Lecture 14 – Data 100, Fall 2023¶

Data 100, Fall 2023

Acknowledgments Page

In [35]:
import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
import warnings
pd.options.mode.chained_assignment = None 
warnings.simplefilter(action='ignore', category=UserWarning)
np.random.seed(42)

Implementing OLS in Code¶

Let's explore the Python syntax for performing ordinary least squares programmatically.

Today, we will work with the penguins dataset, which contains information about penguins observed by field biologists. For simplicity, we will only consider Adelie penguins.

In [36]:
penguins = sns.load_dataset("penguins")
penguins = penguins[penguins["species"] == "Adelie"].dropna()
penguins.head()
Out[36]:
species island bill_length_mm bill_depth_mm flipper_length_mm body_mass_g sex
0 Adelie Torgersen 39.1 18.7 181.0 3750.0 Male
1 Adelie Torgersen 39.5 17.4 186.0 3800.0 Female
2 Adelie Torgersen 40.3 18.0 195.0 3250.0 Female
4 Adelie Torgersen 36.7 19.3 193.0 3450.0 Female
5 Adelie Torgersen 39.3 20.6 190.0 3650.0 Male

We will construct a model to predict a penguin's bill length from its flipper length and body mass. Our model will take the form:

$$\text{bill depth} = \theta_0 + \theta_1 \text{flipper length} + \theta_2 \text{body mass}$$

We start by constructing a design matrix X containing our input features and a target vector y.

In [37]:
# Add a bias column of all ones to `penguins`
penguins["bias"] = np.ones(len(penguins), dtype=int) 

# Define the design matrix, X...
X = penguins[["bias", "flipper_length_mm", "body_mass_g"]].to_numpy()

# ...as well as the target variable, y
Y = penguins[["bill_depth_mm"]].to_numpy()

In our lecture on ordinary least squares, we used the geometry of linear algebra to derive the OLS estimate for the optimal model parameters:

$$\hat{\theta} = (\mathbb{X}^{\top}\mathbb{X})^{-1}\mathbb{X}^{\top}\mathbb{Y}$$

There are three operations we need to perform here: multiplying matrices, taking transposes, and finding inverses.

  • To perform matrix multiplication, use the @ operator
  • To take a transpose, call the .T attribute of an array or DataFrame
  • To compute an inverse, use NumPy's in-built method np.linalg.inv

Putting this all together, we can compute the OLS estimate for the optimal model parameters, stored in the array theta_hat.

In [38]:
theta_hat = np.linalg.inv(X.T @ X) @ X.T @ Y
theta_hat
Out[38]:
array([[1.10029953e+01],
       [9.82848689e-03],
       [1.47749591e-03]])

To make predictions using the fitted model parameters, we apply the model:

$$\hat{\mathbb{Y}} = \mathbb{X} \hat{\theta}$$

In [39]:
y_hat = X @ theta_hat

sklearn¶

sklearn is a popular Python library for building and fitting models. In this lecture, we will walk through the general workflow for writing code for sklearn. While our examples will be focused on linear models, sklearn is highly adaptable for use on other (more complex) kinds of models. We'll see examples of this later in the semester.

There are three steps to creating and using a model in sklearn.

(1) Initialize an instance of the model class

sklearn stores "templates" of useful models for machine learning. We begin the modeling process by making a "copy" of one of these templates for our own use. Model initialization looks like ModelClass(), where ModelClass is the type of model we wish to create.

For now, let's create a linear regression model using LinearRegression().

my_model is now an instance of the LinearRegression class. You can think of it as the "idea" of a linear regression model. We haven't trained it yet, so it doesn't know any model parameters and cannot be used to make predictions. In fact, we haven't even told it what data to use for modeling! It simply waits for further instructions.

In [40]:
import sklearn.linear_model as lm

my_model = lm.LinearRegression()

my_model
Out[40]:
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()

(2) Train the model using .fit

Before the model can make predictions, we will need to fit it to our training data. When we fit the model, sklearn will run gradient descent behind the scenes to determine the optimal model parameters. It will then save these model parameters to our model instance for future use.

All sklearn model classes include a .fit method. This function is used to fit the model. It takes in two inputs: the design matrix, X, and the target variable, y.

