Log-Concave Distribution Estimation

Introduction

Let \(n = 1\) and suppose \(x_i\) are i.i.d. samples from a log-concave discrete distribution on \(\{0,\ldots,K\}\) for some \(K \in {\mathbf Z}_+\). Define \(p_k := {\mathbf {Prob}}(X = k)\) to be the probability mass function. One method for estimating \(\{p_0,\ldots,p_K\}\) is to maximize the log-likelihood function subject to a log-concavity constraint , i.e.,

\[ \begin{array}{ll} \underset{p}{\mbox{maximize}} & \sum_{k=0}^K M_k\log p_k \\ \mbox{subject to} & p \geq 0, \quad \sum_{k=0}^K p_k = 1, \\ & p_k \geq \sqrt{p_{k-1}p_{k+1}}, \quad k = 1,\ldots,K-1, \end{array} \]

where \(p \in {\mathbf R}^{K+1}\) is our variable of interest and \(M_k\) represents the number of observations equal to \(k\), so that \(\sum_{k=0}^K M_k = m\). The problem as posed above is not convex. However, we can transform it into a convex optimization problem by defining new variables \(u_k = \log p_k\) and relaxing the equality constraint to \(\sum_{k=0}^K p_k \leq 1\), since the latter always holds tightly at an optimal solution. The result is

\[ \begin{array}{ll} \underset{u}{\mbox{maximize}} & \sum_{k=0}^K M_k u_k \\ \mbox{subject to} & \sum_{k=0}^K e^{u_k} \leq 1, \\ & u_k - u_{k-1} \geq u_{k+1} - u_k, \quad k = 1,\ldots,K-1. \end{array} \]

Example

We draw \(m = 25\) observations from a log-concave distribution on \(\{0,\ldots,100\}\). We then estimate the probability mass function using the above method and compare it with the empirical distribution.

suppressMessages(suppressWarnings(library(CVXR)))
set.seed(1)
## Calculate a piecewise linear function
pwl_fun <- function(x, knots) {
    n <- nrow(knots)
    x0 <- sort(knots$x, decreasing = FALSE)
    y0 <- knots$y[order(knots$x, decreasing = FALSE)]
    slope <- diff(y0)/diff(x0)
    
    sapply(x, function(xs) {
        if(xs <= x0[1])
            y0[1] + slope[1]*(xs -x0[1])
        else if(xs >= x0[n])
            y0[n] + slope[n-1]*(xs - x0[n])
        else {
            idx <- which(xs <= x0)[1]
            y0[idx-1] + slope[idx-1]*(xs - x0[idx-1])
        }
    })
}
## Problem data
m <- 25
xrange <- 0:100
knots <- data.frame(x = c(0, 25, 65, 100), y = c(10, 30, 40, 15))
xprobs <- pwl_fun(xrange, knots)/15
xprobs <- exp(xprobs)/sum(exp(xprobs))
x <- sample(xrange, size = m, replace = TRUE, prob = xprobs)

K <- max(xrange)
counts <- hist(x, breaks = -1:K, right = TRUE, include.lowest = FALSE,
               plot = FALSE)$counts
library(ggplot2)
ggplot() +
    geom_histogram(mapping = aes(x = x), breaks = -1:K, color = "blue", fill = "orange")

We now solve problem with log-concave constraint.

u <- Variable(K+1)
obj <- t(counts) %*% u
constraints <- list(sum(exp(u)) <= 1, diff(u[1:K]) >= diff(u[2:(K+1)]))
prob <- Problem(Maximize(obj), constraints)
result <- solve(prob)
pmf <- result$getValue(exp(u))

The above lines transform the variables \(u_k\) to \(e^{u_k}\) before calculating their resulting values. This is possible because exp is a member of the CVXR library of atoms, so it can operate directly on a Variable object such as u.

Below are the comparison plots of pmf and cdf.

library(ggplot2)
library(tidyr)
dens <- density(x, bw = "sj")
d <- data.frame(x = xrange, True = xprobs, Optimal = pmf,
                Empirical = approx(x = dens$x, y = dens$y, xout = xrange)$y)
plot.data <- gather(data = d, key = "Type", value = "Estimate", True, Empirical, Optimal,
                    factor_key = TRUE)
ggplot(plot.data) +
    geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
    theme(legend.position = "top")

d <- data.frame(x = xrange, True = cumsum(xprobs),
                Empirical = cumsum(counts) / sum(counts),
                Optimal = cumsum(pmf))
plot.data <- gather(data = d, key = "Type", value = "Estimate", True, Empirical, Optimal,
                    factor_key = TRUE)
ggplot(plot.data) +
    geom_line(mapping = aes(x = x, y = Estimate, color = Type)) +
    theme(legend.position = "top")

From the figures we see that the estimated curve is much closer to the true distribution, exhibiting a similar shape and number of peaks. In contrast, the empirical probability mass function oscillates, failing to be log-concave on parts of its domain. These differences are reflected in the cumulative distribution functions as well.

Session Info

sessionInfo()
## R version 3.4.3 (2017-11-30)
## Platform: x86_64-apple-darwin15.6.0 (64-bit)
## Running under: macOS High Sierra 10.13.3
## 
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] methods   stats     graphics  grDevices datasets  utils     base     
## 
## other attached packages:
## [1] tidyr_0.8.0   ggplot2_2.2.1 CVXR_0.95    
## 
## loaded via a namespace (and not attached):
##  [1] gmp_0.5-13.1      Rcpp_0.12.15      pillar_1.1.0     
##  [4] compiler_3.4.3    plyr_1.8.4        R.methodsS3_1.7.1
##  [7] R.utils_2.6.0     tools_3.4.3       digest_0.6.15    
## [10] bit_1.1-12        evaluate_0.10.1   tibble_1.4.2     
## [13] gtable_0.2.0      lattice_0.20-35   rlang_0.2.0      
## [16] Matrix_1.2-12     yaml_2.1.16       blogdown_0.5.4   
## [19] xfun_0.1          Rmpfr_0.7-0       ECOSolveR_0.4    
## [22] stringr_1.3.0     knitr_1.20        tidyselect_0.2.3 
## [25] rprojroot_1.3-2   bit64_0.9-7       grid_3.4.3       
## [28] glue_1.2.0        R6_2.2.2          rmarkdown_1.8.10 
## [31] bookdown_0.7      purrr_0.2.4       magrittr_1.5     
## [34] backports_1.1.2   scales_0.5.0      htmltools_0.3.6  
## [37] scs_1.1-1         colorspace_1.3-2  labeling_0.3     
## [40] stringi_1.1.6     lazyeval_0.2.1    munsell_0.4.3    
## [43] R.oo_1.21.0

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