What is Linear Regression and How Does it Work?

Linear regression is a statistical method used in machine learning for predictive analysis. It assumes a linear relationship between a dependent (or target) variable and one or more independent (or predictor) variables. The algorithm tries to fit a straight line (or hyperplane in multiple dimensions) to the data points and find the best line that represents the relationship between the dependent and independent variables. The line is represented by a mathematical equation, where the coefficients of the independent variables are the weights that determine the strength of their effect on the dependent variable. The goal of linear regression is to minimize the difference between the actual and predicted values, which is usually done through a process called gradient descent. Linear regression can be applied to both simple linear regression, where only one independent variable is used, and multiple linear regression, where more than one independent variable is used.

The linear regression model provides a sloped straight line representing the relationship between the variables. Consider the below image:

Mathematically, we can represent a linear regression as:

y= a0+a1x+ ε

Here,

Y= Dependent Variable (Target Variable)

X= Independent Variable (predictor Variable)

a0= intercept of the line (Gives an additional degree of freedom)

a1 = Linear regression coefficient (scale factor to each input value).

ε = random error

The values for x and y variables are training datasets for Linear Regression model representation.

Types of Linear Regression

There are two main types of linear regression: simple linear regression and multiple linear regression.

1. Simple Linear Regression: This is a statistical method used to model the relationship between a single independent variable (also called a predictor or explanatory variable) and a dependent variable (also known as the response variable). In simple linear regression, a linear equation is used to model the relationship between the two variables.
2. Multiple Linear Regression: This type of linear regression is used to model the relationship between multiple independent variables and a dependent variable. It is used when there are multiple factors that may contribute to the prediction of the dependent variable. In multiple linear regression, a linear equation is used to model the relationship between the independent variables and the dependent variable.

Both simple and multiple linear regression is widely used in different fields such as finance, economics, and marketing to make predictions and understand the relationship between variables.

Finding the best-fit line:

Finding the best-fit line in linear regression refers to finding the line that best represents the relationship between the independent variable (x) and the dependent variable (y). This line is represented by the equation y = b0 + b1x, where b0 is the y-intercept and b1 is the slope of the line. The goal of linear regression is to find the line that minimizes the difference between the observed values of y and the predicted values of y based on the independent variable x. There are several methods to find the best-fit line including Ordinary Least Squares (OLS), Gradient Descent, and others. The most commonly used method is OLS, which calculates the line that minimizes the sum of the squared differences between the observed and predicted values of y.

Cost function

The cost function in linear regression is a measure of the difference between the actual values and the predicted values, and it helps in finding the best fit line for the given data. The goal of linear regression is to minimize the cost function, which is typically represented as a mean squared error (MSE) between the actual and predicted values. The cost function can be defined mathematically as:

J(θ) = 1/2m * Σ(h(x^(i)) - y^(i))^2

where:

• J(θ) is the cost function.
• m is the number of training examples.
• h(x^(i)) is the predicted value for the i-th training example.
• y^(i) is the actual value for the i-th training example.
• Σ represents the summation over all training examples.

The cost function is a convex function, which means that there is only one global minimum, and we can use optimization algorithms like gradient descent to find the optimal values of the parameters θ that minimize the cost function.

Gradient Descent

Gradient Descent is an optimization algorithm used in linear regression and other machine learning models to minimize the cost function. The basic idea behind gradient descent is to iteratively update the parameters of the model in the direction of the steepest decrease in the cost function until a minimum value is reached.

The algorithm starts with an initial set of parameters and then computes the gradient of the cost function with respect to each parameter. The gradient represents the direction of the steepest increase in the cost function and is used to update the parameters in the opposite direction. The parameters are updated by subtracting the gradient multiplied by a learning rate from the current parameters. This process is repeated until the cost function reaches a minimum value or a stopping criterion is met.

Gradient Descent is an efficient and effective optimization method for linear regression and has been widely used in many machine learning models. It can be applied to both simple and complex cost functions, and it can be used with a variety of optimization techniques, such as batch gradient descent, stochastic gradient descent, and mini-batch gradient descent, to speed up the optimization process.