Logistic Regression I

Let’s visualize the relationship between "GOAL_DIFF" and "WON" using the Seaborn function sns.stripplot. A strip plot automatically introduces a small amount of random noise to jitter the data. Recall that all values in the "WON" column are either 1 (won) or 0 (lost) – if we were to directly plot them without jittering, we would see severe overplotting.

import seaborn as sns
import matplotlib.pyplot as plt

sns.stripplot(data=games, x="GOAL_DIFF", y="WON", orient="h", hue='WON', alpha=0.7)
# By default, sns.stripplot plots 0, then 1. We invert the y axis to reverse this behavior
plt.gca().invert_yaxis();

This dataset is unlike anything we’ve seen before – our target variable contains only two unique values! (Remember that each y value is either 0 or 1; the plot above jitters the y data slightly for ease of reading.)

The regression models we have worked with always assumed that we were attempting to predict a continuous target. If we apply a linear regression model to this dataset, something strange happens.

import sklearn.linear_model as lm

X, Y = games[["GOAL_DIFF"]], games["WON"]
regression_model = lm.LinearRegression()
regression_model.fit(X, Y)

plt.plot(X.squeeze(), regression_model.predict(X), "k")
sns.stripplot(data=games, x="GOAL_DIFF", y="WON", orient="h", hue='WON', alpha=0.7)
plt.gca().invert_yaxis();

The linear regression fit follows the data as closely as it can. However, this approach has a key flaw - the predicted output, $\hat{y}$, can be outside the range of possible classes (there are predictions above 1 and below 0). This means that the output can’t always be interpreted (what does it mean to predict a class of -2.3?).

Our usual linear regression framework won’t work here. Instead, we’ll need to get more creative.

Graph of Averages¶

Back in Data 8, you gradually built up to the concept of linear regression by using the graph of averages. Before you knew the mathematical underpinnings of the regression line, you took a more intuitive approach: you bucketed the $x$ data into bins of common values, then computed the average $y$ for all datapoints in the same bin. The result gave you the insight needed to derive the regression fit.

Let’s take the same approach as we grapple with our new classification task. In the cell below, we 1) bucket the "GOAL_DIFF" data into bins of similar values and 2) compute the average "WON" value of all datapoints in a bin.

Interesting: our result is certainly not like the straight line produced by finding the graph of averages for a linear relationship. We can make two observations:

• All predictions on our line are between 0 and 1

• The predictions are non-linear, following a rough “S” shape

Let’s think more about what we’ve just done.

To find the average $y$ value for each bin, we computed:

This is simply the probability of a datapoint in that bin belonging to Class 1! This aligns with our observation from earlier: all of our predictions lie between 0 and 1, just as we would expect for a probability.

Our graph of averages was really modeling the probability, $p$, that a datapoint belongs to Class 1, or essentially that $\text{Y = 1}$ for a particular value of $\text{x}$.

In logistic regression, we have a new modeling goal. We want to model the probability that a particular datapoint belongs to Class 1 by approximating the S-shaped curve we plotted above. However, we’ve only learned about linear modeling techniques like Linear Regression and OLS.

The Sigmoid Function¶

Let’s start by identifying an S-shaped function that maps ($-\infty$, $\infty$) inputs to (0,1) outputs—the sigmoid ($\sigma$) function. It is also known as the logistic function or the inverse logit function.

As it turns out, the logistic regression model fits a sigmoid function to the data. We can also shift and scale the sigmoid function just as if we were transforming any other function to fit our data:

Note that our input to the sigmoid function here looks exactly like simple linear regression! And just like in ordinary least squares, we’re not limited to one input:

import plotly.graph_objects as go

# Plot a 3D sigmoid surface
x1 = np.linspace(-10, 10, 100)
x2 = np.linspace(-10, 10, 100)
X1, X2 = np.meshgrid(x1, x2)
y_values = 1/(1 + np.exp(-(X1 + X2)))

# Create a 3D surface plot for the sigmoid function
fig = go.Figure(data=[go.Surface(z=y_values, x=X1[0], y=X2[:, 0])])

# Update layout for better visualization
fig.update_layout(
  title="Sigmoid with two inputs",
  scene=dict(
    xaxis_title="X1",
    yaxis_title="X2",
    zaxis_title="\u03C3(X1, X2)"
  ),
  width=800,
  height=600
)

# Reduce tick label size
fig.update_layout(
    scene=dict(
        xaxis=dict(tickfont=dict(size=14)),
        yaxis=dict(tickfont=dict(size=14)),
        zaxis=dict(tickfont=dict(size=14))
    )
)

fig.show()

Below, we fit a logistic regression model to our data. You do not need to worry about what’s going on “under the hood” for now. Just notice how similar the logistic regression model is to our graph of averages.

