transfo examples"
In cellWise: Analyzing Data with Cellwise Outliers

knitr::opts_chunk$set(
  fig.width  = 5,
  fig.height = 5,
  fig.align  = 'center'
)

Introduction

This file contains some examples of the transfo function to robustly transform variables toward central normality

library(cellWise)
library(robustHD) # for the TopGear data

Small toy example

We start with a small toy example, in which standard normal data with one NA is used.

set.seed(1)
X = rnorm(100) 
X[50] = NA
qqnorm(X)

We specify the transformation type as "bestObj", which chooses between Box-Cox and Yeo-Johnson based on the value of the objective function. When there are negative values, Yeo-Johnson is chosen automatically. First, the classical MLE transformation is fit, indicated by the robust = FALSE argument:

ML.out <- transfo(X, type = "bestObj", robust=FALSE)
ML.out$lambdahat

The value of the transformation parameter is close to 1, indicating no transformation is necesarry. Among the other outputs are the value of the objective function, the transformed data, estimates of the location and scale of the transformed data:

ML.out$objective
ML.out$Y[45:55] 
ML.out$muhat[1:20] 
ML.out$sigmahat[1:20] 
qqnorm(ML.out$Y); abline(0,1)

Additionally, the function returns the weights of the observations and the transformation type that was used. The NA observation (number 50) gets weight zero:

plot(ML.out$weights)
ML.out$ttypes[1:20]

We can now repeat the same procedure with the robust estimator. This is specified using the robust = TRUE argument. The results do not differ too much from the classical MLE estimates on this dataset, since there are no outliers.

RewML.out <- transfo(X, type = "bestObj", robust=TRUE)
RewML.out$lambdahat 
RewML.out$objective 
RewML.out$Y[45:55]
RewML.out$muhat 
RewML.out$sigmahat 
qqnorm(RewML.out$Y); abline(0,1)

Using the robust estimator, one large observation (number 61) also gets a weight of 0:

plot(RewML.out$weights) 
X[61] 
RewML.out$ttypes

In order to illustrate the same concepts for the Box-Cox transformation, we take the exponential of the generated data. This yields a sample from a log-normal distribution.

X = exp(X)

We first consider the classical maximum likelihood estimator. We obtain a transformation parameter close to 0, which corresponds with the logarithmic transformation.

ML.out <- transfo(X, type = "BC", robust=FALSE)
ML.out$lambdahat 
ML.out$objective 
ML.out$Y[45:55]  
qqnorm(ML.out$Y); abline(0,1)

The weights are again all equal to 1, except for the NA (number 50) which gets weight zero:

plot(ML.out$weights) 
ML.out$ttypes

We now repeat the example with the robust transformation. The estimated transformation parameter is again fairly close to 0, as expected.

RewML.out <- transfo(X, type = "bestObj", robust=TRUE)
RewML.out$lambdahat 
RewML.out$objective 
RewML.out$Y[45:55] 
qqnorm(RewML.out$Y); abline(0,1)
RewML.out$ttypes

Transform new data and transform back

We briefly illustrate the functions transfo_newdata() and tranfo_transformback. We start with transfo_newdata(). Given the output of transfo(), this function applies the fitted transformation to new data.

set.seed(123); tempraw <- matrix(rnorm(2000), ncol = 2)
tempx <- cbind(tempraw[, 1],exp(tempraw[, 2]))
tempy <- 0.5 * tempraw[, 1] + 0.5 * tempraw[, 2] + 1
x <- tempx[1:900, ]
y <- tempy[1:900]
tx.out <- transfo(x, type = "bestObj")
tx.out$ttypes
tx.out$lambdahats
tx <- tx.out$Y
lm.out <- lm(y ~ tx)
summary(lm.out)
xnew <- tempx[901:1000, ]
xtnew <- transfo_newdata(xnew, tx.out)
yhatnew <- cbind(1, xtnew) %*% lm.out$coefficients 
plot(tempy[901:1000], yhatnew); abline(0, 1)

We now illustrat the transfo_transformback() function. Given a transformation fitted by transfo(), this function applies the inverse of this transformation to the input.

set.seed(123); x <- matrix(rnorm(2000), ncol = 2)
y <- sqrt(abs(0.3 * x[, 1] + 0.5 * x[, 2] + 4))
ty.out <- transfo(y, type = "BC")
ty.out$lambdahats
ty <- ty.out$Y
lm.out <- lm(ty ~ x)
yhat <- transfo_transformback(lm.out$fitted.values, ty.out)
plot(y, yhat); abline(0, 1)

TopGear example

This dataset is part of the robustHD package. We use two of its variables to illustrate the effect outliers can have on estimating a transformation using maximum likelihood. The robust estimators yield more reasonable results.

data(TopGear)

First, we do some preprocessing to avoid zeroes.

