05_ts_chap3 - Outline Stochastic processes Stationary...

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Outline Stochastic processes Stationary processes Autocorrelation function Some useful models Wold Decomposition Some Time-Series Models Haipeng Xing, AMS316, Stony Brook University Outline Stochastic processes Stationary processes Autocorrelation function Some useful models Wold Decomposition Outline 1 Stochastic processes and their properties 2 Stationary processes 3 Some properties of the autocorrelation function 4 Some useful models Purely random processes, random walks, and MA processes Autoregressive processes ARMA, ARIMA and the general linear models 5 The Wold decomposition theorem Haipeng Xing, AMS316, Stony Brook University
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Outline Stochastic processes Stationary processes Autocorrelation function Some useful models Wold Decomposition Stochastic processes and their properties A stochastic process can be described as a statistical phenomenon that evolves in time according to probabilistic laws. Mathematically, a stochastic process is a collection of random variables that are ordered in time and deFned at a set of time points, which may be continuous or discrete. Most statistical problems are concerned with estimating the properties of a population from a sample. In time-series analysis, the order of observations is determined by time and it is usually impossible to make more than one observation at any given time. We may regard the observed time series as just one example of the inFnite set of time series that might have been observed. This inFnite set of time series is called the ensemble , and every member of the ensemble is a possible realization of the stochastic process. Haipeng Xing, AMS316, Stony Brook University Outline Stochastic processes Stationary processes Autocorrelation function Some useful models Wold Decomposition Stochastic processes and their properties A simple way of describing a stochastic process is to give the moments of the process. Denote the random variable at time t by X ( t ) if time is continuous, and by X t if time is discrete. Mean : The mean function μ ( t ) is deFned for all t μ ( t )= E ± X ( t ) ² Variance : The variance function σ 2 ( t ) is deFned for all t σ 2 ( t Var ± X ( t ) ² Autocovariance : We deFne the acv.f. γ ( t 1 ,t 2 ) to e the covariance of X ( t 1 ) with X ( t 2 ) , γ ( t 1 2 E ³± X ( t 1 ) - μ ( t 1 ) ²± X ( t 2 ) - μ ( t 2 ) ²´ Haipeng Xing, AMS316, Stony Brook University
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Outline Stochastic processes Stationary processes Autocorrelation function Some useful models Wold Decomposition Stationary processes A time series is said to be strictly stationary if the joint distribution of X ( t 1 ) ,...,X ( t k ) is the same as the joint distribution of X ( t 1 + τ ) ( t k + τ ) for all t 1 ,...,t k , τ . Strict stationarity implies that for k =1 μ ( t ) μ, σ 2 ( t ) σ 2 ; for k =2 , γ ( τ )= E ±² X ( t ) - μ ³² X ( t + τ ) - μ ³´ = Cov [ X ( t ) ,X ( t + τ )] , which is called the autocovariance coe f cient at lag τ .
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This note was uploaded on 02/28/2012 for the course AMS 316 taught by Professor Xing during the Fall '09 term at SUNY Stony Brook.

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05_ts_chap3 - Outline Stochastic processes Stationary...

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