Implementing an EM Algorithm for Probit Regressions

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

where and are length vectors and the presence of an “intercept” term is absorbed into if desired.

The analyst only has access to a censored version of , namely where the subscript denotes the th observation.

As is common, the censoring is assumed to generate if and otherwise. When we assume , 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 values, we would simply have a linear regression problem and focus on

where the meaning of the matrix notation is assumed.

Because mother nature is not so kind, we have to impute the values. For a given guess of , due to our distributional assumptions about we know that

and

where .

By iterating through these two steps we can eventually recover the desired parameter estimates:

1. impute/augment values
2. estimate 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:

1. Attempt 1: Main Structure
2. Attempt 2: EM with Mistaken Augmentation
3. Attempt 3: EM with Correct Augmentation
4. 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 values.

Attempt 2: EM with Mistaken Augmentation

Updates to the values are different depending on whether or . 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 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 . 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 for loop 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.