Density estimation for data with bounded support

Univariate case with lower bound

x <- rchisq(200, 3)
xgrid <- seq(-2, max(x), length=1000)
f <- dchisq(xgrid, 3)  # true density
dens <- densityMclustBounded(x, lbound = 0)
summary(dens, parameters = TRUE)
## ── Density estimation for bounded data via GMMs ─────────── 
##            
## Boundaries:   x
##       lower   0
##       upper Inf
## 
## Model E (univariate, equal variance) model with 1 component
## on the transformation scale:
## 
##  log-likelihood   n df       BIC       ICL
##       -390.0517 200  3 -795.9983 -795.9983
## 
##                                     x
## Range-power transformation: 0.3715163
## 
## Mixing probabilities:
## 1 
## 1 
## 
## Means:
##         1 
## 0.9191207 
## 
## Variances:
##        1 
## 1.309037
plot(dens, what = "density")
lines(xgrid, f, lty = 2)

plot(dens, what = "density", data = x, breaks = 15)

Univariate case with lower & upper bounds

x <- rbeta(200, 5, 1.5)
xgrid <- seq(-0.1, 1.1, length=1000)
f <- dbeta(xgrid, 5, 1.5)  # true density
dens <- densityMclustBounded(x, lbound = 0, ubound = 1)
summary(dens, parameters = TRUE)
## ── Density estimation for bounded data via GMMs ─────────── 
##            
## Boundaries: x
##       lower 0
##       upper 1
## 
## Model E (univariate, equal variance) model with 1 component
## on the transformation scale:
## 
##  log-likelihood   n df      BIC      ICL
##        120.4011 200  3 224.9072 224.9072
## 
##                                      x
## Range-power transformation: -0.2200992
## 
## Mixing probabilities:
## 1 
## 1 
## 
## Means:
##        1 
## 1.152107 
## 
## Variances:
##         1 
## 0.5212129
plot(dens, what = "density")

plot(dens, what = "density", data = x, breaks = 11)

Bivariate case with lower bounds

x1 <- rchisq(200, 3)
x2 <- 0.5*x1 + sqrt(1-0.5^2)*rchisq(200, 5)
x <- cbind(x1, x2)
dens <- densityMclustBounded(x, lbound = c(0,0))
summary(dens, parameters = TRUE)
## ── Density estimation for bounded data via GMMs ─────────── 
##            
## Boundaries:  x1  x2
##       lower   0   0
##       upper Inf Inf
## 
## Model EVV (ellipsoidal, equal volume) model with 1 component
## on the transformation scale:
## 
##  log-likelihood   n df       BIC       ICL
##       -889.6179 200  7 -1816.324 -1816.324
## 
##                                    x1        x2
## Range-power transformation: 0.3060001 0.2725328
## 
## Mixing probabilities:
## 1 
## 1 
## 
## Means:
##        [,1]
## x1 1.051800
## x2 2.111525
## 
## Variances:
## [,,1]
##           x1        x2
## x1 1.2880789 0.3716033
## x2 0.3716033 0.7138729
plot(dens, what = "BIC")

plot(dens, what = "density")
points(x, cex = 0.3)
abline(h = 0, v = 0, lty = 3)

plot(dens, what = "density", type = "hdr")
abline(h = 0, v = 0, lty = 3)

plot(dens, what = "density", type = "persp")

Suicide data

The data consist in the lengths of 86 spells of psychiatric treatment undergone by control patients in a suicide study (Silverman, 1986).

data("suicide")
dens <- densityMclustBounded(suicide, lbound = 0)
summary(dens, parameters = TRUE)
## ── Density estimation for bounded data via GMMs ─────────── 
##            
## Boundaries: suicide
##       lower       0
##       upper     Inf
## 
## Model E (univariate, equal variance) model with 1 component
## on the transformation scale:
## 
##  log-likelihood  n df       BIC       ICL
##       -497.8204 86  3 -1009.004 -1009.004
## 
##                               suicide
## Range-power transformation: 0.1929267
## 
## Mixing probabilities:
## 1 
## 1 
## 
## Means:
##        1 
## 6.700073 
## 
## Variances:
##        1 
## 7.788326
plot(dens, what = "density", 
     lwd = 2, col = "dodgerblue2",
     data = suicide, breaks = 15, 
     xlab = "Length of psychiatric treatment")
rug(suicide)

