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Introduction

An R package implementing a Projection Pursuit algorithm based on finite Gaussian Mixtures Models for density estimation using Genetic Algorithms (PPGMMGA) to maximise a Negentropy index. The PPGMMGA algorithm provides a method to visualise high-dimensional data in a lower-dimensional space, with special reference to reveal clustering structures.

library(ppgmmga)
##    ___  ___  ___ ___ _  __ _  ___ ____ _
##   / _ \/ _ \/ _ `/  ' \/  ' \/ _ `/ _ `/
##  / .__/ .__/\_, /_/_/_/_/_/_/\_, /\_,_/ 
## /_/  /_/   /___/            /___/       version 1.3

Banknote data

library(mclust)
data("banknote")
X <- banknote[,-1]
Class <- banknote$Status
table(Class)
## Class
## counterfeit     genuine 
##         100         100
clPairs(X, classification = Class, 
        symbols = ppgmmga.options("classPlotSymbols"),
        colors = ppgmmga.options("classPlotColors"))

1-dimensional PPGMMGA

PP1D <- ppgmmga(data = X, d = 1, seed = 1)
PP1D
## Call:
## ppgmmga(data = X, d = 1, seed = 1)
## 
## 'ppgmmga' object containing: 
## [1] "data"       "d"          "approx"     "GMM"        "GA"        
## [6] "Negentropy" "basis"      "Z"
summary(PP1D)
## ── ppgmmga ───────────────────────────── 
## 
## Data dimensions               = 200 x 6 
## Data transformation           = center & scale 
## Projection subspace dimension = 1 
## GMM density estimate          = (VEE,4)
## Negentropy approximation      = UT 
## GA optimal negentropy         = 0.6345935 
## GA encoded basis solution: 
##            x1       x2       x3       x4       x5
## [1,] 3.268902 2.373044 1.051365 0.313128 0.531718
## 
## Estimated projection basis: 
##                 PP1
## Length   -0.0119653
## Left     -0.0934775
## Right     0.1602105
## Bottom    0.5740698
## Top       0.3450346
## Diagonal -0.7189203
## 
## Monte Carlo Negentropy approximation check: 
##                            UT
## Approx Negentropy 0.634593544
## MC Negentropy     0.633614256
## MC se             0.002249545
## Relative accuracy 1.001545559
plot(PP1D)

plot(PP1D, class = Class)

2-dimensional PPGMMGA

PP2D <- ppgmmga(data = X, d = 2, seed = 1)
summary(PP2D)
## ── ppgmmga ───────────────────────────── 
## 
## Data dimensions               = 200 x 6 
## Data transformation           = center & scale 
## Projection subspace dimension = 2 
## GMM density estimate          = (VEE,4)
## Negentropy approximation      = UT 
## GA optimal negentropy         = 1.13624 
## GA encoded basis solution: 
##            x1       x2       x3       x4      x5      x6      x7      x8
## [1,] 2.268667 2.929821 1.061407 1.084929 0.30443 3.85462 0.98329 1.11377
##            x9      x10
## [1,] 0.167174 1.668403
## 
## Estimated projection basis: 
##                 PP1        PP2
## Length   -0.0372687 -0.0718319
## Left      0.0312555 -0.1198116
## Right    -0.1548079  0.0630092
## Bottom   -0.0856931  0.8639049
## Top      -0.1024990  0.4603727
## Diagonal  0.9776601  0.1350576
## 
## Monte Carlo Negentropy approximation check: 
##                            UT
## Approx Negentropy 1.136240194
## MC Negentropy     1.137260367
## MC se             0.003527379
## Relative accuracy 0.999102956
summary(PP2D$GMM)
## ------------------------------------------------------- 
## Density estimation via Gaussian finite mixture modeling 
## ------------------------------------------------------- 
## 
## Mclust VEE (ellipsoidal, equal shape and orientation) model with 4 components: 
## 
##  log-likelihood   n df       BIC       ICL
##       -1191.595 200 51 -2653.405 -2666.898
plot(PP2D$GA)

plot(PP2D)

plot(PP2D, class = Class, drawAxis = FALSE)

