Lead−time demand variance

			Poisson lead−time
Variance	60 80
			Fixed lead−time

Forecast horizon

Fig. 6.6. Lead-time demand variance for an ETS(A,N,N) model with fixed andstochastic lead-times. Here, α = 0.1, σ = 1 and __n = 2.

6.7 Exercises

Exercise 6.1. For the ETS(M,N,N) model, show that

θ_j = _ ² _n (1+ α ² σ ²) ^{j− 1}

and

_ _

v _{+ |} = _ ²(1+ α ² σ ²) ^{h− 1}(1+ σ ²) − 1.

n h n n

Exercise 6.2. For the ETS(A,A,A) model, use (6.23) replacing φ_j by j to showthat

_

v_{n + h|n} = σ ²1+ (h − 1)^_ α ²+ αβh +¹₆ β ² h (2 h − 1)^_

_

+ γh_m { 2 α + γ + βm (h_m + 1) }.

Exercise 6.3. Monthly US 10-year bonds data were forecast with anETS(A,A_d,N) model in Sect. 2.8.1 (p. 28). Find the 95% prediction intervals for this model algebraically and compare the results obtained by simulating 5,000 future sample paths using R.

Exercise 6.4. Quarterly UK passenger vehicle production data were forecastwith an ETS(A,N,A) model in Sect. 2.8.1 (p. 28). Find the 95% prediction intervals for this model algebraically and compare the results obtained by simulating 5,000 future sample paths using R.

Appendix: Derivations

Appendix: Derivations

Derivation of Results for Class 1

The results for Class 1 models are obtained by first noting that all of the linear, homoscedastic ETS models can be written using the following linear state space model, introduced in Chap. 3:

yt = w^_xt− 1+ ε t	(6.19)
xt = F xt− 1+ gε t,	(6.20)

where w^_ is a row vector, g is a column vector, F is a matrix, and x_t is the unobserved state vector at time t. In each case, {ε _t } is NID(0, σ ²).

Let I_k denote the k × k identity matrix, and 0 _k denote a zero vector of length k. Then

• The ETS(A,N,N) model has x_t = _ _t, w = F = 1 and g = α;

• The ETS(A,A_d,N) model has x_t = (__t, b_t) ^_, w^_ = [1 φ ],

F =		φ		and	g =	α	;
	φ	β

• The ETS(A,N,A) model has x_t = (__t, s_t, s_{t− 1},..., s_{t− ( m− 1)}) ^_, w^_ = [10 ^__{m− 1}1],

			0 ^_					α
		m 1
F =		0 m^_^− 1		and g =	γ	;
		0 m− 1 Im−^− 1	0 m− 1				0 m− 1

• The ETS(A,A_d,A) model has x_t = (__t, b_t, s_t, s_{t− 1},..., s_{t− ( m− 1)}) ^_, w^_ = [1 φ 0 ^__{m− 1}1],

φ

0 _

α

m 1

F =

φ

^00 m^_

−^− 1

and g =

β

.

I

m^_

− 1

γ

m

−

m

−

m

−

1 m

−

m

−

The matrices for (A,A,N) and (A,A,A) are the same as for (A,A_d,N) and (A,A_d,A) respectively, but with φ = 1.

Forecast Mean

Let ^mn + h|n = E(^xn + h | ^xn). Then ^mn|n = ^xn and

^mn + h|n = ^{F m}n + h− 1 |n = ^{F 2 m}n + h− 2 |n = · · · = ^{F h m}n|n = ^{F h x}n.

96 6 Prediction Distributions and Intervals

Therefore

µ_{n + h|n} =E(y_{n + h} |x_n) = w^_m_{n + h− 1 |n} = w^_F ^{h− 1} x_n.

Example 6.2: Forecast mean of the ETS(A,Ad,A) model

For the ETS(A,Ad,A) model, w^_ = [1 φ 0 m^_

− 1 1] and

^φj

...

0 φ^j

...

F ^j =

d

+ m

m

^dj + m +

m... d

+

2 m− 1,

m

d

j

,

d

1,

... d

j

,

..

j + m

1, m

j + m, m

.

j +2 m

−

2, m

. ^−

.

. .

.

..

.

.

.

..

.

.

.

^dj +1, m

^dj +2, m

^... ^dj + m, m

where φ_j = φ + φ ² + · · · + φ^j, and d_{k , m} = 1 if k = 0 (mod m) and d otherwise. Therefore,

w^_F ^j = [1, φ_{j +1}, d_{j +1, m} , d_{j +2, m} ,..., d_{j + m , m} ]

and

µn + h|n = _n + φh bn + s_{n−m + h} + _m.

k, m ⁼⁰

(6.21)

The forecast means for the other models can be derived similarly, and are listed in Table 6.2

Дата добавления: 2015-10-24