Understanding Linear Regression with RcppArmadillo

The Mysterious Case of Different Standard Errors

In this article, we’ll delve into the world of linear regression using the popular RcppArmadillo library. Specifically, we’ll explore a puzzling issue where two seemingly identical approaches yield different standard errors.

The problem arises from a misunderstanding in how to handle added columns during matrix operations in RcppArmadillo. In this response, we’ll break down the correct approach, explain the intricacies of linear regression, and provide examples to illustrate the key concepts.

Setting Up the Environment

To tackle this issue, you’ll need to have RcppArmadillo installed on your system. If you’re new to RcppArmadillo, we recommend checking out the official documentation for a comprehensive introduction to the library.

#include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]

using namespace Rcpp;

A Closer Look at Linear Regression

Linear regression is a fundamental concept in statistics and machine learning. The goal of linear regression is to find the best-fitting line that predicts a continuous output variable based on one or more predictor variables.

In this case, we’ll focus on simple linear regression with one intercept term (a constant) and one predictor term (the added column).

The linear regression equation takes the form:

y = β0 + β1 * x + ε

where:

• y is the response variable
• x is the predictor variable
• β0 is the intercept or constant term
• β1 is the slope coefficient of the predictor variable
• ε is the random error term

Creating the Linear Regression Model in RcppArmadillo

Now, let’s create a simple linear regression model using RcppArmadillo. We’ll define two functions: mylm1 and mylm2. Both functions take the same input data (response variable yr and predictor variable Xr) but differ in how they handle added columns.

// [[Rcpp::export]]
List mylm1(NumericVector yr, NumericMatrix Xr) {
    int n = Xr.nrow(), k = Xr.ncol();
    arma::mat X(Xr.begin(), n, k, true);
    Rcout << X << std::endl;
    arma::mat allOne(n, 1, arma::fill::ones);

    // Adding the constant column to X
    X.insert_cols(0, allOne);
    Rcout << X << std::endl;

    // Rest of the code remains the same
}

// [[Rcpp::export]]
List mylm2(NumericVector yr, NumericMatrix Xr) {
    int n = Xr.nrow(), k = Xr.ncol();
    arma::mat X(Xr.begin(), n, k, false);
    Rcout << X << std::endl;

    // Note the difference in handling added columns
    X = X + arma::colvec(arma::fill::ones(n, 1));

    // Rest of the code remains the same
}

Understanding the Issue

The issue arises from how we handle the added column. In mylm2, we directly add a column of ones to the original matrix X. However, in mylm1, we use the insert_cols method to insert the constant column into the existing matrix X.

// [[Rcpp::export]]
List mylm1(NumericVector yr, NumericMatrix Xr) {
    int n = Xr.nrow(), k = Xr.ncol();
    arma::mat X(Xr.begin(), n, k, true);
    Rcout << X << std::endl;
    arma::mat allOne(n, 1, arma::fill::ones);

    // Inserting the constant column into X
    X.insert_cols(0, allOne);
    Rcout << X << std::endl;

    // Rest of the code remains the same
}

The difference lies in how we handle the added column:

• In mylm2, we add a new column to the original matrix X.
• In mylm1, we insert the constant column into the existing matrix X.

This subtle distinction affects the standard errors, which are calculated based on the transformed data.

The Role of Standard Errors in Linear Regression

Standard errors play a crucial role in linear regression as they provide an estimate of the uncertainty associated with the estimated coefficients. In this case, we’re interested in understanding why the two approaches yield different standard errors.

To illustrate the impact of added columns on standard errors, let’s examine how the arma::inv function is used to compute the inverse of the design matrix X.

// Compute the inverse of X
arma::mat X_inv = arma::inv(arma::trans(X)*X);

// Note that X_inv is different between mylm1 and mylm2

The difference in the computed inverse matrices affects the calculation of standard errors, leading to the observed discrepancy.

Conclusion

In this article, we explored a puzzling issue related to linear regression using RcppArmadillo. By understanding how added columns are handled during matrix operations, we can identify and resolve such issues.

To summarize:

• When adding a constant column to the design matrix X, use the insert_cols method to ensure accurate calculations.
• When adding a new column to X, consider the implications on standard errors and be mindful of how arma::inv is used to compute the inverse.
• Familiarize yourself with the fine details of linear regression and RcppArmadillo to tackle complex problems like this one.

By following these guidelines and carefully considering your code, you can avoid similar issues in the future.

Last modified on 2024-03-26