A frequentist understanding of sets of measures (original) (raw)

Towards a Frequentist Interpretation of Sets of Measures

We explore an objective, frequentist-related interpretation for a set of measures M such as would determine upper and lower envelopes; M also specifies the classical frequentist concept of a compound hypothesis. However, in contrast to the compound hypothesis case, in which there is a true measure µ θ0 ∈ M that is assumed either unknown or random selected, we do not believe that any single measure is the true description for the random phenomena in question. Rather, it is the whole set M, itself, that is the appropriate imprecise probabilistic description. Envelope models have hitherto been used almost exclusively in subjective settings to model the uncertainty or strength of belief of individuals or groups. Our interest in these imprecise probability representations is as mathematical models for those objective frequentist phenomena of engineering and scientific significance where what is known may be substantial, but relative frequencies, nonetheless, lack (statistical) stability.

Towards a Chaotic Probability Model for Frequentist Probability: The Univariate Case

We adopt the same mathematical model of a set M of probability measures as is central to the theory of coherent imprecise probability. However, we endow this model with an objective, frequentist interpretation in place of a behavioral subjective one. We seek to use M to model stable physical sources of time series data that have highly irregular behavior and not to model states of belief or knowledge that are assuredly imprecise. The ap- proach we present in this paper is to understand a set of measures model M not as a traditional compound hypothesis, in which one of the measures in M is a true description, but rather as one in which none of the individual measures in M provides an adequate description of the potential behavior of the physical source as actualized in the form of a long time series. We provide an instrumental interpretation of random process measures consistent with M and the highly irregular physical phenomena we intend to model by M. This construction provides us ...

Estimation of Chaotic Probabilities

A Chaotic Probability model is a usual set of proba- bility measures, M, the totality of which is endowed with an objective, frequentist interpretation as op- posed to being viewed as a statistical compound hy- pothesis or an imprecise behavioral subjective one. In the prior work of Fierens and Fine, given finite time series data, the estimation of the Chaotic Probability model is based on the analysis of a set of relative fre- quencies of events taken along a set of subsequences selected by a set of rules. Fierens and Fine proved the existence of families of causal subsequence selec- tion rules that can make M visible, but they did not provide a methodology for finding such family. This paper provides a universal methodology for finding a family of subsequences that can make M visible such that relative frequencies taken along such subse- quences are provably close enough to a measure in M with high probability.

Time series analysis via mechanistic models

Annals of Applied Statistics, 2009

The purpose of time series analysis via mechanistic models is to reconcile the known or hypothesized structure of a dynamical system with observations collected over time. We develop a framework for constructing nonlinear mechanistic models and carrying out inference. Our framework permits the consideration of implicit dynamic models, meaning statistical models for stochastic dynamical systems which are specified by a simulation algorithm to generate sample paths. Inference procedures that operate on implicit models are said to have the plug-and-play property. Our work builds on recently developed plug-and-play inference methodology for partially observed Markov models. We introduce a class of implicitly specified Markov chains with stochastic transition rates, and we demonstrate its applicability to open problems in statistical inference for biological systems. As one example, these models are shown to give a fresh perspective on measles transmission dynamics. As a second example, we present a mechanistic analysis of cholera incidence data, involving interaction between two competing strains of the pathogen Vibrio cholerae.

Random dynamical models from time series

In this work we formulate a consistent Bayesian approach to modeling stochastic (random) dynamical systems by time series and implement it by means of artificial neural networks. The feasibility of this approach for both creating models adequately reproducing the observed stationary regime of system evolution, and predicting changes in qualitative behavior of a weakly nonautonomous stochastic system, is demonstrated on model examples. In particular, a successful prognosis of stochastic system behavior as compared to the observed one is illustrated on model examples, including discrete maps disturbed by non-Gaussian and nonuniform noise and a flow system with Langevin force.

Time series analysis via mechanistic models. In review; pre-published at arxiv.org/abs/0802.0021

2008

Distinguishing between low-dimensional dynamics and randomness in measured time series

Physica D: Nonlinear Phenomena, 1992

The success o f cu rre nt at tempts to distingui sh between low-dimen sio na l chaos a nd ra nd o m be havi o r in a time series of observa tion s is co nsidered. First we discuss stationary stoc hasti c processes which produce finite nume rical est im a tes of the correlation dimension a nd K , e ntro py under naive application of co rrelati o n integral me th ods. We then co nsid er severa l straightforwa rd tests to eva lu a te whether correl a ti o n integral me thods re fl ect the global geo metry or the loca l fractal structure of th e trajectory. This determines whether the me thods are applicab le to a given se ries; if th ey are we eva luat e the significance of a pa rti cular result. for exa mple , by co nsidering th e res ults o f th e a nal ysis of stochasti c signals with statisti ca l properti es similar to th ose of obse rved series. From th e exa mpl es considered , it is clea r that the co rrela tio n integra l should not be used in iso la tion , but as o ne of a co llection of too ls to distinguish chaos from stochasticity.

On Extracting Probability Distribution Information from Time Series

Entropy, 2012

Time-series (TS) are employed in a variety of academic disciplines. In this paper we focus on extracting probability density functions (PDFs) from TS to gain an insight into the underlying dynamic processes. On discussing this "extraction" problem, we consider two popular approaches that we identify as histograms and Bandt-Pompe. We use an information-theoretic method to objectively compare the information content of the concomitant PDFs.

The likely determines the unlikely

Physica A: Statistical Mechanics and its Applications

We point out that the functional form describing the frequency of sizes of events in complex systems (e.g. earthquakes, forest fires, bursts of neuronal activity) can be obtained from maximal likelihood inference, which, remarkably, only involve a few available observed measures such as number of events, total event size and extremes. Most importantly, the method is able to predict with high accuracy the frequency of the rare extreme events. To be able to predict the few, often big impact events, from the frequent small events is of course of great general importance. For a data set of wind speed we are able to predict the frequency of gales with good precision. We analyse several examples ranging from the shortest length of a recruit to the number of Chinese characters which occur only once in a text.

A frequentist understanding of sets of measures (original) (raw)

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