Chapter 31 Regularized Discriminant Analysis

We now use the Sonar dataset from the mlbench package to explore a new regularization method, regularized discriminant analysis (RDA), which combines the LDA and QDA. This is similar to how elastic net combines the ridge and lasso.

31.1 Sonar Data

# this is a temporary workaround for an issue with glmnet, Matrix, and R version 3.3.3
# see here: http://stackoverflow.com/questions/43282720/r-error-in-validobject-object-when-running-as-script-but-not-in-console
library(methods)

library(mlbench)
library(caret)
library(glmnet)
library(klaR)

data(Sonar)

#View(Sonar)

table(Sonar$Class) / nrow(Sonar)

## 
##         M         R 
## 0.5336538 0.4663462

ncol(Sonar) - 1

## [1] 60

31.2 RDA

Regularized discriminant analysis uses the same general setup as LDA and QDA but estimates the covariance in a new way, which combines the covariance of QDA \((\hat{\Sigma}_k)\) with the covariance of LDA \((\hat{\Sigma})\) using a tuning parameter \(\lambda\).

\[ \hat{\Sigma}_k(\lambda) = (1-\lambda)\hat{\Sigma}_k + \lambda \hat{\Sigma} \]

Using the rda() function from the klaR package, which caret utilizes, makes an additional modification to the covariance matrix, which also has a tuning parameter \(\gamma\).

\[ \hat{\Sigma}_k(\lambda,\gamma) = (1 -\gamma) \hat{\Sigma}_k(\lambda) + \gamma \frac{1}{p} \text{tr}(\hat{\Sigma}_k(\lambda)) I \]

Both \(\gamma\) and \(\lambda\) can be thought of as mixing parameters, as they both take values between 0 and 1. For the four extremes of \(\gamma\) and \(\lambda\), the covariance structure reduces to special cases:

• \((\gamma=0, \lambda=0)\): QDA - individual covariance for each group.
• \((\gamma=0, \lambda=1)\): LDA - a common covariance matrix.
• \((\gamma=1, \lambda=0)\): Conditional independent variables - similar to Naive Bayes, but variable variances within group (main diagonal elements) are all equal.
• \((\gamma=1, \lambda=1)\): Classification using euclidean distance - as in previous case, but variances are the same for all groups. Objects are assigned to group with nearest mean.

31.3 RDA with Grid Search

set.seed(1337)
cv_5_grid = trainControl(method = "cv", number = 5)

set.seed(1337)
fit_rda_grid = train(Class ~ ., data = Sonar, method = "rda", trControl = cv_5_grid)
fit_rda_grid

## Regularized Discriminant Analysis 
## 
## 208 samples
##  60 predictor
##   2 classes: 'M', 'R' 
## 
## No pre-processing
## Resampling: Cross-Validated (5 fold) 
## Summary of sample sizes: 167, 166, 166, 167, 166 
## Resampling results across tuning parameters:
## 
##   gamma  lambda  Accuracy   Kappa    
##   0.0    0.0     0.6977933  0.3791172
##   0.0    0.5     0.7644599  0.5259800
##   0.0    1.0     0.7310105  0.4577198
##   0.5    0.0     0.7885017  0.5730052
##   0.5    0.5     0.8271777  0.6502693
##   0.5    1.0     0.7988386  0.5939209
##   1.0    0.0     0.6732869  0.3418352
##   1.0    0.5     0.6780488  0.3527778
##   1.0    1.0     0.6825784  0.3631626
## 
## Accuracy was used to select the optimal model using the largest value.
## The final values used for the model were gamma = 0.5 and lambda = 0.5.

plot(fit_rda_grid)

31.4 RDA with Random Search Search

set.seed(1337)
cv_5_rand = trainControl(method = "cv", number = 5, search = "random")

fit_rda_rand = train(Class ~ ., data = Sonar, method = "rda",
                     trControl = cv_5_rand, tuneLength = 9)
fit_rda_rand

