Periodically Correlated Space-Time Autoregressive Hilbertian Processes
- https://doi.org/10.2991/jsta.d.210525.001How to use a DOI?
- Hilbertian processes, Periodically correlated space-time autoregressive processes, Strong law of large numbers, T-periodic sequences
In this paper, we introduce periodically correlated space-time autoregressive processes with values in Hilbert spaces. The existence conditions and the strong law of large numbers are established. Moreover, we present an estimator for the autocorrelation parameter of such processes.
- © 2021 The Authors. Published by Atlantis Press B.V.
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- This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).
In time series analysis, periodically correlated (PC) processes, which can be categorized in the class of nonstationary harmonizable processes, have been widely used to characterize various real life phenomena, that exhibit some kind of seasonal behavior. Franses , Gardner , Gardner et al.  and Hurd and Miamee  and some other researchers remark the importance of PC processes theoretically and in applied fields, such as metrology, communication, economics, etc. Besides, Hilbertian PC processes of weak type were introduced and studied by Soltani and Shishehbor [5,6]. These processes have interesting time domain and spectral structures.
Among various models in the analysis of time series, autoregressive (AR) models are of great importance. Bosq  generalized the classical AR models to processes with values in Hilbert spaces by introducing autoregressive Hilbertian (ARH) models. In his fundamental work, Bosq  provides basic results on Hilbertian strongly second order AR and moving-average processes. The existence, covariance structure, parameter estimation, strong law of large numbers and central limit theorem are also covered in his book. These models attract the attention of various researchers, such as Mourid , Besse and Cardot , Pumo , Mas [11,12] and Horvath et al. , and are applied drastically in modeling functional time series.
The PC autoregressive Hilbertian process of order one (PCARH) was introduced by Soltani and Hashemi . They studied the structure and existence of PC ARH processes by embedding them into higher dimensions, and provided necessary and sufficient conditions for the existence of these processes. They considered the law of large numbers, the central limit theorem, and also suggested some methods for parameter estimation.
Space-time processes are of great importance in studying spatial processes. The space-time autoregressive moving average (STARMA) models were developed by Pfeifer and Deutsch [15–19]. Processes that can be modeled by the STARMA models are characterized by a single random variable observed at N fixed sites in space. The dependencies between the N time series are incorporated in the model through hierarchical weighting matrices, specified prior to analyzing the data. These weighting matrices should incorporate the relevant physical characteristics of the system into the model. Each of the N time series are simultaneously modeled as linear combinations of past observations and disturbances, as well as weighted past observations and disturbances at neighboring sites.
A STARMA model is formulated as follows:
In the last decades, the technological advances in various fields, such as chemometrics, engineering, finance and medicine, makes it possible to observe samples as curves. In these cases, it is common to assume that the sample has been generated by a stochastic function. To analyze this type of data, it is convenient to use the tools provided by a recent area of statistics, known as functional data analysis (FDA).
Recently, FDA have been developed in the context of spatial statistics (Bosq ; Ruiz-Medina and Salmeron ). Ruiz-Medina [23,24] introduced and studied the structural properties of spatial autoregressive and moving-average Hilbertian processes, which are called SARH and SMAH, respectively, in abbreviation.
In this article, we introduce PC space-time autoregressive Hilbertian (PCSTARH) processes as an extension of Pfeifer and Deutsch's model to an infinite dimensional Hilbert space, and provide a theorem which demonstrates their existence, the strong law of large numbers and parameter estimation. Our methodology is to embed every PCSTARH with period T into a subclass of PCAR processes, and then applied the results of Soltani and Hashemi .
This article is organized as follows. In Section 2, we provide the required definitions and present conditions which guarantees the existence of the model. The strong law of large number of this model is presented in Section 3. In Section 4, we find estimators for autocorrelation operators and prove their consistency and the last section is devoted to some conclusions.
2. PC SPACE-TIME PROCESS
Let stands for a real separable Hilbert space equipped with the scalar product , the norm and the Borel -field . Besides, let denotes the Hilbert space of bounded linear operators on . Consider stands for a probability space. A random variable with values in is an measurable mapping from into . A random variable is called strongly second order if . For the sake of simplicity, we refer to strongly second order random variables with values in as -valued random variables throughout the paper.
For -valued random variables and the covariance and the cross-covariance operators are defined in terms of tensorial product, as follows:
In the first step, let us define PC and PC Hilbertian white noise processes, along with a sequence of T-periodic bounded linear operators.
An -valued stochastic process is said to be PC space-time with period PCS in abbreviation, if
The smallest such T is called the period of the process. If the process is called space-time stationary.
A PC space-time -valued process is called white noise (PCHWN) if it satisfies the following properties:
, where stands for the expected value based on the Bochner integral.
A sequence in is called T-periodic if .
For a bounded linear operator, A, and an matrix, B, we define the multiplication of A and B, C=AB, as an matrix, whose elements are . Here and subsequently, denotes an -valued random variable at time t and site .
In the following, we provide the definition of PC space-time autoregressive Hilbertian process of order one (PCSTARH).
A Hilbertian process is called a PCSTARH(1) with period if it is PC and satisfies
Note that, if we define and as -valued random variables, and as a weight matrix, then we can rewrite (5) as:
In the sequel, we prove that the class of PCSTARH processes can be embedded into the class of PCAR processes. Let us present the following lemma that is crucial in our approach.
The Hilbertian process is also a PCAR process, where and are T-periodic sequences in and W is a weight matrix.
We first show that is a T-periodic sequence in . Since and are T-periodic bounded linear operators, we have
It can be shown that is PCHWN, since
for each , .
Consequently, the proof is completed.
There are integers , such that , where .
If then assumption holds.
It is enough to apply the known inequalitywhere and are positive.
We now state a theorem, concerning existence and uniqueness of the PCSTARH(1,1) process.
Under the assumption , the equation has a unique solution given by
Based on Lemma 2.1, is a PCAR process that can be written as with . Then we can apply Theorem 2.1 of Soltani and Hashemi  to prove that under the assumption , a PCARH process has a unique solution as in Equation (8) and the proof is then completed.
2.1. Strong Law of Large Number
In this section, we prove the strong law of large numbers for PCSTARH processes.
A PCSTARH, , is said to be standard if assumption holds.
Let be a standard PCSTARH(1,1) and be a finite segment of this model. Then, as ,where and .
3. ESTIMATION OF THE AUTOCORRELATION PARAMETERS
Parameter estimation is an important feature of model identification. In this section, the parameters of PCSTARH model are estimated using the method of moment.
Let be a finite segment from , where is a multiple of , and . To estimate the parameters, and , we first estimate .
The classical method of moments provides the following normal equations:
Since is a compact operator, it has the following spectral decomposition
First, it is crucial to note that, since is not necessarily invertible, we can not deduce from (9) that , . A necessary and sufficient condition for to be defined is that , i.e. if and only if
From Equation (9), we have
Then, for any , the derived equation leads to the representation
Equation (13) gives a core idea for the estimation of . Therefore, we estimate , and empirically and substitute them in Equation (13). For this purpose, the estimated eigen elements will be obtained using the empirical covariance operator . Besides, is the empirical counterpart of and is the projector on the space spanned by the first eigenvectors of . Note that by the finite sample, the entire sequence cannot be estimated and just a truncated version can be obtained, which leads to
If grows to infinity by the sample size, the estimator will be consistent. On the other hand, we know that . Hence, it will be a delicate issue to control the behavior of . In fact, a small error in the estimation of can have an enormous impact on (14).
We now turn to estimate and . Let and W be an invertiable matrix, then
To study consistency of estimators, we begin by proving the consistency of . In order to study consistency of the estimators, we need the following assumptions:
is a standard PCARH(1) such that for all .
Following Hashemi and Soltani , the next theorem can be proved.
Suppose that , and hold and are Hilbert Schmidt operators. Then, if for some
Consider the decomposition
It is easy to see that
Under the assumptions of Theorem 3.1, and are consistent estimator.
Our objective in this paper is to introduce a new model for PC space-time data. We first introduce PC space-time autoregressive processes in an Hilbert space and provide conditions for their existence.
Our main aim is to show that there exists a relation between PCSTARH and PCARH and so we can show some properties for PCSTARH models, such as the strong law of large numbers. PCSTARH models have sophisticated statistical structures that open up promising new resources for data modeling strategies. We have focused only on the theoretical setup and leave applied approaches for future research.
CONFLICTS OF INTEREST
The authors declare they have no conflicts of interest.
The authors had the same contribution in designing the model, developing the theory and writing the manuscript.
The author(s) received no financial support for the research, authorship, and/or publication of this article.
The authors would like to express their sincere thanks to the referee for his/her encouragements and valuable comments.
Cite this article
TY - JOUR AU - M. Hashemi AU - J. Mateu AU - A. Zamani PY - 2021 DA - 2021/06/10 TI - Periodically Correlated Space-Time Autoregressive Hilbertian Processes JO - Journal of Statistical Theory and Applications SP - 164 EP - 170 VL - 20 IS - 2 SN - 2214-1766 UR - https://doi.org/10.2991/jsta.d.210525.001 DO - https://doi.org/10.2991/jsta.d.210525.001 ID - Hashemi2021 ER -