Overview of generalized linear models and their implementation in various languages

Generalized Linear Model Overview

The Generalized Linear Model (GLM) is a statistical modeling and machine learning technique used to stochastically model the relationship between response variables (objective variables) and explanatory variables (characteristics). GLM is a generalization of the linear regression model and consists of the following elements

• Probability distribution of the response variable: The GLM specifies the probability distribution that the response variable follows. This can be a normal distribution for continuous values, a binomial distribution for binary values, or a Poisson distribution for count data.
• Link function: A link function is used to relate the predicted value of the response variable to the linear combination of the explanatory variables. The link function is a transformation function that relates the mean or expected value of the response variable to the linear combination of the explanatory variables. As an example, a logit function (inverse logit function) is used in the case of logistic regression.
• Linear predictor: It is the term that represents the linear combination of the explanatory variables. The linear predictor is expressed as a weighted sum of the explanatory variables.

The GLM estimates parameters using the maximum likelihood estimation method described in “Overview of Maximum Likelihood Estimation and Algorithms and Their Implementations” and uses the estimated model to predict or infer new data. GLMs are generally sequentially optimized and can be applied to large data sets.

Algorithm for Generalized Linear Model

There are various algorithms used in generalized linear models that are based on the maximum likelihood estimation method. The maximum likelihood estimation method estimates the values of parameters that maximize the probability that the observed data will be generated, and is often used in GLMs.

The maximum likelihood estimation method finds the optimal parameters by updating the parameters to maximize the objective function (log-likelihood function). The typical algorithms used in GLM include the following.

• Newton-Raphson method: This method updates the log-likelihood function using the second-order derivative, and has the advantage of fast convergence.
• Iteratively Reweighted Least Squares (IRLS): This method applies the least squares method sequentially and is widely used in cases such as Poisson regression and logistic regression.
• Steepest Descent method: This method updates the log-likelihood function in the gradient direction and is simpler than the Newton-Raphson method.
• Markov Chain Monte Carlo (MCMC): This method samples the parameters of the GLM and has the advantage of being applicable to complex distributions.

These are described in detail below.

Newton-Raphson method

The Newton-Raphson method is an iterative algorithm used in optimization problems such as maximum likelihood estimation. The basic algorithm is to update parameter values to maximize (or minimize) the log-likelihood function. The algorithm of the Newton-Raphson method is described below.

1. Initialization: Set the initial values of the parameters.
2. Iteration Steps
   1. Calculate the first derivative (gradient) and second derivative (Hesse matrix) of the log-likelihood function.
   2. Evaluate the gradient and Hesse matrix using the current values of the parameters.
   3. Compute the inverse of the Hesse matrix and obtain the inverse Hesse matrix.
   4. Calculate the update amount of the parameters. The amount of update is the product of the inverse Hesse matrix and the gradient.
   5. Parameter update: Adds the updated amount to the current parameters to obtain the new parameter values.
3. Convergence determination: Check whether the updated parameters satisfy the convergence conditions. In general, convergence is determined when the change in the parameters is sufficiently small or when the change in the log-likelihood function is sufficiently small.
4. If convergence has not occurred, return to the iteration step.

The Newton-Raphson method has the property of converging to a local maximum (or minimum) value of the log-likelihood function, but it may diverge depending on the initial values. It can also be costly in terms of computational effort, since the inverse of the Hesse matrix must be computed. Furthermore, since the Newton-Raphson method is generally applied to convex functions, other optimization methods may be needed for non-convex functions.

For details of the Newton-Raphson method, see “Online Stochastic Optimization and Stochastic Gradient Descent for Machine Learning” and “Quasi-Newton Method as Sequential Optimization for Machine Learning (1) Algorithm Overview,” etc. For Hesse matrices, see “Hesse Matrices and Regularity”

Next, we describe the Iteratively Reweighted Least Squares (IRLS) method.

Iteratively Reweighted Least Squares, IRLS

Iteratively Reweighted Least Squares (IRLS) will be an iterative optimization algorithm often used to estimate generalized linear models (GLMs). It is primarily used in GLMs such as Poisson regression and logistic regression. The basic IRLS algorithm is described below.

