Supervised Learning: Linear Regression

 what is Linear Regression?

Linear regression is a supervised machine learning algorithm used for predicting a continuous outcome variable (also known as a dependent variable) based on one or more predictor variables (also known as independent variables or features). The goal of linear regression is to find the line of best fit that minimizes the sum of the squared differences between the predicted values and the actual values.

Linear regression assumes that there is a linear relationship between the predictor variables and the outcome variable. In other words, it assumes that changes in the predictor variables are directly proportional to changes in the outcome variable.

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

Simple linear regression is used when there is only one predictor variable. The equation for a simple linear regression model is:

Y = b0 + b1*X

Where Y is the outcome variable, X is the predictor variable, b0 is the y-intercept, and b1 is the coefficient for the predictor variable.

Multiple linear regression is used when there are two or more predictor variables. The equation for a multiple linear regression model is:

Y = b0 + b1X1 + b2X2 + ... + bn*Xn

Where Y is the outcome variable, X1, X2, ..., Xn are the predictor variables, b0 is the y-intercept, and b1, b2, ..., bn are the coefficients for the predictor variables.

Linear regression is a widely used and well-understood algorithm that is simple to implement and easy to interpret.

Here is an example of how to find the intercept and coefficients for a simple linear regression model using the method of least squares:

1. Collect the dataset and organize it into two columns: the predictor variable (X) and the outcome variable (Y).

2. Calculate the mean of the predictor variable (X) and the outcome variable (Y).

3. Calculate the slope (b1) using the following formula:

b1 = (Σ(X - Xmean)*(Y - Ymean)) / (Σ(X - Xmean)^2)

4. Calculate the y-intercept (b0) using the following formula:

b0 = Ymean - b1*Xmean

5. The line of best fit is represented by the equation:

Y = b0 + b1*X

[NOTE: b0 and b1 can be found by using the method of least squares to minimize the difference between the predicted values and the actual values of the outcome variable. This can be done using linear regression techniques, such as ordinary least squares (OLS) or gradient descent, and can be implemented using libraries such as scikit-learn in Python. The resulting b0 and b1 values will be the ones that produce the best fit line for the data.]

what is Gradient Descent?

Gradient Descent is an optimization algorithm that is used to minimize the cost function (also known as the loss function or error function) in a linear regression model. The goal of Gradient Descent is to find the values of the coefficients (b0 and b1 in the simple linear regression model) that minimize the cost function.

[ Goal of cost function : Finding the error or cost function is important in linear regression because it helps to measure the difference between the predicted values and the actual values. By minimizing the error or cost function, we can optimize the coefficients (b0 and b1) of the linear regression model to best fit the data. The goal is to find the set of coefficients that result in the lowest error or cost, which will result in the most accurate predictions.]

The Gradient Descent algorithm starts with an initial set of coefficients (b0 and b1) and iteratively updates them in the direction of the negative gradient of the cost function. The negative gradient points in the direction of the steepest decrease in the cost function. The algorithm stops when the cost function reaches a minimum or when a stopping criterion is met.

To find the values of b0 and b1 in a dataset, we use the Gradient Descent algorithm to minimize the cost function. The cost function for linear regression is the mean squared error (MSE), which is the average of the squared differences between the predicted values and the actual values.

The process of finding b0 and b1 using gradient descent involves the following steps:

1. Initialize the coefficients b0 and b1 with random values
2. Compute the predicted values using the current coefficients
3. Compute the gradient of the cost function (MSE) with respect to b0 and b1
4. Update the coefficients b0 and b1 by subtracting a small value (the learning rate) multiplied by the gradient
5. Repeat steps 2-4 until the cost function reaches a minimum or a stopping criterion is met.

Once the algorithm stops, the values of b0 and b1 that minimize the cost function are the estimated coefficients of the linear regression model. These coefficients can then be used to make predictions for new data.

For multiple linear regression, the process is similar but, you would have to calculate the slope for each predictor variable, and use the formula for multiple linear regression.

It's also possible to use python libraries like scikit-learn and statsmodel to find the intercept and coefficients in a dataset. These libraries have built-in functions that can automatically perform linear regression and return the intercept and coefficients.