Class Notes (810,420)
STA457H1 (9)
Zhou Zhou (7)
Lecture

# January 28.docx

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School
University of Toronto St. George
Department
Statistical Sciences
Course
STA457H1
Professor
Zhou Zhou
Semester
Winter

Description
January 28, 2014 White Noise 1. White noise process may not be independent Eg. If Y , 1, Y are ind N(0,1) Let X = 1 , X 1 Y Y2, …, X1= 2 Y …Y n 1 2 n It is easy to see that {X} is notiindependent process We show that X , X are not independent 1 2 Suppose that X , X were independent 1 2 E[X X1] =2E[X ]E[X ] 12 22 RHS = 1 LHS = E[Y Y ] =1E[Y2]E[Y ] = 3 14 22 On the other hand, we will show that {X} is a white noist. E[X] i E[Y ]E[Y1]…E[Y ]2= 0 n 2 2 2 V[X] i E[Y ]E[Y1]…E[Y ] 2 1 n 2 2 2 For any i > j, Cov[X, X] = EiXX]j= E[(Y Y iY)jY Y …Y)] =1E[2 Y …Y i …Y1 2 j 1 2 i i+1 j 2 2 2 = E[Y ]E1Y ]…E[Y 2E[Y ]…E[Y] i 1*1*…*i+1*…*0 = 0 j Removal of seasonal trends 1. Estimate and subtract the seasonal trend Suppose X = S + Yi wheri Y isistationary, S is a seasonal trend witi S = S i i+d for all i, d is known Si We can estimate S by i = 1/#A ∑jϵi X, wheri Aj= {k | kϵZ, 1≤i≤n, k-i can be divided by d} n A i {1, 1+d, 1+2d, 1+3d, …, 1+( ⌈ ⌉ -1)d} d S i It is easy to show, under some mild conditions that -p> S as i -> ∞ for any i ̂ Then X – i S i will be approximately stationary 2. Remove seasonal trend by differencing d Define W = X –IX = (1iB )X i-d i W is weakly stationary i Proof: W = X i X = Si+ Y –i-d – Y i = (S i S ) +i-d – Y i-d Y – Y -i statii-dry i i-d i i-d Goodness of fit tests 1. Make sure that our time series model has modelled the dependence structure of the sequence adequately 2. Tool: Residual analysis 3. Standard: If the residuals are approximately white noise then the model passes the test no trend trend non-constant variance no changing variance 4. Sample ACF plot helps us check no-correlation
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