8.7 Forecasting
105
Here we note that
y i− j y i−k
= γ j−k , where γ k are the auto-covariances encountered
in Sect. 8.3. For a given time series y i , we can estimate the γ j by calculating γ j =
1
n
n
i=1 y i y i− j with unspecified values of y i set to zero. The expression on the lefthand side of (8.33) equals
ˆ
y i+n y i−k
= γ n+k . Using these relations and writing the
previous equation in component form we arrive at
⎛
⎜
⎜
⎜
⎝
γ n
γ n+1
. . .
γ n+m−1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
γ 0 γ 1 . . . γ m−1
γ 1 γ 0 . . . γ m−2
. . .
. . .
. . .
. . .
γ m−1 γ m−2 . . . γ 0
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
α 0
α 1
. . .
α m−1
⎞
⎟
⎟
⎟
⎠
,
(8.34)
which we invert in order to solve for the forecast coefficients α k (n)
⎛
⎜
⎜
⎜
⎝
α 0 (n)
α 1 (n)
. . .
α m−1 (n)
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
γ 0 γ 1 . . . γ m−1
γ 1 γ 0 . . . γ m−2
. . .
. . .
. . .
. . .
γ m−1 γ m−2 . . . γ 0
⎞
⎟
⎟
⎟
⎠
−1 ⎛
⎜
⎜
⎜
⎝
γ n
γ n+1
. . .
γ n+m−1
⎞
⎟
⎟
⎟
⎠
.
(8.35)
Here we included the argument n to the coefficients α k (n) as a reminder that the
coefficients are specific to the forecast distance n. Using these coefficients in (8.31)
will result in a forecast derived from the previously recorded values y k . For the
convenience in further calculations we introduce the following shorthand notation
for the previous equation
α(n) =
−1
γ (n) ,
(8.36)
where α(n) is the column vector as shown in (8.34), γ (n) the column vector on the
right-hand side. The transpose of the vectors will be denoted by a superscripted letter
‘t’. denotes the matrix in (8.34).
Now we will address the expected accuracy of the forecast by calculating the
expectation value of the quadratic deviation of the forecast σ
2
(y i+n )
σ ( ˆ
y i+n )
2
=
⎛
⎝ ˆ
y i+n −
m−1
j=0
α j y i− j
⎞
⎠
⎛
⎝ ˆ
y i+n −
m−1
j=0
y i−k α k
⎞
⎠
(8.37)
=
ˆ
y i+n ˆ
y i+n
− 2
m−1
j=0
α j ˆ
y i+n y i− j +
m−1
j=0
m−1
k=0
α j α k y i− j y i−k
= γ 0 − 2α(n)
t
γ (n) + α(n)
t
α(n) ,
where we used the abbreviations introduced above and, in particular jk = =y i− j y i−k .
Inserting (8.36), we finally obtain
σ ( ˆ
y i+n )
2
= γ 0 − γ (n)
t
−1
γ (n)
(8.38)
105
Here we note that
y i− j y i−k
= γ j−k , where γ k are the auto-covariances encountered
in Sect. 8.3. For a given time series y i , we can estimate the γ j by calculating γ j =
1
n
n
i=1 y i y i− j with unspecified values of y i set to zero. The expression on the lefthand side of (8.33) equals
ˆ
y i+n y i−k
= γ n+k . Using these relations and writing the
previous equation in component form we arrive at
⎛
⎜
⎜
⎜
⎝
γ n
γ n+1
. . .
γ n+m−1
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
γ 0 γ 1 . . . γ m−1
γ 1 γ 0 . . . γ m−2
. . .
. . .
. . .
. . .
γ m−1 γ m−2 . . . γ 0
⎞
⎟
⎟
⎟
⎠
⎛
⎜
⎜
⎜
⎝
α 0
α 1
. . .
α m−1
⎞
⎟
⎟
⎟
⎠
,
(8.34)
which we invert in order to solve for the forecast coefficients α k (n)
⎛
⎜
⎜
⎜
⎝
α 0 (n)
α 1 (n)
. . .
α m−1 (n)
⎞
⎟
⎟
⎟
⎠
=
⎛
⎜
⎜
⎜
⎝
γ 0 γ 1 . . . γ m−1
γ 1 γ 0 . . . γ m−2
. . .
. . .
. . .
. . .
γ m−1 γ m−2 . . . γ 0
⎞
⎟
⎟
⎟
⎠
−1 ⎛
⎜
⎜
⎜
⎝
γ n
γ n+1
. . .
γ n+m−1
⎞
⎟
⎟
⎟
⎠
.
(8.35)
Here we included the argument n to the coefficients α k (n) as a reminder that the
coefficients are specific to the forecast distance n. Using these coefficients in (8.31)
will result in a forecast derived from the previously recorded values y k . For the
convenience in further calculations we introduce the following shorthand notation
for the previous equation
α(n) =
−1
γ (n) ,
(8.36)
where α(n) is the column vector as shown in (8.34), γ (n) the column vector on the
right-hand side. The transpose of the vectors will be denoted by a superscripted letter
‘t’. denotes the matrix in (8.34).
Now we will address the expected accuracy of the forecast by calculating the
expectation value of the quadratic deviation of the forecast σ
2
(y i+n )
σ ( ˆ
y i+n )
2
=
⎛
⎝ ˆ
y i+n −
m−1
j=0
α j y i− j
⎞
⎠
⎛
⎝ ˆ
y i+n −
m−1
j=0
y i−k α k
⎞
⎠
(8.37)
=
ˆ
y i+n ˆ
y i+n
− 2
m−1
j=0
α j ˆ
y i+n y i− j +
m−1
j=0
m−1
k=0
α j α k y i− j y i−k
= γ 0 − 2α(n)
t
γ (n) + α(n)
t
α(n) ,
where we used the abbreviations introduced above and, in particular jk = =y i− j y i−k .
Inserting (8.36), we finally obtain
σ ( ˆ
y i+n )
2
= γ 0 − γ (n)
t
−1
γ (n)
(8.38)