Statistical Inference for MCARMA Processes

Multivariate continuous-time ARMA(p,q) (MCARMA(p,q)) processes are the continuous-time analog of the well-known vector ARMA(p,q) processes. This thesis contributes to the field of statistical inference of MCARMA processes in two ways. In the first part, we study information criteria, which provide a method to select a suitably MCARMA process as a model for given data. The second part of the thesis is concerned with robust estimation of the parameters of MCARMA processes

Functional Clustering of Periodic Transcriptional Profiles through ARMA(p,q)

Background: Gene clustering of periodic transcriptional profiles provides an opportunity to shed light on a variety of biological processes, but this technique relies critically upon the robust modeling of longitudinal covariance structure over time. Methodology: We propose a statistical method for functional clustering of periodic gene expression by modeling the covariance matrix of serial measurements through a general autoregressive moving-average process of order (p,q), the socalled ARMA(p,q). We derive a sophisticated EM algorithm to estimate the proportions of each gene cluster, the Fourier series parameters that define gene-specific differences in periodic expression trajectories, and the ARMA parameters that model the covariance structure within a mixture model framework. The orders p and q of the ARMA process that provide the best fit are identified by model selection criteria. Conclusions: Through simulated data we show that whenever it is necessary, employment of sophisticated covariance structures such as ARMA is crucial in order to obtain unbiased estimates of the mean structure parameters and increased precision of estimation. The methods were implemented on recently published time-course gene expression data in yeast and the procedure was shown to effectively identify interesting periodic clusters in the dataset. The new approach wil

The Effects of Autocorrelation in the Estimation of Process Capability Indices.

The current popularity of the process capability index, a measure of a supplier\u27s ability to meet the product specifications demanded by a customer, has become a matter of some controversy. While admitting the validity of much existing criticism, this research demonstrates that sample estimation of the triple index (Cpl, Cp, Cpu), a variant of the widely used index pair (Cp, Cpk), is equivalent to estimation of the natural parameters (mu, sigma) whenever the measured process characteristic X has an unconditional (marginal) normal probability density function. This includes processes which obey the strictly stationary, normal ARMA( p, q) model. By this extension to stationary normal models beyond ARMA(0, 0), the author shows the continued viability of the process capability index as a decision making tool of wider applicability. Estimators of the indices (Cpl, Cp, Cpu) are studied under conditions of both sample independence and sample autocorrelation. A new method for determining a joint confidence region for the triple index (Cpl, Cp, Cpu) is given. The region presented is, both conceptually and computationally, more direct than previously known approaches

Direct estimation of ARMA model orders using output cumulants

In this paper, the problem of order estimation of ARMA models is treated using third order cumulants of the observed system output only. It is shown that the rank conditions of certain third order cumulants matrices are directly related to the model orders (p, q). These matrices go from being of full rank to being rank deficient as some of their indices cross the correct model orders. This transition in the rank condition is effectively used to estimate the model orders. This method of order estimation does not relay on the knowledge of the model parameters values which is required in many published methods of order estimation. Moreover, the developed technique is immune against contaminating observations noise effects which usually result in over estimation of model orders. Several numerical examples are provided

Direct estimation of ARMA model orders using output cumulants

In this paper, the problem of order estimation of ARMA models is treated using third order cumulants of the observed system output only. It is shown that the rank conditions of certain third order cumulants matrices are directly related to the model orders (p, q). These matrices go from being of full rank to being rank deficient as some of their indices cross the correct model orders. This transition in the rank condition is effectively used to estimate the model orders. This method of order estimation does not relay on the knowledge of the model parameters values which is required in many published methods of order estimation. Moreover, the developed technique is immune against contaminating observations noise effects which usually result in over estimation of model orders. Several numerical examples are provided

A construction of continuous-time ARMA models by iterations of Ornstein-Uhlenbeck processes

We present a construction of a family of continuous-time ARMA processes based on p iterations of the linear operator that maps a Lévy process onto an Ornstein-Uhlenbeck process. The construction resembles the procedure to build an AR(p) from an AR(1). We show that this family is in fact a subfamily of the well-known CARMA(p,q) processes, with several interesting advantages, including a smaller number of parameters. The resulting processes are linear combinations of Ornstein-Uhlenbeck processes all driven by the same L´evy process. This provides a straightforward computation of covariances, a state-space model representation and methods for estimating parameters. Furthermore, the discrete and equally spaced sampling of the process turns to be an ARMA(p, p-1) process. We propose methods for estimating the parameters of the iterated Ornstein-Uhlenbeck process when the noise is either driven by a Wiener or a more general Lévy process, and show simulations and applications to real data.Peer ReviewedPostprint (published version