الفهرس 
INTRODUCTIONOnly 14 pages are availabe for public view
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Abstract
This thesis presented the class of the periodically correlated processes sets up one of
the possible frameworks for description and modeling of such time series. The processes
of this class are non-stationary but many of the concepts of the stationary theory admits
generalization to the periodic case. Actually, there is a duality between the multivariate
stationary processes and the periodically correlated processes which makes the investigation
of these two classes theoretically equivalent.
Fitting a model to time series data usually involves three main steps: model identi cation,
parameter estimation, and diagnostic checking, but we studied the model identi cation.
Model identi cation is to establish an identi cation of a possible model based on an available
realization, i.e., to decide the kind of the model with correct orders.
We generated the inverse of invertible standard multi-companion(IISMC) matrix from the
spectral parameters and then reconstruct the parameters for the required parameterization
of the models. The main idea of the multi-companion method and its inverse for generation
of periodic autoregression models is to generate a multi-companion matrix with the desired
spectrum and extract the parameters of the model from it.
The thesis contains 4 chapters:
Chapter 1: exploring some of important de nitions on an ordinary univariate time series
processes and talked about the concepts of stationarity and the autocovariance function.
Although, we saw that models for time series data that can have many forms which are
three broad classes of practical importance; the autoregressive (AR) models, the moving
average (MA) models and the autoregressive moving average (ARMA) models. Finally, we
studied the multivariate time series and projected the concepts of stationarity on it.
Chapter 2: studied the periodic autoregressive (PAR) models, the periodic moving average
(PMA) models and the periodic autoregressive moving average (PARMA) models and there representations. Also, we discussed the relationship between PC process and its multivariate
stationary process. Finally, we Identi ed the periodic autoregressive moving-average
time series models and we applied some simulated examples with statistical program R which
agrees well with the theoretical results.
Chapter 3: we have a matrix tool called standard multi-companion matrix and studied
a properties of inverse of invertible standard multi-companion matrix then used it to reconstruct
the parameters for the required parameterization of the models when the information
of the standard multi-companion matrix is not enough for the extracting of the parameters
of the model.
Chapter 4: we gave a method for generation of periodically correlated and multivariate
ARMA models whose dynamic characteristics are partially or fully speci ed in terms of
spectral poles and zeroes or their equivalents in the form of eigenvalues and eigenvectors of
associated model matrices. This method uses a reparameterization of the models based on
the spectral decomposition of inverse of invertible standard multi-companion matrices and
their factorization into products of companion matrices.