Building Polynomial Regression

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For this chapter, we have a file named poly.csv. Let's load the file and look at the contents.

So here we have one feature and the target. Let's build a scatter plot.

It is hard to imagine a straight line fitting this data well. So let's build a Polynomial Regression!

Building X̃ matrix

We will once again use the OLS class. Still, we need to create an X̃ matrix. We do it manually by adding a squared Feature column to the DataFrame like this:

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df['Feature_squared'] = df['Feature'] ** 2

But if we want to build a high-degree polynomial regression, that will require adding a lot of columns like this. Luckily Scikit-Learn provides a way to do it less painfully using the PolynomialFeatures class.

fit_transform(X) method expects X to be either 2-d array or pandas DataFrame. If your X is an 1-d numpy array, reshape(-1,1) method will transform it to a 2-d array with the same contents:

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X = X.reshape(-1,1)

If your X is a column from DataFrame, you can use X = df[['col1']] to get a DataFrame instead of pandas Series, which is not suited for fit_transform():

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X = df['Feature']   # X is a pandas Series
X = df[['Feature']]  # X is a pandas DataFrame

So to build an X̃ for the Polynomial Regression of degree n, we would use:

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from sklearn.preprocessing import PolynomialFeatures # Import the class
poly = PolynomialFeatures(n)  # Initialize a PolynomialFeatures object
X_tilde = poly.fit_transform(X)

Note

The PolynomialFeatures class also adds a column with 1s, so you do not need to use sm.add_constant().

Building the Polynomial Regression and making the predictions

Knowing how to get an X̃, we are ready to build the Polynomial Regression the same way as the prior models:

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y = df['Target']
# Prepare X_tilde
X = df[['Feature']]
X_tilde = PolynomialFeatures(n).fit_transform(X)
# Initialize the OLS object and train it
regression_model = sm.OLS(y, X_tilde).fit()

For predicting new values, X_new should be transformed using PolynomialFeatures too.

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X_new_tilde = PolynomialFeatures(n).fit_transform(X_new)
y_pred = regression_model.predict(X_new_tilde)

The following runnable example shows the entire process of building polynomial regression. X_new here is a 1-d array of points between -0.1 and 1.5. They are needed for visualization. And since it is a 1-d array, we should apply reshape(-1,1) method before using it in the PolynomialFeatures class.

Feel free to play with the values of n in the eighth line. You will see how the plot changes depending on the polynomial regression's degree. If you pay attention, you may notice how different the predictions are for feature values lower than 0, or greater than 1.4. That is the subject of the next chapter.