Periodically Correlated Space-Time Autoregressive Hilbertian Processes

M. Hashemi; J. Mateu; A. Zamani

doi:10.2991/jsta.d.210525.001

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Volume 20, Issue 2, June 2021, Pages 164 - 170

Periodically Correlated Space-Time Autoregressive Hilbertian Processes

Authors

M. Hashemi¹^{, *}, J. Mateu², A. Zamani³

¹Department of Statistics, Khansar Campus, University of Isfahan, Isfahan, Iran

²Department of Mathematics, University Jaume I of Castellon, Castellon, Spain

³Department of Statistics, Faculty of Science, Shiraz University, Shiraz, Iran

^*Corresponding author. Email: m.hashemi.t@khn.ui.ac.ir

Corresponding Author

M. Hashemi

Received 23 November 2019, Accepted 13 February 2020, Available Online 10 June 2021.

DOI: 10.2991/jsta.d.210525.001 How to use a DOI?
Keywords: Hilbertian processes; Periodically correlated space-time autoregressive processes; Strong law of large numbers; T-periodic sequences
Abstract: 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.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

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 [1], Gardner [2], Gardner et al. [3] and Hurd and Miamee [4] 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 [7] generalized the classical AR models to processes with values in Hilbert spaces by introducing autoregressive Hilbertian (ARH) models. In his fundamental work, Bosq [7] 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 [8], Besse and Cardot [9], Pumo [10], Mas [11,12] and Horvath et al. [13], and are applied drastically in modeling functional time series.

The PC autoregressive Hilbertian process of order one (PCARH(1)) was introduced by Soltani and Hashemi [14]. 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 N×N 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 (pλ,qm) model is formulated as follows:

yt=∑k=1p∑l=0λkϕkl Wl yt−k−∑k=1q∑l=0mkθkl Wl εt−k+εt(1)

where p, q are the temporal AR and MA lags, λk and mk are the spatial lags, yt is the N × 1 vector of observations at time t at the N sites, Wl is the N × N matrix of weights for spatial order l and, finally, εt is the random disturbance at time t, which is normally distributed. The weighting matrices, in AR and moving-average parts, are the same as Pfeifer and Deutsch [17]. This model has found numerous applications ranging from environmental (Pfeifer and Deutsch [17]; Stoffer [20]) to epidemiological (Pfeifer and Deutsch [15]) and economical (Pfeifer and Bodily [21]) problems.

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 [7]; Ruiz-Medina and Salmeron [22]). Ruiz-Medina [23,24] introduced and studied the structural properties of spatial autoregressive and moving-average Hilbertian processes, which are called SARH(1) and SMAH(1), 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(1,1) with period T into a subclass of PCARHk(1) processes, and then applied the results of Soltani and Hashemi [14].

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 ℋ stands for a real separable Hilbert space equipped with the scalar product 〈.,.〉, the norm ∥.∥ and the Borel σ-field ℬ. Besides, let ℒ(ℋ) denotes the Hilbert space of bounded linear operators on ℋ. Consider (Ω,ℱ,P) stands for a probability space. A random variable with values in ℋ is an ℱ/ℬ measurable mapping from Ω into ℋ. A random variable X is called strongly second order if E‖X‖2<∞. For the sake of simplicity, we refer to strongly second order random variables with values in ℋ as ℋ-valued random variables throughout the paper.

For ℋ-valued random variables X and Y, the covariance and the cross-covariance operators are defined in terms of tensorial product, as follows:

CX(x):=E[(X⊗X)x]=E〈X,x〉X,(2)

CX,Y(x):=E[(X⊗Y)x]=E〈X,x〉Y,(3)

respectively. Note that, throughout this work, Xit will denote an ℋ-valued random variable at time t and site i.

In the first step, let us define PC and PC Hilbertian white noise processes, along with a sequence of T-periodic bounded linear operators.

Definition 2.1.

An ℋ-valued stochastic process {Xit, i=1,⋯,k, t∈ℤ} is said to be PC space-time with period T, PCS in abbreviation, if

CXin,Xim(x)=CXi(n+T),Xi(m+T)(x),(4)

for each n,m∈ℤ,x∈ℋ and some integer T>0.

The smallest such T is called the period of the process. If T=1, the process is called space-time stationary.

Definition 2.2.

A PC space-time ℋ-valued process {εit,i=1,⋯,k,t∈ℤ} is called white noise (PCHWN) if it satisfies the following properties:

E(εit)=0, 0<E∥εit∥2=σit2<∞ for every i=1,⋯,k, t∈ℤ, where E stands for the expected value based on the Bochner integral.
Cεit(x)=Cεi(t+T)(x) for every i=1,⋯,k, t∈ℤ, when x∈ℋ.
Cεit, εjl(x)=0 for all t≠l or i≠j when x∈ℋ.

