[Paper Review] Path prediction of aggregated alpha-stable moving averages using semi-norm representations

Sébastien Fries|arXiv (Cornell University)|Sep 10, 2018

Probabilistic and Robust Engineering Design25 references3 citations

TL;DR

This paper develops a path prediction method for aggregated α-stable moving averages by introducing a semi-norm-based representation on unit cylinders, enabling explicit derivation of the conditional distribution of future paths given large past observations. The key contribution is a novel spectral representation that links tail behavior to pattern identification, valid when the process is sufficiently anticipative, and extends to linear combinations of stable processes with interpretable predictive dynamics in extreme regimes.

ABSTRACT

For (Xt) a two-sided α-stable moving average, this paper studies the conditional distribution of future paths given a piece of observed trajectory when the process is far from its central values. Under this framework, vectors of the form X t=(Xt−m,…,Xt,Xt+1,…,Xt+h), m≥0, h≥1, are multivariate alpha-stable and the dependence between the past and future components is encoded in their spectral measures. A new representation of stable random vectors on unit cylinders -sets {s∈Rm+h+1:∥ s ∥=1} for ∥⋅∥ an adequate semi-norm- is proposed in order to describe the tail behaviour of vectors Xt when only the first m+1 components are assumed to be observed and large in norm. Not all stable vectors admit such a representation and (Xt) will have to be "anticipative enough" for X t to admit one. The conditional distribution of future paths can then be explicitly derived using the regularly varying tails property of stable vectors and has a natural interpretation in terms of pattern identification. The approach extends to processes resulting from the linear combination of stable moving averages and applied to several examples.

Motivation & Objective

• To address the lack of tractable conditional path prediction for anticipative α-stable processes when the process is far from its central values.
• To develop a new representation of multivariate α-stable vectors using a semi-norm that respects past-only conditioning, overcoming limitations of Euclidean norm-based approaches.
• To characterize the conditions under which such a semi-norm representation exists, particularly requiring the process to be sufficiently anticipative.
• To derive the asymptotic conditional distribution of future paths given large past trajectories, enabling predictive inference in extreme regimes.
• To extend the framework to linear combinations of stable moving averages (stable aggregates), demonstrating flexibility in modeling diverse extreme path patterns.

Proposed method

• Proposes a new representation of α-stable random vectors on the unit cylinder {s ∈ ℝ^{m+h+1} : ||s|| = 1}, where ||·|| is a semi-norm satisfying ||(x_{-m},...,x_0, x_1,...,x_h)|| = ||(x_{-m},...,x_0, 0,...,0)||, preserving past-only information.
• Introduces a spectral measure Γ||·|| on the unit cylinder, which characterizes the limiting shape of large stable vectors under the semi-norm, replacing the standard Euclidean spectral measure.
• Establishes that the conditional distribution of future paths, given the past and large norm, converges to a ratio of spectral measures on the cylinder, analogous to (1.1) but adapted to the semi-norm framework.
• Uses the regularly varying tail property of stable vectors to derive the limiting conditional distribution, enabling explicit path prediction under extreme conditions.
• Applies transformation techniques (e.g., TB, T||·||, h1) to relate the new spectral measure to the original one, particularly through the use of the matrix B and its inverse in the transformation of sets.
• Extends the method to stable aggregates via linear combinations of independent α-stable moving averages, showing that such processes admit flexible, diverse path patterns in extreme events.

Experimental results

Research questions

• RQ1 Under what conditions can the conditional distribution of future paths in an α-stable moving average be explicitly derived when the process is far from its center?
• RQ2 Can a semi-norm-based representation on a unit cylinder replace the Euclidean norm in characterizing the limiting shape of large stable vectors, especially when conditioning on past observations only?
• RQ3 What structural properties must the underlying α-stable process possess (e.g., anticipativeness) to admit such a semi-norm representation?
• RQ4 How does the path prediction mechanism based on spectral measures on cylinders compare to existing methods like the spectral process of Basrak and Segers?
• RQ5 To what extent can the framework be generalized to linear combinations of stable moving averages (stable aggregates), and how do their extreme path patterns differ from non-aggregated processes?

Key findings

• The conditional distribution of future paths, given a large past trajectory, converges to a ratio of spectral measures on the semi-norm unit cylinder, providing an explicit predictive law.
• The semi-norm representation exists only if the process is sufficiently anticipative; otherwise, such a representation fails to exist.
• For the case of a 4-dimensional vector, the spectral measure Γ||·||_4 is decomposed into three components: a Dirac mass at the origin, and two parts supported on C||·||_4,1 and C||·||_4,2, respectively.
• When the past is large and the future is conditionally predicted, the limiting probability mass is determined by the overlap of the observed past pattern with the support of the spectral measure on the cylinder.
• In the example with ρ1 and ρ2, the total mass of the spectral measure on B(V0) is shown to be Γ2(V0) + (σα_1 / 2) * |ρ1|^α / (1 - |ρ1|^α), reflecting the contribution of both the past and the tail structure.
• The method successfully generalizes to stable aggregates, where different extreme path patterns can emerge across episodes, unlike non-aggregated processes that repeat the same pattern.