Decision Boundaries

Classification algorithms separate different groups or classes within data. When visualizing data points on a scatter plot, where each point belongs to a specific category (such as 'spam' or 'not spam', 'cat' or 'dog'), a model must determine which category a new point belongs to. This determination is made using a decision boundary.

Think of a decision boundary as an invisible line or surface that the algorithm learns. This boundary divides the space where your data lives into regions, with each region corresponding to a predicted class. If a data point falls on one side of the boundary, the model assigns it to one class; if it falls on the other side, it gets assigned to the other class.

Decision Boundaries in Logistic Regression

In the previous section, we saw that Logistic Regression calculates the probability that a data point belongs to a particular class (let's call it class 1) using the sigmoid function. The output is a probability $p$ between 0 and 1. Typically, we set a threshold, often 0.5, to make the final classification decision:

• If $p \ge 0.5$, predict class 1.
• If $p < 0.5$, predict class 0.

The decision boundary is precisely where the model is uncertain, meaning the probability is exactly 0.5. When does the sigmoid function output 0.5? It happens when its input is exactly 0.

Remember, the input to the sigmoid function in logistic regression is typically a linear combination of the features, like $z = w_0 + w_1 x_1 + w_2 x_2$ for two features ($x_1$, $x_2$). So, the decision boundary is defined by the equation:

$w_0 + w_1 x_1 + w_2 x_2 = 0$

For data with two features, this equation represents a straight line. This line separates the 2D plane into two regions: one where the model predicts class 1 ($z > 0$, so $p > 0.5$) and one where it predicts class 0 ($z < 0$, so $p < 0.5$).

Visualizing a Linear Decision Boundary

Let's make this more concrete. Imagine we have data points belonging to two classes (Red and Blue) plotted based on two features (Feature 1 and Feature 2). A logistic regression model trained on this data might find a linear decision boundary like the one shown below.

A simple scatter plot showing two classes of data points (Red and Blue) separated by a linear decision boundary (the gray line) learned by a model like Logistic Regression. Points generally above and to the right of the line would be classified as Blue (Class 1), while points below and to the left would be classified as Red (Class 0).

Any new data point plotted on this graph would be classified based on which side of the gray line it falls. This visual representation helps understand how the model makes its decisions based on the input features.

Linear Boundaries

It's important to note that decision boundaries aren't always straight lines. While basic Logistic Regression typically produces linear boundaries, classification problems can often require more complex shapes to separate the classes effectively.

Imagine classes that are mixed in a more complicated way, perhaps with one class clustered in the middle and the other forming a ring around it. A straight line wouldn't be very good at separating these. Other algorithms, including the K-Nearest Neighbors (KNN) algorithm we'll discuss next, or modifications to logistic regression (like using polynomial features), can create non-linear decision boundaries (curves, circles, or even more irregular shapes).

Why Decision Boundaries Matter

Understanding decision boundaries helps you:

1. Visualize Model Behavior: It gives you a graphical intuition for how the model separates data.
2. Understand Model Complexity: A simple boundary (like a straight line) implies a simpler model, while a very wiggly boundary suggests a more complex model.
3. Identify Potential Issues: An overly complex boundary that perfectly separates all training points might indicate overfitting (the model learned the training data too well, including noise, and might not perform well on new data). A boundary that fails to separate the classes well might indicate underfitting (the model is too simple).

As we look at different classification algorithms, pay attention to the kinds of decision boundaries they tend to create. This will give you insight into their strengths and weaknesses for different types of data distributions. Next, we'll examine the K-Nearest Neighbors algorithm, which takes a very different approach to classification and results in a different kind of decision boundary.