Definition 2.3.

A sequence {ρt,t∈ℤ} in ℒ(ℋ) is called T-periodic if ρt=ρt+T.

For a bounded linear operator, A, and an n×p matrix, B, we define the multiplication of A and B, C=AB, as an n×p matrix, whose elements are Cij=ABij. Here and subsequently, Xit denotes an ℋ-valued random variable at time t and site i.

In the following, we provide the definition of PC space-time autoregressive Hilbertian process of order one (PCSTARH(1,1)).

Definition 2.4.

A Hilbertian process {Xit,i=1,⋯,k, t∈ℤ} is called a PCSTARH(1) with period T, if it is PC and satisfies

Xit=ϕtXi(t−1)+ψt∑j=1kwij Xj(t−1)+εit,(5)

where {ϕt,t∈ℤ} and {ψt,t∈ℤ} are T-periodic sequences in ℒ(ℋ) as the AR parameters of lag time t,{wij,i=1,⋯,k, j=1,⋯,k} are coefficients that satisfy the following properties:

wij≥0, ∀i,j=1,⋯,kwii=0, ∀i=1,⋯,k∑j=1kwij=1,(6)

and {εit,i=1,⋯,k, t∈ℤ} is a PCHWN.

Note that, if we define Xt:=(X1t,X2t,…,Xkt)′ and ε t:=(ε1t,⋯,εkt)′, as ℋk-valued random variables, and W=(wij) as a k×k weight matrix, then we can rewrite (5) as:

Xt=(ϕtI+ψtW)X(t−1)+ ε t(7)

In the sequel, we prove that the class of PCSTARH(1,1) processes can be embedded into the class of PCARHk(1) processes. Let us present the following lemma that is crucial in our approach.

Lemma 2.1.

The Hilbertian process Xt=(ϕtI+ψtW)X(t−1)+εt is also a PCARHk(1) process, where {ϕt,t∈ℤ} and {ψt,t∈ℤ} are T-periodic sequences in ℒ(ℋ) and W is a weight matrix.

Proof.

We first show that ρt=ϕtI+ψtW is a T-periodic sequence in ℒ(ℋk). Since ϕt and ψt are T-periodic bounded linear operators, we have

ρt+T=ϕt+TI+ψt+TW=ϕtI+ψtW.

It can be shown that ε t is PCHWN, since

E( ε t)=E(ε1t,ε2t,⋯,εkt)′=0,
C ε n+T(x)=C ε n(x), because,
Cεn+T(x)=E〈εn+T,x〉εn+T =E(∑i=1k〈εi(n+T),xi〉ε1(n+T),⋯,∑i=1k〈εi(n+T),xi〉εk(n+T))′ =E(∑i=1k〈εin,xi〉ε1n,⋯,∑i=1k〈εin,xi〉εkn)′=Cεn(x)
C ε n, ε m(x)=0 for each n≠m, x∈ℋk.

Consequently, the proof is completed.

Assumption A₁:

There are integers k0,k1,…,kT−1∈[1,∞), such that ∑i=0T−1∥ρi∥ki<1, where ρi=ϕiI+ψiW.

Corollary 2.1.

If ∑i=0T−12ki(∥ϕi∥ki+∥ψi∥ki∥W∥ki)<1, then assumption A1 holds.

Proof.

It is enough to apply the known inequality

(x+y)p≤2p(xp+yp),

where x,y and p are positive.

We now state a theorem, concerning existence and uniqueness of the PCSTARH(1,1) process.

Theorem 2.1.

Under the assumption A1, the equation Xt=(ϕtI+ψtW)X(t−1)+ ε t has a unique solution given by

XnT+i=∑j=0∞Aj,nT+i ε nT+i−j=∑k=0∞∑l=0T−1[AT,nT+i]kAl,nT+i ε (n−k)T+i−l,(8)

where A0,i=I,A1,i=ρi,A2,i=ρiρi−1,⋯,Ak,i=ρiρi−1⋯ρi−k+1 and ρi=ϕiI+ψiW.

Proof.

Based on Lemma 2.1, Xt is a PCARHk(1) process that can be written as Xt=ρtX(t−1)+ ε t with ρt=ϕtI+ψtW. Then we can apply Theorem 2.1 of Soltani and Hashemi [14] to prove that under the assumption A1, a PCARH(1) 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(1,1) processes.

Definition 2.5.

A PCSTARH(1,1), {Xit,i=1,⋯,k,t∈ℤ}, is said to be standard if assumption A1 holds.

Theorem 2.2.

