Hi, I want to know these two terms in depth.

In a way you can say they are an extension to include and extend the regression based linear models in one definition. A way to define everything in one definition while overcoming some limitations.

Generalized Linear Models are a class of statistical models that are an extension of linear regression. They are designed to handle a broader range of data types and distributional assumptions. Linear regression assumes that the residuals (the differences between predicted and actual values) are normally distributed. Limited in handling non-continuous or non-normally distributed data.

**Generalized Linear Models (GLMs):**

**Flexibility:**GLMs extend the linear regression framework to accommodate a wider range of data types.**Key Components:****Linear Predictor:**Similar to linear regression.**Link Function:**Connects the linear predictor to the mean of the response variable or target.**Distribution Family:**Specifies the type of probability distribution for the response variable.

**Linear Predictor:**$η$

**Link Function $g(μ)$):** Connects the linear predictor to the mean of the response variable.

**Identity Link:**Used for models where the response variable is continuous and follows a normal distribution.**Logit Link:**Commonly used for binary logistic regression.**Log Link:**Suitable for Poisson regression.

**Response Distribution:** Specifies the probability distribution of the response variable (e.g., Normal, Binomial, Poisson).

**Normal Distribution:**Suitable for continuous data.**Binomial Distribution:**Used for binary outcomes.**Poisson Distribution:**Appropriate for count data.

Fitting:

**Choose Distribution:**Based on the nature of the response variable.**Select Link Function:**Depends on the distribution and characteristics of the data.**Estimate Parameters:**Typically done using iterative methods like Maximum Likelihood Estimation (MLE).

Application:

**Binary Classification:**Logistic regression.**Count Data:**Poisson regression.**Categorical Data:**Multinomial logistic regression.

At its core, GLMs extend the classical linear regression model to accommodate various scenarios where linear relationships may not be appropriate. A key strength lies in its adaptability to different types of response variables through the incorporation of three essential components: the linear predictor, the link function, and the probability distribution.

Linear regression serves as a foundational concept for understanding GLMs. In linear regression, the goal is to model the relationship between independent variables and a continuous response variable, assuming a linear relationship. However, this simplicity becomes a limitation when dealing with diverse datasets that may exhibit non-linear patterns or have response variables with distinct characteristics.

Logistic regression, a specific application of GLMs, addresses this limitation when dealing with binary outcomes. Instead of predicting a continuous variable, logistic regression models the probability of an event occurring, typically binary in nature (e.g., pass/fail). The logistic function, or sigmoid curve, serves as the link function, transforming the linear predictor into probabilities. This approach allows for a more nuanced understanding of the relationship between predictors and the probability of an event.

In the context of logistic regression, the response variable represents the probability of the binary outcome. For instance, in predicting student pass/fail based on the number of hours studied, the response variable is the probability of passing the exam. The choice of the logistic function ensures that the predicted probabilities fall within the range of 0 to 1, aligning with the nature of binary outcomes.

Expanding beyond logistic regression, the broader GLM framework accommodates various response variable types, including continuous, binary, and count data. For example, linear regression, a special case of GLMs, continues to be relevant when the response variable is continuous, maintaining the traditional linear relationship between predictors and the mean of the response variable.

The link function in GLMs plays a pivotal role in connecting the linear predictor to the response variable's mean. While logistic regression employs the logistic link for binary outcomes, other link functions exist for different types of responses. The selection of an appropriate link function ensures that the model captures the underlying relationship in the data effectively.

Distributional assumptions are another crucial aspect of GLMs. Different response variable types necessitate distinct probability distributions. In logistic regression, the Bernoulli distribution fits the binary nature of the response variable, modeling the probability of success (pass) and failure (fail). Contrastingly, linear regression assumes a Gaussian (normal) distribution for continuous response variables, reflecting the central limit theorem.

In a practical sense, fitting a GLM involves making informed choices based on the characteristics of the data at hand. When predicting insurance claims, for instance, count data becomes the focus, leading to the selection of the Poisson distribution and its associated link function. The flexibility of the GLM framework allows for the adaptation of models to the specific nature of the data, ensuring robust and accurate predictions.

GLMs offer not only predictive power but also enhanced interpretability. The coefficients derived from GLMs, including logistic and linear regression, provide insights into the impact of predictors on the response variable. In logistic regression, a positive coefficient indicates an increase in the odds of the event occurring, while in linear regression, it signifies a change in the expected value of the response variable.

Generalized Additive Models (GAMs) represent a sophisticated extension of Generalized Linear Models (GLMs), introducing greater flexibility for capturing complex relationships in data. GAMs are particularly useful when the relationships between predictors and the response variable are not strictly linear, allowing for the incorporation of non-linear and non-parametric components. Unlike GLMs, which assume a linear relationship between predictors and the response, GAMs embrace the concept of additivity. The model assumes that the contribution of each predictor to the response is additive but allows for non-linear relationships. GAMs incorporate smoothing functions, often represented by splines or other flexible functions, to capture non-linear trends in the data. These functions smooth out the noise and reveal underlying patterns that might not be apparent in a linear model. The GAM equation includes the additive linear predictors and the smoothing function

More detail can be found in this book: Generalized Linear Models Second Edition

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