1. Initialization: Set initial values for the parameters. Usually, initial values obtained from zero vectors or least squares results are used.
2. Iteration Step:.
   1. Compute linear predictors using the current parameter values.
   2. Compute the residuals from the predictions.
   3. Transform the residuals using the inverse function of the link function.
   4. Apply Weighted Least Squares (WLS) to estimate new parameters using the transformed residuals as weights.
   5. Check for convergence of the estimated parameter values.
3. Convergence Decision: Check whether the value of the estimated parameter satisfies the convergence conditions. In general, convergence is determined when the change in the parameters is sufficiently small or when the change in the log-likelihood function is sufficiently small.
4. If convergence has not been achieved, return to the iterative step.

A feature of IRLS is that the weights are updated at each iteration; the weights are computed using the parameters estimated in the previous iteration, and larger weights are assigned to the more important data points. This has the effect of mitigating the effects of outliers and outliers. In addition, because IRLS uses the least-squares method, it is necessary to solve a linear equation for each iteration. However, in general, efficient methods are used to solve the linear equations, making the algorithm computationally efficient.

For details on IRS, see “Classification (3) Probabilistic Discriminant Functions (Logistic, Softmax Regression) and Local Learning (K-Nearest Neighbor Method, Kernel Density Estimation)” etc.

Next, the Steepest Descent method is described.

Steepest Descent method

The steepest descent method (Steepest Descent method) is one of the iterative optimization methods in optimization problems. It is especially used when the objective function is not convex with respect to the gradient vector. The basic algorithm of the steepest descent method is described below.

1. Initialization: Initial values of parameters are set.
2. Iteration step: Set the initial values of the parameters.
   1. Compute the gradient vector of the objective function.
   2. Set the update direction vector to go in the direction of the gradient vector.
   3. Parameter update: Update parameters according to the update direction vector from the current parameters.
   4. Convergence judgment: Check whether the updated parameters satisfy the convergence conditions.
3. Convergence judgment: Check whether the updated parameters satisfy the convergence conditions. In general, convergence is judged when the amount of change in the parameters is sufficiently small or when the amount of change in the objective function is sufficiently small.
4. If convergence has not occurred, return to the iteration step.

The steepest descent method is a method to find the optimal solution by updating the parameters according to the direction of the gradient vector, but since it proceeds only in the direction of the gradient vector, it may converge to a local optimal solution if the objective function is nonconvex. Also, the steepest descent method tends to converge more slowly than more advanced optimization methods such as the Newton-Raphson method. However, it is a useful method when the computational cost is relatively low and the gradient vector can be easily computed.

For more information on the steepest descent method, see “Basics of Gradient Methods (Linear Search, Coordinate Descent, Steepest Descent, and Error Back Propagation)” etc.

Finally, Markov Chain Monte Carlo (MCMC) is described.

Markov Chain Monte Carlo, MCMC

Markov Chain Monte Carlo (MCMC) will be one of the sampling-based statistical inference methods; MCMC is used for sampling from complex probability distributions, such as Bayesian statistical models and stochastic graphical models. The basic MCMC algorithm is described below.

1. Initialization: Set initial values for sampling. The initial values are related to the parameters or variables to be sampled.
2. Iteration Step:.
   1. Based on the current state, the next state is proposed. Random walk described in “Overview of Random Walks, Algorithms, and Examples of Implementations” and Metropolis-Hastings methods are commonly used for proposal methods.
   2. Calculate the acceptance probability of the proposed state. The acceptance probability is the probability of adopting the proposed state.
   3. Based on the acceptance probability, the proposed state is accepted or rejected. If accepted, it is adopted as the next state. If rejected, the current state is retained as the next state.
   4. Samples of the state are recorded.
3. Convergence Decision: Check whether the sampled state satisfies the convergence conditions. Convergence conditions are set according to the problem, such as sufficient number of iterations, amount of parameter change, etc.
4. If convergence is not achieved, return to the iteration step.

In MCMC, state transitions are treated as a Markov chain. This results in a stochastic process with the property that the next state depends only on the current state and not on any previous state. Due to the properties of Markov chains, MCMC can efficiently sample from probability distributions, and major MCMC methods include the Metropolis-Hastings method and Gibbs sampling.

