Polynomial regression

An obious way to enhance the simple regression model (5) is to add more powers of predictor \(\boldsymbol x\). For example, consider quadratic regression

(10)\[ y = w_0 + w_1 x + w_2 x^2.\]

Now the model has three parameters \(\boldsymbol w = (w_0, w_1, w_2)\), which could be also fitted by optimizing of MSE:

(11)\[\mathcal L(\boldsymbol w) = \frac 1n\sum\limits_{i=1}^n (y_i - w_0 - w_1x_i - w_2x_i^2)^2 \to \min\limits_{\boldsymbol w}.\]

Revisit Boston dataset

The data look quite suitable for a quadratic regression. Let’s do a simple feature engineering and add new feature of squares. Now the design matrix has two columns:

\[\begin{split} \boldsymbol X = [\boldsymbol x\; \boldsymbol x^2] = \begin{pmatrix} x_1 & x_1^2 \\ x_2 & x_2^2 \\ \vdots & \vdots \\ x_n & x_n^2 \end{pmatrix}. \end{split}\]

To fit the linear regression on the new dataset, once again use sklearn library:

import pandas as pd
import numpy as np
from sklearn.linear_model import LinearRegression

boston = pd.read_csv("../ISLP_datsets/Boston.csv")

x = boston['lstat']
y = boston['medv']

LR = LinearRegression()
x_reshaped = x.values.reshape(-1, 1)
x_train = np.hstack([x_reshaped, x_reshaped**2])
LR.fit(x_train, y)
print("intercept:", LR.intercept_)
print("coefficients:", LR.coef_)
print("r-score:", LR.score(x_train, y))
print("MSE:", np.mean((LR.predict(x_train) - y) ** 2))

intercept: 42.86200732816936
coefficients: [-2.3328211   0.04354689]
r-score: 0.6407168971636612
MSE: 30.330520075853713

Our metrics have improved, now plot the graphs:

import matplotlib.pyplot as plt

%config InlineBackend.figure_format = 'svg'
plt.scatter(x, y, s=10, c='b', alpha=0.7)
xs = np.linspace(x.min(), x.max(), num=100)
plt.plot(xs, LR.intercept_ + LR.coef_[0]*xs + LR.coef_[1]*xs**2, c='r', lw=2)
plt.xlabel("lstat")
plt.ylabel("medv")
plt.grid(ls=":");

General case

Of course, the degree of the polynomial can be any number \(m\in\mathbb N\). The model of the polynomial regression is

(12)\[ y = w_0 + w_1 x + w_2 x^2 + \ldots + w_m x^m = \sum\limits_{k=0}^m w_k x^k.\]

Q. How many parameters does this model have?

In case of MSE loss the model is fitted via minimizing the function

(13)\[\mathcal L(\boldsymbol w) = \frac 1n\sum\limits_{i=1}^n \Big(y_i - \sum\limits_{k=0}^m w_k x^k\Big)^2 \to \min\limits_{\boldsymbol w}.\]

Exercises

1. Find the analytical solution of the optimization task (11).

2. Find the feature matrix of the polynomial regression model (12).

TODO

• Provide examples of dataset, where increasing of degreee is beneficial

• Make connection with Runge example from the first chapter

• Add some quizzes

• Add more datasets (may me even simulated)

• Think about cases where the performance of a polynomial model of any degree is poor