We consider a general monotone regression estimation where we allow for independent and dependent regressors. We propose a modification of the classical isotonic least squares estimator and establish its rate of convergence for the integrated L1-loss function. The methodology captures the shape of the data without assuming additivity or a parametric form for the regression function. Furthermore, the degree of smoothing is chosen automatically and no auxiliary tuning is required for the theoretical analysis. Some simulations and two real data illustrations complement the study of the proposed estimator.

We introduce the concept of local dyadic stationarity, to account for nonstationary time series, within the framework of Walsh-Fourier analysis. We define and study time-varying, dyadic, autoregressive, moving average (tvDARMA) models. It is proven that the general tvDARMA process can be approximated locally by either a time-varying dyadic moving average and a time-varying dyadic autoregressive processes.

MOTIVATION: Microarray experiments are now routinely used to collect large-scale time series data, for example to monitor gene expression during the cell cycle. Statistical analysis of this data poses many challenges, one being that it is hard to identify correctly the subset of genes with a clear periodic signature. This has lead to a controversial argument with regard to the suitability of both available methods and current microarray data. METHODS: We introduce two simple but efficient statistical methods for signal detection and gene selection in gene expression time series data. First, we suggest the average periodogram as an exploratory device for graphical assessment of the presence of periodic transcripts in the data. Second, we describe an exact statistical test to identify periodically expressed genes that allows one to distinguish periodic from purely random processes. This identification method is based on the so-called g-statistic and uses the false discovery rate approach to multiple testing. RESULTS: Using simulated data it is shown that the suggested method is capable of identifying cell-cycle-activated genes in a gene expression data set even if the number of the cyclic genes is very small and regardless the presence of a dominant non-periodic component in the data. Subsequently, we re-examine 12 large microarray time series data sets (in part controversially discussed) from yeast, human fibroblast, human HeLa and bacterial cells. Based on the statistical analysis it is found that a majority of these data sets contained little or no statistical significant evidence for genes with periodic variation linked to cell cycle regulation. On the other hand, for the remaining data the method extends the catalog of previously known cell-cycle-specific transcripts by identifying additional periodic genes not found by other methods. The problem of distinguishing periodicity due to generic cell cycle activity and to artifacts from synchronization is also discussed. AVAILABILITY: The approach has been implemented in the R package GeneTS available from http://www.stat.uni-muenchen.de/~strimmer/software.html under the terms of the GNU General Public License.

Abstract Motivation: Microarray experiments are now routinely used to collect large-scale time series data, for example to monitor gene expression during the cell cycle. Statistical analysis of this data poses many challenges, one being that it is hard to identify correctly the subset of genes with a clear periodic signature. This has lead to a controversial argument with regard to the suitability of both available methods and current microarray data. Methods: We introduce two simple but efficient statistical methods for signal detection and gene selection in gene expression time series data. First, we suggest the average periodogram as an exploratory device for graphical assessment of the presence of periodic transcripts in the data. Second, we describe an exact statistical test to identify periodically expressed genes that allows one to distinguish periodic from purely random processes. This identification method is based on the so-called g-statistic and uses the false discovery rate approach to multiple testing. Results: Using simulated data it is shown that the suggested method is capable of identifying cell-cycle-activated genes in a gene expression data set even if the number of the cyclic genes is very small and regardless the presence of a dominant non-periodic component in the data. Subsequently, we re-examine 12 large microarray time series data sets (in part controversially discussed) from yeast, human fibroblast, human HeLa and bacterial cells. Based on the statistical analysis it is found that a majority of these data sets contained little or no statistical significant evidence for genes with periodic variation linked to cell cycle regulation. On the other hand, for the remaining data the method extends the catalog of previously known cell-cycle-specific transcripts by identifying additional periodic genes not found by other methods. The problem of distinguishing periodicity due to generic cell cycle activity and to artifacts from synchronization is also discussed. Availability: The approach has been implemented in the R package GeneTS available from http://www.stat.uni-muenchen.de/~strimmer/software.html under the terms of the GNU General Public License.