CD.out   <- checkDataSet(TopGear)
colnames(CD.out$remX)
# remove the subjective variable `Verdict':
X        <- CD.out$remX[,-12] 
carnames <- TopGear[CD.out$rowInAnalysis, 1:2]
X        <- pmax(X, 1e-8) # avoids zeroes

Now we estimate transformation parameters (takes under a second) using classical MLE as well as the reweighted MLE which is much more robust.

ML.out    <- transfo(X, type = "BC", robust=F)
RewML.out <- transfo(X, type = "BC", robust=T)

Now consider the transformation of the variable MPG.

ML.out$lambdahat[7] 
RewML.out$lambdahat[7]

The classical method estimates a transformation parameter of -0.11, which is close to the log transform. The robust method barely transforms the original variable with a transformation parameter of 0.84, relatively close to 1. The QQplots below suggest that the robust transformation is much more reasonable:

qqnorm(X[, 7], main = "", cex.lab = 1.5, cex.axis = 1.5)

qqnorm(ML.out$Y[, 7], main = "Classical transform",
       cex.lab = 1.5, cex.axis = 1.5)
abline(0, 1)

qqnorm(RewML.out$Y[, 7], main = "Robustly transformed",
       cex.lab = 1.5, cex.axis = 1.5)
abline(0, 1)

Now consider the transformation of the variable Weight.

ML.out$lambdahat[8] 
RewML.out$lambdahat[8]

Here, the situation is reversed: the classical estimator finds a transformation parameter (0.83) relatively close to one, whereas the robust transformation finds a parameter (0.09) close to the log transform. The QQplots below again suggest that the robust transformation is much more reasonable:

qqnorm(X[, 8], main = "Original variable", 
       cex.lab = 1.5, cex.axis = 1.5)

qqnorm(ML.out$Y[, 8], main = "Classical transform", 
       cex.lab = 1.5, cex.axis = 1.5)
abline(0, 1)

qqnorm(RewML.out$Y[, 8], main = "Robustly transformed", 
       cex.lab = 1.5, cex.axis = 1.5)
abline(0, 1)

The classical transformation does not fit as well as it attempts to push the 5 outliers into the fold at the expense of creating skewness in the center of the data.

Glass data example

In this example we study the glass data. It consists of spectra with 750 wavelengths of 180 archaeological glass samples.

data("data_glass")

First, we do some preprocessing to avoid variables with very small scales (in fact, the first 13 variables have mad equal to zero). Afterwards, we focus on the first 500 wavelengths since this is where most of the activity occurs.

X <- as.matrix(data_glass[, -c(1:13)])
X <- X[, 1:500]
Z <- scale(X, center=FALSE, robustbase::colMedians(X))
dim(Z)

Now we estimate transformation parameters using classical MLE as well as the reweighted MLE which is much more robust. This only takes a few seconds for the 500 variables.

ML.out    <- transfo(Z, type = "YJ", robust=F)
RewML.out <- transfo(Z, type = "YJ", robust=T)

We now construct the cellmaps:

indcells_clas = which(abs(ML.out$Y) > sqrt(qchisq(0.99, 1)))
indcells_rob  = which(abs(RewML.out$Y) > sqrt(qchisq(0.99, 1)))
n = dim(ML.out$Y)[1]
d = dim(ML.out$Y)[2]; d
nrowsinblock = 5
rowlabels = rep("", floor(n/nrowsinblock));
ncolumnsinblock = 5
columnlabels = rep("",floor(d/ncolumnsinblock));
columnlabels[3] = "1";
columnlabels[floor(d/ncolumnsinblock)] = "500"

CM_clas = cellMap(ML.out$Y, 
                  nrowsinblock = nrowsinblock, 
                  ncolumnsinblock = ncolumnsinblock,
                  rowblocklabels = rowlabels,
                  columnblocklabels = columnlabels,
                  mTitle = "YJ transformed variables by ML",
                  rowtitle = "", columntitle = "wavelengths",
                  columnangle = 0) 
plot(CM_clas)

