\name{fdrfourier}
\alias{fdrfourier}
\title{Calculation of the false discovery rates (FDR) for periodic expression}
\description{The function calculates the empirical FDR based 
on derived Fourier scores derived by \code{fourierscore} for the 
observed expression and  the comparison with  scores derived for different background model generated by \code{backgroundData}.
}
\usage{fdrfourier(eset,T,times,background.model="rr",N=100,progress=FALSE)}
\arguments{\item{eset}{object of the class \dQuote{ExpressionSet}}
  \item{T}{cycle period}
  \item{times}{time of measurements}
    \item{background.model}{model for generation of background
	     data: \dQuote{rr}- permutation within rows, 
		 \dQuote{gauss}- Gaussian background, 
              \dQuote{ar1}- AR1 models}
         \item{N}{number of generated data sets for the background distribution}
    \item{progress}{if set to TRUE, a progress of calculations is reported}
}

\details{To assess the significance of the Fourier score obtained for the original gene expression time series, the probability has to be calculated of how often such a score would be observed by chance based on the chosen background distribution. The statistical significance is given  by the calculated false discovery rate. It is defined here as the expected proportion of false positives among all genes detected as periodically expressed. Mathematical details can be found in the given reference. }

\value{List with FDR for the features of the eset object (\code{fdr}), 
       and Fourier scores for ExpressionSet object (\code{F}) and 
the background data (\code{F.b}).}

\note{This is the main function of the \code{cycle} package. Note that the calculation of FDR employing empirical background distributions can require considerable time (up to several days for large gene expression data sets).

Importantly, this function evaluates soley the \code{exprs} matrix and 
no information is used from the \code{phenoData}. In particular, 
the ordering of samples (arrays) is the same as the ordering 
of the columns in the \code{exprs} matrix. Also, replicated arrays in the 
\code{exprs} matrix are treated as independent 
i.e. they should be averagered prior to analysis or placed into different
distinct \dQuote{ExpressionSet} objects.} 

\author{Matthias E. Futschik (\url{http://www.cbme.ualg.pt/mfutschik_cbme.html})}

\references{Matthias E. Futschik and Hanspeter Herzel (2008) Are we overestimating the number of cell-cycling genes? The impact of background models on time series analysis, \emph{Bioinformatics}, 24(8):1063-1069
}

\examples{
if (interactive()){

set.seed(1)
data(yeast) # loading the reduced CDC28 yeast set (from the Mfuzz package)

# Data preprocessing 
yeast <- filter.NA(yeast) # filters genes with more than 25% of the expression values missing 
yeast  <- fill.NA(yeast) # for illustration only; rather use knn method for 
yeast <- standardise(yeast)
# 
T.yeast <- 85   # cell cycle period (t=85min)
times.yeast <-  pData(yeast)$time  # time of measurements
#
yeast.test <- yeast[1:600,] # To speed up the example
#

NN <- 50 # number of generated background models
         # Here, a small number was chosen for demonstration purpose.
	 # For the actual analysis, rather set N = 1000 
 
# Calculation of FDRs
# i) based on random permutation as background model
fdr.rr <- fdrfourier(eset=yeast.test,T=T.yeast,
                    times=times.yeast,background.model="rr",N=NN,progress=TRUE)
# ii) based on Gaussian distribution 
fdr.g <- fdrfourier(eset=yeast.test,T=T.yeast,
                   times=times.yeast,background.model="gauss",N=NN,progress=TRUE)
# iii) based on AR(1) models as background
fdr.ar1 <- fdrfourier(eset=yeast.test,T=T.yeast,
                     times=times.yeast,background.model="ar1",N=NN,progress=TRUE)


# Number of significant genes based on diff. background models
sum(fdr.rr$fdr < 0.1) 
sum(fdr.g$fdr < 0.1)
sum(fdr.ar1$fdr < 0.1)

# Plot top scoring gene
plot(times.yeast,exprs(yeast.test)[order(fdr.ar1$fdr)[1],],type="o",
     xlab="Time",ylab="Expression",
     main=paste(featureNames(yeast.test)[order(fdr.ar1$fdr)[1]],"-- FDR:",
       fdr.ar1$fdr[order(fdr.ar1$fdr)[1]]))

# List significant genes
fdr.ar1$fdr[which(fdr.ar1$fdr < 0.1)]
}
}

\keyword{htest}
\keyword{ts}