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Given a set of p predictor variables and a response variable, multiple linear regression uses a method known as least squares to minimize the sum of squared residuals (RSS):

RSS = Σ(y_i – ŷ_i)²

where:

• Σ: A greek symbol that means sum
• y_i: The actual response value for the i^th observation
• ŷ_i: The predicted response value based on the multiple linear regression model

However, when the predictor variables are highly correlated then multicollinearitycan become a problem. This can cause the coefficient estimates of the model to be unreliable and have high variance.

One way to avoid this problem is to instead use principal components regression, which finds M linear combinations (known as “principal components”) of the original p predictors and then uses least squares to fit a linear regression model using the principal components as predictors.

This tutorial provides a step-by-step example of how to perform principal components regression in Python.

Step 1: Import Necessary Packages

First, we’ll import the necessary packages to perform principal components regression (PCR) in Python:

import numpy as np
import pandas as pd
import matplotlib.pyplotas plt
from sklearn.preprocessingimport scale 
from sklearn import model_selection
from sklearn.model_selectionimport RepeatedKFold
from sklearn.model_selection import train_test_split
from sklearn.decompositionimport PCA
from sklearn.linear_modelimport LinearRegression
from sklearn.metricsimport mean_squared_error

Step 2: Load the Data

For this example, we’ll use a dataset called mtcars, which contains information about 33 different cars. We’ll use hp as the response variable and the following variables as the predictors:

• mpg
• disp
• drat
• wt
• qsec

The following code shows how to load and view this dataset:

#define URL where data is located
url = "https://raw.githubusercontent.com/Statology/Python-Guides/main/mtcars.csv"

#read in data
data_full = pd.read_csv(url)

#select subset of data
data = data_full[["mpg", "disp", "drat", "wt", "qsec", "hp"]]

#view first six rows of data
data[0:6]


        mpg	disp	drat	wt	qsec	hp
0	21.0	160.0	3.90	2.620	16.46	110
1	21.0	160.0	3.90	2.875	17.02	110
2	22.8	108.0	3.85	2.320	18.61	93
3	21.4	258.0	3.08	3.215	19.44	110
4	18.7	360.0	3.15	3.440	17.02	175
5	18.1	225.0	2.76	3.460	20.22	105

Step 3: Fit the PCR Model

The following code shows how to fit the PCR model to this data. Note the following:

• pca.fit_transform(scale(X)): This tells Python that each of the predictor variables should be scaled to have a mean of 0 and a standard deviation of 1. This ensures that no predictor variable is overly influential in the model if it happens to be measured in different units.
• cv = RepeatedKFold(): This tells Python to use k-fold cross-validation to evaluate the performance of the model. For this example we choose k = 10 folds, repeated 3 times.

#define predictor and response variables
X = data[["mpg", "disp", "drat", "wt", "qsec"]]
y = data[["hp"]]

#scale predictor variables
pca = PCA()
X_reduced = pca.fit_transform(scale(X))

#define cross validation method
cv = RepeatedKFold(n_splits=10, n_repeats=3, random_state=1)

regr = LinearRegression()
mse = []

# Calculate MSE with only the intercept
score = -1*model_selection.cross_val_score(regr,
           np.ones((len(X_reduced),1)), y, cv=cv,
           scoring='neg_mean_squared_error').mean()    
mse.append(score)

# Calculate MSE using cross-validation, adding one component at a time
for i in np.arange(1, 6):
    score = -1*model_selection.cross_val_score(regr,
               X_reduced[:,:i], y, cv=cv, scoring='neg_mean_squared_error').mean()
    mse.append(score)
    
# Plot cross-validation results    
plt.plot(mse)
plt.xlabel('Number of Principal Components')
plt.ylabel('MSE')
plt.title('hp')

The plot displays the number of principal components along the x-axis and the test MSE (mean squared error) along the y-axis.

From the plot we can see that the test MSE decreases by adding in two principal components, yet it begins to increase as we add more than two principal components.

Thus, the optimal model includes just the first two principal components.

We can also use the following code to calculate the percentage of variance in the response variable explained by adding in each principal component to the model:

np.cumsum(np.round(pca.explained_variance_ratio_, decimals=4)*100)

array([69.83, 89.35, 95.88, 98.95, 99.99])

We can see the following:

• By using just the first principal component, we can explain 69.83% of the variation in the response variable.
• By adding in the second principal component, we can explain 89.35% of the variation in the response variable.

Note that we’ll always be able to explain more variance by using more principal components, but we can see that adding in more than two principal components doesn’t actually increase the percentage of explained variance by much.

Step 4: Use the Final Model to Make Predictions

We can use the final PCR model with two principal components to make predictions on new observations.

The following code shows how to split the original dataset into a training and testing set and use the PCR model with two principal components to make predictions on the testing set.

#split the dataset into training (70%) and testing (30%) sets
X_train,X_test,y_train,y_test = train_test_split(X,y,test_size=0.3,random_state=0) 

#scale the training and testing data
X_reduced_train = pca.fit_transform(scale(X_train))
X_reduced_test = pca.transform(scale(X_test))[:,:1]

#train PCR model on training data 
regr = LinearRegression()
regr.fit(X_reduced_train[:,:1], y_train)

#calculate RMSE
pred = regr.predict(X_reduced_test)
np.sqrt(mean_squared_error(y_test, pred))

40.2096

We can see that the test RMSE turns out to be 40.2096. This is the average deviation between the predicted value for hp and the observed value for hp for the observations in the testing set.

The complete Python code use in this example can be found here.