| Title: | Significance of periodic expression pattern in time-series data |
|---|---|
| Description: | Package for assessing the statistical significance of periodic expression based on Fourier analysis and comparison with data generated by different background models |
| Authors: | Matthias Futschik <[email protected]> |
| Maintainer: | Matthias Futschik <[email protected]> |
| License: | GPL-2 |
| Version: | 1.59.0 |
| Built: | 2024-07-24 04:43:18 UTC |
| Source: | https://github.com/bioc/cycle |
Calculation of the autocorrelation coefficients genes and variance of corresponding random variables to fit gene expression time series by AR1 processes
ar1analysis(eset)ar1analysis(eset)
eset |
object of the class “ExpressionSet” |
List of fitted autocorrelation coefficients (alpha)
for ExpressionSet features and variance (sigma2)
of corresponding random variables obtained using the ar function
of the stats package.
Note that this function evaluates soley the exprs matrix and
no information is used from the phenoData. In particular,
the ordering of samples (arrays) is the same as the ordering
of the columns in the exprs matrix. Also, replicated arrays in the
exprs matrix are treated as independent
i.e. they should be averagered prior to analysis or placed into different
distinct “ExpressionSet” objects.
data(yeast) # loading the reduced CDC28 yeast set (from the Mfuzz package) # Data preprocessing if (interactive()){ data(yeast) 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 replacing missing values tmp <- ar1analysis(yeast) # fits AR1 process autocorrelation coefficients plot(density(tmp$alpha),main="Autocorrelation") }data(yeast) # loading the reduced CDC28 yeast set (from the Mfuzz package) # Data preprocessing if (interactive()){ data(yeast) yeast <- filter.NA(yeast) # filters genes with more than 25% of the expression values missing yeast <- fill.NA(yeast) # for illustration only; rather use knn method for replacing missing values tmp <- ar1analysis(yeast) # fits AR1 process autocorrelation coefficients plot(density(tmp$alpha),main="Autocorrelation") }
The function generates background expression sets using different methods (permutation within rows, Gaussian distribution, auto-regressive models)
backgroundData(eset,model=c("rr", "gauss", "ar1"))backgroundData(eset,model=c("rr", "gauss", "ar1"))
eset |
object of the class “ExpressionSet” |
model |
model for generation of background data: “rr”- random permutation, “gauss”- Gaussian and “ar1”- AR1 models |
Microarray data comprise the measurements of transcript levels for many thousands of genes. Due to the large number of genes, it can be expected that some genes show periodicity simply by chance. To assess therefore the significance of periodic signals, it is necessary first to define what distribution of signals can be expected if the studied process exhibits no true periodicity. In statistical terms this is equivalent with the definition of a null hypothesis of non-periodic expression.
The most simple model for non-periodic expression is based on randomization of the observed times series. A background distribution can then be constructed by (repeated) random permutation of the sequentially ordered measurements in the experiment. This background model is used here if model="rr" is chosen.
Alternatively, non-periodic expression can be derived using a statistical model. A conventional approach is based on the assumption of data normality and to use the normal distribution. This background model is chosen if model="gauss".
However, these two approaches neglect the fact that time series data exhibit generally a considerable autocorrelation i.e. correlation between successive measurements. Therefore, neither the assumptions of data normality nor for randomizations may hold. As demonstrated for yeast cell cycle data (Bioinformatics 2008), this failure can substantially interfere with the significance testing, and that neglecting autocorrelation can potentially lead to a considerable overestimation of the number of periodically expressed genes.
A more suitable model is based on autoregressive processes of order one (AR(1)),
for which the value of the time-dependent variable X depends on its previous value up to a normally distributed random variable Z. Such model is used here for the setting of
model="ar1". The autocorrelation of X and variance of Z is estimated for each feature of the ExpressionSet object separately. Mathematical details can be found in the given reference.
It is important to note in this context, that AR(1) processes cannot capture periodic patterns except for alternations with period two. Since Z is a random variable, we can readily generate a collection of time series with the same autocorrelation as in the original data set. Therefore, although AR(1) processes constitute random processes, they allow us to construct a background distribution that captures the autocorrelation structure of original gene expression time series without fitting the potentially included periodic pattern.
ExpressionSet object with expression data generated by the chosen background model
Note that this function evaluates soley the exprs matrix and
no information is used from the phenoData. In particular,
the ordering of samples (arrays) is the same as the ordering
of the columns in the exprs matrix. Also, replicated arrays in the
exprs matrix are treated as independent
i.e. they should be averagered prior to analysis or placed into different
distinct “ExpressionSet” objects.
