Lecture 16 – Data 100, Fall 2023¶

Data 100, Fall 2023

Acknowledgments Page

In [1]:
import seaborn as sns
import pandas as pd
import matplotlib.pyplot as plt
import numpy as np

import warnings
warnings.filterwarnings('ignore')

In this lecture, we will work with the vehicles dataset.

In [2]:
vehicles = sns.load_dataset("mpg").rename(columns={"horsepower":"hp"}).dropna().sort_values("hp")
vehicles.head()
Out[2]:
mpg cylinders displacement hp weight acceleration model_year origin name
19 26.0 4 97.0 46.0 1835 20.5 70 europe volkswagen 1131 deluxe sedan
102 26.0 4 97.0 46.0 1950 21.0 73 europe volkswagen super beetle
326 43.4 4 90.0 48.0 2335 23.7 80 europe vw dasher (diesel)
325 44.3 4 90.0 48.0 2085 21.7 80 europe vw rabbit c (diesel)
244 43.1 4 90.0 48.0 1985 21.5 78 europe volkswagen rabbit custom diesel

We will attempt to predict a car's "mpg" from transformations of its "hp".

In [3]:
X = vehicles[["hp"]]
X["hp^2"] = vehicles["hp"]**2
X["hp^3"] = vehicles["hp"]**3
X["hp^4"] = vehicles["hp"]**4

Y = vehicles["mpg"]

Test Sets¶

To perform a train-test split, we can use the train_test_split function of the sklearn.model_selection module.

In [4]:
from sklearn.model_selection import train_test_split

# `test_size` specifies the proportion of the full dataset that should be allocated to testing.
# `random_state` makes our results reproducible for educational purposes.
# shuffle is True by default and randomizes the data before splitting.

X_train, X_test, Y_train, Y_test = train_test_split(X, Y, 
                                                    test_size=0.2, 
                                                    random_state=100, 
                                                    shuffle=True)

print(f"Size of full dataset: {X.shape[0]} points")
print(f"Size of training set: {X_train.shape[0]} points")
print(f"Size of test set: {X_test.shape[0]} points")

Size of full dataset: 392 points
Size of training set: 313 points
Size of test set: 79 points

We then fit the model using the training set...

In [5]:
import sklearn.linear_model as lm

model = lm.LinearRegression()

model.fit(X_train, Y_train)
Out[5]:
LinearRegression()
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
On GitHub, the HTML representation is unable to render, please try loading this page with nbviewer.org.
LinearRegression()

...and evaluate its performance by making predictions on the test set. Notice that the model performs more poorly on the test data it did not encounter during training.

In [6]:
from sklearn.metrics import mean_squared_error

train_error = mean_squared_error(Y_train, model.predict(X_train))
test_error = mean_squared_error(Y_test, model.predict(X_test))

print(f"Training error: {train_error}")
print(f"Test error: {test_error}")

Training error: 18.34167355487827
Test error: 21.527475812703674

Validation Sets¶

To assess model performance on unseen data, then use this information to finetune the model, we introduce a validation set. You can imagine this as us splitting the training set into a validation set and a "mini" training set.

In [7]:
# Split X_train further into X_train_mini and X_val.
X_train_mini, X_val, Y_train_mini, Y_val = train_test_split(X_train, Y_train, test_size=0.2, random_state=100)

print(f"Size of original training set: {X_train.shape[0]} points")
print(f"Size of mini training set: {X_train_mini.shape[0]} points")
print(f"Size of validation set: {X_val.shape[0]} points")

Size of original training set: 313 points
Size of mini training set: 250 points
Size of validation set: 63 points

In the cell below, we repeat the experiment from Lecture 14: fit several models of increasing complexity, then compute their errors. Here, we find the model's errors on the validation set to understand how model complexity influences performance on unseen data.

In [8]:
fig, ax = plt.subplots(1, 3, dpi=200, figsize=(12, 3))

for order in [2, 3, 4]:
    model = lm.LinearRegression()
    model.fit(X_train_mini.iloc[:, :order], Y_train_mini)
    val_predictions = model.predict(X_val.iloc[:, :order])
    
    output = X_val.iloc[:, :order]
    output["y_hat"] = val_predictions
    output = output.sort_values("hp")
    
    ax[order-2].scatter(X_val["hp"], Y_val, edgecolor="white", lw=0.5)
    ax[order-2].plot(output["hp"], output["y_hat"], "tab:red")
    ax[order-2].set_title(f"Model with degree {order}")
    ax[order-2].set_xlabel("hp")
    ax[order-2].set_ylabel("mpg")
    ax[order-2].annotate(f"Validation MSE: {np.round(mean_squared_error(Y_val, val_predictions), 3)}", (90, 30))

plt.subplots_adjust(wspace=0.3);

Let's repeat this process:

  1. Fit an degree-x model to the mini training set
  2. Evaluate the fitted model's MSE when making predictions on the validation set

We use the model's performance on the validation set as a guide to selecting the best combination of features. We are not limited in the number of times we use the validation set – we just never use this set to fit the model.

