Filtering problem for sequences with periodically stationary multiseasonal increments with spectral densities allowing canonical factorizations

We consider a stochastic sequence $\xi(m)$ with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. The filtering problem is solved for this type of sequences based on observations with a periodically stationary noise. When spectral densities are known and allow the canonical factorizations, we derive the mean square error and the spectral characteristics of the optimal estimate of the functional $A{\xi}=\sum_{k=0}^{\infty}{a}(k) {\xi}(-k)$. Formulas that determine the least favourable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimate of the functional are proposed in the case where the spectral densities are not known, but some sets of admissible spectral densities are given.


Introduction
Non-stationary time series models have found wide-ranging applications in economics, finance, climatology, air pollution, signal processing.A fundamental example is a general multiplicative model, known as SARIM A(p, d, q) × (P, D, Q) s (Seasonal Autoregressive Integrated Moving Average), introduced in the book by Box and Jenkins et al. [4].This model incorporates both integrated and seasonal factors.It is described by the equation where ε t is a sequence of independent and identically distributed (i.i.d.) random variables, ψ(z), θ(z) are polynomials of p and q degrees, respectively, with roots outside the unit circle, and where Ψ(z) and Θ(z) are two polynomials of degrees P and Q, respectively, with roots outside the unit circle.The parameters d and D can take fractional values, resulting to what is known as seasonal ARFIMA, or SARFIMA model.The process described by equation ( 1) is stationary and invertible when |d + D| < 1/2 and |D| < 1/2.One application of seasonal ARFIMA models to the analysis of monetary aggregates used by the U.S. Federal Reserve is demonstrated in the work of Porter-Hudak [32]. . . ,d r ) ∈ (N * ) r , s = (s 1 , s 2 , . . ., s r ) ∈ (N * ) r and µ = (µ 1 , µ 2 , . . . ,µ r ) ∈ (N * ) r or ∈ (Z \ N) r ; n(γ) := r i=1 µ i s i d i .Here N * = N \ {0}.The explicit representation of the coefficients e γ (k) is given in [23].Within the article, δ lp denotes Kronecker symbols, n l = n! l!(n−l)! .Definition 1 ( [23]) For a stochastic sequence ξ(m), m ∈ Z, the sequence is called a stochastic generalized multiple (GM) increment sequence of differentiation order d with a fixed seasonal vector s ∈ (N * ) r and a varying step µ ∈ (N * ) r or µ ∈ (Z \ N) r .
The theory of (wide sense) stationary stochastic sequences describes second order random variables η(m), m ∈ Z, such that the mean value a = Eη(m 0 ) and the covariance function γ(h) = Cov(η(m 0 ), η(m 0 + h)) are finite and do not depend on m 0 .The following definition describes the stationarity of the increment sequence χ   exist for all m 0 , m, µ, µ 1 , µ 2 and do not depend on m 0 .The function c s (m; µ 1 , µ 2 ) is called a structural function of the stationary GM increment sequence (of a stochastic sequence with stationary GM increments).The stochastic sequence ξ(m), m ∈ Z determining the stationary GM increment sequence χ 2) is called a stochastic sequence with stationary GM increments (or GM increment sequence of order d).

Example 1
Consider an increment operator χ In this case the SARIMA time series (1) can be modeled by a GM increment sequence x m = χ (d+D) (1,1),(1,s) (ξ(m)) with the step µ = (1, 1), which is defined as an ARMA model where ε m is a sequence of i.i.d.random variables, and Ψ(z s )ψ(z) and Θ(z s )θ(z) are two polynomials with roots outside the unit circle.

Definition and spectral representation of stochastic sequences with periodically stationary GM increment
In this subsection, we present definition, justification and a brief review of the spectral theory of stochastic sequences with periodically stationary multiple seasonal increments, introduced in [23].

Definition 3
A stochastic sequence ξ(m), m ∈ Z is called a stochastic sequence with periodically stationary (periodically correlated) GM increments with period T if the mathematical expectations exist for every m, k, µ 1 , µ 2 and T > 0 is the least integer for which these equalities hold.
Theorem 1 1.The mean value and the structural function of the vector-valued stochastic stationary GM increment sequence χ where (iλ − 2πik j /s j ) dj , c is a vector, F (λ) is the matrix-valued spectral function of the stationary stochastic sequence χ The vector c and the matrix-valued function F (λ) are determined uniquely by the GM increment sequence χ 2. The stationary vector-valued GM increment sequence χ where ⃗ Z ξ (d) (λ) = {Z p (λ)} T p=1 is a (vector-valued) stochastic process with uncorrelated increments on [−π, π) connected with the spectral function F (λ) by the relation

