Important ProcessesModels Purely Random Process and Random Walk

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Important Processes/Models, (Purely Random Process and Random Walk Model) Sadia Qamar

Important Processes/Models, (Purely Random Process and Random Walk Model) Sadia Qamar

Filters/Operators Important Stochastic Processes Ø A purely Random Process Ø Random Walk Ø Moving

Filters/Operators Important Stochastic Processes Ø A purely Random Process Ø Random Walk Ø Moving Average Process Ø Autoregressive Process Ø A mixed autoregressive-moving average Process

Filters and Operators Filters Through filters, from one series, another series is generated. xt

Filters and Operators Filters Through filters, from one series, another series is generated. xt Filter 1 Weights {aj} acts on {xt} to generate {yt}. Ø Differencing Ø Backward Ø Forward yt Filter 2 zt Weights {bj} acts on {yt} to generate {zt}. Linear Filter Model

Operators Differencing Operator ØUsed to get successive differences. Ø is used as notation. Ø

Operators Differencing Operator ØUsed to get successive differences. Ø is used as notation. Ø xt+1 = xt+1 – xt Ø 2 xt+2 = xt+2 – xt+1 Ø For seasonal data differencing can attain stationarity. Backward Operator Ø Used to get previous observations. Ø B is used as notation. Ø Bxt = xt-1 Ø B 2 x t = xt-2 Ø It is used in mathematical operations. Forward Operator Ø Used to get following observations. Ø F is used as notation. Ø Fxt = xt+1 Ø F 2 xt = xt+2 Ø It is used in mathematical operations.

Purely Random Process Ø A discrete process {Zt} is called a purely random process

Purely Random Process Ø A discrete process {Zt} is called a purely random process if the random variables Zt are a sequence of mutually independent, identically distributed variables. Ø The process has constant mean and variance. Ø Ø Ø This process is some times called white noise by engineers.

Random Walk Suppose that {Zt} is a discrete purely random process with mean µ

Random Walk Suppose that {Zt} is a discrete purely random process with mean µ and variance σZ 2. A process {Xt} is said to be a ‘ random walk ‘if Xt = Xt-1 + Zt t = 0, 1, 2, . . So that and X 1 = X 0 + Z 1 X 1 = Z 1 X 2 = X 1 + Z 2 X 2 = Z 1 + Z 2 X 3 = X 2 + Z 3 X 3 = Z 1 + Z 2 + Z 3. . . t Xt = Zi Xt Random Walk t å i =1 Which shows that the process is not stationary. But is stationary. 1

Random Walk Continued Random Walk with drift. Ø A simple extension of the random

Random Walk Continued Random Walk with drift. Ø A simple extension of the random walk process. Ø This process accounts for a trend. Ø The process {Xt} is a random walk with drift if Ø The process on average will tend to move upward for d > 0. Figure Xt Random walk with drift t