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Logistic regression is a classification algorithm used to assign observations to a discrete set of classes. Unlike linear regression which outputs continuous number values, logistic regression transforms its output using the logistic sigmoid function to return a probability value which can then be mapped to two or more discrete classes. Given data on time spent studying and exam scores. Linear Regression and logistic regression can predict different things:. We have two features hours slept, hours studied and two classes: In order to map predicted values to probabilities, we use the sigmoid function.
The function maps any real value into another value between 0 and 1. In machine learning, we use sigmoid to map predictions to probabilities. Our current prediction function returns a probability score between 0 and 1.
For example, if our threshold was. If our prediction was. For logistic regression with multiple classes we could select the class with the highest predicted probability.
Using our knowledge of sigmoid functions and decision boundaries, we can now write a prediction function. As the probability gets closer to 1, our model is more confident that the observation is in class 1. This time however we will transform the output using the sigmoid function to return a probability value between 0 and 1. If the model returns.
If our decision boundary was. We wrap the sigmoid function over the same prediction function we used in multiple linear regression. Squaring this prediction as we do in MSE results in a non-convex function with many local minimums.
If our cost function has many local minimums, gradient descent may not find the optimal global minimum. Cross-entropy loss can be divided into two separate cost functions: These smooth monotonic functions  always increasing or always decreasing make it easy to calculate the gradient and minimize cost.
The key thing to note is the cost function penalizes confident and wrong predictions more than it rewards confident and right predictions! The corollary is increasing prediction accuracy closer to 0 or 1 has diminishing returns on reducing cost due to the logistic nature of our cost function. In both cases we only perform the operation we need to perform.
To minimize our cost, we use Gradient Descent just like before in Linear Regression. Machine learning libraries like Scikit-learn hide their implementations so you can focus on more interesting things!
One of the neat properties of the sigmoid function is its derivative is easy to calculate. Michael Neilson also covers the topic in chapter 3 of his book. Notice how this gradient is the same as the Mean Squared Error gradient, the only difference is the hypothesis function. Our training code is the same as we used for linear regression. Accuracy measures how correct our predictions were. In this case we simple compare predicted labels to true labels and divide by the total.
Another helpful technique is to plot the decision boundary on top of our predictions to see how our labels compare to the actual labels. This involves plotting our predicted probabilities and coloring them with their true labels.
Basically we re-run binary classification multiple times, once for each class. Then we take the class with the highest predicted value. Linear Regression and logistic regression can predict different things: Linear regression predictions are continuous numbers in a range. Logistic Regression could help use predict whether the student passed or failed.
Logistic regression predictions are discrete only specific values or categories are allowed. Studied Slept Passed 4. Math One of the neat properties of the sigmoid function is its derivative is easy to calculate. Calculate gradient average 2. Multiply by learning rate 3. For each class… Predict the probability the observations are in that single class.
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