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```
# /*
# This is an R script containing R markdown comments. It can be run as is in R.
# To generate a document containing the formatted R code, R output and markdown
# click the "Compile Notebook" button in R Studio, or run the command
# rmarkdown::render() - see http://rmarkdown.rstudio.com/r_notebook_format.html
# */
#' ---
#' title: "Analysis of SRBCT Data"
#' output: pdf_document
#' author: ""
#' date: Requires "sda" in version 1.3.2 (January 2014) or later
#' ---
#'
#' # Load "sda" package and create SRBCT data set
library("sda")
#' Load data set from Khan et al. (2001):
data(khan2001)
#' Create data set containing only the SRBCT samples:
del.idx = which( khan2001$y == "non-SRBCT" )
srbct.x = khan2001$x[-del.idx,]
srbct.y = factor(khan2001$y[-del.idx])
dim(srbct.x)
#' Four subtypes of cancer:
levels(srbct.y)
#' Divide into training and test data
Xtrain = srbct.x[1:63,]
Ytrain = srbct.y[1:63]
Xtest = srbct.x[64:83,]
Ytest = srbct.y[64:83]
#' \newpage
#'
#' # Diagonal Discriminant Analysis (DDA)
#' In DDA correlation among predictors is assumed to be zero, i.e. a diagonal
#' covariance matrix is used.
#'
#' ## Step 1 - feature ranking
#' As there are more than two groups in the response there are three different
#' ways to obtain a summary test statistic to rank genes:
#' a) ranking by averaged squared t-scores across the four groups
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, diagonal=TRUE, ranking.score="avg")
sum( ra[, "lfdr"]< 0.80) # 97 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 145 genes according to HC criterion
#' b) ranking by maximum of squared t-scores across the four groups
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, diagonal=TRUE, ranking.score="max")
sum( ra[, "lfdr"]< 0.80) # 78 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 121 genes according to HC criterion
#' c) ranking by mutual information (weighted sum of squared t-scores)
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, diagonal=TRUE, ranking.score="entropy")
sum( ra[, "lfdr"]< 0.80) # 99 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 158 genes according to HC criterion
#' here we pick the top 99 genes of option c)
#+ fig.height=8
plot(ra, top=99, main="The 99 Top Ranking Genes", ylab="Gene ID")
#' Select these 99 variables:
idx = ra[1:99,"idx"]
Xtrain2 = Xtrain[,idx]
Xtest2 = Xtest[,idx]
#'
#' ## Step 2 - training the classifier
#' Learn DDA predictor:
sda.fit = sda(Xtrain2, Ytrain, diagonal=TRUE)
#'
#' ## Step 3 - prediction
#' Predict class labels from test data and compare with known labels:
dim(Xtest2)
predict(sda.fit, Xtest2)
ynew = predict(sda.fit, Xtest2)$class
#' Number of missclassified test samples:
sum(ynew != Ytest)
# /* 1 */
#' \newpage
#'
#' # Linear Discriminant Analysis (LDA)
#' In LDA correlation among predictors is taken into account.
#'
#' ## Step 1 - feature ranking
#' As there are more than two groups in the response there are three different
#' ways to obtain a summary test statistic to rank genes:
#' a) ranking by averaged squared cat-scores across the four groups
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, ranking.score="avg")
sum( ra[, "lfdr"]< 0.80) # 93 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 143 genes according to HC criterion
#' b) ranking by maximum of squared cat-scores across the four groups
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, ranking.score="max")
sum( ra[, "lfdr"]< 0.80) # 156 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 194 genes according to HC criterion
#' c) ranking by mutual information (weighted sum of squared cat-scores)
#+ fig.height=5
ra = sda.ranking(Xtrain, Ytrain, fdr=TRUE, plot.fdr=TRUE, ranking.score="entropy")
sum( ra[, "lfdr"]< 0.80) # 97 genes included in classifier (by FNDR control)
which.max( ra[, "HC"] ) # 140 genes according to HC criterion
#' here we pick the top 97 genes of option c)
#+ fig.height=8
plot(ra, top=97, main="The 97 Top Ranking Genes", ylab="Gene ID")
#' Select these 97 variables:
idx = ra[1:97,"idx"]
Xtrain2 = Xtrain[,idx]
Xtest2 = Xtest[,idx]
#'
#' ## Step 2 - training the classifier
#' Learn LDA predictor:
sda.fit = sda(Xtrain2, Ytrain)
#'
#' ## Step 3 - prediction
#' Predict class labels from test data and compare with known labels:
dim(Xtest2)
predict(sda.fit, Xtest2)
ynew = predict(sda.fit, Xtest2)$class
#' Number of missclassified test samples:
sum(ynew != Ytest)
# /* 0 */
#' \newpage
#'
#' # Estimate prediction accuracy using crossvalidation
#' Using crossvalidation we can estimate the prediction error
#' from the training data set alone.
library("crossval")
#' Setup prediction function: estimate the accuracy of a predictor with a fixed number of predictors (note
#' this takes into account the uncertainty in estimating the variable ordering).
predfun = function(Xtrain, Ytrain, Xtest, Ytest, numVars, diagonal=FALSE,
ranking.score="entropy")
{
# estimate ranking and determine the best numVars variables
ra = sda.ranking(Xtrain, Ytrain, verbose=FALSE, diagonal=diagonal,
fdr=FALSE, ranking.score=ranking.score)
selVars = ra[,"idx"][1:numVars]
# fit and predict
sda.out = sda(Xtrain[, selVars, drop=FALSE], Ytrain, diagonal=diagonal,
verbose=FALSE)
ynew = predict(sda.out, Xtest[, selVars, drop=FALSE], verbose=FALSE)$class
# compute accuracy
acc = mean(Ytest == ynew)
return(acc)
}
#' Our setup for crossvalidation:
K = 10 # number of folds
B = 20 # number of repetitions
#' Crossvalidation estimate of accuracy for
#' LDA using the top 100 features ranked by CAT scores
#' (combined across groups using "entropy" for overall ranking):
set.seed(12345)
cv.lda100 = crossval(predfun, Xtrain, Ytrain, K=K, B=B, numVars=100,
diagonal=FALSE, verbose=FALSE)
cv.lda100$stat
# /* 1 */
#'
#' ## Comparison of LDA / DDA and "entropy" and "max" options
#' LDA using the top 10 features ranked by CAT scores
#' (combined across groups using "entropy" for overall ranking):
set.seed(12345)
cv.lda10 = crossval(predfun, Xtrain, Ytrain, K=K, B=B, numVars=10,
diagonal=FALSE, verbose=FALSE)
cv.lda10$stat
# /* 0.9909762 */
#' DDA using the top 10 features ranked by t scores
#' (combined across groups using "entropy" for overall ranking):
set.seed(12345)
cv.dda10 = crossval(predfun, Xtrain, Ytrain, K=K, B=B, numVars=10,
diagonal=TRUE, verbose=FALSE)
cv.dda10$stat
# /* 0.9643869 */
#' DDA using the top 10 features ranked by t scores,
#' (combined across groups using "max" for overall ranking, as in PAM):
set.seed(12345)
cv.dda10b = crossval(predfun, Xtrain, Ytrain, K=K, B=B, numVars=10,
diagonal=TRUE, ranking.score="max", verbose=FALSE)
cv.dda10b$stat
# /* 0.9585595 */
#' **Conclusions**:
#'
#' 1. LDA/CAT score ranking performs petter than DDA/t-score ranking.
#' 2. "entropy" is better as group summary than "max".
```

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