Let's start by fitting a model with just one feature: the flipper length. We create a design matrix X by pulling out the "flipper_length_mm" column from the DataFrame. Notice that we use double brackets to extract this column. Why double brackets instead of just single brackets? The .fit method, by default, expects to receive 2-dimensional data – some kind of data that includes both rows and columns. Writing penguins["flipper_length_mm"] would return a 1D Series, causing sklearn to error. We avoid this by writing penguins[["flipper_length_mm"]] to produce a 2D DataFrame.

In [41]:
# .fit expects a 2D data design matrix, so we use double brackets to extract a DataFrame
X = penguins[["flipper_length_mm"]]
y = penguins["bill_depth_mm"]

my_model.fit(X, y)
Out[41]:
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 in just three lines of code, our model has run gradient descent to determine the optimal model parameters! Our single-feature model takes the form:

$$\text{bill depth} = \theta_0 + \theta_1 \text{flipper length}$$

Note that LinearRegression will automatically include an intercept term.

The fitted model parameters are stored as attributes of the model instance. my_model.intercept_ will return the value of $\hat{\theta}_0$ as a scalar. my_model.coef_ will return all values $\hat{\theta}_1, \hat{\theta}_1, ...$ in an array. Because our model only contains one feature, we see just the value of $\hat{\theta}_1$ in the cell below.

In [42]:
# The intercept term, theta_0
my_model.intercept_
Out[42]:
7.297305899612306
In [43]:
# All parameters theta_1, ..., theta_p
my_model.coef_
Out[43]:
array([0.05812622])

(3) Use the fitted model to make predictions

Now that the model has been trained, we can use it to make predictions! To do so, we use the .predict method. .predict takes in one argument, the design matrix that should be used to generate predictions. To understand how the model performs on the training set, we would pass in the training data. Alternatively, to make predictions on unseen data, we would pass in a new dataset that wasn't used to train the model.

Below, we call .predict to generate model predictions on the original training data. As before, we use double brackets to ensure that we extract 2-dimensional data.

In [44]:
y_hat = my_model.predict(penguins[["flipper_length_mm"]])

print(f"The RMSE of the model is {np.sqrt(np.mean((y-y_hat)**2))}")

The RMSE of the model is 1.1549363099239012

What if we wanted a model with two features?

$$\text{bill depth} = \theta_0 + \theta_1 \text{flipper length} + \theta_2 \text{body mass}$$

We repeat this three-step process by intializing a new model object, then calling .fit and .predict as before.

In [45]:
# Step 1: initialize LinearRegression model
two_feature_model = lm.LinearRegression()

# Step 2: fit the model
X_two_features = penguins[["flipper_length_mm", "body_mass_g"]]
y = penguins["bill_depth_mm"]

two_feature_model.fit(X_two_features, y)

# Step 3: make predictions
y_hat_two_features = two_feature_model.predict(X_two_features)

print(f"The RMSE of the model is {np.sqrt(np.mean((y-y_hat_two_features)**2))}")

The RMSE of the model is 0.9881331104079044

Feature Engineering¶

Feature engineering is the process of applying feature functions to generate new features for use in modeling. In this lecture, we will discuss:

  • One-hot encoding
  • Polynomial features

To do so, we'll work with our familiar tips dataset.

In [46]:
tips = sns.load_dataset("tips")
tips.head()
Out[46]:
total_bill tip sex smoker day time size
0 16.99 1.01 Female No Sun Dinner 2
1 10.34 1.66 Male No Sun Dinner 3
2 21.01 3.50 Male No Sun Dinner 3
3 23.68 3.31 Male No Sun Dinner 2
4 24.59 3.61 Female No Sun Dinner 4

One-Hot Encoding¶

One-hot encoding is a feature engineering technique to generate numeric feature from categorical data. For example, we can use one-hot encoding to incorporate the day of the week as an input into a regression model.

Suppose we want to use a design matrix of three features – the total_bill, size, and day – to predict the tip.

In [47]:
X_raw = tips[["total_bill", "size", "day"]]
y = tips["tip"]

Because day is non-numeric, we will apply one-hot encoding before fitting a model.