Consider another vector-valued stochastic sequence with the stationary GM increments
, where ⃗ η(m) is a vector-valued stationary stochastic sequence, uncorrelated with ⃗ ξ(m), with the spectral representation where , is a stochastic process with uncorrelated increments, that corresponds to the spectral function G(λ) [12].The stochastic stationary GM increment χ  π).Therefore, in the case where the spectral functions F (λ) and G(λ) have the spectral densities f (λ) and g(λ), the spectral density For a regular stationary GM increment sequence χ µ,s ( ⃗ ξ(m)) [24], there exists an innovation sequence Representation ( 7) is called a canonical moving average representation of the stochastic stationary GM increment sequence χ Based on moving average representation (7) define Then the following relation holds true: In the following the one-sided moving average representation (7) and relation (8) are used for finding the mean square optimal estimates of unobserved values of vector-valued sequences with stationary GM increments.

Hilbert space projection method of filtering
3.1.Filtering of vector-valued stochastic sequence with stationary GM increments Consider a vector-valued stochastic sequence ⃗ ξ(m) with stationary GM increments constructed from transformation (3) and a vector-valued stationary stochastic sequence ⃗ η(m) uncorrelated with the sequence ⃗ ξ(m).Let the stationary GM increment sequence χ µ,s ( ⃗ ξ(m)) = 0, E⃗ η(m) = 0 and µ > 0. Filtering problem.Consider the problem of mean square optimal linear estimation of the functional which depends on unobserved values of a stochastic sequence ⃗ ξ(k) = {ξ p (k)} T p=1 with stationary GM increments.Estimates are based on observations of the sequence ⃗ ζ(k) = ⃗ ξ(k) + ⃗ η(k) at points k = 0, −1, −2, . ... We suppose that the conditions on coefficients ⃗ a(k and the minimality condition on the spectral densities f (λ) and g(λ) are satisfied.The second condition (11) is the necessary and sufficient one under which the mean square error of the optimal estimate of functional A ⃗ ξ is not equal to 0. Any linear estimate A ⃗ ξ of the functional A ⃗ ξ allows the representation [24] A where ⃗ h µ (λ) = {h p (λ)} T p=1 is the spectral characteristic of the estimate A⃗ η.In the Hilbert space L 2 (p), define a subspace Define the following matrix-valued Fourier coefficients: Define the matrices S µ , P µ and Q by the matrix-valued entries The solution to the filtering problem is described by the following theorem in terms of Fourier coefficients Stat., Optim.Inf.Comput.Vol. 12, March 2024

Theorem 2 ([24])
A solution A ⃗ ξ to the filtering problem for the linear functional A ⃗ ξ of the values of a vector-valued stochastic sequence ⃗ ξ(m) with stationary GM increments under conditions ( 10) and ( 11) is calculated by formula (12).The spectral characteristic ⃗ h µ (λ) and the value of the mean square error ∆(f, g; A ⃗ ξ) are calculated by the formulas where and Remark 2 The filtering problem in the presence of fractional integration is considered in [24].

Filtering based on factorizations of the spectral densities
The main goal of the article is to derive the classical and minimax estimates of the functional A ⃗ ξ in terms of the coefficients of the canonical factorizations of the spectral densities f (λ), g(λ) and f (λ Let the following canonical factorizations take place Define the matrix-valued function Ψ µ (e −iλ ) = {Ψ µ,ij (e −iλ )} j=1,T i=1,q by the equation where E q is an identity q × q matrix.One can check that the following factorization takes place Remark 3 The following Lemmas provide representations of P µ and P −1 µ S µ a µ , which contain coefficients of factorizations ( 16) - (18).

Lemma 2
Let factorizations ( 16) and ( 17) take place.Define by e µ (m) = Θ ⊤ µ S µ a µ m , m ≥ 0, the mth element of the vector where Z µ (j), j ∈ Z, are defined as Define the linear operators G − , G + , Φ + in the space ℓ 2 by matrices with the matrix entries And the linear operator Φ in the space ℓ 2 determined by a matrix with the matrix entries ( Φ) k,j = ϕ ⊤ (k − j) for 0 ≤ j ≤ k, ( Φ) k,j = 0 for 0 ≤ k < j.
Define also the coefficients , where coefficients ⃗ a −µ (k) are calculated by formula (13), and the vectors The following theorem describes a solution to the filtering problem in the case when the spectral densities f (λ) and g(λ) admit canonical factorizations ( 16) - (18) .

