# Generalized Linear Models

## Understanding Generalized Linear Models (GLMs)

Generalized linear models, commonly known as GLMs, define the relationship between the dependent variable, Y, and the random error value, using diverse distributions from the exponential family such as the Binomial, Normal, Poisson, Inverse Gaussian, and Gamma. GLMs are different from traditional linear models as they assume that the dependent variable corresponds to the exponential family of distributions.

These models represent the expected value of the response variable. Depending on the projected spread of dependent variables, different link functions are employed to convert g(µ) into the result value, which is then modeled using several regression models. If the dependent variable follows a normal distribution, the identity function acts as the link function.

## Benefits of GLMs

GLMs are beneficial for predicting the value of a response variable wherein the returning variable, Y, and the error term (ϵ), both adhere to a normal distribution X, given as a predictor value. The average value in a normal distribution is a composite of the weights (W) and predictor (X), along with the standard deviation. Examples of generalized models include generalized linear regression and ANOVA models.

The link function in GLMs machine learning is an identity function, which transforms the probability of outcomes of a categorical dependent variable into an uninterrupted continuous scale. This makes the linear regression model apt for establishing the relationship between the predictors and the response.

## Training Regression Models with GLMs

In the process of training regression models, it is essential to comprehend that the average of the dependent variable values is being predicted as opposed to the actual variables. When the response variable Y follows a normal distribution, the total of the weights and the predictor variable can equate to Y’s expected value. The identity function operates as a link function in the linear regression models, linking the average of the predicted value of the Y response variable and the summation of weights and predictor variables. Consequently, g(E(Y)) is represented as Ypredicted.

The usage of GLMs is dependent on the theoretical dispersion of the response variables. For instance, logistic regression works well when the response variable is linked to a binary outcome; Poisson regression with a log-link is apt for response variables showing quantities or relative frequencies, and the GammaRegressor can be used for high and skewed response variables.

## Conclusion

In summary, GLMs are a powerful tool capable of interpreting response variables using various distributions such as Gamma, binomial, and Tweedie. Python’s SkLearn package offers classes for designing GLM models based on particular probability distributions and response variables. The response variable can be modeled as a linear amalgamation of weights and predictors, given that the dependent variable and standard error are part of the exponential family of distributions.

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