The OneHotEncoder class of sklearn (documentation) offers a quick way to perform one-hot encoding. You will explore its use in detail in Lab 7. For now, recognize that we follow a very similar workflow to when we were working with the LinearRegression class: we initialize a OneHotEncoder object, fit it to our data, then use .transform to apply the fitted encoder.

In [48]:
from sklearn.preprocessing import OneHotEncoder

# Initialize a OneHotEncoder object
ohe = OneHotEncoder()

# Fit the encoder
ohe.fit(tips[["day"]])

# Use the encoder to transform the raw "day" feature
encoded_day = ohe.transform(tips[["day"]]).toarray()
encoded_day_df = pd.DataFrame(encoded_day, columns=ohe.get_feature_names_out())

encoded_day_df.head()
Out[48]:
day_Fri day_Sat day_Sun day_Thur
0 0.0 0.0 1.0 0.0
1 0.0 0.0 1.0 0.0
2 0.0 0.0 1.0 0.0
3 0.0 0.0 1.0 0.0
4 0.0 0.0 1.0 0.0

The OneHotEncoder has converted the categorical day feature into four numeric features! Recall that the tips dataset only included data for Thursday through Sunday, which is why only four days of the week appear.

Let's join this one-hot encoding to the original data to form our featurized design matrix. We drop the original day column so our design matrix only includes numeric values.

In [49]:
X = X_raw.join(encoded_day_df).drop(columns="day")

X.head()
Out[49]:
total_bill size day_Fri day_Sat day_Sun day_Thur
0 16.99 2 0.0 0.0 1.0 0.0
1 10.34 3 0.0 0.0 1.0 0.0
2 21.01 3 0.0 0.0 1.0 0.0
3 23.68 2 0.0 0.0 1.0 0.0
4 24.59 4 0.0 0.0 1.0 0.0

Now, we can use sklearn's LinearRegression class to fit a model to this design matrix.

In [50]:
ohe_model = lm.LinearRegression(fit_intercept=False) #Tell sklearn to not add an additional bias column. Why?
ohe_model.fit(X, y)

pd.DataFrame({"Feature":X.columns, "Model Coefficient":ohe_model.coef_}).set_index("Feature")
Out[50]:
Model Coefficient
Feature
total_bill 0.092994
size 0.187132
day_Fri 0.745787
day_Sat 0.621129
day_Sun 0.732289
day_Thur 0.668294

Polynomial Features¶

Consider the vehicles dataset, which includes information about cars.

In [51]:
vehicles = sns.load_dataset("mpg").dropna().rename(columns = {"horsepower": "hp"}).sort_values("hp")
vehicles.head()
Out[51]:
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

Suppose we want to use the hp (horsepower) of a car to predict its mpg (gas mileage in miles per gallon). If we visualize the relationship between these two variables, we see a non-linear curvature. Fitting a linear model to these variables results in a high (poor) value of RMSE.

$$\hat{y} = \theta_0 + \theta_1 (\text{hp})$$

In [52]:
X = vehicles[["hp"]]
y = vehicles["mpg"]

hp_model = lm.LinearRegression()
hp_model.fit(X, y)
hp_model_predictions = hp_model.predict(X)

sns.scatterplot(data=vehicles, x="hp", y="mpg")

plt.plot(vehicles["hp"], hp_model_predictions, c="tab:red");

print(f"MSE of model with (hp) feature: {np.mean((y-hp_model_predictions)**2)}")

MSE of model with (hp) feature: 23.943662938603108
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To capture the non-linear relationship between the variables, we can introduce a non-linear feature: hp squared. Our new model is:

$$\hat{y} = \theta_0 + \theta_1 (\text{hp}) + \theta_2 (\text{hp}^2)$$

In [53]:
X = vehicles[["hp"]]
X.loc[:, "hp^2"] = vehicles["hp"]**2

hp2_model = lm.LinearRegression()
hp2_model.fit(X, y)
hp2_model_predictions = hp2_model.predict(X)

sns.scatterplot(data=vehicles, x="hp", y="mpg")

plt.plot(vehicles["hp"], hp2_model_predictions, c="tab:red");

print(f"MSE of model with (hp^2) feature: {np.mean((y-hp2_model_predictions)**2)}")

MSE of model with (hp^2) feature: 18.984768907617216
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What if we take things further and add even more polynomial features?

