Konstantinos Fokianos

Identifying periodically expressed transcripts in microarray time series data

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.

Log-linear Poisson autoregression

We consider a log-linear model for time series of counts. This type of model provides a framework where both negative and positive association can be taken into account. In addition time dependent covariates are accommodated in a straightforward way. We study its probabilistic properties and maximum likelihood estimation. It is shown that a perturbed version of the process is geometrically ergodic, and, under some conditions, it approaches the non-perturbed version. In addition, it is proved that the maximum likelihood estimator of the vector of unknown parameters is asymptotically normal with a covariance matrix that can be consistently estimated. The results are based on minimal assumptions and can be extended to the case of log-linear regression with continuous exogenous variables. The theory is applied to aggregated financial transaction time series. In particular, we discover positive association between the number of transactions and the volatility process of a certain stock.

Interventions in INGARCH Processes

We study the problem of intervention effects generating various types of outliers in a linear count time series model. This model belongs to the class of observation driven models and extends the class of Gaussian linear time series models within the exponential family framework. Studies about effects of covariates and interventions for count time series models have largely fallen behind due to the fact that the underlying process, whose behavior determines the dynamics of the observed process, is not observed. We suggest a computationally feasible approach to these problems, focusing especially on the detection and estimation of sudden shifts and outliers. To identify successfully such unusual events we employ the maximum of score tests, whose critical values in finite samples are determined by parametric bootstrap. The usefulness of the proposed methods is illustrated using simulated and real data examples.

Some recent progress in count time series

We review some regression models for the analysis of count time series. These models have been the focus of several investigations over the last years, but only recently simple conditions for stationarity and ergodicity were worked out in detail. This advancement makes possible the development of the maximum-likelihood estimation theory under minimal assumptions.

QUASI-LIKELIHOOD INFERENCE FOR NEGATIVE BINOMIAL TIME SERIES MODELS

We study inference and diagnostics for count time series regression models that include a feedback mechanism. In particular, we are interested in negative binomial processes for count time series. We study probabilistic properties and quasi‐likelihood estimation for this class of processes. We show that the resulting estimators are consistent and asymptotically normally distributed. These facts enable us to construct probability integral transformation plots for assessing any assumed distributional assumptions. The key observation in developing the theory is a mean parameterized form of the negative binomial distribution. For transactions data, it is seen that the negative binomial distribution offers a better fit than the Poisson distribution. This is an immediate consequence of the fact that transactions can be represented as a collection of individual activities that correspond to different trading strategies.

On Comparing Several Spectral Densities

We investigated the problem of testing equality among spectral densities of several independent stationary processes. Our main methodological contribution is the introduction of a novel semiparametric log-linear model that links all of the spectral densities under consideration. This model is motivated by the asymptotic properties of the periodogram ordinates and specifies that the logarithmic ratio of G − 1 spectral density functions with respect to the Gth is linear in some parameters. Then the problem of testing equality of several spectral density functions is reduced to a parametric problem. Under this assumption, the large-sample theory of the maximum likelihood estimator was studied, and it was found that the estimator is asymptotically normal even when the model is misspecified. The development of the asymptotic theory is based on a new contrast function that might be useful for other spectral domain time series problems. The results are applicable to a variety of models, including linear and nonlinear processes. Simulations and data analysis support further the theoretical findings.

Clustering of biological time series by cepstral coefficients based distances

Clustering of stationary time series has become an important tool in many scientific applications, like medicine, finance, etc. Time series clustering methods are based on the calculation of suitable similarity measures which identify the distance between two or more time series. These measures are either computed in the time domain or in the spectral domain. Since the computation of time domain measures is rather cumbersome we resort to spectral domain methods. A new measure of distance is proposed and it is based on the so-called cepstral coefficients which carry information about the log spectrum of a stationary time series. These coefficients are estimated by means of a semiparametric model which assumes that the log-likelihood ratio of two or more unknown spectral densities has a linear parametric form. After estimation, the estimated cepstral distance measure is given as an input to a clustering method to produce the disjoint groups of data. Simulated examples show that the method yields good results, even when the processes are not necessarily linear. These cepstral-based clustering algorithms are applied to biological time series. In particular, the proposed methodology effectively identifies distinct and biologically relevant classes of amino acid sequences with the same physicochemical properties, such as hydrophobicity.

Interventions in log-linear Poisson autoregression:

We consider the problem of estimating and detecting outliers in count time series data following a log-linear observation driven model. Log-linear models for count time series arise naturally because they correspond to the canonical link function of the Poisson distribution. They yield both positive and negative dependence, and covariate information can be conveniently incorporated. Within this framework, we establish test procedures for detection of unusual events (‘interventions’) leading to different kinds of outliers, we implement joint maximum likelihood estimation of model parameters and outlier sizes and we derive formulae for correcting the data for detected interventions. The effectiveness of the proposed methodology is illustrated with two real data examples. The first example offers a fresh data analytic point of view towards the polio data. Our methodology identifies different forms of outliers in these data by an observation-driven model. The second example deals with some campylobacterosis data which we analyzed in a previous communication, by a different model. The results are reconfirmed by the new model that we put forward in this communication. The reliability of the procedure is verified using artificial data examples.

A goodness-of-fit test for Poisson count processes

We are studying a novel class of goodness-of-fit tests for parametric count time series regression models. These test statistics are formed by considering smoothed versions of the empirical process of the Pearson residuals. Our construction yields test statistics which are consistent against Pitman’s local alternatives and they converge weakly at the usual parametric rate. To approximate the asymptotic null distribution of the test statistics, we propose a parametric bootstrap method and we study its properties. The methodology is applied to simulated and real data.

On Combining Instruments

Abstract Suppose two instruments I0 and I1 measure the same quantity with the same resolution, where it is known that I0 is more reliable. The second, I1, is assumed a distortion of I0 in some sense. A statistical method is outlined where 1) the information from both I0 and I1 is combined to increase the reliability of I0 and 2) the distortion is quantified. An example is given in terms of shipborne precipitation radar and a spaceborne radiometer, both measuring rain rate.