Implementing an EM Algorithm for Probit Regressions
Jonathan Olmsted — written Oct 1, 2014 — source
Users new to the Rcpp family of functionality are often impressed with the performance gains that can be realized, but struggle to see how to approach their own computational problems. Many of the most impressive performance gains are demonstrated with seemingly advanced statistical methods, advanced C++–related constructs, or both. Even when users are able to understand how various demonstrated features operate in isolation, examples may implement too many at once to seem accessible.
The point of this Gallery article is to offer an example application that performs well (thanks to the Rcpp family) but has reduced statistical and programming overhead for some users. In addition, rather than simply presenting the final product, the development process is explicitly documented and narrated.
Motivating Example: Probit Regression
As an example, we will consider estimating the parameters the standard Probit regression model given by
\[y_i^* = x_i'\beta + \epsilon_i\]where \(x_i\) and \(\beta\) are length \(K\) vectors and the presence of an “intercept” term is absorbed into \(x_i\) if desired.
The analyst only has access to a censored version of \(y_i^*\), namely \(y_i\) where the subscript \(i\) denotes the \(i\) th observation.
As is common, the censoring is assumed to generate \(y_i = 1\) if \(y_i^* \geq 0\) and \(y_i = 0\) otherwise. When we assume \(\epsilon_i \sim N(0, 1)\), the problem is just the Probit regression model loved by all.
To make this concrete, consider a model of voter turnout using the dataset provided by the Zelig R package.
library("Zelig")
data("turnout")
head(turnout)race age educate income vote 1 white 60 14 3.3458 1 2 white 51 10 1.8561 0 3 white 24 12 0.6304 0 4 white 38 8 3.4183 1 5 white 25 12 2.7852 1 6 white 67 12 2.3866 1
dim(turnout)[1] 2000 5
Our goal will be to estimate the parameters associated with the variables income, educate, and age. Since there is nothing special about this dataset, standard methods work perfectly well.
fit0 <- glm(vote ~ income + educate + age,
data = turnout,
family = binomial(link = "probit")
)
fit0
Call: glm(formula = vote ~ income + educate + age, family = binomial(link = "probit"),
data = turnout)
Coefficients:
(Intercept) income educate age
-1.6824 0.0994 0.1067 0.0169
Degrees of Freedom: 1999 Total (i.e. Null); 1996 Residual
Null Deviance: 2270
Residual Deviance: 2030 AIC: 2040
Using fit0 as our baseline, the question is how can we recover these estimates
with an Rcpp-based approach. One answer is implement the EM-algorithm in C++
snippets that can be processed into R-level functions; that’s what we will
do. (Think of this as a Probit regression analog to
the linear regression example — but with fewer features.)
EM Algorithm: Intuition
For those unfamiliar with the EM algorithm, consider the Wikipedia article and a denser set of Swarthmore lecture notes.
The intuition behind this approach begins by noticing that if mother nature revealed the \(y_i^*\) values, we would simply have a linear regression problem and focus on
\[\widehat{\beta} = (X'X)^{-1} X'y^*\]where the meaning of the matrix notation is assumed.
Because mother nature is not so kind, we have to impute the \(y_i^*\) values. For a given guess of \(\widehat{\beta}\), due to our distributional assumptions about \(\epsilon_i\) we know that
\[E\left[y_i^* \big\vert y_i=1, \widehat{\beta}\right] = \mu_i + \frac{\phi(-\mu_i)}{1 - \Phi(-\mu_I)}\]and
\[E\left[y_i^* \big\vert y_i=0, \widehat{\beta}\right] = \mu_i - \frac{\phi(-\mu_i)}{\Phi(-\mu_I)}\]where \(\mu_i = x_i'\hat{\beta}\).
By iterating through these two steps we can eventually recover the desired parameter estimates:
- impute/augment \(y_i^*\) values
- estimate \(\widehat{\beta}\) given the data augmentation
Our Implementations
To demonstrate implementation of the EM algorithm for a Probit regression model using Rcpp-provided functionality we consider a series of steps.
These are:
- Attempt 1: Main Structure
- Attempt 2: EM with Mistaken Augmentation
- Attempt 3: EM with Correct Augmentation
- Attempt 4: EM with Correct Augmentation in Parallel
These steps are not chosen because each produces useful output (from the perspective of parameters estimation), but because they mirror milestones in a development process that benefits new users: only small changes are made at a time.
To begin, we prepare our R-level data for passage to our eventual C++-based functions.
mY <- matrix(turnout$vote)
mX <- cbind(1,
turnout$income,
turnout$educate,
turnout$age
)Attempt 1: Main Structure
The first milestone will be to mock up a function em1 that is exported to
create an R-level function of the same name. The key features here are that we
have defined the function to
- accept arguments corresponding to likely inputs
- create containers for the to-be-computed values,
- outline the main loop of the code for the EM iterations, and
- return various values of interest in a list
Users new to the Rcpp process will benefit from return List objects in the
beginning. They allow you rapidly return new and different values to the R-level
for inspection.
