Relevant states and memory in Markov chain bootstrapping and simulation

Abstract

Markov chain theory is proving to be a powerful approach to bootstrap highly nonlinear time series. In this work we provide a method to estimate the memory of a Markov chain (i.e. its order) and to identify its relevant states. In particular the choice of memory lags and the aggregation of irrelevant states are obtained by looking for regularities in the transition probabilities. Our approach is based on an optimization model. More specifically we consider two competing objectives that a researcher will in general pursue when dealing with bootstrapping:preserving the "structural" similarity between the original and the simulated series and assuring a controlled diversification of the latter. A discussion based on information theory is developed to define the desirable properties for such optimal criteria. Two numerical tests are developed to verify the effectiveness of the method proposed here.MSC classification: 60J10, 60J20, 60J22, 62B10, 62F40, 91G60.

Approximating Markov Chains for Bootstrapping and Simulation

“…In this way the resulting groups emerge as the relevant states, that is the states which significantly influence the conditional distribution of the process. In this work we also extend theoretically the analysis in Cerqueti et al, (2010Cerqueti et al, ( , 2012Cerqueti et al, ( , 2013 by introducing L p norm based distance measures. We also show that the minimization of the objective function represented by the distance measure of the partitions, which is based on the transition probabilities of the states, corresponds to the minimization of the information loss function in the sense of Kolmogorov (1965).…”

A mixed integer linear program to compress transition probability matrices in Markov chain bootstrapping

“…Constraint (9) forces at least one variable t γ to be equal to 1, thus guaranteeing that the diameter of the optimal partition is at least γ . Cerqueti et al (2010) discuss the coherence of some distance indicators to informationtype criteria advanced by Kolmogorov (1965). According to Remark 1, the dissimilarity measure used in the objective function of model (12) respects the conditions of Kolmogorov (1965).…”

“…It is, in fact, a distance indicator between the rows of matrix M which takes values in the interval [0, 2] (see Cerqueti et al 2010). As we will argue below, the dissimilarity d uv can be viewed as a proxy for the "cost" of putting k-states α u (k) and α v (k) in the same class of a partition, and, indeed, the objective function of our optimization model is based on it.…”

“…For example, a heuristic approach was adopted by Cerqueti et al (2013), who proposed a Tabu Search procedure for the problem of simultaneously partitioning the state space of a continuous-valued process and identifying the order of the process. Their approach starts by modeling a continuous-valued process as "Markov switching regimes" (see, e.g., Hamilton 1996;Jeanne and Masson 2000;Hamilton 2005), and identifying the "relevant" states defined by the thresholds that mark the passage from a regime to another (see also Cerqueti et al 2010). Their idea was to partition the set of rows of the transition probability matrix (associated to the discretized version of the process) to reduce its size and yet minimizing the information loss that may derive from matrix compression.…”

A stratified bootstrapping approach to assessing the success of TQM implementation in Peruvian companies

“…Proposed initially by Efron (1979), bootstrapping is a technique that consists of resampling some given observations for the purposes of obtaining a good estimation of statistical properties of the original population. The bootstrapping approach allows to preserve the 'structural' similarity between the original and resampled data sets, while also ensuring a controlled diversification of the same (Cerqueti et al, 2017). For information regarding procedure details and methodological contributions, the reader is referred to the studies by Freedman (1984), Freedman and Peters (1984), and Efron andTibshirani (1986, 1993).…”