# We'll discuss the `LogisticRegression` class next time
xs = np.linspace(-0.3, 0.3)

logistic_model = lm.LogisticRegression(C=20)
logistic_model.fit(X, Y)
predicted_prob = logistic_model.predict_proba(xs[:, np.newaxis])[:, 1]

sns.stripplot(data=games, x="GOAL_DIFF", y="WON", orient="h", alpha=0.5, hue="WON", legend=False)
plt.plot(xs, predicted_prob, c="k", lw=3, label="Logistic regression model")
plt.plot(win_rates_by_bin.index, win_rates_by_bin, lw=2, c="tab:red", label="Graph of averages")
plt.legend(loc="upper left")
plt.gca().invert_yaxis();

Formalizing the Logistic Regression Model¶

Let’s now generalize our model to multiple inputs rather than just one or two inputs. The probability of a datapoint belonging to Class 1 is:

To predict a probability using the logistic regression model, we:

1. Compute a linear combination of the features, $x^{\top}\theta$

2. Apply the sigmoid activation function, $\sigma(x^{\top} \theta)$.

Our main takeaway from this section is that we can:

• Model the probability $P(Y = 1 | x)$ using a sigmoid transformation of a linear model

Indeed, the estimated probability that the response is 1 given the features $x$, is equal to the logistic function $\sigma()$ at the value $x^{\top}\theta$:

More commonly, the logistic regression model is written as:

With our logistic regression model formalized, we have 2 natural questions about the parameters that follow:

1. How do we interpret the $\theta$’s?

2. How do we estimate the $\theta$’s?

We’ll dive deep into answering the first question in the next section, and answer the second question in the next lecture.

Log-Odds and Linearity¶

Interpreting Logistic Regression Coefficients¶

Let’s focus on the simple logistic regression model with just one input:

Recall that in simple linear regression ($y = \theta_0 + \theta_1 x$), we interpret $\theta_1$ as the change in $y$ for a one-unit increase in $x$. However, in logistic regression, the change in the predicted outcome is not constant for every one-unit increase in $x$ since the sigmoid function is non-linear (not a straight line). To make interpretation a little easier, we can isolate the linear combination $\theta_0 + \theta_1$ using some algebra:

We have just derived that as $x$ increases by one, $\log \left( \frac{P(Y=1 \mid x)}{1 - P(Y=1 \mid x)} \right)$ increases by $\theta_1$. Does this really have a meaningful interpretation?

To help with our interpretation, consider the expression: “Cal has 10 to 1 odds of winning the game.” These $10:1$ odds means that $P(\text{Cal wins}) = 10 \cdot P(\text{Cal loses})$. The odds is defined as the probability of a datapoint Y belonging to Class 1 divided by the probability of it belonging to Class 0.

Great! Now we have a much more meaningful interpretation of the logistic regression coefficients as log-odds:

• For every one-unit increase in $x$, $\text{log}(\text{Odds}(Y=1|x))$ increases by $\theta_1$.

Linearity¶

To summarize, probability is a non-linear function of the features because $P(Y=1|x) =\sigma(x^{\top}\theta)$ applies a non-linear sigmoid transformation to $x^{\top}\theta$. However, we have just found that the log-odds are a linear function with respect to the $\theta$’s. Namely, $\text{log}(\text{Odds}(Y=1|x)) = x^{\top}\theta$. Indeed, this means there is linearity in the logistic regression model! This linearity in the log-odds leads to a linear decision boundary between Class 0 and Class 1, which we will explore more in the next section.

Decision Boundaries¶

In logistic regression, we model the probability that a datapoint belongs to Class 1.

In this lecture, we developed the logistic regression model to predict that probability, but the model itself does not classify whether a prediction is Class 0 or Class 1.

A decision rule tells us how to convert model outputs into classifications. We commonly make decision rules by specifying a threshold, $T$. If the predicted probability is greater than or equal to $T$, predict Class 1. Otherwise, predict Class 0.

The threshold is often set to $T = 0.5$, but not always. We’ll discuss why we might want to use other thresholds $T \neq 0.5$ later in this lecture and the next.

Using our decision rule, we can define a decision boundary as the “line” that splits the data into classes based on its features. For logistic regression, since we are working in $p$ dimensions, the decision boundary is a hyperplane—a linear combination of the features in $p$-dimensions—and we can recover it from the log-odds.

For instance, suppose we decide to classify all new data points with $P(Y=1|x) \geq 0.5$ as Class 1, and all points with $P(Y=1|x) < 0.5$ as Class 0. In other words, we choose 0.5 as our chosen probability threshold. We can calculate our log-odds as follows:

This is the same as labeling all points with $\text{log}(\text{Odds}(Y=1|x)) \geq 0$ as Class 1, and all points with $\text{log}(\text{Odds}(Y=1|x)) < 0$ as Class 0. Since $\text{log}(\text{Odds}(Y=1|x)) = x^{\top}\theta$, we have just found that

is our linear decision boundary for classification given a threshold of 0.5.