# include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp ;
// [[Rcpp::export()]]
List em1 (const arma::mat y,
const arma::mat X,
const int maxit = 10
) {
// inputs
const int N = y.n_rows ;
const int K = X.n_cols ;
// containers
arma::mat beta(K, 1) ;
beta.fill(0.0) ; // initialize betas to 1
arma::mat eystar(N, 1) ;
eystar.fill(0) ;
// algorithm
for (int it = 0 ; it < maxit ; it++) { // EM iterations
// NEXT STEP: implement algorithm
}
// returns
List ret ;
ret["N"] = N ;
ret["K"] = K ;
ret["beta"] = beta ;
ret["eystar"] = eystar ;
return(ret) ;
}We know that this code does not produce estimates of anything. Indeed, that is
by design. Neither the beta nor eystar elements of the returned list are
ever updated after they are initialized to 0.
However, we can see that much of the administrative work for a working implementation is complete.
fit1 <- em1(y = mY,
X = mX,
maxit = 20
)
fit1$beta
[,1]
[1,] 0
[2,] 0
[3,] 0
[4,] 0
head(fit1$eystar)
[,1]
[1,] 0
[2,] 0
[3,] 0
[4,] 0
[5,] 0
[6,] 0
Having verified that input data structures and output data structures are “working” as expected, we turn to updating the \(y_i^*\) values.
Attempt 2: EM with Mistaken Augmentation
Updates to the \(y_i^*\) values are different depending on whether \(y_i=1\) or \(y_i=0\). Rather than worrying about correctly imputing the unobserved propensities, we will use dummy values of 1 and -1 as placeholders. Instead, the focus is on building on the necessary conditional structure of the code and looping through the update step for every observation.
Additionally, at the end of each imputation step (the E in EM) we update the \(\beta\) estimate with the least squares estimate (the M in EM).
# include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp ;
// [[Rcpp::export()]]
List em2 (const arma::mat y,
const arma::mat X,
const int maxit = 10
) {
// inputs
const int N = y.n_rows ;
const int K = X.n_cols ;
// containers
arma::mat beta(K, 1) ;
beta.fill(0.0) ; // initialize betas to 0
arma::mat eystar(N, 1) ;
eystar.fill(0) ;
// algorithm
for (int it = 0 ; it < maxit ; it++) {
arma::mat mu = X * beta ;
// augmentation step
for (int n = 0 ; n < N ; n++) {
if (y(n, 0) == 1) { // y = 1
// NEXT STEP: fix augmentation
eystar(n, 0) = 1 ;
}
if (y(n, 0) == 0) { // y = 0
// NEXT STEP: fix augmentation
eystar(n, 0) = -1 ;
}
}
// maximization step
beta = (X.t() * X).i() * X.t() * eystar ;
}
// returns
List ret ;
ret["N"] = N ;
ret["K"] = K ;
ret["beta"] = beta ;
ret["eystar"] = eystar ;
return(ret) ;
}This code, like that in Attempt 1, is syntactically fine. But, as we know, the
update step is very wrong. However, we can see that the updates are happening as
we’d expect and we see non-zero returns for the beta element and the eystar
element.
fit2 <- em2(y = mY,
X = mX,
maxit = 20
)
fit2$beta
[,1]
[1,] -0.816273
[2,] 0.046065
[3,] 0.059481
[4,] 0.009085
head(fit2$eystar)
[,1]
[1,] 1
[2,] -1
[3,] -1
[4,] 1
[5,] 1
[6,] 1
Attempt 3: EM with Correct Augmentation
With the final logical structure of the code built out, we will now correct the
data augmentation. Specifically, we replace the assignment of 1 and -1 with the
expectation of the unobservable values \(y_i^*\). Rather than muddy our EM
function (em3()) with further arithmetic, we sample call the C++ level
functions f() and g() which were included prior to our definition of
em3().
But, since these are just utility functions needed internally by em3(), they
are not tagged to be exported (via // [[Rcpp::export()]]) to the R level.
As it stands, this is a correct implementation (although there is room for improvement).
# include <RcppArmadillo.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp ;
double f (double mu) {
double val = ((R::dnorm(-mu, 0, 1, false)) /
(1 - R::pnorm(-mu, 0, 1, true, false))
) ;
return(val) ;
}
double g (double mu) {
double val = ((R::dnorm(-mu, 0, 1, false)) /
(R::pnorm(-mu, 0, 1, true, false))
) ;
return(val) ;
}
// [[Rcpp::export()]]
List em3 (const arma::mat y,
const arma::mat X,
const int maxit = 10
) {
// inputs
const int N = y.n_rows ;
const int K = X.n_cols ;
// containers
arma::mat beta(K, 1) ;
beta.fill(0.0) ; // initialize betas to 0
arma::mat eystar(N, 1) ;
eystar.fill(0) ;
// algorithm
for (int it = 0 ; it < maxit ; it++) {
arma::mat mu = X * beta ;
// augmentation step
// NEXT STEP: parallelize augmentation step
for (int n = 0 ; n < N ; n++) {
if (y(n, 0) == 1) { // y = 1
eystar(n, 0) = mu(n, 0) + f(mu(n, 0)) ;
}
if (y(n, 0) == 0) { // y = 0
eystar(n, 0) = mu(n, 0) - g(mu(n, 0)) ;
}
}
// maximization step
beta = (X.t() * X).i() * X.t() * eystar ;
}
// returns
List ret ;
ret["N"] = N ;
ret["K"] = K ;
ret["beta"] = beta ;
ret["eystar"] = eystar ;
return(ret) ;
}fit3 <- em3(y = mY,
X = mX,
maxit = 100
)head(fit3$eystar)
[,1]
[1,] 1.3910
[2,] -0.6599
[3,] -0.7743
[4,] 0.8563
[5,] 0.9160
[6,] 1.2677
Second, notice that this output is identical to the parameter estimates (the
object fit0) from our R level call to the glm() function.