3-dimensional PPGMMGA

PP3D <- ppgmmga(data = X, d = 3, 
                center = TRUE, scale = FALSE, 
                gatype = "gaisl", 
                options = ppgmmga.options(numIslands = 2),
                seed = 1)
summary(PP3D)
## ── ppgmmga ───────────────────────────── 
## 
## Data dimensions               = 200 x 6 
## Data transformation           = center 
## Projection subspace dimension = 3 
## GMM density estimate          = (VVE,3)
## Negentropy approximation      = UT 
## GA optimal negentropy         = 1.16915 
## GA encoded basis solution: 
##            x1      x2       x3       x4       x5       x6       x7       x8
## [1,] 4.274545 2.47064 1.055677 1.022896 0.851247 4.924235 1.982288 2.039161
##            x9      x10  ...       x14      x15
## [1,] 1.939208 2.210582       1.548995 2.489197
## 
## Estimated projection basis: 
##                 PP1        PP2        PP3
## Length   -0.3145939  0.5612330 -0.5201907
## Left     -0.1472768 -0.1498109 -0.3297848
## Right     0.3043823  0.5008715 -0.3739875
## Bottom    0.2818318  0.3353769  0.4238383
## Top       0.3062895  0.4589957  0.3562206
## Diagonal -0.7832300  0.2975690  0.4174266
## 
## Monte Carlo Negentropy approximation check: 
##                           UT
## Approx Negentropy 1.16914962
## MC Negentropy     1.17493505
## MC se             0.00430878
## Relative accuracy 0.99507596
plot(PP3D$GA)

plot(PP3D)

plot(PP3D, class = Class)

plot(PP3D, dim = c(1,2))

plot(PP3D, dim = c(1,3), class = Class)

# A rotating 3D plot can be obtained using
if(!require("msir")) install.packages("msir")
msir::spinplot(PP3D$Z, markby = Class, 
               pch.points = c(20,17),
               col.points = ppgmmga.options("classPlotColors")[1:2])


References

Scrucca L, Serafini A (2019). “Projection pursuit based on Gaussian mixtures and evolutionary algorithms.” Journal of Computational and Graphical Statistics, 28(4), 847–860. https://doi.org/10.1080/10618600.2019.1598871.


sessionInfo()
## R version 4.3.0 (2023-04-21)
## Platform: x86_64-apple-darwin20 (64-bit)
## Running under: macOS Ventura 13.6
## 
## Matrix products: default
## BLAS:   /Library/Frameworks/R.framework/Versions/4.3-x86_64/Resources/lib/libRblas.0.dylib 
## LAPACK: /Library/Frameworks/R.framework/Versions/4.3-x86_64/Resources/lib/libRlapack.dylib;  LAPACK version 3.11.0
## 
## 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] mclust_6.0.1 ppgmmga_1.3  knitr_1.44  
## 
## loaded via a namespace (and not attached):
##  [1] sass_0.4.6        utf8_1.2.4        generics_0.1.3    GA_3.2.3         
##  [5] stringi_1.7.12    digest_0.6.33     magrittr_2.0.3    evaluate_0.22    
##  [9] grid_4.3.0        iterators_1.0.14  fastmap_1.1.1     foreach_1.5.2    
## [13] rprojroot_2.0.3   jsonlite_1.8.7    purrr_1.0.1       fansi_1.0.5      
## [17] scales_1.2.1      codetools_0.2-19  jquerylib_0.1.4   cli_3.6.1        
## [21] rlang_1.1.1       crayon_1.5.2      munsell_0.5.0     withr_2.5.1      
## [25] cachem_1.0.8      yaml_2.3.7        tools_4.3.0       memoise_2.0.1    
## [29] dplyr_1.1.2       colorspace_2.1-0  ggplot2_3.4.3     vctrs_0.6.4      
## [33] R6_2.5.1          lifecycle_1.0.3   stringr_1.5.0     fs_1.6.3         
## [37] pkgconfig_2.0.3   desc_1.4.2        pkgdown_2.0.7     bslib_0.4.2      
## [41] pillar_1.9.0      gtable_0.3.4      glue_1.6.2        Rcpp_1.0.11      
## [45] xfun_0.40         tibble_3.2.1      tidyselect_1.2.0  rstudioapi_0.15.0
## [49] farver_2.1.1      htmltools_0.5.6   rmarkdown_2.22    labeling_0.4.3   
## [53] compiler_4.3.0