Racial data

This dataset provides the proportion of white student enrollment in 56 school districts in Nassau County (Long Island, New York), for the 1992-1993 school year (Simonoff 1996, Sec. 3.2).

data("racial")
x <- racial$PropWhite
dens <- densityMclustBounded(x, lbound = 0, ubound = 1)
summary(dens, parameters = TRUE)
## ── Density estimation for bounded data via GMMs ─────────── 
##            
## Boundaries: x
##       lower 0
##       upper 1
## 
## Model E (univariate, equal variance) model with 1 component
## on the transformation scale:
## 
##  log-likelihood  n df      BIC      ICL
##         42.4598 56  3 72.84355 72.84355
## 
##                                     x
## Range-power transformation: 0.3869476
## 
## Mixing probabilities:
## 1 
## 1 
## 
## Means:
##        1 
## 2.795429 
## 
## Variances:
##        1 
## 5.253254
plot(dens, what = "density", 
     lwd = 2, col = "dodgerblue2",
     data = x, breaks = 15, 
     xlab = "Proportion of white student enrolled in schools")
rug(x)

Modal clustering using MEM algorithm for Gaussian mixtures

Simulated datasets

data(Baudry_etal_2010_JCGS_examples, package = "mclust")
GMM <- Mclust(ex4.1)
plot(GMM, what = "classification")

MEM <- MclustMEM(GMM)
summary(MEM)
## ── Modal EM for GMMs ─────────────────── 
## 
## Data dimensions = 600 x 2 
## Mclust model    = EEV,6 
## MEM iterations  = 17 
## Number of modes = 4 
## 
## Modes:
##                X1          X2
## mode1  8.06741504 -0.01772230
## mode2  8.07370160  4.98485099
## mode3  1.10622966  4.97230749
## mode4 -0.01639289  0.06464381
## 
## Modal clustering:
##   1   2   3   4 
## 118 122 228 132
plot(MEM)

plot(MEM, addPoints = FALSE)

GMM <- Mclust(ex4.4.2)
plot(GMM, what = "classification")

MEM <- MclustMEM(GMM)
summary(MEM)
## ── Modal EM for GMMs ─────────────────── 
## 
## Data dimensions = 300 x 3 
## Mclust model    = EVI,3 
## MEM iterations  = 15 
## Number of modes = 2 
## 
## Modes:
##              X1        X2        X3
## mode1 0.7964915 0.7444244 0.4547285
## mode2 0.4996361 0.5014374 0.4957522
## 
## Modal clustering:
##   1   2 
##  78 222
plot(MEM)

plot(MEM, addDensity = FALSE)

Entropy estimation

Simulated data

Univariate Gaussian

EntropyGauss(1)       # population entropy
## [1] 1.418939
x = rnorm(1000)       # generate sample
EntropyGauss(var(x))  # sample entropy assuming Gaussian distribution
## [1] 1.384565
mod = densityMclust(x, plot = FALSE)
EntropyGMM(mod)       # GMM-based entropy estimate
## [1] 1.384065
plot(mod, what = "density", data = x, breaks = 31); rug(x)

Univariate Mixed-Gaussian
Consider the mixed-Gaussian distribution \(f(x) = 0.5 \times N(-2,1) + 0.5 \times N(2,1)\), whose entropy is 2.051939 in the population.

cl = rbinom(1000, size = 1, prob = 0.5)
x = ifelse(cl == 1, rnorm(1000, 2, 1), rnorm(1000, -2, 1))   # generate sample
mod = densityMclust(x, plot = FALSE)
EntropyGMM(mod)       # GMM-based entropy estimate
## [1] 2.037679
plot(mod, what = "density", data = x, breaks = 31); rug(x)

Multivariate Chi-squared
Consider a 10-dimensional independent \(\chi^2\) distribution, whose entropy is 24.23095 in the population.

x = matrix(rchisq(1000*10, df = 5), nrow = 1000, ncol = 10)
mod1 = densityMclust(x, plot = FALSE)
EntropyGMM(mod1)      # GMM-based entropy estimate, not too bad but...
## [1] 25.01403
mod2 = densityMclustBounded(x, lbound = rep(0,10))
EntropyGMM(mod2)      # much more accurate
## [1] 24.22231

Faithful data

data(faithful)
mod = densityMclust(faithful, plot = FALSE)
EntropyGMM(mod)       # GMM-based entropy estimate
## [1] 4.140889
# or provide the data and fit GMM implicitly
EntropyGMM(faithful)
## [1] 4.140889

