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mvnormDPD examples

A. Ghosh, C. Agostinelli and A. Basu

2026-08-24

Introduction

This file contains some examples of the use of the function mvnormDPD. It reproduces all the figures and tables in section C “Performances under benchmark Gaussian datasets” of A. Ghosh, C. Agostinelli and A. Basu (2026) A Composite Divergence Approach to Robust Multivariate Estimation under Cellwise and Casewise Contamination, arXiv:2608.18914, https://arxiv.org/abs/2608.18914.

library("mvdpd")
library("cellWise")
library("robustbase")
library("dplyr")

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library("ggplot2")
library("knitr")

Functions and wrappers

loo.analysis <- function(X, method) {
  p <- ncol(X)
  var_names <- colnames(X)
  eigen_res <- list()
  mu_res <- list()
  md_res <- list()
  ## Leave-One-Out
  for (i in 1:nrow(X)) {
    Xb <- X[-i,]
    res <- method(Xb)
    mu_res[[i]] <- data.frame(
      LeaveOut = i,
      Method = res$method,
      Parameter = var_names,
      Estimate = res$mu
    )
    rownames(mu_res[[i]]) <- var_names
  
    eigen_res[[i]] <- data.frame(
      LeaveOut = i,
      Method = res$method,
      Parameter = paste0("e", 1:p),
      Eigenvalues = eigen(res$Sigma)$values
    )
    rownames(eigen_res[[i]]) <- paste0("e", 1:p)
  
    md_res[[i]] <- data.frame(
      LeaveOut = i,
      Method = res$method,
      Observation = paste0("Obs", seq_len(nrow(X))),
      Distance = sqrt(mahalanobis(X,
        center = res$mu,
        cov = res$Sigma))
    )
  }  
  res <- list(mu=mu_res, eigen=eigen_res, mahalanobis=md_res)
  return(res)
}

ML <- function(X, ...) {
  list(mu=colMeans(X), Sigma=cov(X), method="ML/MCL")
}

MCD <- function(X, ...) { 
  mcd <- covMcd(X, ...)
  list(mu=mcd$center, Sigma=mcd$cov, method="MCD")
}

CELLMCD <- function(X, ...) { 
  cellmcd <- cellMCD(X, checkPars=list(silent=TRUE), ...)
  list(mu=cellmcd$mu, Sigma=cellmcd$S, method="CellMCD")
}

MDPD <- function(X, beta, ...) { 
  mdpd <- mvnormDPD(X, beta=beta, method="multivariate", ...)
  list(mu=mdpd$mu, Sigma=mdpd$Sigma, method=paste0("MDPD(", beta, ")"))
}

CDPD <- function(X, beta, ...) { 
  cdpd <- mvnormDPD(X, beta=beta, method="composite", ...)
  list(mu=cdpd$mu, Sigma=cdpd$Sigma, method=paste0("CDPD(", beta, ")"))
}

perform.analysis <- function(X) {  
  resML <- loo.analysis(X, method=ML)
  resMCD <- loo.analysis(X, method=MCD)
  resCELLMCD <- loo.analysis(X, method=CELLMCD)
  resMDPD1 <- loo.analysis(X, method=function(X) MDPD(X, beta=0.1))
  resMDPD3 <- loo.analysis(X, method=function(X) MDPD(X, beta=0.3))
  resMDPD5 <- loo.analysis(X, method=function(X) MDPD(X, beta=0.5))
  resCDPD1 <- loo.analysis(X, method=function(X) CDPD(X, beta=0.1))
  resCDPD3 <- loo.analysis(X, method=function(X) CDPD(X, beta=0.3))
  resCDPD5 <- loo.analysis(X, method=function(X) CDPD(X, beta=0.5))
  mu_df <- bind_rows(resML$mu, resMCD$mu, resCELLMCD$mu, 
    resMDPD1$mu, resMDPD3$mu, resMDPD5$mu,
    resCDPD1$mu, resCDPD3$mu, resCDPD5$mu)
  eigen_df <- bind_rows(resML$eigen, resMCD$eigen, resCELLMCD$eigen, 
    resMDPD1$eigen, resMDPD3$eigen, resMDPD5$eigen,
    resCDPD1$eigen, resCDPD3$eigen, resCDPD5$eigen)

  md_df <- bind_rows(resML$mahalanobis, 
    resMCD$mahalanobis, resCELLMCD$mahalanobis, 
    resMDPD1$mahalanobis, resMDPD3$mahalanobis, resMDPD5$mahalanobis,
    resCDPD1$mahalanobis, resCDPD3$mahalanobis, resCDPD5$mahalanobis)
  
