Contributed by : Prabhu Ram.

What is a Generalized Linear Model and what is the difference between a Generalized Linear Model and a General Linear Model ?

## A generalized linear model is what it is.

The Generalized Linear Model was developed in 1972 by John Nelder and Robert Wedderburn. It is an umbrella term that covers many other models that allow the response variable y to have an error distribution other than a normal distribution. Linear Regression, Logistic Regression, and Poisson Regression are included in the models.

In a Linear Regression Model, the response ( aka dependent/target ) variable is expressed as a linear function/ linear combination of all the predictors. The relationship between the response and predictors is linear. A straight line is what we can use to visualize the relationship. The distribution of the response variable should be normal. We are building a model.

Even though the underlying relationship is not linear, GLM models allow us to build a linear relationship between the response and predictors. The link function links the response variable to a linear model. The error distribution of the response variable is different from Linear Regression models. An exponential family of distribution is assumed to follow the errors in the response variable. There are normal, binomial, and Poisson distributions. The name Generalized Linear Models was chosen since we are trying to generalize a linear regression model.

## Why did GLM happen ?

The Linear Regression model is not suitable.

- The relationship between X and y is not linear. There is a relationship between them. As X increases, y increases as well.

- Homoscedasticity in Linear Regression is not constant and varies with X.
- The response variable is not continuous. Linear Regression only applies the normal distribution of the response variable to continuous data. The linear regression model predicts negative values for the corresponding response variable if we try to build a model on a y variable.

The response is either 0 or 1 in the graph. When X 5000, y is 0, and when X > 5000, y is 1.

Consider a linear model.

An example of a mobile price in an e-commerce platform.

12500 + 1 is the price. The screen size is 5 and the battery backup is less than 4 hours.

There is data available.

- The price of the phone.
- The screen size is in inches.
- Is the battery backup less than 4 hours ?

The price of the mobile increases by 1 if the screen size increases by 1 unit. 5 times the default price, keeping the intercept and battery backups constant. The mobile price is reduced by three times the default price if the battery backup is less than 4 hours. The mobile price is unaffected if the battery backup is less than 4 hours. The default price is indicated by the intercept 12500. This model is valid.

However, if we get a model.

12500 is 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 888-739-5110 The screen size is 5 and the battery backup is less than 4 hours.

The model is saying the price of the phone increases by three times if the battery backup is less than 4 hours. We know this is not correct. There will be less demand for such mobiles. When compared to the current range of mobiles with the latest features, these are going to be very less expensive. The relationship between the two variables is not linear, but we are trying to express it as a linear relationship. An invalid model is built.

If we are trying to predict if a particular phone will be sold or not, using the same independent variables, but the target is we are trying to predict if the phone will sell or not, so it has only single outcomes.

We get a model like that using Linear Regression.

There were 12500 sales. The screen size is 5 and the battery backup is less than 4 hours.

The output of a linear regression model is continuous value, so it does n’t tell us if the mobile will be sold or not. It is possible to get both negative and positive values. It does n’t translate to our actual objective of whether phones with some specifications based on predictors will sell or not.

A negative value means nothing if we are trying to see the number of sales of this mobile in the next month. The minimum value is 0 and the positive value is related to the count of the sales. We do n’t care if the count is a negative value.

## There are assumptions of GLM.

Similar to Linear Regression Model, there are some basic assumptions for Generalized Linear Models. Most of the assumptions are similar to Linear Regression models.

- Data should be random and independent.
- The response variable does not need to be distributed normally. Binomial, multinomial, normal.
- The transformed response variable is dependent on the independent variables and does not need to have a linear relationship with them.

Logistic Regression Equation has log odds of 0+1X1+2X2.

The regression coefficients are 0,1,2 and X1, X2.

- I can apply feature engineering on the Independent variable. Instead of taking the original raw independent variables, variable transformation can be done, and the transformed independent variables can be used to build the GLM model.
- Homoscedasticity. Not being satisfied is the need for constant variance. The independent variables can affect the response variable Error variance.
- Errors do not need to be distributed.

## There are components of GLM.

GLM has 3 components.

- TheLinear Predictor is a systematic component.

The linear combination of Predictors and regression coefficients is what it is.

1X1+2X2

- Link function

It specifies the link between random and systematic components. The expected/predicted value of the response relates to the linear combination of predictor variables.

- Random component distribution.

The family of distributions is where the probability distribution comes from.

Normal distribution, binomial distribution, and poisson distribution are included in the family of distributions.

The table of Probability Distribution has a Link function.

Probability Distribution | Link Function |

Normal Distribution | Identity function |

Binomial Distribution | Logit/Sigmoid function |

Poisson Distribution | Log function (aka log-linear, log-link) |

## There are different generalized linear models.

The models in the family are commonly used.

- Linear Regression is used for continuous outcomes with normal distribution.

The mean expected value is a function of the explanatory variables. The simplest link function is identity link function.

The model is called Simple Linear Regression if there is only one predictor. The model is called Multiple Linear Regression if there are 2 explanatory variables.

There is a simple linear regression.

Multiple linear regressions are defined as 0+1X1+2X2.

Continuous response is what it is.

Predictors can be continuous or categorical.

Errors are distributed normally.

- Logistic regression with binomial distribution.

The log odds are a combination of the explanatory variables. The link function is called logit. The Logistic or Sigmoid function returns the probability as the output.

There are log odds of 0+1X1+2X2.

There are only 2 outcomes for the response variable.

Predictors can be continuous or categorical.

The image is from https : //en. There is a website called wikipedia. There is a function in the org.

- There is a regression for count based outcomes.

The explanatory variables are expressed as a linear combination of count values. The link function is called log link.

log is related to 0+11+2

The average value of the count variable is.

Each unit of time and space has a response variable.

Predictors can be continuous or categorical.

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## There is a difference between the generalized linear model and the general linear model.

General Linear Models is a special case of Generalized Linear Models. Normal linear regression models have a continuous response variable. It has many statistical models such as Single Linear Regression, Multiple Linear Regression, Ancova, Manova, Mancova, t-test and F-test. The residuals/errors are assumed to follow a normal distribution. The generalized linear model allows residuals to have other distributions.

## Is it possible that Generalized Linear Models have correlated data ?

Data should n’t be correlated with each other for generalized linear models. The model performance will not be reliable if the data is correlated. Data that has auto-correlation in it is unsuitable for GLMs on time series data. The Generalized Estimating Equations ( GEEs ) model and the Generalized Linear Mixed Models ( GLMMs ) model have been developed to consider the correlation in the data.

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