Multi-scale and hidden resolution time series models

We introduce a class of multi-scale models for time series. The novel framework couples standard linear models at different levels of resolution via stochastic links across scales. Jeffrey’s rule of conditioning is used to revise the implied distributions and ensure that the probability distributions at the different levels are strictly compatible. This results in a new class of models for time series with three key characteristics: this class exhibits a variety of autocorrelation structures based on a parsimonious parameterization, it has the ability to combine information across levels of resolution, and it also has the capacity to emulate long memory processes. The potential applications of such multi-scale models include problems in which it is of interest to develop consistent stochastic models across time-scales and levels of resolution, in order to coherently combine and integrate information arising at different levels of resolution. Bayesian estimation based on MCMC analysis and forecasting based on simulation are developed. One application to the analysis of the flow of a river illustrates the new class of models and its utility.

[1]  L. Telser,et al.  Discrete Samples and Moving Sums in Stationary Stochastic Processes , 1967 .

[2]  James Davidson,et al.  Modelling Political Popularity: an Analysis of Long‐range Dependence in Opinion Poll Series , 1997 .

[3]  Phillip A. Arkin,et al.  Analyses of Global Monthly Precipitation Using Gauge Observations, Satellite Estimates, and Numerical Model Predictions , 1996 .

[4]  Dani Gamerman,et al.  Temporal Aggregation in Dynamic Linear Models , 1997 .

[5]  Mike West,et al.  Markov Random Field Models for High-Dimensional Parameters in Simulations of Fluid Flow in Porous Media , 2002, Technometrics.

[6]  Peter F. Craigmile.  Approximate wavelet-based simulation of long memory processes , 2005 .

[7]  Sailes K. Sengupta,et al.  Bayesian Inference in Wavelet-Based Models , 2002, Technometrics.

[8]  P. Diaconis,et al.  Updating Subjective Probability , 1982 .

[9]  Takeshi Amemiya,et al.  The Effect of Aggregation on Prediction in the Autoregressive Model , 1972 .

[10]  S. Chib,et al.  Bayes inference in regression models with ARMA (p, q) errors , 1994 .

[11]  Michael R. Chernick,et al.  Wavelet Methods for Time Series Analysis , 2001, Technometrics.

[12]  Nouna Kettaneh,et al.  Statistical Modeling by Wavelets , 1999, Technometrics.

[13]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[14]  Jonathan H. Wright,et al.  Semiparametric estimation of long-memory volatility dependencies: The role of high-frequency data , 2000 .

[15]  Achi Brandt,et al.  Multiscale Algorithm for Atmospheric Data Assimilation , 1997, SIAM J. Sci. Comput..

[16]  Gregory W. Wornell,et al.  A Karhunen-Loève-like expansion for 1/f processes via wavelets , 1990, IEEE Trans. Inf. Theory.

[17]  Pilar Loreto Iglesias,et al.  Conditioning on uncertain event: Extensions to bayesian inference , 2002 .

[18]  P. Green.  Reversible jump Markov chain Monte Carlo computation and Bayesian model determination , 1995 .

[19]  G. Wornell.  Wavelet-based representations for the 1/f family of fractal processes , 1993, Proc. IEEE.

[20]  Soosung Hwang.  THE EFFECTS OF SYSTEMATIC SAMPLING AND TEMPORAL AGGREGATION ON DISCRETE TIME LONG MEMORY PROCESSES AND THEIR FINITE SAMPLE PROPERTIES , 2000, Econometric Theory.

[21]  T. Nijman,et al.  Temporal Aggregation of GARCH Processes. , 1993 .

[22]  C. Granger,et al.  Modeling volatility persistence of speculative returns: A new approach , 1996 .

[23]  Mike West,et al.  Multi-scale Modeling of 1-D Permeability Fields , 2002 .

[24]  F. Palm,et al.  Missing observations in the dynamic regression model , 1984 .

[25]  G. Shafer.  Jeffrey's Rule of Conditioning , 1981, Philosophy of Science.

[26]  R. Jeffrey.  Probability and the Art of Judgment , 1992 .

[27]  Michael A. West,et al.  Bayesian Forecasting and Dynamic Models (2nd edn) , 1997, J. Oper. Res. Soc..

[28]  C. Granger.  Long memory relationships and the aggregation of dynamic models , 1980 .

[29]  A. Walden,et al.  Wavelet Analysis and Synthesis of Stationary Long-Memory Processes , 1996 .

[30]  P. Craigmile.  Simulating a class of stationary Gaussian processes using the Davies–Harte algorithm, with application to long memory processes , 2003 .