Theorem 3
Suppose that condition (10) is fulfilled and the spectral functions F (λ) and G(λ) of the stochastic sequences ⃗ ξ(m) and ⃗ η(m) have the spectral densities f (λ) and g(λ) admitting canonical factorizations ( 16) - (18).A solution A ⃗ ξ to the filtering problem for the linear functional A ⃗ ξ of the values of a vector-valued stochastic sequence ⃗ ξ(m) with stationary GM increments is calculated by formula (12).The spectral characteristic ⃗ h µ (λ) is calculated by the formulas where The the value of the mean square error ∆(f, g; A ⃗ ξ) is calculated by the formulas

Proof
See Appendix.

Remark 4
The following factorizations hold true: The filtering problem for the functional A N ⃗ ξ is solved directly by Theorem 3 by putting ⃗ a(k) = ⃗ 0 for k > N .To solve the filtering problem for the pth coordinate of the single vector ⃗ ξ(−N ), we put ⃗ a(N The following corollaries take place.

Filtering of stochastic sequences with periodically stationary GM increment
Consider the filtering problem for the functionals which depend on the unobserved values of a stochastic sequence ξ(m) with periodically stationary GM increments.
In the same way, the functional Aη is represented as Making use of the introduced notations and statements of Theorem 2 we can claim that the following theorem holds true.
The functional A M ξ can be represented in the form where N = [ M T ], the sequence ⃗ ξ(m) is determined by formula ( 26), An estimate of a single unobserved value ξ(−M ), M ≥ 0 of a stochastic sequence ξ(m) with periodically stationary GM increments is obtained by making use of the notations ξ(−M ) = ξ p (−N ) = ( ⃗ δ p ) ⊤ ⃗ ξ(N ), N = [ M T ], p = M + 1 − N T .We can conclude that the following corollaries hold true.

Corollary 3
Let a stochastic sequence ξ(m) with periodically stationary GM increments and a stochastic periodically stationary sequence η(m) generate by formulas ( 26) and ( 28) vector-valued stochastic sequences ⃗ ξ(m) and ⃗ η(m).A solution A M ξ to the filtering problem for the functional A M ξ = A N ⃗ ξ under condition (11) is calculated by formula (21) for the coefficients ⃗ a(m), 0 ≤ m ≤ N , defined in (29).The spectral characteristic and the value of the mean square error of the estimate A M ξ are calculated by formulas ( 19) and ( 20) respectively.

Corollary 4
Let a stochastic sequence ξ(m) with periodically stationary GM increments and a stochastic periodically stationary sequence η(m) generate by formulas ( 26) and ( 28) vector-valued stochastic sequences ⃗ ξ(m) and ⃗ η(m).A solution ξ(−M ) to the filtering problem for an unobserved value ξ (11) is calculated by formula (23).The spectral characteristic and the value of the mean square error of the estimate ξ(−M ) are calculated by formulas (19) and ( 20) or ( 24) and ( 25) respectively.

Solutions of the problem of estimating the functionals A ⃗
ξ and A N ⃗ ξ constructed from unobserved values of the stochastic sequence ⃗ ξ(m) with stationary GM increments χ In this section, we study the case where the complete information about the spectral density matrices is not available, while some sets of admissible spectral densities D = D f × D g is known.The minimax approach of estimation of the functionals from unobserved values of stochastic sequences is considered, which consists in finding an estimate that minimizes the maximal values of the mean square errors for all spectral densities from a class D simultaneously.This method will be applied for the concrete classes of spectral densities.
The proceed with the stated problem, we recall the following definitions [26].

Definition 4
For a given class of spectral densities D = D f × D g , the spectral densities f 0 (λ) ∈ D f , g 0 (λ) ∈ D g are called the least favourable densities in the class D for optimal linear filtering of the functional Aξ if the following relation holds true

Definition 5
For a given class of spectral densities D = D f × D g the spectral characteristic ⃗ h 0 (λ) of the optimal estimate of the functional Aξ is called minimax (robust) if the following relations hold true Taking into account the introduced definitions and the relations derived in the previous sections we can verify that the following lemmas hold true.

Lemma 3
The spectral densities f 0 ∈ D f , g 0 ∈ D g which admit the canonical factorizations ( 8), ( 16) and ( 17) are least favourable densities in the class D for the optimal linear filtering of the functional A ⃗ ξ based on observations of the sequence ⃗ ξ(m) + ⃗ η(m) at points m = 0, −1, −2, . . .if the matrix coefficients of the canonical factorizations ( 16) and (17) determine a solution of the constrained optimization problem

Lemma 4
The spectral density g 0 ∈ D g which admits the canonical factorizations ( 16), (17) with the known spectral density f (λ) is the least favourable in the class D g for the optimal linear filtering of the functional Aξ based on observations of the sequence ⃗ ξ(m) + ⃗ η(m) at points m = 0, −1, −2, . . .if the matrix coefficients of the canonical factorizations