For more information on MCMC methods, see “Markov Chain Monte Carlo (MCMC) Methods and Bayesian Estimation.

GLM Extension

One caveat in using GLM is that since GLM is a generalization of the linear model, a different approach is needed when modeling the non-linearity of the data. Extension methods for handling nonlinear relationships in this GLM include the following

• Adding polynomial terms: The simplest way to introduce nonlinearity into a linear model is to add polynomial terms for the predictors, which can capture nonlinear relationships by adding, for example, a squared term (x^2) or a cross term (x1*x2) for the predictor x.
• Extending the linear regression model: Another way to extend the linear regression model to nonlinear relationships is through a basis function expansion. A basis function is a function that transforms a predictor into a nonlinear function. Polynomial basis functions, trigonometric basis functions, and spline basis functions can be used. In addition, basis function expansions can be used to model more flexible nonlinear relationships.
• Logistic regression extension: The logistic regression model is a type of generalized linear model that uses a sigmoid function to model probabilities in the range 0 to 1. Link functions other than sigmoid functions can also be used to model nonlinear relationships. These include probit link functions and complementary log-log link functions.
• Decision tree-based models: Decision tree-based models, such as decision trees and random forests, are powerful methods for modeling nonlinear relationships. These models can automatically capture nonlinear interactions and nonlinear effects of predictors.

These extension methods are used as a means to make GLMs more flexible to model nonlinear relationships. In addition to nonlinearization, other extensions to GLM include the following.

• Mixed Effects Models: Mixed effects models are used to model group and hierarchical structures within a data set. By incorporating random effects among individuals and fixed effects among groups, they allow for clustering of data and modeling that accounts for hierarchical variation.
• Hierarchical Bayesian Models: Hierarchical Bayesian models are an extension of GLMs within the framework of Bayesian statistics that model parameter uncertainty by introducing a hierarchical structure as a prior distribution and estimating a posterior distribution that fits the data. Hierarchical Bayesian models are useful not only for parameter estimation, but also for model selection and computing predictive distributions.
• Zero-Inflated Models: Zero-inflated models are used when the data have an excess of zeros. This can be done, for example, by combining a negative binomial distribution and a zero-inflated Poisson distribution to model the process of generating zero and non-zero observations.
• Zero-Truncated Models: Zero-truncation models are used when zeros are present in the data but no zero occurrences are observed. An appropriate probability distribution is selected to model the distribution of this zero-unobserved data.
• Count Regression Models: Count regression models are used for regression modeling of count data. These include Poisson regression models and negative binomial regression models.

GLM Application Examples

GLM is used in a variety of application areas. Some examples of GLM applications are listed below.

• Logistic regression: Used in binary and multiclass classification problems where the objective variable belongs to a binary or multiclass class. This applies, for example, to the problem of predicting whether a customer will make a purchase or not, or to the classification of positives and negatives in the diagnosis of a disease.
• Poisson regression: Used in modeling count data. This can be, for example, the problem of predicting the number of events that will occur within a certain period of time, or the number of visitors to a website.
• Linear regression: Used for problems that involve predicting a continuous-valued objective variable. This is the case, for example, for predicting home prices or sales.
  Poisson Mixture Models: Used when count data are generated from multiple components. This can be the case, for example, in detecting anomalous events or modeling topics in a document.
• Modeling Bernoulli or binomial distributions: Also used when modeling binary data where the probability of success is not constant. This applies, for example, to modeling purchasing behavior or predicting click-through rates.
• Modeling gamma and exponential distributions: Also used for modeling positive continuous variables. This can be used, for example, for modeling product longevity or risk assessment.

Python implementation of GLM

Various libraries are available for the Python implementation of GLM. Among them, the most common and widely used are “statsmodels” and “scikit-learn”. Examples of GLM implementations using each library are shown below.