fit3$beta
[,1]
[1,] -1.68241
[2,] 0.09936
[3,] 0.10667
[4,] 0.01692
fit0
Call: glm(formula = vote ~ income + educate + age, family = binomial(link = "probit"),
data = turnout)
Coefficients:
(Intercept) income educate age
-1.6824 0.0994 0.1067 0.0169
Degrees of Freedom: 1999 Total (i.e. Null); 1996 Residual
Null Deviance: 2270
Residual Deviance: 2030 AIC: 2040
Attempt 4: EM with Correct Augmentation in Parallel
With a functional implementation complete as em3(), we know turn to the second
order concern: performance. The time required to evaluate our function can be
reduced from the perspective of a user sitting at a computer with idle cores.
Although the small size of these data don’t necessitate parallelization, the E step is a natural candidate for being parallelized. Here, the parallelization relies on OpenMP. See here for other examples of combining Rcpp and OpenMP or here for a different approach.
Sys.setenv("PKG_CXXFLAGS" = "-fopenmp")
Sys.setenv("PKG_LIBS" = "-fopenmp")Aside from some additional compiler flags, the changes to our new implementation
in em4() are minimal. They are:
- include the additional header
- mark the
forloop for parallelization with a#pragma
# include <RcppArmadillo.h>
# include <omp.h>
// [[Rcpp::depends(RcppArmadillo)]]
using namespace Rcpp ;
double f (double mu) {
double val = ((R::dnorm(-mu, 0, 1, false)) /
(1 - R::pnorm(-mu, 0, 1, true, false))
) ;
return(val) ;
}
double g (double mu) {
double val = ((R::dnorm(-mu, 0, 1, false)) /
(R::pnorm(-mu, 0, 1, true, false))
) ;
return(val) ;
}
// [[Rcpp::export()]]
List em4 (const arma::mat y,
const arma::mat X,
const int maxit = 10,
const int nthr = 1
) {
// inputs
const int N = y.n_rows ;
const int K = X.n_cols ;
omp_set_num_threads(nthr) ;
// containers
arma::mat beta(K, 1) ;
beta.fill(0.0) ; // initialize betas to 0
arma::mat eystar(N, 1) ;
eystar.fill(0) ;
// algorithm
for (int it = 0 ; it < maxit ; it++) {
arma::mat mu = X * beta ;
// augmentation step
#pragma omp parallel for
for (int n = 0 ; n < N ; n++) {
if (y(n, 0) == 1) { // y = 1
eystar(n, 0) = mu(n, 0) + f(mu(n, 0)) ;
}
if (y(n, 0) == 0) { // y = 0
eystar(n, 0) = mu(n, 0) - g(mu(n, 0)) ;
}
}
// maximization step
beta = (X.t() * X).i() * X.t() * eystar ;
}
// returns
List ret ;
ret["N"] = N ;
ret["K"] = K ;
ret["beta"] = beta ;
ret["eystar"] = eystar ;
return(ret) ;
}This change should not (and does not) result in any change to the calculations being done. However, if our algorithm involved random number generation, great care would need to be taken to ensure our results were reproducible.
fit4 <- em4(y = mY,
X = mX,
maxit = 100
)
identical(fit4$beta, fit3$beta)[1] TRUE
Finally, we can confirm that our parallelization was “successful”. Again, because there is really no need to parallelize this code, performance gains are modest. But, that it indeed runs faster is clear.
library("microbenchmark")
microbenchmark(seq = (em3(y = mY,
X = mX,
maxit = 100
)
),
par = (em4(y = mY,
X = mX,
maxit = 100,
nthr = 4
)
),
times = 20
)Unit: milliseconds expr min lq mean median uq max neval cld seq 32.94 33.01 33.04 33.03 33.07 33.25 20 b par 11.16 11.20 11.35 11.26 11.29 13.16 20 a
Wrap-Up
The purpose of this lengthy gallery post is neither to demonstrate new functionality nor the computational feasibility of cutting-edge algorithms. Rather, it is to explicitly walk through a development process similar that which new users can benefit from using while using a very common statistical problem, Probit regression.
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