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Introduction to Time Series Analysis. Lecture 8. 1. Review: Linear prediction, projection in Hilbert space. 2. Forecasting and backcasting. 3. Prediction operator. 4. Partial autocorrelation function. 1 Linear prediction Given X 1 , X 2 , . . . , X n , the best linear predictor X n n+m = α 0 + n i=1 α i X i of X n+m satisfies the prediction equations E _ X n+m −X n n+m _ = 0 E __ X n+m −X n n+m _ X i ¸ = 0 for i = 1, . . . , n. This is a special case of the projection theorem.
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  Introduction to Time Series Analysis. Lecture 8. 1. Review: Linear prediction, projection in Hilbert space.2. Forecasting and backcasting.3. Prediction operator.4. Partial autocorrelation function. 1  Linear prediction Given  X  1 ,X  2 ,...,X  n , the best linear predictor X  nn + m  =  α 0  + n  i =1 α i X  i of   X  n + m  satisfies the  prediction equations E  X  n + m  − X  nn + m   = 0 E  X  n + m  − X  nn + m  X  i   = 0  for  i  = 1 ,...,n .This is a special case of the  projection theorem . 2  Projection theorem If  H  is a Hilbert space, M is a closed subspace of  H ,and  y  ∈ H ,then there is a point  Py  ∈ M (the  projection of   y  on M )satisfying1.   Py − y  ≤  w − y  2.   y − Py,w   = 0 for  w  ∈ M .  y y−PyPy M 3  Projection theorem for linear forecasting Given  1 ,X  1 ,X  2 ,...,X  n  ∈  r.v.s  X   :  E X  2 <  ∞  ,choose  α 0 ,α 1 ,...,α n  ∈ R so that  Z   =  α 0  +  ni =1 α i X  i  minimizes E ( X  n + m  − Z  ) 2 .Here,   X,Y    =  E ( XY  ) , M  =  { Z   =  α 0  +  ni =1 α i X  i  :  α i  ∈ R }  = ¯ sp { 1 ,X  1 ,...,X  n } , and y  =  X  n + m . 4
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