## Introduction to Classification and Logistic Regression

Supervised machine learning is a type of artificial intelligence that is used to predict future outcomes based on previous data. One subfield of supervised learning is classification.

Classification algorithms take a set of input data and sort it into different categories, known as classes. This article will delve deeper into what classification is and how logistic regression is a common method used in classification problems.

### What is Classification?

Classification is a type of supervised learning in machine learning, where the goal is to predict which class a particular input falls under.

For example, we might want to classify emails as either spam or not spam, or diagnose a patient as having a certain disease or not. The input variables, or features, can be numerical or categorical, and from these features, the algorithm will learn the relationship between them and the output class.

In classification problems, the output variable is discrete, meaning it can only take on a finite number of values, as opposed to continuous variables that can take on any number of values. There are two main categories of classification: binary classification, where there are only two classes, and multiclass classification, where there are more than two classes.

## Logistic Regression Overview

Logistic regression is a type of linear classifier that is commonly used for binary classification problems. It was first introduced by statistician David Cox in 1958 and has since become one of the most widely used algorithms in data science.

One of the defining features of logistic regression is its use of the sigmoid function, which maps any input value to a value between 0 and 1. The sigmoid function has an S-shaped curve that is suitable for classification tasks, as it can determine the probability of an input variable belonging to a certain class.

### The equation for the sigmoid function is:

f(x) = 1 / (1 + e^-x)

where x is the input value and e is the mathematical constant, approximately equal to 2.71828. The output of the sigmoid function is always between 0 and 1, and values close to 0 indicate a low probability of belonging to a certain class, while values close to 1 indicate a high probability.

In logistic regression, the input features are combined linearly and fed into the sigmoid function to predict the probability of output classes. This makes logistic regression a linear classifier, meaning it separates different classes by drawing a straight line or hyperplane through the input feature space.

To train a logistic regression model, the maximum likelihood estimation (MLE) method is used. MLE estimates the parameters of the model that best fit the training data by maximizing the likelihood of observing the training data given the model parameters.

## Math Prerequisites

### The Sigmoid Function

In order to fully understand logistic regression, there are two mathematical concepts that are important to know: the sigmoid function and natural logarithm. The sigmoid function, as mentioned before, is an S-shaped curve that maps any input value to a value between 0 and 1.

It is given by the equation f(x) = 1 / (1 + e^-x), where x is the input value. The sigmoid function is used in logistic regression to map the output of the linear function to a probability score.

### The Natural Logarithm Function

The natural logarithm function, or log function, is a mathematical function that is used to find the power to which a certain number, e, must be raised to produce a given value. The notation for the natural logarithm is ln(x), and it is often abbreviated as log(x) in programming languages such as Python.

The natural logarithm is useful in logistic regression, as it is used to define the likelihood function, which is the probability of observing the training data given the model parameters.

## Conclusion

In conclusion, machine learning is a fascinating field that is becoming increasingly important in today’s data-driven world, and one of its subfields is classification. Logistic regression is a popular method used for classification problems, and it works by combining input features linearly and applying the sigmoid function to get a probability score.

In order to fully grasp logistic regression, it is important to understand the sigmoid function and the natural logarithm function. With this knowledge, we can make more informed decisions based on data and build better predictive models.

## Problem Formulation for Binary Classification

Binary classification is a type of machine learning, where the goal is to classify input data into one of two possible categories, or classes. Some examples of binary classification problems include identifying whether a customer will make a purchase or not, predicting if a patient will develop a particular disease or not, or determining if a loan applicant is likely to default or not.

In this article, we will discuss the problem formulation for binary classification and the logistic regression function. We will also explore how to train a logistic regression model using maximum likelihood estimation.

### Independent/Dependent Variables

In a binary classification problem, there are two types of variables: the independent variables or predictors, and the dependent variable or response. The independent variables are the input data that we use to predict the output variable, which is the response.

The independent variables can be continuous or categorical, and they should be relevant to the classification problem we are trying to solve. The dependent variable, on the other hand, is the output we are trying to predict.

In binary classification, the dependent variable can take on only two values, typically represented as 0 or 1. The 0 denotes the negative class and 1 denotes the positive class.

## Logistic Regression Function

Logistic regression is a type of linear classifier that is commonly used for binary classification problems. It models the probability of the dependent variable taking a certain value, given the values of the independent variables.

The logistic regression function is a mathematical equation that uses the inputs to predict the probability of the output variable being 1. The logistic regression function is defined by:

P(y=1 | x) = 1 / (1 + e^-z)

where P(y=1 | x) is the conditional probability that the output variable y is 1 given the input variables x; e is Euler’s number, approximately equal to 2.71828; and z is a linear combination of the input variables and their associated coefficients, that is:

z = b0 + b1 x1 + b2 x2 + …

+ bk xk

where b0 is the intercept, or bias term, and b1 … bk are the coefficients that define the relationship between the independent variables and the dependent variable.

The logistic regression function uses the sigmoid function, which maps any input value to a value between 0 and 1, to ensure that the predicted probabilities fall within the range of 0 to 1. The sigmoid function is given by:

f(z) = 1 / (1 + e^-z)

where z is the linear combination of the input variables and their coefficients, as defined above.

### Predicted Probability

The output of the logistic regression function is a predicted probability that the dependent variable takes the value of 1 given the input values. This probability can be interpreted as the likelihood that a certain data point belongs to the positive class.

The logistic regression function can predict probabilities over a range of input values. To make a final prediction for a binary classification problem, a threshold probability value is chosen.

For example, if the threshold probability is set at 0.5, any predicted probability greater than 0.5 is classified as the positive class, while any predicted probability less than 0.5 is classified as the negative class.

### Model Training/Fitting

To train a logistic regression model, we need to fit the coefficients b0, b1, b2, etc., to the data.

We do this by using a training set of data that consists of input values and their associated output values. The objective of training is to find the coefficients that minimize the difference between the predicted probabilities and the actual output values.

The method used to train a logistic regression model is called maximum likelihood estimation (MLE). MLE estimates the values of the coefficients that maximize the likelihood of observing the training data given the model parameters.

The likelihood function is defined as the probability of observing the training data given the model parameters. The log-likelihood function (LLF) is often used instead, as it is easier to compute.

The goal is to maximize the LLF with respect to the model parameters. Once the model is trained, we can use it to predict the output for new data points.

The trained model will be able to accurately predict the probability of a dependent variable taking a certain value, given the independent variables.

## Conclusion

Binary classification is a important and widely used field in machine learning, especially in business, medicine, and finance. Logistic regression is a popular algorithm for binary classification because it is simple to understand and implement.

By understanding the problem formulation for binary classification, the logistic regression function, and model training using maximum likelihood estimation, we can effectively build and tune our logistic regression models for binary classification problems. In conclusion, binary classification is an essential component of machine learning that involves predicting whether an input belongs to one of two categories.

Logistic regression, a type of linear classifier, is a commonly used algorithm for binary classification, using the sigmoid function to calculate the probabilities of an input belonging to a certain class. Maximum likelihood estimation is used to train a logistic regression model by finding the coefficients that minimize the difference between predicted and actual output values.

By understanding the problem formulation, logistic regression function, and model training, we can build better predictive models and make more informed decisions in fields such as business, medicine, and finance.