1. Example GLM implementation using statsmodels:.

import statsmodels.api as sm

# Prepare data for explanatory and objective variables
X = ...  # Data for explanatory variables
y = ...  # Objective variable data

# Definition of GLM Model
glm_model = sm.GLM(y, X, family=sm.families.)

# Model Fitting
glm_result = glm_model.fit()

# View Model Summary
print(glm_result.summary())

# Calculation of Predicted Values
y_pred = glm_result.predict(X_new)

In the above code, X and y are assumed to contain data for the explanatory and objective variables, respectively. The <distribution> part specifies the probability distribution to be used (for example, sm.families.Poisson() or sm.families.Binomial()).

2. Example implementation of GLM using scikit-learn (version 1.0 or later):.

from sklearn.linear_model import PoissonRegressor, LogisticRegression

# Prepare data for explanatory and objective variables
X = ...  # Data for explanatory variables
y = ...  # Objective variable data

# Definition and adaptation of the GLM model
if :
    glm_model = LogisticRegression(fit_intercept=True)
else:
    glm_model = PoissonRegressor(fit_intercept=True)
    
glm_model.fit(X, y)

# Calculation of Predicted Values
y_pred = glm_model.predict(X_new)

In the above code, X and y are assumed to contain data for the explanatory and objective variables, respectively. The <classification_problem> part should be True for binary classification problems and False otherwise. scikit-learn uses the PoissonRegressor and LogisticRegression classes to implement GLMs for count data and binary classification.

Implementation of GLM by R

The implementation of GLM with R is widely used as a powerful tool for statistical analysis; R has a wealth of libraries and packages specialized for statistical analysis, and GLM can be easily implemented. Below is an example of a GLM implementation using R.

1. Example of GLM implementation using stats package:.

# Load the necessary libraries
library(stats)

# Prepare data for explanatory and objective variables
X <- ...  # Data for explanatory variables
y <- ...  # Objective variable data

# Definition and adaptation of the GLM model
glm_model <- glm(y ~ X, family=)

# View Model Summary
summary(glm_model)

# Calculation of Predicted Values
X_new <- ...  # Explanatory variables for new data
y_pred <- predict(glm_model, newdata=X_new)

In the above code, X and y are assumed to contain data for the explanatory and objective variables, respectively. The <distribution> part specifies the probability distribution to be used (e.g., poisson(), binomial(), etc.).

2. Example implementation of GLM for L1 regularization using glmnet package:.

# Load the necessary libraries
library(glmnet)

# Prepare data for explanatory and objective variables
X <- ...  # Data for explanatory variables
y <- ...  # Objective variable data

# Definition and adaptation of the GLM model
glmnet_model <- glmnet(X, y, family=, alpha=1)

# View Model Summary
print(glmnet_model)

# Calculation of Predicted Values
X_new <- ...  # Explanatory variables for new data
y_pred <- predict(glmnet_model, newx=X_new)

In the above code, X and y are assumed to contain data for the explanatory and objective variables, respectively. The <distribution> part specifies the probability distribution to be used (e.g., gaussian, poisson, etc.). alpha parameter controls the rate of L1 regularization.

Implementation of GLM by Clojure

Clojure is a functional programming language that runs on the JVM and features seamless integration with Java. The following is an example of GLM implementation using Clojure.

First, add the following dependencies to the project.clj file

:dependencies [[incanter "1.5.7"]]

Next, implement GLM with the following Clojure code.

(ns glm-example
  (:require [incanter.stats :as stats]))

(def X [[1 2 3]
        [1 3 4]
        [1 4 5]
        [1 5 6]])

(def y [1 2 3 4])

(def glm-model (stats/glm y X :family stats/gaussian-link-function))

;; Display of model parameters
(println "Coefficients:")
(println (:coefficients glm-model))

;; Calculation of Predicted Values
(def X-new [[1 6 7]])

(println "Predicted values:")
(println (stats/glm-predict glm-model X-new))

In the above code, X and y are assumed to contain data for the explanatory and objective variables, respectively. glm function is used to fit y and X to the GLM model. The :family parameter specifies the probability distribution to be used (e.g., stats/gaussian-link-function, stats/poisson-link-function, etc.). In addition, use (:coefficients glm-model) to display the parameters of the GLM model, and use the glm-predict function to calculate the predictions, passing the new explanatory variable data X-new.