  mu_var <- mu_df %>%
    group_by(Method, Parameter) %>%
    summarise(
      Variance = var(Estimate),
      .groups = "drop"
    )

  eigen_var <- eigen_df %>%
    group_by(Method, Parameter) %>%
    summarise(
      Variance = var(Eigenvalues),
      .groups = "drop"
    )

  md_var <- md_df %>%
    group_by(Method, Observation) %>%
    summarise(
      Variance = var(Distance),
      .groups = "drop"
    )
  res <- list(mu_df=mu_df, mu_var=mu_var,
    eigen_df=eigen_df, eigen_var=eigen_var,
    md_df=md_df,  md_var=md_var)
  return(res)
}

plot.results <- function(object) {
  # Boxplot of the location estimates
  mu_df_gg <- ggplot(object$mu_df, aes(x=Method, y=Estimate, fill=Parameter)) +
    geom_boxplot(position = position_dodge(0.8), width = 0.7) +
    scale_x_discrete(name="Method", limits=c("CellMCD", "ML/MCL", "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")) +
    scale_y_continuous(name=expression(paste("Leave-one-out ",hat(mu)))) + 
    theme(
      axis.text = element_text(size = 12, face="bold"),
      axis.title.x = element_text(size = 14, face="bold"),
      axis.title.y = element_text(size = 16, face="bold"),
      legend.text = element_text(size = 12, face="bold"),
      legend.title = element_text(size = 14, face="bold")
    )

  # Boxplot of variances of the location estimates
  mu_var_gg <- ggplot(object$mu_var, aes(x = Method, y = Variance)) +
    geom_boxplot(fill = "grey85", width = 0.6) +
    geom_point(aes(color = Parameter), size = 3, 
      position = position_jitter(width = 0.08)) +
    scale_x_discrete(name="Method",
      limits=c("CellMCD", "ML/MCL", "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")) +
    scale_y_continuous(name=expression(paste("Variances of leave-one-out ",
      hat(mu)))) + 
    theme(
      axis.text = element_text(size = 12, face="bold"),
      axis.title.x = element_text(size = 14, face="bold"),
      axis.title.y = element_text(size = 16, face="bold"),
      legend.text = element_text(size = 12, face="bold"),
      legend.title = element_text(size = 14, face="bold")
    )

  # Boxplot of the eigenvalues of Scatter estimates
  eigen_df_gg <- ggplot(object$eigen_df, aes(x=Method,
    y=Eigenvalues, fill=Parameter)) +
    geom_boxplot(position = position_dodge(0.8), width = 0.7) +
    scale_x_discrete(name="Method", limits=c("CellMCD", "ML/MCL",
      "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")) +
    scale_y_continuous(name=expression(paste("Leave-one-out eigenvalues of ",
      hat(Sigma)))) + 
    theme(
      axis.text = element_text(size = 12, face="bold"),
      axis.title.x = element_text(size = 14, face="bold"),
      axis.title.y = element_text(size = 16, face="bold"),
      legend.text = element_text(size = 12, face="bold"),
      legend.title = element_text(size = 14, face="bold")
    )

  # Boxplot of variances of the eigenvalues of Scatter estimates
  eigen_var_gg <- ggplot(object$eigen_var, aes(x = Method, y = Variance)) +
    geom_boxplot(fill = "grey85", width = 0.6) +
    geom_point(aes(color = Parameter), size = 3, 
      position = position_jitter(width = 0.08)) +
    scale_x_discrete(name="Method", limits=c("CellMCD",
      "ML/MCL", "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")) +
    scale_y_continuous(name=expression(paste("Variances of leave-one-out 
      eigenvalues of ",hat(Sigma)))) + 
    theme(
      axis.text = element_text(size = 12, face="bold"),
      axis.title.x = element_text(size = 14, face="bold"),
      axis.title.y = element_text(size = 16, face="bold"),
      legend.text = element_text(size = 12, face="bold"),
      legend.title = element_text(size = 14, face="bold")
    )
  res <- list(mu_df_gg=mu_df_gg, eigen_df_gg=eigen_df_gg,
    mu_var_gg=mu_var_gg, eigen_var_gg=eigen_var_gg)
  return(res)
}

Alcohol (n=44, d=7)

data(alcohol)
X <- as.matrix(alcohol)
X <- transfo(X)$Y

##  
##  The input data has 44 rows and 7 columns.

resAlcohol <- perform.analysis(X)
plotAlcohol <- plot.results(resAlcohol)