For a model with 2 features, the decision boundary is a line in terms of its features. To make it easier to visualize, we’ve included an example of a 1-dimensional and a 2-dimensional decision boundary below. Notice how the decision boundary predicted by our logistic regression model perfectly separates the points into two classes. Here the color is the predicted class, rather than the true class.

In real life, however, that is often not the case, and we often see some overlap between points of different classes across the decision boundary. The true classes of the 2D data are shown below:

As you can see, the decision boundary predicted by our logistic regression does not perfectly separate the two classes. There’s a “muddled” region near the decision boundary where our classifier predicts the wrong class. What would the data have to look like for the classifier to make perfect predictions?

[BONUS] Maximum Likelihood Estimation¶

It may have seemed like we pulled cross-entropy loss out of thin air. How did we know that taking the negative logarithms of our probabilities would work so well? It turns out that cross-entropy loss is justified by probability theory.

The following section is out of scope, but is certainly an interesting read!

Building Intuition: The Coin Flip¶

To build some intuition for logistic regression, let’s look at an introductory example to classification: the coin flip. Suppose we observe some outcomes of a coin flip (1 = Heads, 0 = Tails).

A reasonable model is to assume all flips are IID (independent and identically distributed). In other words, each flip has the same probability of returning a 1 (or heads). Let’s define a parameter $\theta$, the probability that the next flip is a heads. We will use this parameter to inform our decision for $\hat y$ (predicting either 0 or 1) of the next flip. If $\theta \ge 0.5, \hat y = 1, \text{else } \hat y = 0$.

You may be inclined to say 0.5 is the best choice for $\theta$. However, notice that we made no assumption about the coin itself. The coin may be biased, so we should make our decision based only on the data. We know that exactly $\frac{4}{10}$ of the flips were heads, so we might guess $\hat \theta = 0.4$. In the next section, we will mathematically prove why this is the best possible estimate.

Likelihood of Data¶

Let’s call the result of the coin flip a random variable $Y$. This is a Bernoulli random variable with two outcomes. $Y$ has the following distribution:

$p$ is unknown to us. But we can find the $p$ that makes the data we observed the most likely.

The probability of observing 4 heads and 6 tails follows the binomial distribution.

We define the likelihood of obtaining our observed data as a quantity proportional to the probability above. To find it, simply multiply the probabilities of obtaining each coin flip.

The technique known as maximum likelihood estimation finds the $p$ that maximizes the above likelihood. You can find this maximum by taking the derivative of the likelihood, but we’ll provide a more intuitive graphical solution.

More generally, the likelihood for some Bernoulli($p$) random variable $Y$ is:

Equivalently, this can be written in a compact way:

• When $y = 1$, this reads $P(Y=y) = p$

• When $y = 0$, this reads $P(Y=y) = (1-p)$

In our example, a Bernoulli random variable is analogous to a single data point (e.g., one instance of a basketball team winning or losing a game). All together, our games data consists of many IID Bernoulli($p$) random variables. To find the likelihood of independent events in succession, simply multiply their likelihoods.

As with the coin example, we want to find the parameter $p$ that maximizes this likelihood. Earlier, we gave an intuitive graphical solution, but let’s take the derivative of the likelihood to find this maximum.

At a first glance, this derivative will be complicated! We will have to use the product rule, followed by the chain rule. Instead, we can make an observation that simplifies the problem.

Finding the $p$ that maximizes

is equivalent to the $p$ that maximizes

This is because $\text{log}$ is a strictly increasing function. It won’t change the maximum or minimum of the function it was applied to. From $\text{log}$ properties, $\text{log}(a*b)$ = $\text{log}(a) + \text{log}(b)$. We can apply this to our equation above to get:

We can add a constant factor of $\frac{1}{n}$ out front. It won’t affect the $p$ that maximizes our likelihood.

One last “trick” we can do is change this to a minimization problem by negating the result. This works because we are dealing with a concave function, which can be made convex.

Now let’s say that we have data that are independent with different probability $p_i$. Then, we would want to find the $p_1, p_2, \dots, p_n$ that maximize

Setting up and simplifying the optimization problems as we did above, we ultimately want to find:

For logistic regression, $p_i = \sigma(x^{\top}\theta)$. Plugging that in, we get:

This is exactly our average cross-entropy loss minimization problem from before!

Why did we do all this complicated math? We have shown that minimizing cross-entropy loss is equivalent to maximizing the likelihood of the training data.

• By minimizing cross-entropy loss, we are choosing the model parameters that are “most likely” for the data we observed.

Note that this is under the assumption that all data is drawn independently from the same logistic regression model with parameter $\theta$. In fact, many of the model + loss combinations we’ve seen can be motivated using MLE (e.g., OLS, Ridge Regression, etc.). In probability and ML classes, you’ll get the chance to explore MLE further.