Supervised Learning(Part-1)

 Supervised Learning


Logistic Regression

 It is used when the dependent variable(target) is categorical.

For example,

  • To predict whether an email is spam (1) or (0)
  • Whether the tumor is malignant (1) or not (0)

Simple Logistic Regression

Model

Output = 0 or 1

Hypothesis => Z = WX + B

hΘ(x) = sigmoid (Z)

Sigmoid Function

If ‘Z’ goes to infinity, Y(predicted) will become 1 and if ‘Z’ goes to negative infinity, Y(predicted) will become 0.

Types of Logistic Regression

1. Binary Logistic Regression: The categorical response has only two 2 possible outcomes. Example: Spam or Not

2. Multinomial Logistic Regression: Three or more categories without ordering. Example: Predicting which food is preferred more (Veg, Non-Veg, Vegan)

3. Ordinal Logistic Regression: Three or more categories with ordering. Example: Movie rating from 1 to 5

Decision Boundary

To predict which class a data belongs, a threshold can be set. Based upon this threshold, the obtained estimated probability is classified into classes.

Say, if predicted value ≥ 0.5, then classify email as spam else as not spam.

Decision boundary can be linear or non-linear. Polynomial order can be increased to get complex decision boundary.

Cost Function

Linear regression uses mean squared error as its cost function. If this is used for logistic regression, then it will be a non-convex function of parameters (theta). Gradient descent will converge into global minimum only if the function is convex.

Figure 5: Convex and non-convex cost function

Cost function explanation

Figure 6: Cost Function part 1
Figure 7: Cost Function part 2

Simplified cost function

Figure 8: Simplified Cost Function

Why this cost function?

Figure 9: Maximum Likelihood Explanation part-1
Figure 10: Maximum Likelihood Explanation part-2

This negative function is because when we train, we need to maximize the probability by minimizing loss function. Decreasing the cost will increase the maximum likelihood assuming that samples are drawn from an identically independent distribution.

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