The cell below fits models of increasing complexity and computes their MSEs.

In [54]:
def mse(predictions, observations):
    return np.mean((observations - predictions)**2)

# Add hp^3 and hp^4 as features to the data
X["hp^3"] = vehicles["hp"]**3
X["hp^4"] = vehicles["hp"]**4

# Fit a model with order 3
hp3_model = lm.LinearRegression()
hp3_model.fit(X[["hp", "hp^2", "hp^3"]], vehicles["mpg"])
hp3_model_predictions = hp3_model.predict(X[["hp", "hp^2", "hp^3"]])

# Fit a model with order 4
hp4_model = lm.LinearRegression()
hp4_model.fit(X[["hp", "hp^2", "hp^3", "hp^4"]], vehicles["mpg"])
hp4_model_predictions = hp4_model.predict(X[["hp", "hp^2", "hp^3", "hp^4"]])

# Plot the models' predictions
fig, ax = plt.subplots(1, 3, dpi=200, figsize=(12, 3))

predictions_dict = {0:hp2_model_predictions, 1:hp3_model_predictions, 2:hp4_model_predictions}

for i in predictions_dict:
    ax[i].scatter(vehicles["hp"], vehicles["mpg"], edgecolor="white", lw=0.5)
    ax[i].plot(vehicles["hp"], predictions_dict[i], "tab:red")
    ax[i].set_title(f"Model with order {i+2}")
    ax[i].set_xlabel("hp")
    ax[i].set_ylabel("mpg")
    ax[i].annotate(f"MSE: {np.round(mse(vehicles['mpg'], predictions_dict[i]), 3)}", (120, 40))

plt.subplots_adjust(wspace=0.3);
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Complexity and Overfitting¶

What we saw above was the phenomenon of model complexity – as we add additional features to the design matrix, the model becomes increasingly complex. Models with higher complexity have lower values of training error. Intuitively, this makes sense: with more features at its disposal, the model can match the observations in the trainining data more and more closely.

We can run an experiment to see this in action. In the cell below, we fit many models of progressively higher complexity, then plot the MSE of predictions on the training set. The code used (specifically, the Pipeline and PolynomialFeatures functions of sklearn) is out of scope.

The order of a polynomial model is the highest power of any term in the model. An order 0 model takes the form $\hat{y} = \theta_0$, while an order 4 model takes the form $\hat{y} = \theta_0 + \theta_1 x + \theta_2 x^2 + \theta_3 x^3 + \theta_4 x^4$.

In [55]:
from sklearn.pipeline import Pipeline
from sklearn.preprocessing import PolynomialFeatures

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

    pipelined_model.fit(dataset[["hp"]], dataset["mpg"])
    return mse(dataset['mpg'], pipelined_model.predict(dataset[["hp"]]))

errors = [fit_model_dataset(degree, vehicles) for degree in range(0, 8)]
MSEs_and_k = pd.DataFrame({"k": range(0, 8), "MSE": errors})

plt.plot(range(0, 8), errors)
plt.xlabel("Model Complexity (degree of polynomial)")
plt.ylabel("Training MSE");

def plot_degree_k_model(k, MSEs_and_k, axs):
    pipelined_model = Pipeline([
        ('poly_transform', PolynomialFeatures(degree = k)),
        ('regression', lm.LinearRegression(fit_intercept = True))    
    ])
    pipelined_model.fit(vehicles[["hp"]], vehicles["mpg"])
    
    row = k // 4
    col = k % 4
    ax = axs[row, col]
    
    sns.scatterplot(data=vehicles, x='hp', y='mpg', ax=ax)
    
    x_range = np.linspace(45, 210, 100).reshape(-1, 1)
    ax.plot(x_range, pipelined_model.predict(pd.DataFrame(x_range, columns=['hp'])), c='tab:red', linewidth=2)
    
    ax.set_ylim((0, 50))
    mse_str = f"MSE: {MSEs_and_k.loc[k, 'MSE']:.4}\nDegree: {k}"
    ax.text(130, 35, mse_str, dict(size=14))

fig = plt.figure(figsize=(15, 6), dpi=150)
axs = fig.subplots(nrows=2, ncols=4)

for k in range(8):
    plot_degree_k_model(k, MSEs_and_k, axs)
fig.subplots_adjust(wspace=0.4, hspace=0.3)
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As the model increases in polynomial degree (that is, it increases in complexity), the training MSE decreases, plateauing at roughly ~18.

