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Supercharge Your Analysis with Generalized Linear Regression in Python

This blog discusses the class of regression model known as generalized linear regression, which will aid data scientists in understanding this model in depth and using it as a training model. A type of regression model that may simulate relationships between response variables and predictor factors is known as generalized linear regression. While classic linear regression presupposes a linear relationship between the response variable and predictor variables, the generalized linear regression model is more flexible and enables non-linear interactions utilizing statistical distribution. In the parts that follow, this blog offers a comprehensive knowledge of generalized linear regression in Python.

I. Introduction to Generalized Linear Regression in Python

It is possible to describe response variables using non-normal error distributions and non-linear relationships between the predictor and response variables using generalized linear regression, an extension of the linear regression model. The generalized linear regression model, which outperforms many other regression models in terms of statistical power, can handle a broad variety of data distributions and kinds. In the subsections that follow, this part introduces regression analysis and generalized linear regression.

A. Brief overview of regression analysis

A statistical modelling method called regression analysis is employed to investigate the connection between one or more independent variables and a dependent variable. Regression models seek to better anticipate outcomes by estimating the magnitude and direction of the relationship between the variables. Investigating the strength of the link between response variables and predictor factors is the goal of regression analysis. To improve forecasts, it measures the strength of the association between variables.

Simple linear regression, multiple linear regression, polynomial linear regression, logistic regression, Poisson regression, ridge regression, and lasso regression are the seven distinct methods of regression analysis. The linearity, independence, residual normality, and lack of multicollinearity are among the assumptions made by the regression analysis model. Violating these assumptions affects the validity of results that requires appropriate remedies. Building a regression model involves the selection of an appropriate independent variable that determines the functional form of the relationship and estimates the model parameters.

This can be done through techniques such as OLS, MLE, or regularization methods. Regression analysis suffers from multicollinearity due to the assumption of linear relationships and can be affected through outliers while failing to establish causality on their own. The regression analysis model can be implemented in various fields such as economics, social science, marketing, finances, healthcare, etc. This makes regression analysis a powerful technique for understanding relationships among the variables and making predictions accordingly.

B. Introduce Generalized Linear Regression (GLR)

As we discussed earlier, GLR assumes a non-linear relationship between the predictor and response variables, as opposed to an ordinary linear regression model that follows a linear relationship. As compared to the ordinary linear regression which assumes to follow a normal distribution, with the relationship between predictor and response variables to be linear, the GLR model specifies the link function and error distribution for the appropriate response variable which relaxes the assumptions for the relationship. GLR is capable of handling different types of response variables such as continuous, count, binary, and duration data. Selecting an appropriate error distribution and link function helps GLR model non-linear relations, accommodate data having a non-constant variance, and handle overdispersion.

GLR is mostly implemented through statistical software packages and provides functions that are designed for generalized linear models. these packages are able to estimate model parameters, calculate confidence intervals, make predictions on the basis of the fitted model, and perform hypothesis tests.

There are several features of GLR that make it better than the common linear regression such as the flexibility to model a wide range of response and predictor variables. It provides interpretability of the relationship between the predictor and response variables of the model. GLR is more robust to the anomalies and outliers in data allowing for non-normal distributions of response variables. GLR is scalable as it can be used for large datasets and complex models and are easy to understand and use. It also allows hypothesis testing and statistical inference that is useful for multiple applications and helps us understand the significance of the relationship between the variables. GLR can be regularized for reducing overfitting for improving the model’s performance through techniques such as lasso, ridge, or elastic net regression.

Some of the disadvantages of using GLR models is that it makes assumptions about the distribution of response variables which may not always hold. It is challenging for GLR to specify the correct underlying statistical distribution which results in incorrect prediction. GLR is prone to overfitting in case of the model is complex or has many predictor variables. GLR is a powerful and flexible tool to model the relationship between response and predictor variables with higher flexibility than traditional linear regression models.

