# Regression

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## Definition of 'Regression'

In statistics, regression analysis is a statistical process for estimating the relationships between variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the value of a dependent variable (Y) is influenced by the values of one or more independent variables (X).

Regression analysis is used in many fields, including business, economics, and social science. It can be used to predict future outcomes, to understand the relationships between variables, and to make informed decisions.

There are many different types of regression analysis, each with its own strengths and weaknesses. The most common type of regression analysis is linear regression, which assumes that the relationship between the dependent variable and the independent variables is linear. Other types of regression analysis include logistic regression, polynomial regression, and multiple regression.

Linear regression is the simplest type of regression analysis. It assumes that the relationship between the dependent variable and the independent variables is linear. This means that the change in the dependent variable is proportional to the change in the independent variable.

The equation for a linear regression model is:

Y = b0 + b1X1 + b2X2 + ... + bnXn

where:

Y is the dependent variable

b0 is the intercept

b1, b2, ..., bn are the coefficients of the independent variables

X1, X2, ..., Xn are the independent variables

The intercept is the value of Y when all of the independent variables are equal to zero. The coefficients of the independent variables represent the change in Y for a one-unit change in each independent variable.

For example, consider the following linear regression model:

Y = 10 + 2X

In this model, the intercept is 10 and the coefficient of X is 2. This means that when X is equal to 0, Y is equal to 10. For every one-unit increase in X, Y increases by 2 units.

Linear regression is a powerful tool for understanding the relationships between variables. However, it is important to note that linear regression makes the assumption that the relationship between the dependent variable and the independent variables is linear. If this assumption is not met, then the results of the linear regression model may be inaccurate.

Logistic regression is a type of regression analysis that is used to predict the probability of an event occurring. It is often used in marketing and sales to predict the likelihood of a customer purchasing a product or service.

The equation for a logistic regression model is:

P = 1 / (1 + e^(-(b0 + b1X1 + b2X2 + ... + bnXn)))

where:

P is the probability of the event occurring

b0 is the intercept

b1, b2, ..., bn are the coefficients of the independent variables

X1, X2, ..., Xn are the independent variables

The intercept is the value of P when all of the independent variables are equal to zero. The coefficients of the independent variables represent the change in P for a one-unit change in each independent variable.

For example, consider the following logistic regression model:

P = 1 / (1 + e^(-(10 + 2X)))

In this model, the intercept is 10 and the coefficient of X is 2. This means that when X is equal to 0, P is equal to 1 / (1 + e^(-10)) = 0.000045. For every one-unit increase in X, P increases by 2 / (1 + e^(-10)) = 0.0009.

Logistic regression is a powerful tool for predicting the probability of an event occurring. However, it is important to note that logistic regression makes the assumption that the relationship between the dependent variable and the independent variables is logistic. If this assumption is not met, then the results of the logistic regression model may be inaccurate.

Polynomial regression is a type of regression analysis that is used to model relationships between variables that are not linear. It is often used in cases where the relationship between the dependent variable and the independent variables is curved.

The equation for a polynomial regression model is:

Y = b0 + b1X + b2X^2 + ... + bnX^n

where:

Y is the dependent variable

b0 is the intercept

b1, b2, ..., bn are the coefficients of the independent variables

X is the independent variable

n is the degree of the polynomial

The intercept is the value of Y when X is equal to 0.

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