## Regularized Discriminant Analysis 
## 
## 208 samples
##  60 predictor
##   2 classes: 'M', 'R' 
## 
## No pre-processing
## Resampling: Cross-Validated (5 fold) 
## Summary of sample sizes: 167, 166, 166, 167, 166 
## Resampling results across tuning parameters:
## 
##   gamma       lambda      Accuracy   Kappa    
##   0.07399023  0.99371759  0.7795587  0.5556383
##   0.14604362  0.33913968  0.8508711  0.6996434
##   0.24540405  0.92379666  0.8418118  0.6798190
##   0.28111731  0.97238848  0.8226481  0.6405093
##   0.33131745  0.98132543  0.8083624  0.6123279
##   0.37327926  0.19398230  0.7981417  0.5930526
##   0.45386562  0.82735873  0.8272938  0.6501074
##   0.94763002  0.02522857  0.7742160  0.5399326
##   0.97943029  0.24657940  0.7454123  0.4839854
## 
## Accuracy was used to select the optimal model using the largest value.
## The final values used for the model were gamma = 0.1460436 and lambda
##  = 0.3391397.

ggplot(fit_rda_rand)

31.5 Comparison to Elastic Net

set.seed(1337)
fit_elnet_grid = train(Class ~ ., data = Sonar, method = "glmnet",
                       trControl = cv_5_grid, tuneLength = 10)

set.seed(1337)
fit_elnet_int_grid = train(Class ~ . ^ 2, data = Sonar, method = "glmnet",
                           trControl = cv_5_grid, tuneLength = 10)

31.6 Results

get_best_result = function(caret_fit) {
  best_result = caret_fit$results[as.numeric(rownames(caret_fit$bestTune)), ]
  rownames(best_result) = NULL
  best_result
}

knitr::kable(rbind(
  get_best_result(fit_rda_grid),
  get_best_result(fit_rda_rand)))

gamma	lambda	Accuracy	Kappa	AccuracySD	KappaSD
0.5000000	0.5000000	0.8271777	0.6502693	0.0970937	0.1962078
0.1460436	0.3391397	0.8508711	0.6996434	0.0887983	0.1797382

knitr::kable(rbind(
  get_best_result(fit_elnet_grid),
  get_best_result(fit_elnet_int_grid)))

alpha	lambda	Accuracy	Kappa	AccuracySD	KappaSD
0.1	0.0350306	0.8271777	0.6501991	0.0416190	0.0881929
0.1	0.0561881	0.8364692	0.6687824	0.0751608	0.1540228

31.7 External Links

• Random Search for Hyper-Parameter Optimization - Paper justifying random tuning parameter search.
• Random Hyperparameter Search - Details on random tuning parameter search in caret.

31.8 RMarkdown

The RMarkdown file for this chapter can be found here. The file was created using R version 4.0.2 and the following packages:

• Base Packages, Attached

## [1] "stats"     "graphics"  "grDevices" "utils"     "datasets"  "methods"  
## [7] "base"

• Additional Packages, Attached

## [1] "klaR"    "MASS"    "glmnet"  "Matrix"  "caret"   "ggplot2" "lattice"
## [8] "mlbench"

• Additional Packages, Not Attached

##  [1] "splines"      "foreach"      "prodlim"      "shiny"        "highr"       
##  [6] "stats4"       "yaml"         "ipred"        "pillar"       "glue"        
## [11] "pROC"         "digest"       "promises"     "colorspace"   "recipes"     
## [16] "htmltools"    "httpuv"       "plyr"         "timeDate"     "pkgconfig"   
## [21] "labelled"     "haven"        "questionr"    "bookdown"     "purrr"       
## [26] "xtable"       "scales"       "later"        "gower"        "lava"        
## [31] "tibble"       "combinat"     "generics"     "farver"       "ellipsis"    
## [36] "withr"        "nnet"         "survival"     "magrittr"     "crayon"      
## [41] "mime"         "evaluate"     "nlme"         "forcats"      "class"       
## [46] "tools"        "data.table"   "hms"          "lifecycle"    "stringr"     
## [51] "munsell"      "compiler"     "e1071"        "rlang"        "grid"        
## [56] "iterators"    "rstudioapi"   "miniUI"       "labeling"     "rmarkdown"   
## [61] "gtable"       "ModelMetrics" "codetools"    "reshape2"     "R6"          
## [66] "lubridate"    "knitr"        "dplyr"        "fastmap"      "shape"       
## [71] "stringi"      "Rcpp"         "vctrs"        "rpart"        "tidyselect"  
## [76] "xfun"