Figure S1

plotAlcohol$mu_df_gg

## Warning: Removed 1232 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotAlcohol$mu_var_gg

## Warning: Removed 28 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 28 rows containing missing values or values outside the scale range
## (`geom_point()`).
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plotAlcohol$eigen_df_gg

## Warning: Removed 1232 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotAlcohol$eigen_var_gg

## Warning: Removed 28 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 28 rows containing missing values or values outside the scale range
## (`geom_point()`).
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Milk (n=68, d=8)

data(milk)
X <- as.matrix(milk)
X <- transfo(X)$Y

##  
##  The input data has 86 rows and 8 columns.

resMilk <- perform.analysis(X)
plotMilk <- plot.results(resMilk)

Figure S2

plotMilk$mu_df_gg

## Warning: Removed 2752 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotMilk$mu_var_gg

## Warning: Removed 32 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 32 rows containing missing values or values outside the scale range
## (`geom_point()`).
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plotMilk$eigen_df_gg

## Warning: Removed 2752 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotMilk$eigen_var_gg

## Warning: Removed 32 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 32 rows containing missing values or values outside the scale range
## (`geom_point()`).
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Bushfire (n=38, d=5)

data(bushfire)
X <- as.matrix(bushfire)
X <- transfo(X)$Y

##  
##  The input data has 38 rows and 5 columns.

resBushfire <- perform.analysis(X)
plotBushfire <- plot.results(resBushfire)

Figure S3

plotBushfire$mu_df_gg

## Warning: Removed 760 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotBushfire$mu_var_gg

## Warning: Removed 20 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 20 rows containing missing values or values outside the scale range
## (`geom_point()`).
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plotBushfire$eigen_df_gg

## Warning: Removed 760 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).
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plotBushfire$eigen_var_gg

## Warning: Removed 20 rows containing missing values or values outside the scale range
## (`stat_boxplot()`).

## Warning: Removed 20 rows containing missing values or values outside the scale range
## (`geom_point()`).
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Toxicity (n=38, d=10)

Transformed data

data(toxicity)
X <- as.matrix(toxicity)
X <- transfo(X)$Y

##  
##  The input data has 38 rows and 10 columns.

betas <- c(0, 0.1, 0.3, 0.5)
d <- ncol(X)
pairs <- which(upper.tri(matrix(0, d, d)), arr.ind=TRUE)
result <-matrix(NA, nrow = d*(d+3)/2, ncol=length(betas))
for (i in seq_along(betas)){
  res <- mvnormDPD(X, betas[i], method = "composite")
  cor_matrix <- res$rho
  result[,i] <- c(res$mu, res$sigma,
    cor_matrix[upper.tri(cor_matrix, diag = FALSE)])
}
colnames(result) <- c("ML/MCL", "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")
rownames(result) <- c(paste0("mu_", 1:10), paste0("sigma2_", 1:10), paste0("rho_", pairs[,1], pairs[,2]))

