Traditional Problem Definition - Computational Intelligence in Time Series Forecasting - page 42

1

nd

J ˆ() d ( t d x )(() x t x )

¦

n

t 1

is the corresponding sample autocovariance function for - n < d < n .
In contrast with the autocorrelation function, which is infinite in extent, the
partial autocorrelation function

is described in terms of N non-zero

autocorrelation functions

N

U ¦ IU , i 1, 2,..., N ;

i jN i j

j 1

which can be described in a compact form by the Yule-Walker equation

5 , I U

NN N

using the vectors

T

> @

UUU U , , ...,

1 2 N

N

T

> @

III I , , ...,

N N 1 N 2 NN

and the matrix

ª º

1 UU U ...

1 2 N 1

« »

« »

UUU 1 ...

5 «

1 1 N 2

»

N

« ... ... ... ...

... 1

»

« »

UU

« »

¬ ¼

N 1 N 2

Solving the Yule-Walker equation for N = 1, 2, 3, …, one gets

1 UU

1 1

U 1 U

1 U

U U

1 2

1

UUU

2 1 3

1 2

I U

1 ,

I , I ,

11 22

33

1 U

1 UU

1

1 2

U 1

U 1 U

1

1 1

UU 1

2 1

etc . The last equality, N I , represents the partial autocorrelation function . Here,
the sample autocorrelation function for an AR(1) process should have mixed