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I wanted to use IRLS to fit a simple logistic regression. The following code does so for all columns of X simultaneously. That is, it fits y ~ mu + b Xj for j= 1…p.
I start by writing a function to fit a single vector x. The code is based on the IRLS algorithm for logistic regression, as described in the book “Elements of Statistical Learning” by Hastie, Tibshirani and Friedman (2009), section 4.4.1, written in a way that it makes it easy to vectorize/repeat over the columns of a matrix X via matrix multiplications. (See below)
logistic_IRLS_simple <- function(x, y, max_iter = 100, tolerance = 1e-6, lambda=0) {
# Initialize coefficients
mu <- 0
beta <- 0
converged <- FALSE
for (iter in 1:max_iter) {
eta <- mu + x * beta # Linear predictor
pi <- exp(eta) / (1 + exp(eta)) # Predicted probabilities
# Weights for IRLS
w <- pi * (1 - pi)
# Working response variable
z <- eta + (y - pi) / (pi * (1 - pi))
# Weighted least squares update
#These are the elements of the X'X matrix (a b) (c d)
a = sum(w)
b = sum(w*x)
c = sum(w*x)
d = sum(w*x^2) + lambda
wz = sum(w*z)
wxz = sum(w*x*z)
new_mu <- (d*wz - b*wxz) / (a*d - b*c)
new_beta <- (-c*wz + a*wxz) / (a*d - b*c)
# Check for convergence
if (all(abs(new_beta - beta) < tolerance)) {
converged = TRUE
break
}
beta <- new_beta
mu <- new_mu
}
# Return fitted coefficients
return(list(mu = mu, beta = beta, converged = converged, iter=iter))
}Now I want to test the function with some simulated data. Simulate a simple logistic regression:
set.seed(1)
n <- 10000
x <- rnorm(n)
eta <- 5 + 2*x
pi <- exp(eta) / (1 + exp(eta))
y <- rbinom(n, 1, pi)
logistic_IRLS_simple(x, y)$mu
[1] 5.043687
$beta
[1] 2.046827
$converged
[1] TRUE
$iter
[1] 9glm(y~x, family = binomial)
Call: glm(formula = y ~ x, family = binomial)
Coefficients:
(Intercept) x
5.044 2.047
Degrees of Freedom: 9999 Total (i.e. Null); 9998 Residual
Null Deviance: 2994
Residual Deviance: 1998 AIC: 2002And compare with glmnet. Note that glmnet requires two variables so I add an intercept. One thing I don’t understand: I tried setting “intercept=FALSE” but and not penalizing the intercept but it produced different results, which was unexpected.
library(glmnet)Loading required package: MatrixLoaded glmnet 4.1-8fit = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=1/n)
fit$beta2 x 1 sparse Matrix of class "dgCMatrix"
s0
.
x 2.031983logistic_IRLS_simple(x, y, lambda=1)$mu
[1] 5.024606
$beta
[1] 2.033247
$converged
[1] TRUE
$iter
[1] 9fit = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=10/n)
fit$beta2 x 1 sparse Matrix of class "dgCMatrix"
s0
.
x 1.920453logistic_IRLS_simple(x, y, lambda=10)$mu
[1] 4.872961
$beta
[1] 1.923797
$converged
[1] TRUE
$iter
[1] 9fit2 = glmnet(cbind(rep(1,n),x), y, family="binomial",alpha=0, lambda=10/n, penalty.factor = c(0,1), intercept = FALSE)
coef(fit2)3 x 1 sparse Matrix of class "dgCMatrix"
s0
(Intercept) .
.
x 0.2065842coef(fit)3 x 1 sparse Matrix of class "dgCMatrix"
s0
(Intercept) 4.868390
.
x 1.920453I now write a function that repeats the above for all columns of X simultaneously via vector/matrix operations.
logistic_IRLS_simple_matrix <- function(X, y, max_iter = 100, tolerance = 1e-6, lambda = 0) {
# Initialize coefficients
p <- ncol(X)
mu <- rep(0,p)
beta <- rep(0,p)
converged <- FALSE
for (iter in 1:max_iter) {
eta <- t(mu + t(X) * beta) # Linear predictor
pi <- exp(eta) / (1 + exp(eta)) # Predicted probabilities
# Weights for IRLS
w <- pi * (1 - pi)
# Working response variable
z <- eta + (y - pi) / (pi * (1 - pi))
# Weighted least squares update
#These are the elements of the X'X matrix (a b) (c d)
a = colSums(w)
b = colSums(w*X)
c = b
d = colSums(w*X^2) + lambda
wz = colSums(w*z)
wxz = colSums(w*X*z)
new_mu <- (d*wz - b*wxz) / (a*d - b*c)
new_beta <- (-c*wz + a*wxz) / (a*d - b*c)
# Check for convergence
if (all(abs(new_beta - beta) < tolerance)) {
converged = TRUE
break
}
beta <- new_beta
mu <- new_mu
}
# Return fitted coefficients
return(list(mu = mu, beta = beta, converged = converged, iter=iter))
}Now I test the function with some simulated data. It seems to work.
set.seed(1)
n <- 10000
X <- cbind(rnorm(n), rnorm(n))
eta <- 10 + 2*X[,1] + 3*X[,2]
pi <- exp(eta) / (1 + exp(eta))
y <- rbinom(n, 1, pi)
glm(y~X[,1], family = binomial)
Call: glm(formula = y ~ X[, 1], family = binomial)
Coefficients:
(Intercept) X[, 1]
6.084 1.427
Degrees of Freedom: 9999 Total (i.e. Null); 9998 Residual
Null Deviance: 764.1
Residual Deviance: 639.4 AIC: 643.4glm(y~X[,2], family = binomial)
Call: glm(formula = y ~ X[, 2], family = binomial)
Coefficients:
(Intercept) X[, 2]
7.214 2.211
Degrees of Freedom: 9999 Total (i.e. Null); 9998 Residual
Null Deviance: 764.1
Residual Deviance: 521.9 AIC: 525.9logistic_IRLS_simple_matrix(X, y)$mu
[1] 6.083537 7.213586
$beta
[1] 1.427305 2.211099
$converged
[1] TRUE
$iter
[1] 10
sessionInfo()R version 4.2.1 (2022-06-23)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Big Sur ... 10.16
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] glmnet_4.1-8 Matrix_1.6-4
loaded via a namespace (and not attached):
[1] Rcpp_1.0.12 pillar_1.9.0 compiler_4.2.1 bslib_0.6.1
[5] later_1.3.2 jquerylib_0.1.4 git2r_0.33.0 workflowr_1.7.1
[9] iterators_1.0.14 tools_4.2.1 digest_0.6.33 lattice_0.22-5
[13] jsonlite_1.8.8 evaluate_0.23 lifecycle_1.0.4 tibble_3.2.1
[17] pkgconfig_2.0.3 rlang_1.1.2 foreach_1.5.2 cli_3.6.2
[21] rstudioapi_0.15.0 yaml_2.3.8 xfun_0.41 fastmap_1.1.1
[25] stringr_1.5.1 knitr_1.45 fs_1.6.3 vctrs_0.6.5
[29] sass_0.4.8 rprojroot_2.0.4 grid_4.2.1 glue_1.6.2
[33] R6_2.5.1 fansi_1.0.6 survival_3.5-7 rmarkdown_2.25
[37] magrittr_2.0.3 whisker_0.4.1 splines_4.2.1 codetools_0.2-19
[41] promises_1.2.1 htmltools_0.5.7 shape_1.4.6 httpuv_1.6.13
[45] utf8_1.2.4 stringi_1.8.3 cachem_1.0.8