C. Importance of GLR in machine learning

Generalized linear regression plays a very important role in machine learning and provides several advantages to the user. Some of the reasons for GLR being useful are:

  1. Modeling flexibility: GLR provides a wide range of error distributions and response variables which provides flexibility and helps the modeling of complex relations among the predictor and response variables along with the non-linear associations. Selecting an appropriate error distribution and link function allows GLR to handle different types of data such as count, duration data, and binary data.
  2. Handling non-normality and heteroscedasticity: GLR relaxes all the assumptions of linear regression and allows the modeling of response variables through heteroscedastic and non-normal error distribution, which is required for the accurate capturing and analysis of data through different distributional characteristics.
  3. Robustness towards outliers: GLR provides robust estimates even when influential observations or outliers are present and using appropriate error distributions can help GLR downweight the impact of outliers to produce reliable parameter estimates.
  4. Predictive modeling and inference: GLR can be used for both predictive modeling and statistical inference which allows the making of predictions on the basis of a fitted model, conducting hypothesis tests or a confidence interval of the model parameter, and generating uncertainty estimates. This helps enable comprehensive analysis of the relationship between variables providing a solid foundation for decision-making processes.
  5. Interpretable parameter estimates: the interpretable parameter estimates provided by GLR, quantifies the relationship between the predictor and response variables which can be used for understanding the direction and magnitude of effects of predictors for the response which aids in the interpretation and explanation of the behavior of the models.

Thus, GLR is an important concept in machine learning whose importance lies in the ability to handle different types of data, model non-linear relationships, and handle non-normality to provide interpretable results. Thus, GLR serves as a foundation for advanced modeling techniques and a wide range of problems can be addressed through GLR which can help enhance the accuracy and reliability of machine learning models.

II. Understanding Generalized Linear Regression in Python

In the previous section, we have seen the usefulness of Generalized linear regression in machine learning with a brief introduction to the model and an overview of regression analysis. It helped us understand the concept of a model and its advantages in the field of machine learning. To understand a few more concepts of GLR let us dive deep into its concepts and limitations which will help us achieve more knowledge of this model.

A. Linear regression limitations

The Generalized Linear Regression in Python is a better version of the linear regression model which aims to provide better results as compared to the linear regression model. thus, the Generalized linear regression model was designed to help with these limitations of linear regression. Apart from the advantages of using linear regression, there are several limitations of this model which need to be considered and this sub-section discusses these limitations here:

  1. Independence of error: The linear regression model assumes error or residuals to be independent of each other, but the real-world data violates this assumption due to the presence of dependence or autocorrelation among the observations. This violation leads to incorrect standard errors and unreliable results.
  2. Assumption of linearity: Linear regression assumes residuals or errors to be independent of each other and it might be violated due to the presence of dependence among observations.
  3. Outliers or influential observations: Linear regression is sensitive to outliers and influential observations that bias the coefficient estimates and affect the model fit.
  4. Homoscedasticity assumption: Homoscedasticity means having constant variance and linear regression shows homoscedasticity across all levels of the independent variables. If the variance varies with predictors this means that model is not representing the data accurately.
  5. Casual inference challenges: Linear Regression identifies association among variables with limitations in establishing causality. The observed relationship might be influenced by confounding variables or omitted variable bias which makes it necessary to use experimental designs or quasi-experimental methods for making causal inferences.

There are other limitations to the linear regression model such as multicollinearity, linear relationships, and sensitivity to influential observations which makes it inferior for data scientists to use it for solving complex machine learning problems. It is important for us to be aware of these limitations while using linear regression and these limitations are what gave birth to the generalized linear regression model which is an improvement of the linear model.