In [9]:
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures

def fit_model_dataset(degree):
    pipelined_model = Pipeline([
            ('polynomial_transformation', PolynomialFeatures(degree)),
            ('linear_regression', lm.LinearRegression())    
        ])

    pipelined_model.fit(X_train_mini[["hp"]], Y_train_mini)
    return mean_squared_error(Y_val, pipelined_model.predict(X_val[["hp"]]))

errors = [fit_model_dataset(degree) for degree in range(0, 18)]
MSEs_and_k = pd.DataFrame({"k": range(0, 18), "MSE": errors})

plt.figure(dpi=120)
plt.plot(range(0, 18), errors)
plt.xlabel("Model Complexity (degree of polynomial)")
plt.ylabel("Validation MSE")
plt.xticks(range(0, 18));
In [10]:
MSEs_and_k.rename(columns={"k":"Degree"}).set_index("Degree")
Out[10]:
MSE
Degree
0 54.024495
1 17.798108
2 12.415902
3 12.196206
4 12.192597
5 11.938316
6 12.168635
7 11.955918
8 11.936344
9 12.018955
10 12.037131
11 12.118490
12 15.757697
13 18.727578
14 21.780936
15 24.325657
16 26.277495
17 27.808360

From this model selection process, we might choose to create a model with degree 8.

In [11]:
print(f'Polynomial degree with lowest validation error: {MSEs_and_k.sort_values("MSE").head(1)["k"].values}')

Polynomial degree with lowest validation error: [8]

After this choice has been finalized, and we are completely finished with the model design process, we finally assess model performance on the test set. We typically use the entire training set (both the "mini" training set and validation set) to fit the final model.

In [12]:
# Update our training and test sets to include all polynomial features between 5 and 9
for degree in range(5, 9):
    X_train[f"hp^{degree}"] = X_train["hp"]**degree
    X_test[f"hp^{degree}"] = X_test["hp"]**degree

final_model = lm.LinearRegression()
final_model.fit(X_train, Y_train)

print(f"Test MSE of final model: {mean_squared_error(Y_test, final_model.predict(X_test))}")

Test MSE of final model: 20.80824130943735

Cross-Validation¶

The validation set gave us an opportunity to understand how the model performs on a single set of unseen data. The specific validation set we drew was fixed – we used the same validation points every time.

It's possible that we may have, by random chance, selected a set of validation points that was not representative of other unseen data that the model might encounter (for example, if we happened to have selected all outlying data points for the validation set).

Different train/validation splits lead to different validation errors:

In [13]:
for i in range(1, 4):
    X_train_mini, X_val, Y_train_mini, Y_val = train_test_split(X_train, Y_train, test_size=0.2)
    model = lm.LinearRegression()
    model.fit(X_train_mini, Y_train_mini)
    y_hat = model.predict(X_val)
    print(f"Val error from train/validation split #{i}: {mean_squared_error(y_hat, Y_val)}")

Val error from train/validation split #1: 19.689592552823704
Val error from train/validation split #2: 12.251918187375107
Val error from train/validation split #3: 13.326933792260284

To apply cross-validation, we use the KFold class of sklearn.model_selection. KFold will return the indices of each cross-validation fold. Then, we iterate over each of these folds to designate it as the validation set, while training the model on the remaining folds.

In [14]:
from sklearn.model_selection import KFold
np.random.seed(25) # Ensures reproducibility of this notebook

# n_splits sets the number of folds to create
kf = KFold(n_splits=5, shuffle=True)
validation_errors = []

for train_idx, valid_idx in kf.split(X_train):
    # Split the data
    split_X_train, split_X_valid = X_train.iloc[train_idx], X_train.iloc[valid_idx]
    split_Y_train, split_Y_valid = Y_train.iloc[train_idx], Y_train.iloc[valid_idx]

    # Fit the model on the training split
    model.fit(split_X_train, split_Y_train)

    error = mean_squared_error(model.predict(split_X_valid), split_Y_valid)

    validation_errors.append(error)

print(f"Cross-validation error: {np.mean(validation_errors)}")

Cross-validation error: 19.19580505014516

Regularization¶

L1 (LASSO) Regularization¶

To apply L1 regularization, we use the Lasso model class of sklearn. Lasso functions just like LinearRegression. The difference is that now the model will apply a regularization penalty. We specify the strength of regularization using the alpha parameter, which is equivalent to $\frac{1}{\lambda}$ from our objective function formulation.

