# This Tutorial talks about basics of Linear regression by discussing in depth about the concept of Linearity and Which type of linearity is desirable. ## What is the meaning of the term Linear ?

In Linear Regression the term linear is understood in 2 ways -

1. Linearity in variables
2. Linearity in parameters

Linear regression however always means linearity in parameters , irrespective of linearity in explanatory variables.

A linear regression for 2 variables is represented mathematically as ( u is the error term )-

Y = B1 + B2X + u          Or

Y = B1 + B2X ² + u

Here the variable X can be non linear i.e X or X² and still we can consider this as a linear regression. However if our parameters are not linear i.e say the regression equation is

Y = B1² + B2²X + u

then this can not be said to represent a linear regression equation.

## Linear Regression Models

 Model linear in parameters? Model linear in variables? Yes No Yes Linear Model Linear Model No Non Linear Model Non Linear Model

## Linearity in predictor variables - Xi

A function Y = f(x) is said to be linear in X if X appears with a power or index of 1 only. i.e the  terms such as x2, Γx, and so on are excluded or  if x is not multiplied or divided by any other variable.

## Linearity in parameters - Bi

Y is linearly related to X if the rate of change of Y with respect to X (dY/dX) is independent of the value of X.

A function is said to be linear in the parameter, say, B1, if B1 appears with a power of 1 only and is not multiplied or divided by any other parameter (for eg B1 x B2 , or B2 / B1)

To reiterate again - For purpose of Linear regression we are only concerned about linearity of parameters B1, B2 .... and not the actual variables X1, X2 ....

## Non Linear Models

• Some models may look non linear in the parameters but are inherently or intrinsically linear.
• This is because with suitable transformations they can be made linear in parameters.
• However, if these cannot be linearized, these are called intrinsically non linear regression models
• When we say ‘non linear regression model’ we mean that it is intrinsically non linear.

## Example

For  Log(Yi) = Log(B1) + B2 Log(Xi) + u

B2 is Linear but B1 is non-linear but if we transform α = Log(B1) then the model

Log(Yi) = α + B2 Log(Xi) + u

is linear in α and B2 as parameters. Implying we can make the regression equation linear in parameters using a simple transformation

For other cases we may not have an easy way to transform parameters to their linear form and such equations are hence treated as intrinsically non-linear and are NOT modeled using linear regression

This tutorial was originally posted here

Next in the series :

Reference : Based on Lectures by Dr. Manish Sinha. ( Associate Prof. SCMHRD )

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Tags: Data, Learning, Machine, Regression, Science

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