nice and smooth behaviour. They have also good options to control the extrapolation, which defaults to continue with a constant. Note that most often, you would rather increase the number of knots but keep degree=3.

In order to give more insights into the generated feature bases, we plot all columns of both transformers separately.

fig, axes = plt.subplots(ncols=2, figsize=(16, 5))
pft = PolynomialFeatures(degree=3).fit(X_train)
axes[0].plot(x_plot, pft.transform(X_plot))
axes[0].legend(axes[0].lines, [f"degree {n}" for n in range(4)])
axes[0].set_title("PolynomialFeatures")

splt = SplineTransformer(n_knots=4, degree=3).fit(X_train)
axes[1].plot(x_plot, splt.transform(X_plot))
axes[1].legend(axes[1].lines, [f"spline {n}" for n in range(6)])
axes[1].set_title("SplineTransformer")

# plot knots of spline
knots = splt.bsplines_[0].t
axes[1].vlines(knots[3:-3], ymin=0, ymax=0.8, linestyles="dashed")
plt.show()

In the left plot, we recognize the lines corresponding to simple monomials from x**0 to x**3. In the right figure, we see the six B-spline basis functions of degree=3 and also the four knot positions that were chosen during fit. Note that there are degree number of additional knots each to the left and to the right of the fitted interval. These are there for technical reasons, so we refrain from showing them. Every basis function has local support and is continued as a constant beyond the fitted range. This extrapolating behaviour could be changed by the argument extrapolation.

Periodic Splines¶

In the previous example we saw the limitations of polynomials and splines for extrapolation beyond the range of the training observations. In some settings, e.g. with seasonal effects, we expect a periodic continuation of the underlying signal. Such effects can be modelled using periodic splines, which have equal function value and equal derivatives at the first and last knot. In the following case we show how periodic splines provide a better fit both within and outside of the range of training data given the additional information of periodicity. The splines period is the distance between the first and last knot, which we specify manually.

Periodic splines can also be useful for naturally periodic features (such as day of the year), as the smoothness at the boundary knots prevents a jump in the transformed values (e.g. from Dec 31st to Jan 1st). For such naturally periodic features or more generally features where the period is known, it is advised to explicitly pass this information to the SplineTransformer by setting the knots manually.

def g(x):
    """Function to be approximated by periodic spline interpolation."""
    return np.sin(x) - 0.7 * np.cos(x * 3)


y_train = g(x_train)

# Extend the test data into the future:
x_plot_ext = np.linspace(-1, 21, 200)
X_plot_ext = x_plot_ext[:, np.newaxis]

lw = 2
fig, ax = plt.subplots()
ax.set_prop_cycle(color=["black", "tomato", "teal"])
ax.plot(x_plot_ext, g(x_plot_ext), linewidth=lw, label="ground truth")
ax.scatter(x_train, y_train, label="training points")

for transformer, label in [
    (SplineTransformer(degree=3, n_knots=10), "spline"),
    (
        SplineTransformer(
            degree=3,
            knots=np.linspace(0, 2 * np.pi, 10)[:, None],
            extrapolation="periodic",
        ),
        "periodic spline",
    ),
]:
    model = make_pipeline(transformer, Ridge(alpha=1e-3))
    model.fit(X_train, y_train)
    y_plot_ext = model.predict(X_plot_ext)
    ax.plot(x_plot_ext, y_plot_ext, label=label)

ax.legend()
fig.show()

fig, ax = plt.subplots()
knots = np.linspace(0, 2 * np.pi, 4)
splt = SplineTransformer(knots=knots[:, None], degree=3, extrapolation="periodic").fit(
    X_train
)
ax.plot(x_plot_ext, splt.transform(X_plot_ext))
ax.legend(ax.lines, [f"spline {n}" for n in range(3)])
plt.show()