In [15]:
import sklearn.linear_model as lm

lasso_model = lm.Lasso(alpha=0.1) # In sklearn, alpha represents the lambda hyperparameter
lasso_model.fit(X_train, Y_train) 

lasso_model.coef_
Out[15]:
array([-5.14100640e-01,  1.16422594e-03,  2.70209864e-06, -8.05153574e-10,
       -2.78280269e-11, -1.02040718e-13, -5.44295812e-17,  1.83589942e-18])

To increase the strength of regularization (decrease model complexity), we increase the $\lambda$ hyperparameter by changing alpha.

In [16]:
lasso_model_large_lambda = lm.Lasso(alpha=10)
lasso_model_large_lambda.fit(X_train, Y_train) 

lasso_model_large_lambda.coef_
Out[16]:
array([-0.00000000e+00, -3.37446532e-03,  1.31817186e-05,  1.71062658e-08,
       -2.44893438e-11, -2.11314339e-13, -5.38994214e-16,  7.05457777e-19])

Notice that these model coefficients are very small (some are effectively 0). This reflects L1 regularization's tendency to set the parameters of unimportant features to 0. We can use this in feature selection.

The features in our dataset are on wildly different numerical scales. To see this, compare the values of hp to the values of hp^8.

In [17]:
X_train.head()
Out[17]:
hp hp^2 hp^3 hp^4 hp^5 hp^6 hp^7 hp^8
72 150.0 22500.0 3375000.0 5.062500e+08 7.593750e+10 1.139062e+13 1.708594e+15 2.562891e+17
89 150.0 22500.0 3375000.0 5.062500e+08 7.593750e+10 1.139062e+13 1.708594e+15 2.562891e+17
92 158.0 24964.0 3944312.0 6.232013e+08 9.846580e+10 1.555760e+13 2.458100e+15 3.883799e+17
124 180.0 32400.0 5832000.0 1.049760e+09 1.889568e+11 3.401222e+13 6.122200e+15 1.101996e+18
88 137.0 18769.0 2571353.0 3.522754e+08 4.826172e+10 6.611856e+12 9.058243e+14 1.240979e+17

In order for the feature hp to contribute in any meaningful way to the model, LASSO is "forced" to allocate disproportionately much of its parameter "budget" towards assigning a large value to the model parameter for hp. Notice how the parameter for hp is much, much greater in magnitude than the parameter for hp^8.

In [18]:
pd.DataFrame({"Feature":X_train.columns, "Parameter":lasso_model.coef_})
Out[18]:
Feature Parameter
0 hp -5.141006e-01
1 hp^2 1.164226e-03
2 hp^3 2.702099e-06
3 hp^4 -8.051536e-10
4 hp^5 -2.782803e-11
5 hp^6 -1.020407e-13
6 hp^7 -5.442958e-17
7 hp^8 1.835899e-18

We typically scale data before regularization such that all features are measured on the same numeric scale. One way to do this is by standardizing the data such that it has mean 0 and standard deviation 1.

In [19]:
# Center the data to have mean 0
X_train_centered = X_train - X_train.mean() 

# Scale the centered data to have SD 1
X_train_standardized = X_train_centered/X_train_centered.std()

X_train_standardized.head()
Out[19]:
hp hp^2 hp^3 hp^4 hp^5 hp^6 hp^7 hp^8
72 1.135297 0.984896 0.775312 0.553371 0.354272 0.194283 0.074957 -0.009505
89 1.135297 0.984896 0.775312 0.553371 0.354272 0.194283 0.074957 -0.009505
92 1.339125 1.231354 1.041813 0.815907 0.594634 0.402529 0.248159 0.130244
124 1.899651 1.975127 1.925462 1.773459 1.560109 1.324804 1.094889 0.885230
88 0.804077 0.611709 0.399115 0.207724 0.058991 -0.044537 -0.110553 -0.149359

When we re-fit a LASSO model, the coefficients are no longer as uneven in magnitude as they were before.

In [20]:
lasso_model_scaled = lm.Lasso(alpha=0.1)
lasso_model_scaled.fit(X_train_standardized, Y_train)
lasso_model_scaled.coef_
Out[20]:
array([-9.31789105,  0.        ,  0.        ,  2.89288682,  0.65909948,
        0.        ,  0.        ,  0.        ])

L2 (Ridge) Regression¶

We perform ridge regression using sklearn's Ridge class.

In [21]:
ridge_model = lm.Ridge(alpha=0.1)
ridge_model.fit(X_train_standardized, Y_train)

ridge_model.coef_
Out[21]:
array([-16.85961652,   3.26398097,   9.1167183 ,   4.53790201,
        -2.32110639,  -5.6066523 ,  -3.15831859,   4.75104822])