Equivalent Deterministic and Stochastic Models for Periodic Time Series
O'connor, Michael John
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https://hdl.handle.net/2142/70137
Description
Title
Equivalent Deterministic and Stochastic Models for Periodic Time Series
Author(s)
O'connor, Michael John
Issue Date
1986
Department of Study
Mechanical Engineering
Discipline
Mechanical Engineering
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Operations Research
Abstract
Many time series encountered in engineering applications are strongly periodic in the sense that they exhibit a recurrent pattern every S time units. Outdoor temperature, electrical consumption, the roughness of machined surfaces, and machine tool chatter vibration are a few examples. This periodic behavior is caused by the dynamics of the underlying physical systems that give rise to the time series. Engineers are interested in analyzing these time series in order to characterize, forecast, and/or control the underlying physical systems.
This thesis identifies mathematically equivalent stochastic representations of a general class of deterministic and deterministic plus stochastic components in a periodic time series. A periodic component is represented deterministically by a sum of sinusoids, possibly damped, and stochastically by an autoregressive-moving average (ARMA) model. The deterministic and stochastic representation of a periodic component are equivalent in the sense that they yield point by point identical time series when driven by the same innovation process. The identification of equivalent stochastic representations of a general class of deterministic and deterministic plus stochastic models extends and generalizes results that have been previously reported.
The equivalent stochastic representation of a determinism can be used to develop good initial parameter estimates for ARMA models of periodic time series. This significantly reduces a long standing problem with the estimation of high order ARMA models. Stochastic modeling of a periodic component also provides a method for quantitatively determining whether a periodicity is deterministic or stochastic. If the fitted ARMA model results in an equivalent stochastic representation of a determinism, then the periodicity is deterministic, otherwise it is stochastic.
These concepts are applied to the on-line control of machining chatter. Equivalent stochastic representations of periodic components of machining force are used to develop good initial parameter estimates for the off-line development of an ARMA model that contains both the deterministic and the stochastic structure of the machining process. A stochastic filter, derived from the combined ARMA model, is used to filter the original time series of machining force. The residual series is then modeled on-line with a low order ARMA model.
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