Lemma 5
The spectral density f 0 ∈ D f which admits the canonical factorizations ( 8), ( 16) with the known spectral density g(λ) is the least favourable spectral density in the class D f for the optimal linear filtering of the functional A ⃗ ξ based on observations of the sequence ⃗ ξ(m) + ⃗ η(m) at points m = 0, −1, −2, . . .if matrix coefficients of the canonical factorization are determined by the equation Ψ 0 µ (e −iλ )Θ 0 µ (e −iλ ) = E q and a solution {ψ 0 µ (k) : k ≥ 0} of the constrained optimization problem The more detailed analysis of properties of the least favorable spectral densities and the minimax-robust spectral characteristics shows that the minimax spectral characteristic h 0 and the least favourable spectral densities f 0 and g 0 form a saddle point of the function ∆(h; f, g) on the set H D × D. The saddle point inequalities where the functional ∆( ⃗ h µ (f 0 , g 0 ); f, g) is calculated by the formula where Stat., Optim.Inf.Comput.Vol.The constrained optimization problem ( 33) is equivalent to the unconstrained optimization problem where δ(f, g|D) is the indicator function of the set D, namely δ(f, g|D The condition 0 ∈ ∂∆ D (f 0 , g 0 ) characterizes a solution (f 0 , g 0 ) of the stated unconstrained optimization problem.This condition is the necessary and sufficient condition that the point (f 0 , g 0 ) belongs to the set of minimums of the convex functional ∆ D (f, g) [27,34].Thus, it allows us to find the equalities for the least favourable spectral densities in some special classes of spectral densities D.
The form of the functional ∆( ⃗ h µ (f 0 , g 0 ); f, g) is suitable for application of the Lagrange method of indefinite multipliers to the constrained optimization problem (33).Thus, the complexity of the problem is reduced to finding the subdifferential of the indicator function of the set of admissible spectral densities.We illustrate the solving of the problem (34) for concrete sets admissible spectral densities in the following subsections.A semi-uncertain filtering problem, when the spectral density f (λ) is known and the spectral density g(λ) belongs to in class D g , is considered as well.

Least favorable spectral density in classes
Consider the minimax filtering problem for the functional A ⃗ ξ for sets of admissible spectral densities D k 0 , k = 1, 2, 3, 4 of the sequence with GM increments ⃗ ξ(m) where p, p k , k = 1, T are given numbers, P, B 1 are given positive-definite Hermitian matrices, and sets of admissible spectral densities D U V , k = 1, 2, 3, 4 for the stationary noise sequence ⃗ η(m) where g 1 (λ) = {g 1 ij (λ)} T i,j=1 is a fixed spectral density, B 2 is a given positive-definite Hermitian matrix, δ, δ k , k = 1, T , δ j i , i, j = 1, T , are given numbers.The condition 0 ∈ ∂∆ D (f 0 , g 0 ) implies the following equations which determine the least favourable spectral densities for these given sets of admissible spectral densities.
For the first set of admissible spectral densities D 1 f 0 × D 1 g1δ : where ⃗ α f , β ij are Lagrange multipliers, functions |γ ij (λ)| ≤ 1 and For the second set of admissible spectral densities D 2 f 0 × D 2 g1δ we have equations where α 2 f , β 2 are Lagrange multipliers, function |γ 2 (λ)| ≤ 1 and For the third set of admissible spectral densities D 3 f 0 × D 3 g1δ we have equations where α 2 f k , β 2 k are Lagrange multipliers, δ kl are Kronecker symbols, functions γ 2 k (λ) ≤ 1 and For the fourth set of admissible spectral densities D 4 f 0 × D 4 g1δ we have equations where The following theorem holds true.

Conclusions
In this article, we presented a solution of the filtering problem for stochastic sequences with periodically stationary GM increments, introduced in the article by Luz and Moklyachuk [23].We proposed a solution of the filtering problem in terms of coefficients of canonical factorizations of the spectral densities of the involved stochastic sequences.The results obtained in [24] are based on the Fourier transformations of the spectral densities.
In the case where the spectral densities of sequences are not exactly known while the sets of admissible spectral densities are specified (spectral uncertainty), the minimax-robust approach to filtering problem was applied.We described the minimax (robust) estimates of the functionals and relations determining the least favourable spectral densities and the minimax spectral characteristics of the optimal estimates of linear functionals for a list of specific classes of admissible spectral densities.Z µ (m + j)⃗ a −µ (j)e iλm .
(d) µ,s ( ⃗ ξ(m)) having the spectral density matrix f (λ) based on its observations with stationary noise ⃗ ξ(m) + ⃗ η(m) at points m = 0, −1, −2, . . .are proposed in Theorem 3 and Corollary 1 in the case where the spectral density matrices f (λ) and g(λ) of the basic sequence and the noise are exactly known.