B. Components of Generalized Linear Regression

The Generalized Linear Regression model has a lot of components that help it make better predictions of the model and helps in improving the results of this model. Some of the components of GLR include response variable, linear predictor, link function, error distribution,

1. Random component

The random component in Generalized Linear Regression is the random part of the model that represents variability or randomness in the response variable and cannot be explained through the systematic part of the model that includes link function and linear predictor. The random component is specified through an error distribution method that characterizes the distributional properties of the response variable and determines the probability distribution through which the response variable is drawn. The form of error distribution depends on the nature of the response variable and if the response variable is binary or follows count distribution, then the random component gets modeled through the specification of appropriate error distribution for these variables.

The random component is important in GLR as captures variability that isn’t accounted for in the systematic part of the model. Considering the random component helps GLR perform a more realistic representation of data for the proper estimation of model parameters.

2. Systematic component

The systematic component refers to the deterministic part of a model representing the relationship between the expected value of the response variable and predictor variable which is denoted by η (eta) and is represented as a linear combination of predictor variables. The systematic component is defined as:

η = β₀ + β₁x₁ + β₂x₂ + … + βₚxₚ,

here β₀, β₁, β₂, …, βₚ are the regression coefficients and η is the linear predictor with X representing the predictor variables and p being the number of predictor variables.

The systematic component is linear in predictor variables and the relationship between the expected value of the response variable and the linear predictor is determined through the link function which maps the range of linear predictor to that of the response variable, establishing the relationship between them. The form of the systematic component is dependent on the modeling objective and the relationship between the predictor variable and the response variable. The estimation of the regression coefficients with the estimation of parameters of random components allows for the construction of a GLR model that enables predictions, inference, and analysis of relationships between variables.  

3. Link function

The link function is an important key component in GLR establishing the relationship between linear predictors and the expected value of the response variable which maps the range of linear predictors to that of the range of response variables. The link function transforms the linear predictor and ensures that the predicted values fall within the appropriate range of response variables. It is denoted as g(μ) and connects linear predictors to the parameters of error distribution which allows the modeling of different types of response variables.

Choosing a link function depends on the characteristics of response variables and its modeling objectives where different link functions are suitable for different forms of response variables. Some of the most common link functions are identity link, logit link, log link, and inverse link functions. Choosing a link function affects the interpretation of regression coefficients and shapes the relationship between predictor and response variables which makes it important for the selection of a link function that is appropriate for specific modeling tasks and distributional characteristics of the response variable.

C. Types of GLR

There are different types of Generalized Linear Regression in Python that are commonly used for different tasks and in different problems. In this section, we have tried to introduce some of the important forms of GLR such as logistic regression, Poisson regression, negative binomial regression, and gamma regression.

1. Logistic regression: It is a form of regression model that is commonly used for modeling the binary or categorical response variables and is suited for dichotomous outcomes or binary outcomes like yes/no. The response variable in logistic regression follows binary distribution where the relationship between the predictors and response variable is modeled using the logit link function and this logit link function transforms the linear predictor to a log-odds scale that allows for the prediction of success probabilities. The logistic regression can be modeled as:

  • logit(p) = β₀ + β₁x₁ + β₂x₂ + … + βₚxₚ,

where logit(p) is logarithm of odds of success and β₀, β₁, β₂, …, βₚ denotes regression coefficients. The X denotes predictor variables and p is the number of predictor variables.

The logistic regression model estimates regression coefficients using maximum likelihood estimation that finds parameter values maximizing the likelihood of observed data through the given model.

2. Poisson regression: It is a type of GLR model that models count data and is designed for situations in which the response variable represents the count of events. The response variables in Poisson regression follow Poisson distribution, a discrete probability distribution that models count data. This distribution assumes that the mean and variance of response variables are equal. The Poisson regression model uses the maximum likelihood function to find parameter values and maximizes the likelihood of observed data through the given model. The estimated coefficients are used for the prediction of the expected counts for new observations on the basis of predictor values.

Interpretation of Poisson regression coefficients is similar to linear regression where coefficients represent expected change in log count associated with a one-unit change in the predictor variable and holds other variables as a constant. Poisson regression provides statistical significance tests measures of model fit and confidence intervals like the deviance or AIC. This model is particularly useful when the response variable represents assumptions and the count of Poisson distribution.

3. Negative binomial regression: It is a form of GLR model that is used for the analysis of over-dispersed count data and is an extension of Poisson regression that assumes the mean and variance of response variables are equal. The count data often exhibits more variability than it can be handled by the Poisson regression and the Negative binomial regression addresses this issue through relaxation of the assumption of equal mean and variance. This allows greater flexibility in modeling count data with overdispersion in which the variance is larger than the mean. The response variable follows Negative binomial distribution which is probability distribution accounting for overdispersion. The mean and dispersion parameter is characterized in the Negative binomial regression where the mean represents the expected count and the dispersion parameter control the level of overdispersion in data. The Negative binomial regression is used when count data exhibit overdispersion and variance exceeds the mean.

4. Gamma regression: The gamma regression is a form of Generalized Linear Regression in Python that is used for modeling continuous response variables and follows gamma distribution which is a positively skewed distribution for modeling non-negative continuous data with a right-skewed shape. The response variable follows gamma distribution characterized by shape and rate parameters. The shape determines the shape of distribution while the rate controls the scale of distribution. The gamma regression assumes the mean is related to predictor variables through the logarithmic link function which ensures predicted values fall within the positive range of response variables. 

D. Model assumptions and validation

The Generalized Linear Regression in Python model follows some assumptions to ensure the validity and reliability of the model. Some of the key assumptions for the GLR model are:

  1. Independence: The observations need to be independent of each other and the errors and residuals must not be correlated. Dependence on data can lead to biased parameter estimates and inaccurate inference.
  2. Linearity: The relationship between the predictor variable and linear predictor must be linear which ensures that the model captures the relationship between predictors and response variables adequately.
  3. Distributional assumptions: GLR assumes a specific distributional form of response variable where choosing the distribution depends on the nature of the normal distribution.
  4. Homoscedasticity: The variance of response variables need to be constant across all levels of predictor variables and the term Homoscedasticity means that spread of residuals needs to be consistent across a range of predictor values.

For validation purposes, Generalized Linear Regression in Python uses some diagnostic techniques and assesses the assumptions through the following techniques:

  1. The goodness of Fit tests: The statistical tests are able to evaluate the chosen distribution fits of data.
  2. Residual analysis: Examining residuals helps assess the assumption of linearity and independence for residual plots.
  3. Cross-validation: Splitting of data into training and validation sets helps assess the model’s performance on unseen data and techniques such as k-fold cross-validation provides an estimate of the model’s predictive ability to identify overfitting issues.
  4. Influence and outlier analysis: It is important to identify influential observations and outliers as they heavily influence the model’s parameter estimates.
  5. Model comparison: Comparing different models helps in assessing the adequacy of chosen GLR model through techniques like likelihood ratio tests.

Through careful examination of these diagnostic techniques and addressing the violations identified, we can easily validate our GLR model so that its reliability and applicability are maintained.

Applications of Generalized Linear Regression in Python

Applications of Generalized Linear Regression in Python

Implementation of Generalized Linear Regression in Python

Dataset collection– This dataset offers valuable data on the probability of customer default, which makes it perfect for developing and testing predictive models like generalized linear regression.

Link– https://www.kaggle.com/datasets/laotse/credit-risk-dataset

Conclusion

In this blog, we have introduced the Generalized Linear Regression in Python model as an extension of linear regression and a better regression analysis model. We have tried to explore Generalized linear regression through an overview of regression analysis, introducing GLR and discussing the importance of GLR in machine learning. We also discussed the limitations of linear regression which helps us understand the value and requirements of Generalized linear regression. The key components of GLR are discussed along with four different forms of the GLR model and then we discuss the model assumptions and validations of GLR. This blog will have an impact on data scientists and will help them work with GLR by providing them with the necessary information needed to model the algorithm.

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