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Robust interpolation of sequences with periodically stationary multiplicative seasonal increments

Maksym Luz, Mykhailo Moklyachuk

TL;DR

This work develops a robust interpolation framework for stochastic sequences that possess periodically stationary generalized multiple (GM) increments, combining cyclostationary and long-memory seasonal patterns. It employs a Hilbert-space projection (Kolmogorov) approach to derive optimal linear estimates of functionals based on observations with periodically stationary noise, and extends the classical theory to both stationary and fractional GM increments. When spectral densities are uncertain, the authors formulate a minimax (robust) scheme, characterize least-favorable densities and minimax spectral characteristics, and provide concrete sets of admissible spectral-density classes with corresponding equations to determine the robust solutions. The results yield explicit mean-square error expressions and spectral-characteristic formulas, enabling robust interpolation of non-stationary, seasonally structured processes, with potential applications in econometrics, climatology, and signal processing where periodic long-memory behavior is present.

Abstract

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

Robust interpolation of sequences with periodically stationary multiplicative seasonal increments

TL;DR

This work develops a robust interpolation framework for stochastic sequences that possess periodically stationary generalized multiple (GM) increments, combining cyclostationary and long-memory seasonal patterns. It employs a Hilbert-space projection (Kolmogorov) approach to derive optimal linear estimates of functionals based on observations with periodically stationary noise, and extends the classical theory to both stationary and fractional GM increments. When spectral densities are uncertain, the authors formulate a minimax (robust) scheme, characterize least-favorable densities and minimax spectral characteristics, and provide concrete sets of admissible spectral-density classes with corresponding equations to determine the robust solutions. The results yield explicit mean-square error expressions and spectral-characteristic formulas, enabling robust interpolation of non-stationary, seasonally structured processes, with potential applications in econometrics, climatology, and signal processing where periodic long-memory behavior is present.

Abstract

We consider stochastic sequences with periodically stationary generalized multiple increments of fractional order which combines cyclostationary, multi-seasonal, integrated and fractionally integrated patterns. We solve the interpolation problem for linear functionals constructed from unobserved values of a stochastic sequence of this type based on observations of the sequence with a periodically stationary noise sequence. For sequences with known matrices of spectral densities, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal interpolation of the functionals. Formulas that determine the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear interpolation of the functionals are proposed in the case where spectral densities of the sequences are not exactly known while some sets of admissible spectral densities are given.

Paper Structure

This paper contains 11 sections, 10 theorems, 129 equations.

Table of Contents

  1. Stochastic sequences with periodically stationary generalized multiple increments
  2. Preliminary notations and definitions
  3. Definition and spectral representation of stochastic sequence with periodically stationary GM increment
  4. Sequences with GM fractional increments
  5. Hilbert space projection method of interpolation
  6. Interpolation of vector-valued stochastic sequences with stationary GM increments
  7. Interpolation of stochastic sequences with periodically stationary GM increments
  8. Minimax (robust) method of interpolation
  9. Least favorable spectral density in classes $\mathcal{D}_0\times \mathcal{D}_{\varepsilon}$
  10. Least favorable spectral density in classes $\mathcal{D}_{1\delta} \times \mathcal{D}_{V}^{U}$
  11. Conclusions

Key Result

Theorem 1.1

1. The mean value and structural function of the vector-valued stochastic stationary GM increment sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$ can be represented in the form where $c$ is a vector, $F(\lambda)$ is the matrix-valued spectral function of the stationary stochastic sequence $\chi_{\overline{\mu},\overline{s}}^{(d)}(\vec{\xi}(m))$. The vector $c$ and the matrix-va

Theorems & Definitions (19)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Definition 1.3
  • Theorem 1.1
  • Definition 1.4
  • Lemma 1.1
  • Theorem 1.2
  • Example 1.1
  • Lemma 2.1
  • ...and 9 more