probFDA (version 1.0.1)

pfda: A Probabilistic Version of Fisher Linear Discriminant Analysis

Description

Probabilistic Fisher discriminant analysis (pFDA) is a probabilistic version of the popular and powerful Fisher linear discriminant analysis for dimensionality reduction and classification. PFDA overcomes the known limitations of FDA in the contexts of label noise and sparse labeled data. To this end, pFDA relaxes the homoscedastic assumption on the class covariance matrices and adds a term to explicitly model the non-discriminative information. The pFDA method works at least as well as the traditional FDA method (even better in most cases) in standard situations and it clearly improves the modeling and the prediction when the dataset is subject to label noise and/or sparse labels. The practitioner may therefore replace without prejudice FDA by pFDA for its daily use.

Usage

pfda(Y, cls, model = "AkjBk", crit = "bic", cv.fold = 10, kernel = "", display = FALSE)

Arguments

Y

The data.

cls

The labels.

model

The model name. Possible model names are 'DkBk', 'DkB', 'DBk', 'DB', 'AkjBk', 'AkjB', 'AkBk', 'AkB', 'AjBk', 'AjB', 'ABk', 'AB' or 'all'. The 'all' option allows to try all models. The default is 'AkjBk'.

crit

The model selection criteria. Possible criteria are BIC ('bic') or cross-validation ('cv'). The default is 'bic'.

cv.fold

In case of cross-validation, the number of cross-validation folds.

kernel

The kernel option allows to do the classification in a feature space generated by a kernel. This could be especially useful when n < p. The possible options are '', 'linear', 'sigmoid' or 'rbf'. The default is no kernel ('').

display

A display option. The default is FALSE.

Value

A list of class pfda is returned:

prms

All estimated model parameters.

V

The estimated loading matrix.

bic

The BIC value.

ll

The log-likelihood value.

K

The estimated loading matrix.

References

C. Bouveyron and C. Brunet, Probabilistic Fisher discriminant analysis: A robust and flexible alternative to Fisher discriminant analysis, Neurocomputing, vol. 90 (1), pp. 12-22, 2012.

See Also

lda

Examples

palette(c("#E41A1C","#377EB8","#4DAF4A"))

# Simulation of data
n = 900; p = 25
n1 = 1/3*n; n2 = 1/3*n; n3 = 1/3*n; 
S1 = diag(2)
S2 = rbind(c(1,-0.95),c(-0.95,1))
S3 = rbind(c(2,0),c(0,0.05))
m1 = c(0,0); m2 = c(0,2); m3 = c(2,0)
X = rbind(mvrnorm(n1,m1,S1),mvrnorm(n2,m2,S2),mvrnorm(n3,m3,S3))
Q = qr.Q(qr(mvrnorm(p,mu=rep(0,p),Sigma=diag(25,p))))
B = mvrnorm(nrow(X),rep(0,p-2),0.1*diag(rep(p-2,p-2)))
X = crossprod(t(cbind(X,B)),Q)
cls = rep(c(1,2,3),c(n1,n2,n3))

# Cross-validation
nbrep = 10
CCR = matrix(NA,2,nbrep)
for (i in 1:nbrep){
  ind = sample(n)[1:(3/5*n)]
  lda.c = lda(X[ind,],cls[ind])
  res = predict(lda.c,X[-ind,])
  CCR[1,i] = sum(res$cl==cls[-ind])/length(cls[-ind])
  prms = pfda(X[ind,],cls[ind],model=c('DkBk','DB','AkB','AB'),crit='bic',display=TRUE)
  res = predict(prms,X[-ind,])
  CCR[2,i] = sum(res$cl==cls[-ind])/length(cls[-ind])
}

# Display results
split.screen(c(2,1))
split.screen(c(1,3), screen = 1)
screen(3)
plot(predict(princomp(X)),col=cls,pch=(17:19)[cls],main='PCA')
screen(4)
plot(crossprod(t(X),lda(X,cls)$scaling),col=cls,pch=(17:19)[cls],main='LDA')
screen(5)
plot(crossprod(t(X),pfda(X,cls,model='DkBk')$V),col=cls,pch=(17:19)[cls],main='PFDA',
  xlab='LD1',ylab='LD2')
screen(2)
boxplot(t(CCR),names=c('LDA','PFDA'),col=c(1,2),ylab="CCR",
  main='CV correct classification rate')