Description

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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