Skew normal mixture
Last updated: 2020-02-25
Checks: 7 0
Knit directory: RcppComputingClub/
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Objectives
Normal mixture models are a popular method of distribution based clustering. Despite this, in practice they often don’t work well as slight departures from normality lead to poor model fit. One way to remedy this is to use a more robust distribution for clustering, such as adding additional parameters for kurtosis (i.e. t-distribution) or skewness.
A finite dimensional mixture of skew-normal distributions assumes data \(y = (y_1, \ldots, y_n) \in R^n\) are a sample from a probability density function of the form \[ f_{SN}(y; \xi, \omega^2, \alpha) = \frac{2}{\omega} \phi\left( \frac{y - \xi}{\omega}\right) \Phi(\alpha \omega^{-1}(y - \xi)) \] where \(\alpha\) is a skewness parameter.
Full conditionals are available for the proper parameter transformations and Gibbs sampling is still feasible. See ‘Bayesian inference for finite mixtures of univariate and multivariate skew-normal and skew-t distributions.’ by Früwirth-Schnatter, Pyne (2010) for derivations and greater detail.
# Load packages
library(tidyverse)── Attaching packages ────────────────────────────────── tidyverse 1.2.1 ──✔ ggplot2 3.2.1 ✔ purrr 0.3.2
✔ tibble 2.1.3 ✔ dplyr 0.8.3
✔ tidyr 1.0.0 ✔ stringr 1.4.0
✔ readr 1.3.1 ✔ forcats 0.4.0── Conflicts ───────────────────────────────────── tidyverse_conflicts() ──
✖ dplyr::filter() masks stats::filter()
✖ dplyr::lag() masks stats::lag()library(devtools)Loading required package: usethis#install_github("scristia/SkewNormalMix")
load_all("~/Software/SkewNormalMix")Loading SkewNormalMixLoading required package: RcppLoading required package: RcppArmadillolibrary(sn)Loading required package: stats4
Attaching package: 'sn'The following object is masked from 'package:SkewNormalMix':
dsnThe following object is masked from 'package:stats':
sdlibrary(mvtnorm)
library(msm)
library(MASS)
Attaching package: 'MASS'The following object is masked from 'package:dplyr':
selectlibrary(gtools)
library(truncnorm)# simulated data
omega <- c(4, 1)
omega2 <- omega^2
alpha <- c(-3, 0)
mu <- c(0, 4)
xx <- c(rsn(5000, mu[1], omega[1], alpha[1]), rsn(8000, mu[2], omega[2], alpha[2]))
xx <- xx[sample.int(8000)]
par(bg="white")
plot(density(xx), type="l")
n <- length(xx)
##transformations
delta <- alpha/sqrt(1+alpha^2)
Ey <- mu+omega2*delta*sqrt(2/3.1415)
psi <- omega*delta
sigma2 <- omega2*(1-delta^2)
K = 2
nsim=10000
burnin <- 1:500R implementation of function:
set.seed(4321)
res = skewnormal.gibbs(xx, K=K, nsim=nsim)
mus <- colMeans(res$MU[-burnin, ])
omegas <- colMeans(res$OMEGA[-burnin, ])
alphas <- colMeans(res$ALPHA[-burnin, ])
etas <- colMeans(res$ETA[-burnin, ])Rcpp implementation of function:
