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"""
===================
Isotonic Regression
===================
An illustration of the isotonic regression on generated data (non-linear
monotonic trend with homoscedastic uniform noise).
The isotonic regression algorithm finds a non-decreasing approximation of a
function while minimizing the mean squared error on the training data. The
benefit of such a non-parametric model is that it does not assume any shape for
the target function besides monotonicity. For comparison a linear regression is
also presented.
The plot on the right-hand side shows the model prediction function that
results from the linear interpolation of thresholds points. The thresholds
points are a subset of the training input observations and their matching
target values are computed by the isotonic non-parametric fit.
"""
# Author: Nelle Varoquaux <nelle.varoquaux@gmail.com>
# Alexandre Gramfort <alexandre.gramfort@inria.fr>
# License: BSD
import matplotlib.pyplot as plt
import numpy as np
from matplotlib.collections import LineCollection
from sklearn.isotonic import IsotonicRegression
from sklearn.linear_model import LinearRegression
from sklearn.utils import check_random_state
n = 100
x = np.arange(n)
rs = check_random_state(0)
y = rs.randint(-50, 50, size=(n,)) + 50.0 * np.log1p(np.arange(n))
# %%
# Fit IsotonicRegression and LinearRegression models:
ir = IsotonicRegression(out_of_bounds="clip")
y_ = ir.fit_transform(x, y)
lr = LinearRegression()
lr.fit(x[:, np.newaxis], y) # x needs to be 2d for LinearRegression
# %%
# Plot results:
segments = [[[i, y[i]], [i, y_[i]]] for i in range(n)]
lc = LineCollection(segments, zorder=0)
lc.set_array(np.ones(len(y)))
lc.set_linewidths(np.full(n, 0.5))
fig, (ax0, ax1) = plt.subplots(ncols=2, figsize=(12, 6))
ax0.plot(x, y, "C0.", markersize=12)
ax0.plot(x, y_, "C1.-", markersize=12)
ax0.plot(x, lr.predict(x[:, np.newaxis]), "C2-")
ax0.add_collection(lc)
ax0.legend(("Training data", "Isotonic fit", "Linear fit"), loc="lower right")
ax0.set_title("Isotonic regression fit on noisy data (n=%d)" % n)
x_test = np.linspace(-10, 110, 1000)
ax1.plot(x_test, ir.predict(x_test), "C1-")
ax1.plot(ir.X_thresholds_, ir.y_thresholds_, "C1.", markersize=12)
ax1.set_title("Prediction function (%d thresholds)" % len(ir.X_thresholds_))
plt.show()
# %%
# Note that we explicitly passed `out_of_bounds="clip"` to the constructor of
# `IsotonicRegression` to control the way the model extrapolates outside of the
# range of data observed in the training set. This "clipping" extrapolation can
# be seen on the plot of the decision function on the right-hand.
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