Sequential Detection of an Abrupt Change in a Random Sequence with Unknown Initial State

James Falt; Steven D. Blostein

Home/Archive/arXiv

arXiv · 1401.6044

Sequential Detection of an Abrupt Change in a Random Sequence with Unknown Initial State

James Falt·Steven D. Blostein

View original source ↗Read source document ↗Export metadata ↓

Abstract

The problem of sequentially detecting an abrupt change in a sequence of independent and identically distributed (IID) random variables is addressed. Whereas previous approaches assume a known probability density function (PDF) at the start of the sequence, the problem addressed is the detection of a single change in distribution among a finite number of known 'equal-energy' PDFs, but where the initial and final distributions are not known a priori. A Bayesian multiple hypothesis approach is proposed where (i) unlike previous threshold policies, the minimum cost hypothesis is tracked through time, (ii) under an exponential delay-cost function that satisfies an upper bound determined by the distances between hypotheses, the probability of detecting a change from an incorrect initial distribution asymptotically vanishes with time, (iii) computation is recursive and constant per unit time, and (iv) the unknown initial state gives rise to unavoidable incorrect detections that be made to vanish with a constant test threshold with negligibly small effect on correct detection delay for change times beyond a lower bound. Simulations illustrate the analysis and reveal that average delay approaches that of the optimal CUSUM test after an initial transient period determined by an incorrect detection probability constraint.

Explore related subjects

math.ST stat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

James Falt, Steven D. Blostein. 2017-12-08. Sequential Detection of an Abrupt Change in a Random Sequence with Unknown Initial State. https://arxiv.org/abs/1401.6044

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Transitional Conditional Independence

Statistical models contain variables that are not random: parameters, treatments, environments, design points. Ordinary conditional independence cannot express relations involving such variables. To apply it one must first put a distribution on them, and that changes the meaning of the statement. This paper introduces transitional conditional independence. It relates three variables on a Markov kernel $K(W|T)$ with non-stochastic input $T$, and is defined by a single factorization: \[ X\perp\!\!\perp_{K(W|T)} Y |Z \quad :\iff \quad \exists\, Q(X|Z):\; K(X,Y,Z|T) = Q(X|Z)\otimes K(Y,Z|T).\] The relation asserts a Markov kernel $Q(X|Z)$ that is the same for every input $t$. It therefore yields a factorization rather than an almost-sure identity between conditional expectations, and it needs no distribution on the input space. The relation is asymmetric. We show that the asymmetry is essential: symmetrizing it destroys the statements it was built to make. We prove left and right versions of all separoid rules except Symmetry. Ten of them hold on arbitrary measurable spaces, the remaining ones under one condition on the spaces involved, and we give criteria for when Symmetry itself holds. We axiomatize the resulting structure and show that it arises from any symmetric separoid by a shift. We give several applications. Ancillarity, sufficiency and adequacy become factorizations that hold pointwise in the parameter, without a prior and without null sets; the theorems of Fisher--Neyman and of Basu take this form. The invariance hypothesis of invariant prediction, $Y \perp\!\!\perp E | X_S$, receives its intended meaning: one kernel predicts $Y$ from $X_S$ in every environment $E$. And Bayesian networks with non-stochastic input nodes satisfy a directed global Markov property whose graphical id-separation criterion returns a factorization of Markov kernels, on arbitrary input spaces.

math.ST↗

arXiv2026

Choosing the penalty in nonparametric regression: short and long-range dependence

In this work, we study the one-dimensional regression problem under random design and Gaussian errors. Our framework is very general: we make no prior assumptions about the design (which may be nonstationary and exhibit short or long-range dependence), nor do we assume that the errors are homoscedastic. We examine in detail the cases where the error process exhibits short or long-range dependence. We adopt a least-squares penalized strategy using piecewise polynomials to estimate the regression function, following the framework of Baron, Birg{é} and Massart [1999]. We derive explicit penalties, up to calibration constants, to obtain adaptive estimators for which we establish risk bounds. Since these penalties depend on the dependence properties of the error process, which are unknown in practice, we propose several adaptations of the dimension jump calibration algorithm to make our procedures fully data-driven.

math.ST↗

arXiv2026

Information geometry of emergent symmetry quotients and their weak unfoldings

A regular observed statistical model may converge to a limit in which a previously identifiable signed parameter becomes identifiable only modulo a reflection. We study the local information geometry of this transition. For a twice differentiable Hellinger embedding with an exact limiting reflection, the observed displacement is forced into the two-jet form $\varepsilonλJ_-+λ^2J_+/2$, up to higher-order terms. The mixed jet restores the sign away from the symmetric face, whereas the even jet is the first tangent inherited by the quotient. After nuisance elimination, a positive Gram determinant yields a nondegenerate cross-cap two-jet. The associated local asymptotic theory has three regimes governed by $τ_n=\sqrt n\,\varepsilon_n^2$: regular signed LAN, a critical curved Gaussian subexperiment, and a quotient regime with the $n^{-1/4}$ signed scale. We prove that the same parabolic critical experiment persists for predictive likelihoods along a single stationary dependent trajectory. The limiting quotient has a regular Fisher metric in the invariant coordinate, while its pullback degenerates in the signed coordinate. For a solvable CIR--OU benchmark motivated by coherent sea-clutter observations, we derive the quotient Fisher metric and curvature explicitly and show that the curvature is strictly negative. We also determine the restricted holonomy of the full Amari family: $\operatorname{Hol}_0(\nabla^{(a)})=SO(2)$ for $a=0$, whereas $\operatorname{Hol}_0(\nabla^{(a)})=GL^+(2,\mathbb R)$ for $a\neq0$. The results separate the intrinsic geometry of the limiting quotient from the transverse geometry of its weak unfolding.

math.ST↗