Table S1

kable(result)
ML/MCL CDPD(0.1) CDPD(0.3) CDPD(0.5)
mu_1 -0.0779019 -0.0776080 -0.0852228 -0.0675296
mu_2 0.0000000 0.0164793 0.0580872 0.0952169
mu_3 0.0000000 0.0194761 0.0651009 0.1234987
mu_4 0.0000000 0.0243500 0.0771907 0.1321183
mu_5 -0.0705141 -0.0601355 -0.0394666 -0.0242888
mu_6 0.0000000 -0.0030182 -0.0076627 -0.0135830
mu_7 0.0000000 0.0082279 0.0223343 0.0205170
mu_8 0.0000000 0.0061019 0.0122583 0.0104286
mu_9 0.0000000 -0.0081548 -0.0255715 -0.0484908
mu_10 0.0000000 0.0185714 0.0608116 0.1080782
sigma2_1 0.8868514 1.1918660 1.2001601 1.1658771
sigma2_2 0.7368421 1.0066978 1.0232597 1.0092182
sigma2_3 0.7368421 1.0381067 1.1452420 1.2164303
sigma2_4 0.7368421 1.0210980 1.0971169 1.1507239
sigma2_5 0.8561501 1.1303067 1.0858797 1.0122745
sigma2_6 0.7368421 1.0259148 1.1060055 1.1706685
sigma2_7 0.7368421 0.9993916 1.0177457 1.0182493
sigma2_8 0.7368421 1.0254819 1.1116006 1.1860707
sigma2_9 0.7368421 1.0194204 1.0885973 1.1465985
sigma2_10 0.7368421 1.0571674 1.2131411 1.3600295
rho_12 0.8322388 0.8628886 0.9152557 0.9250386
rho_13 0.1131656 0.1196088 0.1370754 0.1562779
rho_23 0.2808114 0.2835286 0.2756927 0.2609005
rho_14 0.2952595 0.3218801 0.3858360 0.4427930
rho_24 0.4860845 0.4980747 0.5072281 0.5148794
rho_34 0.7236445 0.7414810 0.7722202 0.7998671
rho_15 0.4804663 0.4974810 0.5271693 0.5386596
rho_25 0.7615118 0.7569288 0.7388229 0.7193244
rho_35 0.6083289 0.6015518 0.5778396 0.5411152
rho_45 0.5926599 0.5786725 0.5443224 0.5066433
rho_16 0.7319192 0.7317449 0.7285395 0.7260053
rho_26 0.8571748 0.8566828 0.8368186 0.8218142
rho_36 0.0833179 0.0769015 0.0595573 0.0375573
rho_46 0.2272637 0.2210286 0.2121922 0.2111919
rho_56 0.6249486 0.6311714 0.6456317 0.6690204
rho_17 0.5647034 0.5885093 0.6302105 0.6790042
rho_27 0.8278655 0.8341521 0.8337057 0.8556621
rho_37 0.1745410 0.1651261 0.1398045 0.1112415
rho_47 0.2450245 0.2383051 0.2193476 0.2019986
rho_57 0.8196564 0.8235258 0.8184323 0.8091686
rho_67 0.8333421 0.8502673 0.8622851 0.8645118
rho_18 -0.1301424 -0.1226917 -0.1033393 -0.0792777
rho_28 -0.0359887 -0.0276313 -0.0104282 0.0251174
rho_38 -0.5998177 -0.6165466 -0.6389156 -0.6561084
rho_48 -0.5783522 -0.6171167 -0.6806671 -0.7294232
rho_58 -0.0083235 -0.0111258 -0.0168985 -0.0251884
rho_68 0.1943763 0.2116480 0.2427001 0.2652904
rho_78 0.4185087 0.4295602 0.4313967 0.4263750
rho_19 -0.3173434 -0.3047845 -0.2793955 -0.2451218
rho_29 -0.2886486 -0.2808820 -0.2347194 -0.1693658
rho_39 -0.6278474 -0.6317284 -0.6386160 -0.6525587
rho_49 -0.6453400 -0.6821225 -0.7577854 -0.8116444
rho_59 -0.2901517 -0.2722216 -0.2269667 -0.1763799
rho_69 0.0583660 0.0769415 0.1208629 0.1873380
rho_79 0.1406515 0.1651036 0.2103375 0.2433964
rho_89 0.8850675 0.8954339 0.8989220 0.9048678
rho_110 0.2277795 0.2233137 0.2190017 0.2146313
rho_210 0.3587042 0.3606548 0.3444764 0.3231322
rho_310 0.8612577 0.8696825 0.8780543 0.8789530
rho_410 0.8013970 0.8091754 0.8243439 0.8419843
rho_510 0.4966583 0.4971298 0.4958759 0.4985985
rho_610 0.0677998 0.0593673 0.0455917 0.0313605
rho_710 0.1012489 0.1016181 0.0967266 0.0908839
rho_810 -0.6822559 -0.7077478 -0.7347179 -0.7490808
rho_910 -0.6834668 -0.6871258 -0.6932972 -0.6997174

Original data (without transformation)

data(toxicity)
X <- as.matrix(toxicity)
betas <- c(0, 0.1, 0.3, 0.5)
d <- ncol(X)
pairs <- which(upper.tri(matrix(0, d, d)), arr.ind=TRUE)
result <-matrix(NA, nrow = d*(d+3)/2, ncol=length(betas))
for (i in seq_along(betas)){
  res <- mvnormDPD(X, betas[i], method = "composite")
  cor_matrix <- res$rho
  result[,i] <- c(res$mu, res$sigma,
    cor_matrix[upper.tri(cor_matrix, diag = FALSE)])
}
colnames(result) <- c("ML/MCL", "CDPD(0.1)", "CDPD(0.3)", "CDPD(0.5)")
rownames(result) <- c(paste0("mu_", 1:10), paste0("sigma2_", 1:10), paste0("rho_", pairs[,1], pairs[,2]))