Matthias E. Futschik (http://www.cbme.ualg.pt/mfutschik_cbme.html)
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, Bioinformatics, 24(8):1063-1069
The function calculates the empirical FDR based
on derived Fourier scores derived by fourierscore for the
observed expression and the comparison with scores derived for different background model generated by backgroundData.
fdrfourier(eset,T,times,background.model="rr",N=100,progress=FALSE)fdrfourier(eset,T,times,background.model="rr",N=100,progress=FALSE)
eset |
object of the class “ExpressionSet” |
T |
cycle period |
times |
time of measurements |
background.model |
model for generation of background data: “rr”- permutation within rows, “gauss”- Gaussian background, “ar1”- AR1 models |
N |
number of generated data sets for the background distribution |
progress |
if set to TRUE, a progress of calculations is reported |
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.
List with FDR for the features of the eset object (fdr),
and Fourier scores for ExpressionSet object (F) and
the background data (F.b).
This is the main function of the 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 exprs matrix and
no information is used from the phenoData. In particular,
the ordering of samples (arrays) is the same as the ordering
of the columns in the exprs matrix. Also, replicated arrays in the
exprs matrix are treated as independent
i.e. they should be averagered prior to analysis or placed into different
distinct “ExpressionSet” objects.
Matthias E. Futschik (http://www.cbme.ualg.pt/mfutschik_cbme.html)
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, Bioinformatics, 24(8):1063-1069
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)] }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)] }
The function calculates fourier score for a chosen periodicity.
fourierscore(eset,T,times=seq_len(ncol(eset)))fourierscore(eset,T,times=seq_len(ncol(eset)))
eset |
object of the class “ExpressionSet” |
T |
length of chosen period |
times |
measurement times (with the index of the column as default) |
The Fourier score can be used to detect periodic signals.
The closer a gene's expression follows a (possibly shifted) cosine curve of period T, the larger is the Fourier score. Mathematical details can be found in the given reference. The function fourierscore calculates the Fourier scores for all features of an ExpressionSet object.
Fourier scores for all features (genes) of eset
Note that this function evaluates soley the exprs matrix and
no information is used from the phenoData. In particular,
the ordering of samples (arrays) is the same as the ordering
of the columns in the exprs matrix. Also, replicated arrays in the
exprs matrix are treated as independent
i.e. they should be averagered prior to analysis or placed into different
distinct “ExpressionSet” objects.
Matthias E. Futschik (http://www.cbme.ualg.pt/mfutschik_cbme.html)
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, Bioinformatics, 24(8):1063-1069
# Data preprocessing if (interactive()){ set.seed(1) data(yeast) # loading the reduced CDC28 yeast set (from the Mfuzz package) yeast <- yeast[1:600,] # for illustration 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)$time # time of measurements F <- fourierscore(yeast,T = T.yeast, times= times.yeast) # calculates the Fourier scores for yeast genes # Plot highest scoring gene plot(times.yeast,exprs(yeast)[order(-F)[1],],type="o", xlab="Time",ylab="Expression", main=featureNames(yeast)[order(-F)[1]]) # Plot lowest scoring gene plot(times.yeast,exprs(yeast)[order(F)[1],],type="o", xlab="Time",ylab="Expression", main=featureNames(yeast)[order(F)[1]]) }# Data preprocessing if (interactive()){ set.seed(1) data(yeast) # loading the reduced CDC28 yeast set (from the Mfuzz package) yeast <- yeast[1:600,] # for illustration yeast <- filter.NA(yeast) # filters genes with more than 25% of the expression values missing yeast <- fill.NA(yeast) # for illustration only; rather use knn method for yeast <- standardise(yeast) T.yeast <- 85 # cell cycle period (t=85min) times.yeast <- pData(yeast)$time # time of measurements F <- fourierscore(yeast,T = T.yeast, times= times.yeast) # calculates the Fourier scores for yeast genes # Plot highest scoring gene plot(times.yeast,exprs(yeast)[order(-F)[1],],type="o", xlab="Time",ylab="Expression", main=featureNames(yeast)[order(-F)[1]]) # Plot lowest scoring gene plot(times.yeast,exprs(yeast)[order(F)[1],],type="o", xlab="Time",ylab="Expression", main=featureNames(yeast)[order(F)[1]]) }