In fact, it is a mathematical fact that if we create a polynomial model with degree $n-1$, we can perfectly model a set of $n$ points. For example, a set of 5 points can be perfectly modeled by a degree 4 model.

In [56]:
np.random.seed(101)

fig, ax = plt.subplots(1, 3, dpi=200, figsize=(12, 3))

for i in range(0, 3):
    points = 3*np.random.uniform(size=(5, 2))

    polynomial_model = Pipeline([
                ('polynomial_transformation', PolynomialFeatures(4)),
                ('linear_regression', lm.LinearRegression())    
            ])

    polynomial_model.fit(points[:, [0]], points[:, 1])

    ax[i].scatter(points[:, 0], points[:, 1])

    xs = np.linspace(0, 3)
    ax[i].plot(xs, polynomial_model.predict(xs[:, np.newaxis]), c="tab:red");
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You may be tempted to always design models with high polynomial degree – after all, we know that we could theoretically achieve perfect predictions by creating a model with enough polynomial features.

It turns out that the examples we looked at above represent a somewhat artificial scenario: we trained our model on all the data we had available, then used the model to make predictions on this very same dataset. A more realistic situation is when we wish to apply our model on unseen data – that is, datapoints that it did not encounter during the model fitting process.

Suppose we obtain a random sample of 6 datapoints from our population of vehicle data. We want to train a model on these 6 points and use it to make predictions on unseen data (perhaps cars for which we don't already know the true mpg).

In [57]:
np.random.seed(100)

sample_6 = vehicles.sample(6)

sns.scatterplot(data=sample_6, x="hp", y="mpg")
plt.ylim(-35, 50)
plt.xlim(0, 250);
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If we design a model with polynomial degree 5, we can make perfect predictions on this sample of training data.

In [58]:
degree_5_model = Pipeline([
                ('polynomial_transformation', PolynomialFeatures(5)),
                ('linear_regression', lm.LinearRegression())    
            ])

degree_5_model.fit(sample_6[["hp"]], sample_6["mpg"])
xs = np.linspace(0, 250, 1000)
degree_5_model_predictions = degree_5_model.predict(xs[:, np.newaxis])

plt.plot(xs, degree_5_model_predictions, c="tab:red")
sns.scatterplot(data=sample_6, x="hp", y="mpg", s=50)
plt.ylim(-35, 50)
plt.xlim(0, 250);
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However, when we reapply this fitted model to the full population of data, it fails to capture the major trends of the dataset.

In [59]:
plt.plot(xs, degree_5_model_predictions, c="tab:red")
sns.scatterplot(data=vehicles, x="hp", y="mpg", s=50)
plt.ylim(-35, 50);
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The model has overfit to the data used to train it. It has essentially "memorized" the six datapoints used during model fitting, and does not generalize well to new data.

Complex models tend to be more sensitive to the data used to train them. The variance of a model refers to its tendency to vary depending on the training data used during model fitting. It turns out that our degree-5 model has very high model variance. If we randomly sample new sets of datapoints to use in training, the model varies erratically.

In [33]:
np.random.seed(100)

fig, ax = plt.subplots(1, 3, dpi=200, figsize=(12, 3))

for i in range(0, 3):
    sample = vehicles.sample(6)

    polynomial_model = Pipeline([
                ('polynomial_transformation', PolynomialFeatures(5)),
                ('linear_regression', lm.LinearRegression())    
            ])

    polynomial_model.fit(sample[["hp"]], sample["mpg"])

    ax[i].scatter(sample[["hp"]], sample["mpg"])

    xs = np.linspace(50, 210, 1000)
    ax[i].plot(xs, polynomial_model.predict(xs[:, np.newaxis]), c="tab:red")
    ax[i].set_ylim(-80, 100)
    ax[i].set_xlabel("hp")
    ax[i].set_ylabel("mpg")
    ax[i].set_title(f"Resample #{i+1}")
    
fig.tight_layout();
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