Used very often for forecasting
(e.g. stock market).

Given a time-series
yn
y
n
, assumed to be produced by an auto-regressive (AR)
(all-pole) system:
yn=−∑
k
=1M
a
k
yn−k+un
y
n
k
1
M
a
k
y
n
k
u
n
where
un
u
n
is a white Gaussian noise sequence which is
stationary and has zero mean.

To determine the model parameters
a
k
a
k
minimizing the variance of the prediction error, we
seek

min
a
k
a
k
Eyn+∑
k
=1M
a
k
yn−k2=min
a
k
a
k
Ey2n+2∑
k
=1M
a
k
ynyn−k+∑
k
=1M
a
k
yn−k∑
l
=1M
a
l
yn−l=min
a
k
a
k
Ey2n+2∑
k
=1M
a
k
Eynyn−k+∑
k
=1M∑
l
=1M
a
k
a
l
Eyn−kyn−l
a
k
y
n
k
1
M
a
k
y
n
k
2
a
k
y
n
2
2
k
1
M
a
k
y
n
y
n
k
k
1
M
a
k
y
n
k
l
1
M
a
l
y
n
l
a
k
y
n
2
2
k
1
M
a
k
y
n
y
n
k
k
1
M
l
1
M
a
k
a
l
y
n
k
y
n
l

(1)
The mean of
yn
y
n
is zero.

ε2=r0+2(
r1r2r3...rM
)
a
1
a
2
a
3
⋮
a
M
+(
a
1
a
2
a
3
...
a
M
)(
r0r1r2...rM−1
r1r0r1...⋮
r2r1r0...⋮
⋮⋮⋮⋱⋮
rM−1.........r0
)
ε
2
r
0
2
r
1
r
2
r
3
...
r
M
a
1
a
2
a
3
⋮
a
M
a
1
a
2
a
3
...
a
M
r
0
r
1
r
2
...
r
M
1
r
1
r
0
r
1
...
⋮
r
2
r
1
r
0
...
⋮
⋮
⋮
⋮
⋱
⋮
r
M
1
...
...
...
r
0

(2)
∂ε2∂
a
=2r+2Ra
a
ε
2
2
r
2
R
a

(3)
Setting

Equation 3 equal to zero yields:

Ra=−r
R
a
r
These are called the

Yule-Walker equations. In
practice, given samples of a sequence

yn
y
n
, we estimate

rn
r
n
as

rn
^=1N∑
k
=0N−nynyn+k≃Eykyn+k
r
n
1
N
k
0
N
n
y
n
y
n
k
y
k
y
n
k
which is extremely similar to the deterministic least-squares
technique.