Iris data

data(iris)
mod = densityMclust(iris[,1:4], plot = FALSE)
EntropyGMM(mod)       # GMM-based entropy estimate
## [1] 1.438173

Volatility analysis of financial log-returns

Gold price 2023 data

data(gold)
head(gold)
##         date   log.returns
## 1 2023-01-03  0.0109308628
## 2 2023-01-04  0.0070955464
## 3 2023-01-05 -0.0097625244
## 4 2023-01-06  0.0158964701
## 5 2023-01-09  0.0045492333
## 6 2023-01-10 -0.0005875467

# GMM modeling 
mod = GMMlogreturn(gold$log.returns)
summary(mod)
## ── Log-returns density estimation via Gaussian finite mixture modeling ─────────
## Model: GMM(V,2)
## Prior: defaultPrior()
## 
##  log-likelihood   n df    BIC Entropy
##          852.46 250  5 1677.3 -3.4098
## 
## Mixture parameters:
##      Prob       Mean     StDev
## 1 0.52433 0.00029069 0.0045432
## 2 0.47567 0.00073238 0.0107633
## 
## Marginal statistics:
##        Mean     StDev Skewness Kurtosis      VaR       ES
##  0.00050079 0.0081227 0.058714   4.5584 0.012877 0.017929
plot(mod, what = "BIC")

plot(mod, what = "density", data = gold$log.returns)

plot(mod, what = "diagnostic")


# compare to single Gaussian model
mod1 = GMMlogreturn(gold$log.returns, G = 1)
y0 = extendrange(mod$data, f = 0.1)
y0 = seq(min(y0), max(y0), length = 1000)
plot(mod, what = "density", data = gold$log.returns, col = "steelblue",
     xlab = "Gold price log-returns", ylab = "Density")
lines(y0, predict(mod1, what = "dens", newdata = y0), col = "red3")
legend("topright", legend = c("Gaussian", "GMM"), lty = c(1,1),
       col = c("red3", "steelblue"), inset = 0.02)

References

Scrucca L., Fraley C., Murphy T. B. and Raftery A. E. (2023) Model-Based Clustering, Classification, and Density Estimation Using mclust in R. Chapman & Hall/CRC, ISBN: 978-1032234953, https://mclust-org.github.io/book/

Scrucca L., Fop M., Murphy T. B. and Raftery A. E. (2016) mclust 5: clustering, classification and density estimation using Gaussian finite mixture models. The R Journal 8/1, pp. 289-317. https://doi.org/10.32614/RJ-2016-021

Scrucca L. (2019) A transformation-based approach to Gaussian mixture density estimation for bounded data, Biometrical Journal, 61:4, 873–888. https://doi.org/10.1002/bimj.201800174

Scrucca L. (2021) A fast and efficient Modal EM algorithm for Gaussian mixtures. Statistical Analysis and Data Mining, 14:4, 305–314. https://doi.org/10.1002/sam.11527

Robin S. and Scrucca L. (2023) Mixture-based estimation of entropy. Computational Statistics & Data Analysis, 177, 107582. https://doi.org/10.1016/j.csda.2022.107582

sessionInfo()
## R version 4.4.1 (2024-06-14)
## Platform: aarch64-apple-darwin20
## Running under: macOS 15.0.1
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib;  LAPACK version 3.12.0
## 
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## 
## time zone: Europe/Rome
## tzcode source: internal
## 
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## 
## other attached packages:
## [1] mclustAddons_0.9.1 mclust_6.1.2       knitr_1.48        
## 
## loaded via a namespace (and not attached):
##  [1] doParallel_1.0.17 cli_3.6.3         rlang_1.1.4       xfun_0.47        
##  [5] highr_0.11        jsonlite_1.8.9    htmltools_0.5.8.1 rngtools_1.5.2   
##  [9] sass_0.4.9        rmarkdown_2.28    evaluate_1.0.0    jquerylib_0.1.4  
## [13] fastmap_1.2.0     yaml_2.3.10       foreach_1.5.2     lifecycle_1.0.4  
## [17] doRNG_1.8.6       compiler_4.4.1    codetools_0.2-20  Rcpp_1.0.13      
## [21] rstudioapi_0.16.0 digest_0.6.37     R6_2.5.1          parallel_4.4.1   
## [25] bslib_0.8.0       tools_4.4.1       iterators_1.0.14  cachem_1.1.0

A quick tour of mclustAddons

Luca Scrucca

12 Nov 2024

Introduction