Logistic Regression is one of the foundational algorithms for binary classification. At its core, Logistic Regression models the probability that a given instance belongs to a particular category. Let’s delve into its mechanism:
1. Linear Transformation
The first step in Logistic Regression is to compute a weighted sum of the input features (similar to linear regression):
Where:
- x₁, x₂, …, xₙ are the input features.
- β₁, β₂, …, βₙ are the parameters of the model (intercept and coefficients respectively).
- z is the output of the linear transformation
2. Logistic Function
Instead of returning the linear combination of inputs directly as the prediction (like in linear regression), Logistic Regression passes this value through the logistic (sigmoid) function:
The logistic function maps any input into a value between 0 and 1, which can be interpreted as the probability of the instance belonging to the positive class.
3. Binary Classification Decision
Given the output probability from the logistic function:
- If σ(z) ≥ 0.5, the predicted class is 1 (or the positive class).
- If σ(z) < 0.5, the predicted class is 0 (or the negative class).
The threshold of 0.5 is standard, but it can be adjusted depending on the specific problem or based on other criteria like precision-recall trade-offs.
4. Model Training: Maximum Likelihood Estimation
While the prediction process is relatively straightforward, training the logistic regression model (i.e., estimating the parameters) requires a bit more machinery.
For training, Logistic Regression uses Maximum Likelihood Estimation (MLE) to find the set of parameters (coefficients) that maximizes the likelihood of the observed data. Intuitively, MLE adjusts the parameters such that if you run the model on your training data, the predicted probabilities will be as close as possible to the observed outcomes.
Key Takeaways:
- Linear Transformation: The input features are combined linearly using weights or coefficients, similar to linear regression.
- Logistic Function: The results of the linear combination are passed through the sigmoid function, which squishes values between 0 and 1.
- Classification Decision: A threshold (commonly 0.5) is used to decide the final class label.
- Training with MLE: The model is trained to find the parameters that maximize the likelihood of the observed data.
The beauty of Logistic Regression lies in its simplicity, interpretability, and foundational role in understanding more complex algorithms. While newer algorithms can offer better accuracy for certain tasks, Logistic Regression remains a go-to for binary classification problems, especially when interpretability is a priority.