Let {Xit,i=1,⋯,k,t∈ℤ} be a standard PCSTARH(1,1) and Xi,0,Xi,1,⋯,Xi,NT−1 be a finite segment of this model. Then, as N→∞,

n14(logn)βSi,n(X)n→a.s0, β>12,

where Si,n(X)=∑t=0n−1Xit and n=NT.

Proof.

By defining Xt=(X1t,X2t,⋯,Xkt)′, Xt is a PCARHk(1) process and, using Theorem 2.2 of Soltani and Hashemi [14], we have

n14(logn)βSn(X)n→a.s0, β>12.

and the proof is completed.

3. ESTIMATION OF THE AUTOCORRELATION PARAMETERS

Parameter estimation is an important feature of model identification. In this section, the parameters of PCSTARH(1,1) model are estimated using the method of moment.

Let X0,⋯,Xn−1 be a finite segment from Xt=(ϕtI+ψtW)X(t−1)+ ε t, where n is a multiple of T, n=NT, and Xt=(X1t,X2t,…,Xkt)′. To estimate the parameters, ϕt and ψt, we first estimate ρt=ϕtI+ψtW.

The classical method of moments provides the following normal equations:

Dl−1=ρlCl−1, l=1,⋯,T,(9)

where

ρl=ϕlI+ψlW,(10)

Cl−1=E(Xl−1⊗Xl−1),Dl−1 =E(Xl⊗Xl−1).(11)

Since Cl−1 is a compact operator, it has the following spectral decomposition

Cl−1=∑m∈ℕλm,l−1(em,l−1⊗em,l−1), ∑λm,l−1≤∞,(12)

where (λm,l−1)m⩾1 is a sequence of the positive eigenvalues of Cl−1 and (em,l−1)m⩾1 is a complete orthonormal system in ℋk. We define πm,l−1 as the associated sequence of projections, hence, πm,l−1=em,l−1⊗em,l−1 and Πkn,l−1=∑j=1knπj,l−1.

First, it is crucial to note that, since Cl−1−1 is not necessarily invertible, we can not deduce from (9) that ρl=Dl−1Cl−1−1, l=1,⋯,T. A necessary and sufficient condition for Cl−1−1 to be defined is that Ker(Cl−1)=0, i.e. Cl−1(x)=0 if and only if x=0.

From Equation (9), we have

Dl−1(ej,l−1)=ρlCl−1(ej,l−1)=λj,l−1ρl(ej,l−1)

Then, for any x∈ℋk, the derived equation leads to the representation

ρl(x)=ρl(∑j=1∞〈x,ej,l−1〉 ej,l−1)=∑j=1∞Dl−1(ej,l−1)λj,l−1〈x,ej,l−1〉.(13)

Equation (13) gives a core idea for the estimation of ρl. Therefore, we estimate Dl−1, λj,l−1 and ej,l−1 empirically and substitute them in Equation (13). For this purpose, the estimated eigen elements (λ^j,l−1,e^j,l−1)1⩽j⩽n will be obtained using the empirical covariance operator C^l−1=1N∑k=0N−1Xl−1+kT⊗Xl−1+kT. Besides, π^j,l−1 is the empirical counterpart of πj,l−1 and Π˜kn,l−1=∑j=1knπ^j,l−1 is the projector on the space spanned by the kn first eigenvectors of C^l−1. Note that by the finite sample, the entire sequence (λj,l−1,ej,l−1) cannot be estimated and just a truncated version can be obtained, which leads to

ρ^l(x)=∑j=1knD^l−1(e^j,l−1)λ^j,l−1〈x,e^j,l−1〉.(14)

If kn grows to infinity by the sample size, the estimator ρ^l will be consistent. On the other hand, we know that λj,l−1→0. Hence, it will be a delicate issue to control the behavior of 1λ^j,l−1. In fact, a small error in the estimation of λj,l−1 can have an enormous impact on (14).

We now turn to estimate ϕl and ψl. Let B=(1,0,⋯,0)′ and W be an invertiable matrix, then

ϕl=B′ρlB (15)

ψl=W−1(ρl−ϕl I).(16)

To study consistency of estimators, we begin by proving the consistency of ρ^. In order to study consistency of the estimators, we need the following assumptions:

Assumption B₁:

χ={Xn; n∈ℤ} is a standard PCARH(1) such that E∥Xn∥4≤∞ for all n∈ℤ.

Assumption B₂:

λ1,l−1≥λ2,l−1≥⋯≥0.

Assumption B₃:

λ^kn,l−1≥0, a.s.

Following Hashemi and Soltani [25], the next theorem can be proved.

Theorem 3.1.