set.seed(4321)
res2 <- skewNormalCpp(r=xx, K=K, nsim=nsim)
mus2 <- colMeans(res2$MU[-burnin, ])
omegas2 <- colMeans(res2$OMEGA[-burnin, ])
alphas2 <- colMeans(res2$ALPHA[-burnin, ])
etas2 <- colMeans(res2$ETA[-burnin, ])Check posterior fit:
Check truth:
| Parameter | Truth | Estimate (R) | Estimate (Rcpp) |
|---|---|---|---|
| mu1 | 0.000 | -0.05 | -0.05 |
| mu2 | 4.000 | 4.03 | 3.95 |
| omega1 | 4.000 | 3.96 | 3.96 |
| omega2 | 1.000 | 1.05 | 1.04 |
| alpha1 | -3.000 | -3.00 | -2.98 |
| alpha2 | 0.000 | -0.04 | 0.07 |
| eta1 | 0.615 | 0.62 | 0.62 |
| eta2 | 0.385 | 0.38 | 0.38 |
Benchmarking
| test | replications | elapsed | relative | user.self | sys.self | user.child | sys.child | |
|---|---|---|---|---|---|---|---|---|
| 2 | skewnormal.gibbs(xx, K = 2, nsim = 200, thin = 1) | 3 | 10.485 | 6.114 | 9.546 | 0.928 | 0 | 0 |
| 1 | skewNormalCpp(r = xx, K = 2, nsim = 200) | 3 | 1.715 | 1.000 | 1.532 | 0.182 | 0 | 0 |
R version 3.5.2 (2018-12-20)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.5
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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] stats4 stats graphics grDevices utils datasets methods
[8] base
other attached packages:
[1] rbenchmark_1.0.0 truncnorm_1.0-8
[3] gtools_3.8.1 MASS_7.3-51.4
[5] msm_1.6.8 mvtnorm_1.0-11
[7] sn_1.5-5 SkewNormalMix_1.0
[9] RcppArmadillo_0.9.850.1.0 Rcpp_1.0.3
[11] devtools_2.2.1 usethis_1.5.1
[13] forcats_0.4.0 stringr_1.4.0
[15] dplyr_0.8.3 purrr_0.3.2
[17] readr_1.3.1 tidyr_1.0.0
[19] tibble_2.1.3 ggplot2_3.2.1
[21] tidyverse_1.2.1 nvimcom_0.9-81
loaded via a namespace (and not attached):
[1] nlme_3.1-141 fs_1.3.1 lubridate_1.7.4
[4] httr_1.4.1 rprojroot_1.3-2 numDeriv_2016.8-1.1
[7] tools_3.5.2 backports_1.1.5 R6_2.4.0
[10] lazyeval_0.2.2 colorspace_1.4-1 withr_2.1.2
[13] tidyselect_0.2.5 prettyunits_1.0.2 mnormt_1.5-6
[16] processx_3.4.1 compiler_3.5.2 git2r_0.26.1
[19] cli_1.1.0 rvest_0.3.4 expm_0.999-4
[22] xml2_1.2.2 desc_1.2.0 scales_1.0.0
[25] callr_3.3.2 digest_0.6.21 rmarkdown_1.16
[28] pkgconfig_2.0.3 htmltools_0.4.0 sessioninfo_1.1.1
[31] highr_0.8 rlang_0.4.0 readxl_1.3.1
[34] rstudioapi_0.10 generics_0.0.2 jsonlite_1.6
[37] magrittr_1.5 Matrix_1.2-17 munsell_0.5.0
[40] lifecycle_0.1.0 stringi_1.4.3 whisker_0.4
[43] yaml_2.2.0 pkgbuild_1.0.6 grid_3.5.2
[46] promises_1.1.0 crayon_1.3.4 lattice_0.20-38
[49] haven_2.1.1 splines_3.5.2 hms_0.5.1
[52] zeallot_0.1.0 knitr_1.25 ps_1.3.0
[55] pillar_1.4.2 pkgload_1.0.2 glue_1.3.1
[58] evaluate_0.14 remotes_2.1.0 modelr_0.1.5
[61] vctrs_0.2.0 httpuv_1.5.2 testthat_2.2.1
[64] cellranger_1.1.0 gtable_0.3.0 assertthat_0.2.1
[67] xfun_0.10 broom_0.5.2 later_1.0.0
[70] survival_2.44-1.1 memoise_1.1.0 workflowr_1.6.0
[73] ellipsis_0.3.0