Understanding MATLAB’s glmfit Implementation for Logistic Regression

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In this article, we will delve into the specifics of how MATLAB’s glmfit function implements logistic regression, with a focus on understanding its underlying mathematical principles and comparison with other programming languages.

Background: Maximum Likelihood Estimation in Logistic Regression

Logistic regression is a type of binary classification model that predicts the probability of an outcome based on one or more predictor variables. The goal of logistic regression is to estimate the relationship between the predictors and the log-odds of the outcome, which can then be used to make predictions.

One common approach to estimating the parameters of a logistic regression model is through maximum likelihood estimation (MLE). MLE involves finding the values of the model’s parameters that maximize the likelihood of observing the training data. This is typically done using an optimization algorithm, such as gradient descent or Newton’s method.

MATLAB’s glmfit Function

The glmfit function in MATLAB is a general-purpose linear regression solver that can be used to fit a wide range of models, including logistic regression. When used for logistic regression, glmfit estimates the parameters of the model using maximum likelihood estimation.

To understand how glmfit implements logistic regression, let’s consider an example:

[b_lr, dev, stats] = glmfit(x', y, 'binomial', 'link', 'logit');

In this code snippet, we are fitting a binary logistic regression model to the data in x' and y. The 'binomial' option specifies that we want to fit a binomial logistic regression model, while the 'link' option specifies that we want to use the logit link function. Finally, the 'logit' option specifies that we want to estimate the parameters using maximum likelihood estimation.

Comparison with Other Programming Languages

As mentioned in the original question, the author is interested in understanding why MATLAB’s glmfit implementation produces different results compared to equivalent implementations in Python and R.

To investigate this further, let’s consider how these programming languages implement logistic regression. In Python, we can use the statsmodels library to fit a binomial logistic regression model:

import statsmodels.api as sm

glm_logit = sm.GLM(yvec.T, Xmat, family=sm.families.Binomial()).fit()

In R, we can use the glm function with the family=binomial(logit) option to fit a binomial logistic regression model:

glm.out = glm(Data ~ ONI + Percentiles, family=binomial(logit), data=df)

As we can see, these implementations are similar, but not identical. This highlights the importance of understanding how each programming language’s implementation differs.

Treatment of Singular or Almost Singular Cases

One potential difference between MATLAB’s glmfit and equivalent implementations in Python and R is how they treat singular or almost singular cases.

In matrix algebra, a matrix $X$ is said to be singular if its determinant is zero. When a matrix is singular, the system of equations $X\beta = y$ may not have a unique solution.

To address this issue, some programming languages use regularization techniques, such as LASSO or ridge regression, to penalize large coefficients and promote sparsity in the model. Others may use iterative methods, such as Newton’s method, to find a solution that converges to the maximum likelihood estimate.

Numerical Precision Problems

Another potential difference between MATLAB’s glmfit and equivalent implementations in Python and R is how they handle numerical precision problems.

In high-dimensional data, the number of parameters in the model can be large, which can lead to numerical instability. This can manifest as issues with convergence or optimization, or even crashes due to overflow.

To address this issue, some programming languages use techniques such as gradient clipping or regularization to prevent overfitting and promote stability in the model.

Conclusion

In conclusion, MATLAB’s glmfit implementation for logistic regression is based on maximum likelihood estimation. While it produces different results compared to equivalent implementations in Python and R, these differences can often be attributed to differences in numerical precision problems, treatment of singular or almost singular cases, or coding mistakes.

To replicate the results produced by MATLAB’s glmfit, we should carefully examine the implementation details of each programming language, including the choice of optimization algorithm, regularization techniques, and handling of numerical precision problems.

Example Code

Here is an example code snippet that demonstrates how to fit a binomial logistic regression model using MATLAB’s glmfit function:

function b = glmfit(x, y, family, link, sol)
  % Define the design matrix X
  X = [ones(size(x, 1), 1); x];
  
  % Define the response vector y
  y = y';
  
  % Define the model parameters
  a = zeros(1, size(y, 2));
  b = zeros(1, size(X, 2));
  
  % Solve for the model parameters using MLE
  [a, b] = glmfit(x', y, family, link, sol);
  
  % Plot the predicted probabilities
  z = X * a + b;
  plot(z);
end

This code snippet defines a function glmfit that takes in the design matrix $X$, response vector $y$, model parameters $a$ and $b$, and optimization solver sol. It then solves for the model parameters using maximum likelihood estimation and plots the predicted probabilities.

We can use this function to fit a binomial logistic regression model as follows:

x = [ones(10, 1); 2; 3; ...];
y = [0; 1; 1; ...];
glmfit(x, y, 'binomial', 'logit');

This code snippet defines the design matrix $X$ and response vector $y$, and then calls the glmfit function to fit a binomial logistic regression model. The resulting model parameters are plotted as a histogram.

Last modified on 2024-01-11