Suppose that B1, B2 and B3 hold and ρl,l=1,⋯,T, are Hilbert Schmidt operators. Then, if for some β≥1,

λkn,l−1−1∑j=1∞aj,l−1=O(n14logn)β,(17)

we obtain

‖ρ^l−ρl‖→a.s0, l=1,⋯,T,(18)

where aj,l−1=22max[(λj−1,l−1−λj,l−1)−1,(λj,l−1−λj+1,l−1)−1] if j⩾2, and a1,l−1=22(λ1,l−1−λ2,l−1)−1.

Proof.

Consider the decomposition

(ρ^l−ρl)(x) =[ρ^l(x)−ρlΠkn, ℓ−1(x)] +[ρlΠkn, ℓ−1(x)−ρlΠ˜kn, ℓ−1(x)] +[ρlΠ˜kn, ℓ−1(x)−ρl(x)] :=a^l(x)+b^l(x)+c^l(x),(19)

and put

α^l=sup‖x‖≤1‖a^l(x)‖, β^l=sup‖x‖≤1‖b^l(x)‖, γ^l=sup‖x‖≤1‖c^l(x)‖.

It is easy to see that

α^l→a⋅s0,β^l→a⋅s0,γ^l→a⋅s0,(20)

and the proof is completed. For more details, see Theorem (3.3) of Hashemi and Soltani [25].

Next corollary states the consistency of ϕ and ψ estimators, define in (15) and (16).

Corollary 3.1.

Under the assumptions of Theorem 3.1, ϕ^l and ψ^l are consistent estimator.

4. CONCLUSION

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(1,1) and PCARH(1) and so we can show some properties for PCSTARH(1,1) 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.

AUTHORS' CONTRIBUTIONS

The authors had the same contribution in designing the model, developing the theory and writing the manuscript.

Funding Statement

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ACKNOWLEDGMENTS

The authors would like to express their sincere thanks to the referee for his/her encouragements and valuable comments.

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3.W.A. Gardner, A. Napolitano, and L. Paura, Signal Process., Vol. 86, 2006, pp. 639-697.

4.H.L. Hurd and A. Miamee, Periodically Correlated Random Sequences: Spectral Theory and Practice, Wiley, Hoboken, NJ, USA, 2007.

5.A.R. Soltani and Z. Shishehbor, Acta Math. Hung., Vol. 80, 1998, pp. 265-270.

6.A.R. Soltani and Z. Shishehbor, J. Georgian Math., Vol. 6, 1999, pp. 91-98.

7.D. Bosq, Linear Processes in Function Spaces: Theory and Applications, Berlin, Germany, Springer, 2000.

8.T. Mourid, Acad. Sci., Vol. 317, 1993, pp. 1167-1172.

9.P. Besse and H. Cardot, Canad. J. Stat., Vol. 24, 1996, pp. 467-487.

10.B. Pumo, Stat. Inference Stoch. Process., Vol. 3, 1999, pp. 1-13.

11.A. Mas, Stochastic Process Appl., Vol. 99, 2002, pp. 117-135.

12.A. Mas, J. Multivariate Anal., Vol. 98, 2007, pp. 1231-1261.

13.L. Horvath, M. Huskova, and P. Kokoszka, J. Multivariate Anal., Vol. 101, 2010, pp. 352-367.

14.A.R. Soltani and M. Hashemi, Stat. Inference Stochastic Process., Vol. 14, 2011, pp. 177-188.

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Journal: Journal of Statistical Theory and Applications
Volume-Issue: 20 - 2
Pages: 164 - 170
Publication Date: 2021/06/10
ISSN (Online): 2214-1766
ISSN (Print): 1538-7887
DOI: 10.2991/jsta.d.210525.001 How to use a DOI?
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

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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  - 10.2991/jsta.d.210525.001
ID  - Hashemi2021
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Journal of Statistical Theory and Applications

Periodically Correlated Space-Time Autoregressive Hilbertian Processes

1. INTRODUCTION

2. PC SPACE-TIME PROCESS

Definition 2.1.

Definition 2.2.

Definition 2.3.

Definition 2.4.

Lemma 2.1.

Proof.

Assumption A1:

Corollary 2.1.

Proof.

Theorem 2.1.

Proof.

2.1. Strong Law of Large Number

Definition 2.5.

Theorem 2.2.

Proof.

3. ESTIMATION OF THE AUTOCORRELATION PARAMETERS

Assumption B1:

Assumption B2:

Assumption B3:

Theorem 3.1.

Proof.

Corollary 3.1.

4. CONCLUSION

CONFLICTS OF INTEREST

AUTHORS' CONTRIBUTIONS

Funding Statement

ACKNOWLEDGMENTS

REFERENCES

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Assumption A₁:

Assumption B₁:

Assumption B₂:

Assumption B₃: