arXiv subjects

Kevin McGoff

Publications and source records attributed to Kevin McGoff.

29 records · Page 2Linked to original sources

arXiv2018

Variational analysis of inference from dynamical systems

We introduce and study a variational framework for the analysis of empirical risk based inference for dynamical systems and ergodic processes. The analysis applies to a two-stage estimation procedure in which (i) the trajectory of an observed (but unknown) system is fit to a trajectory from a known reference system by minimizing cumulative per-state loss, and (ii) a parameter estimate is obtained from the initial state of the best fit reference trajectory. We show that the empirical risk of the best fit trajectory converges almost surely to a constant that can be expressed in a variational form as the minimal expected loss over dynamically invariant couplings (joinings) of the observed and reference systems. Moreover, we establish that the family of joinings minimizing the expected loss is convex and compact, and that it fully characterizes the asymptotic behavior of the estimated parameters, addressing both identifiability and misspecification. The two-stage estimation framework and associated variational analysis apply to a broad family of empirical risk miminization procedures for dependent observations. To illustrate this, we apply variational analysis to the well studied problems of maximum likelihood and non-linear regression, and then undertake an extended analysis of system identification from quantized trajectories subject to noise, a problem of interest in dynamics, where the models themselves exhibit dynamical behavior across time.

math.DS↗

arXiv2018

Empirical risk minimization and complexity of dynamical models

A dynamical model consists of a continuous self-map $T: \mathcal{X} \to \mathcal{X}$ of a compact state space $\mathcal{X}$ and a continuous observation function $f: \mathcal{X} \to \mathbb{R}$. This paper considers the fitting of a parametrized family of dynamical models to an observed real-valued stochastic process using empirical risk minimization. The limiting behavior of the minimum risk parameters is studied in a general setting. We establish a general convergence theorem for minimum risk estimators and ergodic observations. We then study conditions under which empirical risk minimization can effectively separate the signal from the noise in an additive observational noise model. The key, necessary condition in the latter results is that the family of dynamical models has limited complexity, which is quantified through a notion of entropy for families of infinite sequences. Close connections between entropy and limiting average mean widths for stationary processes are established.

math.ST↗

arXiv2017

Factor maps and embeddings for random $\mathbb{Z}^d$ shifts of finite type

For any $d \geq 1$, random $\mathbb{Z}^d$ shifts of finite type (SFTs) were defined in previous work of the authors. For a parameter $α\in [0,1]$, an alphabet $\mathcal{A}$, and a scale $n \in \mathbb{N}$, one obtains a distribution of random $\mathbb{Z}^d$ SFTs by randomly and independently forbidding each pattern of shape $\{1,\dots,n\}^d$ with probability $1-α$ from the full shift on $\mathcal{A}$. We prove two main results concerning random $\mathbb{Z}^d$ SFTs. First, we establish sufficient conditions on $α$, $\mathcal{A}$, and a $\mathbb{Z}^d$ subshift $Y$ so that a random $\mathbb{Z}^d$ SFT factors onto $Y$ with probability tending to one as $n$ tends to infinity. Second, we provide sufficient conditions on $α$, $\mathcal{A}$ and a $\mathbb{Z}^d$ subshift $X$ so that $X$ embeds into a random $\mathbb{Z}^d$ SFT with probability tending to one as $n$ tends to infinity.

math.DS↗

arXiv2014

Consistency of maximum likelihood estimation for some dynamical systems

We consider the asymptotic consistency of maximum likelihood parameter estimation for dynamical systems observed with noise. Under suitable conditions on the dynamical systems and the observations, we show that maximum likelihood parameter estimation is consistent. Our proof involves ideas from both information theory and dynamical systems. Furthermore, we show how some well-studied properties of dynamical systems imply the general statistical properties related to maximum likelihood estimation. Finally, we exhibit classical families of dynamical systems for which maximum likelihood estimation is consistent. Examples include shifts of finite type with Gibbs measures and Axiom A attractors with SRB measures.

math.ST↗

arXiv2014

Coalescence and meeting times on $n$-block Markov chains

We consider finite state, discrete-time, mixing Markov chains $(V,P)$, where $V$ is the state space and $P$ is transition matrix. To each such chain $(V,P)$, we associate a sequence of chains $(V_n,P_n)$ by coding trajectories of $(V,P)$ according to their overlapping $n$-blocks. The chain $(V_n,P_n)$, called the $n$-block Markov chain associated to $(V,P)$, may be considered an alternate version of $(V,P)$ having memory of length $n$. Along such a sequence of chains, we characterize the asymptotic behavior of coalescence times and meeting times as $n$ tends to infinity. In particular, we define an algebraic quantity $L(V,P)$ depending only on $(V,P)$, and we show that if the coalescence time on $(V_n,P_n)$ is denoted by $C_n$, then the quantity $\frac{1}{n} \log C_n$ converges in probability to $L(V,P)$ with exponential rate. Furthermore, we fully characterize the relationship between $L(V,P)$ and the entropy of $(V,P)$.

math.PR↗

arXiv2014

Random $\mathbb{Z}^d$-shifts of finite type

In this work we consider an ensemble of random $\mathbb{Z}^d$-shifts of finite type ($\mathbb{Z}^d$-SFTs) and prove several results concerning the behavior of typical systems with respect to emptiness, entropy, and periodic points. These results generalize statements made in \cite{McGoff} regarding the case $d=1$. Let $\mathcal{A}$ be a finite set, and let $d \geq 1$. For $n$ in $\mathbb{N}$ and $α$ in $[0,1]$, define a random subset $ω$ of $\mathcal{A}^{[1,n]^d}$ by independently including each pattern in $\mathcal{A}^{[1,n]^d}$ with probability $α$. Let $X_ω$ be the (random) $\mathbb{Z}^d$-SFT built from the set $ω$. For each $α\in [0,1]$ and $n$ tending to infinity, we compute the limit of the probability that $X_ω$ is empty, as well as the limiting distribution of entropy of $X_ω$. Furthermore, we show that the probability of obtaining a nonempty system without periodic points tends to zero. For $d>1$, the class of $\mathbb{Z}^d$-SFTs is known to contain strikingly different behavior than is possible within the class of $\mathbb{Z}$-SFTs. Nonetheless, the results of this work suggest a new heuristic: typical $\mathbb{Z}^d$-SFTs have similar properties to their $\mathbb{Z}$-SFT counterparts.

math.DS↗

arXiv2014

On intrinsic ergodicity of factors of $\mathbb{Z}^d$ subshifts

It is well-known that any $\mathbb{Z}$ subshift with the specification property has the property that every factor is intrinsically ergodic, i.e., every factor has a unique factor of maximal entropy. In recent work, other $\mathbb{Z}$ subshifts have been shown to possess this property as well, including $β$-shifts and a class of $S$-gap shifts. We give two results that show that the situation for $\mathbb{Z}^d$ subshifts with $d >1 $ is quite different. First, for any $d>1$, we show that any $\mathbb{Z}^d$ subshift possessing a certain mixing property must have a factor with positive entropy which is not intrinsically ergodic. In particular, this shows that for $d>1$, $\mathbb{Z}^d$ subshifts with specification cannot have all factors intrinsically ergodic. We also give an example of a $\mathbb{Z}^2$ shift of finite type, introduced by Hochman, which is not even topologically mixing, but for which every positive entropy factor is intrinsically ergodic.

math.DS↗

arXiv2012

Statistical inference for dynamical systems: a review

The topic of statistical inference for dynamical systems has been studied extensively across several fields. In this survey we focus on the problem of parameter estimation for non-linear dynamical systems. Our objective is to place results across distinct disciplines in a common setting and highlight opportunities for further research.

math.ST↗

arXiv2012

Random subshifts of finite type

Let $X$ be an irreducible shift of finite type (SFT) of positive entropy, and let $B_n(X)$ be its set of words of length $n$. Define a random subset $ω$ of $B_n(X)$ by independently choosing each word from $B_n(X)$ with some probability $α$. Let $X_ω$ be the (random) SFT built from the set $ω$. For each $0\leq α\leq1$ and $n$ tending to infinity, we compute the limit of the likelihood that $X_ω$ is empty, as well as the limiting distribution of entropy for $X_ω$. For $α$ near 1 and $n$ tending to infinity, we show that the likelihood that $X_ω$ contains a unique irreducible component of positive entropy converges exponentially to 1. These results are obtained by studying certain sequences of random directed graphs. This version of "random SFT" differs significantly from a previous notion by the same name, which has appeared in the context of random dynamical systems and bundled dynamical systems.

math.PR↗

arXiv2009

Orders of accumulation of entropy on manifolds

For a continuous self-map $T$ of a compact metrizable space with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the theory of entropy structure and symbolic extensions. Given any compact manifold $M$ and any countable ordinal $\al$, we construct a continuous, surjective self-map of $M$ having order of accumulation of entropy $\al$. If the dimension of $M$ is at least 2, then the map can be chosen to be a homeomorphism.

math.DS↗

arXiv2009

Orders of accumulation of entropy

For a continuous map $T$ of a compact metrizable space $X$ with finite topological entropy, the order of accumulation of entropy of $T$ is a countable ordinal that arises in the context of entropy structure and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if $M$ is a metrizable Choquet simplex, we bound the ordinals that appear as the order of accumulation of entropy of a dynamical system whose simplex of invariant measures is affinely homeomorphic to $M$. These bounds are given in terms of the Cantor-Bendixson rank of $\overline{\ex(M)}$, the closure of the extreme points of $M$, and the relative Cantor-Bendixson rank of $\overline{\ex(M)}$ with respect to $\ex(M)$. We also address the optimality of these bounds.

math.DS↗