Table S2

kable(result)
ML/MCL CDPD(0.1) CDPD(0.3) CDPD(0.5)
mu_1 -0.1557895 -0.1755423 -0.2067308 -0.2356793
mu_2 1.6668421 1.6348539 1.6644823 1.6129076
mu_3 0.6489342 0.6444936 0.9038378 0.8953938
mu_4 4.3442105 4.3855271 4.6532383 4.6340566
mu_5 17.1908263 17.1778033 17.2250379 17.2222028
mu_6 3.1191263 2.9800565 2.8412039 2.6743840
mu_7 34.2747368 33.8515598 33.4611542 33.1061037
mu_8 1.4457632 1.4490532 1.4488896 1.4530081
mu_9 38.1447368 38.2452858 32.9937452 33.2497057
mu_10 6.8075526 1.4589594 1.4566000 1.4568976
sigma2_1 0.1228241 0.1648587 0.1612789 0.1671877
sigma2_2 1.3394439 1.8511166 1.9776046 2.1979217
sigma2_3 0.1480713 0.2084506 0.0110871 0.0129568
sigma2_4 0.5302261 0.6078094 0.0633034 0.0703075
sigma2_5 0.3379609 0.4466209 0.4212268 0.4359295
sigma2_6 6.9743331 9.2346262 9.3477467 9.5926329
sigma2_7 114.7460143 151.4431450 148.8149667 155.4520783
sigma2_8 0.0003998 0.0005507 0.0006251 0.0006260
sigma2_9 69.6179164 87.8339734 2.0839588 2.5158479
sigma2_10 31.9733432 0.0002870 0.0004049 0.0004736
rho_12 0.8229998 0.8420820 0.9193282 0.9428273
rho_13 -0.0961598 -0.0990495 0.5117906 0.5249982
rho_23 0.1509435 0.1610507 0.4244994 0.4204528
rho_14 -0.0024717 0.0575391 0.5727432 0.6138614
rho_24 0.3692350 0.3608647 0.4810635 0.4884879
rho_34 0.5684439 0.5881605 0.8623857 0.8576459
rho_15 0.4348867 0.4529446 0.5219870 0.6006378
rho_25 0.7514938 0.7481595 0.7289631 0.7301583
rho_35 0.5595866 0.5669460 0.3834806 0.3956621
rho_45 0.6442904 0.6160204 0.3245595 0.3067424
rho_16 0.7225736 0.7364638 0.7433900 0.7535805
rho_26 0.8691874 0.8596619 0.8493307 0.8322227
rho_36 0.1144352 0.0905973 0.0555203 0.0248107
rho_46 0.1407676 0.1265196 0.0915442 0.0856556
rho_56 0.5896459 0.5852197 0.5896458 0.6041467
rho_17 0.5743438 0.5997138 0.6396515 0.6895283
rho_27 0.8261782 0.8285386 0.8310871 0.8338219
rho_37 0.2103176 0.1986373 0.0006108 -0.0022758
rho_47 0.2523277 0.2197944 -0.0341496 -0.0119319
rho_57 0.7600946 0.7652160 0.7677910 0.7817935
rho_67 0.8920588 0.8922827 0.8856304 0.8894553
rho_18 -0.0432016 -0.0738186 -0.1325072 -0.2293875
rho_28 -0.0044649 -0.0008877 0.0143843 -0.0479339
rho_38 -0.3695229 -0.3694220 -0.6553771 -0.6197531
rho_48 -0.3416130 -0.4258811 -0.7628937 -0.7947009
rho_58 -0.0051047 -0.0071040 -0.0199242 -0.0553162
rho_68 0.2452322 0.2129706 0.2087165 0.1227525
rho_78 0.4332253 0.4028166 0.3843388 0.3094479
rho_19 -0.4622954 -0.4468673 0.1305310 0.0057667
rho_29 -0.5045037 -0.4794080 0.1150527 0.0354298
rho_39 -0.2017107 -0.2118809 -0.7010589 -0.7171404
rho_49 -0.4978286 -0.5733830 -0.4688995 -0.5425924
rho_59 -0.3915145 -0.3413993 -0.2678894 -0.2898946
rho_69 -0.2061051 -0.1632387 0.3772019 0.2951604
rho_79 -0.0600534 -0.0015708 0.2462240 0.2019507
rho_89 0.7446842 0.7449957 0.6435060 0.6369507
rho_110 0.3979104 -0.2413620 -0.2300900 -0.2111631
rho_210 0.5067174 0.0279085 0.0040479 0.0203725
rho_310 0.6019403 0.1496060 -0.2810203 -0.2372763
rho_410 0.4916877 -0.0891624 -0.6077092 -0.5921160
rho_510 0.5575355 0.3309931 0.2882333 0.3018304
rho_610 0.2760859 0.1245816 0.0745546 0.0597530
rho_710 0.1906814 0.4462666 0.4026010 0.3748937
rho_810 -0.5628562 0.8275556 0.7752512 0.8035794
rho_910 -0.5080758 0.6227186 0.0289097 0.0766466

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.