CM_rob = cellMap(RewML.out$Y, 
                 nrowsinblock = nrowsinblock, 
                 ncolumnsinblock = ncolumnsinblock, 
                 rowblocklabels = rowlabels,
                 columnblocklabels = columnlabels,
                 mTitle = "YJ transformed variables by RewML",
                 rowtitle = "", columntitle = "wavelengths",
                 columnangle = 0) 
plot(CM_rob)

# pdf("Glass_YJ_ML_RewML.pdf",width=8,height=8)
# gridExtra::grid.arrange(CM_clas, CM_rob,ncol=1)
# dev.off()

DPOSS data example

As a last example, we analyze the DPOSS data. This is a random subset of 20'000 stars from the Digitized Palomar Sky Survey described by Odewahn et al (1998).

data("data_dposs") # in package cellWise
n = nrow(data_dposs); n 
ncol(data_dposs)

There are lots of missing values in this dataset. Therefore, we first do some preprocessing to select the band of wavelengths which contain the fewest missing values.

missmat = is.na(data_dposs)
sizemat = nrow(missmat)*ncol(missmat); sizemat 
100*sum(as.vector(missmat))/sizemat 

missrow = length(which(rowSums(missmat) > 0))
100*missrow/nrow(missmat) 

# Missingness by band:

# F band:
300*sum(as.vector(missmat[,1:7]))/sizemat           
100*length(which(rowSums(missmat[,1:7]) > 0))/20000 

# J band:
300*sum(as.vector(missmat[,8:14]))/sizemat           
100*length(which(rowSums(missmat[,8:14]) > 0))/20000  

# N band:
300*sum(as.vector(missmat[,15:21]))/sizemat           
100*length(which(rowSums(missmat[,15:21]) > 0))/20000  

# So typically the whole band is missing or not.
# We focus on the J band which has the most available rows.

indx = which(rowSums(missmat[,8:14]) ==0)
dpossJ = data_dposs[indx,8:14]
dim(dpossJ)

A quick exploration of the skewness in the data shows that there is skewness in both directions:

par(mfrow = c(1, 1))
boxplot(scale(dpossJ))

Now we fit the Yeo-Johnson transformation robustly and transform the data with the estimated parameters. The analysis shows that there are both lambdas larger than 1 and smaller than 1, corresponding with the skewness in both directions.

transfoJ_YJ <- transfo(dpossJ, type = "YJ", robust = T)
plot(transfoJ_YJ$lambdahat) 
dpossJ_YJ = transfoJ_YJ$Y

We now apply cellwise robust PCA to further analyze the data. We perform the PCA using both the original and the transformed data.

DDCPars = list(fastDDC=F,fracNA=0.5)
MacroPCAPars = list(DDCpars=DDCPars,scale=TRUE,silent=T)
MacroPCAdpossJ = MacroPCA(dpossJ,k=4,MacroPCApars=MacroPCAPars) 
MacroPCAdpossJ_YJ = MacroPCA(dpossJ_YJ,k=4,MacroPCApars=MacroPCAPars)

Let's make a plot of the scores of YJ-transformed data:

MacroPCAdpossJ_YJ$scores[, 1] <- -MacroPCAdpossJ_YJ$scores[, 1] 
cols <- rep("black", dim(MacroPCAdpossJ$scores)[1])
cols[c(98, 10894)] <- "darkorange"
cols[which(MacroPCAdpossJ_YJ$SD > 14)] <- "skyblue3"
cols[which(MacroPCAdpossJ_YJ$SD > 25)] <- "firebrick"

# pdf("dpossJ_scores_YJ.pdf")
pairs(MacroPCAdpossJ_YJ$scores,gap=0,main="",pch=19,col=cols)
# dev.off()

We finally show the outlier maps on the transformed and the untransformed data:

# pdf("dpossJ_outliermap_YJ.pdf",height=5,width=5)
outlierMap(MacroPCAdpossJ_YJ, title="Outlier map of transformed data", col=cols, labelOut=FALSE)
# dev.off()

# pdf("dpossJ_outliermap_rawdata.pdf",height=5,width=5)
outlierMap(MacroPCAdpossJ, title="Outlier map of raw data", col=cols, labelOut=FALSE)
# dev.off()