UK passenger motor vehicle production Overseas visitors to Australia 2 страница

To distinguish the models with additive and multiplicative errors, we add an extra letter to the front of the method notation. The triplet (E,T,S) refers to the three components: error, trend and seasonality. So the model ETS(A,A,N) has additive errors, additive trend and no seasonality—in other words, this is Holt’s linear method with additive errors. Similarly, ETS(M,M_d,M) refers to a model with multiplicative errors, a damped multiplicative trend and multiplicative seasonality. The notation ETS(·, ·, ·) helps in remembering the order in which the components are specified. ETS can also be considered an abbreviation of E xponen T ial S moothing.

Once a model is specified, we can study the probability distribution of future values of the series and find, for example, the conditional mean

^{y ˆ} t + h|t

^{y ˆ} t + h|t

Table 2.1. Formulae for recursive calculations and point forecasts.

18

Trend

N

A

^Ad

M

N

__t = αy_t + (1 − α) __{t− 1}

^{y ˆ} t + h|t = ^_t

__t = αy_t + (1 − α)(__{t− 1} + b_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) b_{t− 1}

^{y ˆ} t + h|t = ^_t + ^hbt

__t = αy_t + (1 − α)(__{t− 1} + φb_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) φb_{t− 1}

^{y ˆ} t + h|t = ^_t + ^φh ^bt

__t = αy_t + (1 − α) __{t− 1} b_{t− 1}

b_t = β^∗ (__t / __{t− 1}) + (1 − β^∗) b_{t− 1}

^{y ˆ} t + h|t = ^_t ^b_t^h

Seasonal

A

__t = α (y_t − s_t−m) + (1 − α) __{t− 1} s_t = γ (y_t − __{t− 1}) + (1 − γ) s_t−m

^{y ˆ} t + h|t = ^_t + ^s_t−m + _h ⁺ _m

__t = α (y_t − s_t−m) + (1 − α)(__{t− 1} + b_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) b_{t− 1}

s_t = γ (y_t − __{t− 1} − b_{t− 1}) + (1 − γ) s_t−m ^{y ˆ} t + h|t = ^_t + ^hbt + ^s_t−m + _h ⁺ _m

__t = α (y_t − s_t−m) + (1 − α)(__{t− 1} + φb_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) φb_{t− 1}

s_t = γ (y_t − __{t− 1} − φb_{t− 1}) + (1 − γ) s_t−m ^{y ˆ} t + h|t = ^_t + ^φh ^bt + ^s_t−m + _h ⁺ _m

__t = α (y_t − s_t−m) + (1 − α) __{t− 1} b_{t− 1}

b_t = β^∗ (__t / __{t− 1}) + (1 − β^∗) b_{t− 1}

s_t = γ (y_t − __{t− 1} b_{t− 1}) + (1 − γ) s_t−m

^{y ˆ} t + h|t = ^_t ^bh + ^s ₋ +⁺ t t m h_m

M

__t = α (y_t / s_t−m) + (1 − α) __{t− 1} s_t = γ (y_t / __{t− 1}) + (1 − γ) s_t−m

^{y ˆ} t + h|t = ^_t ^s_t−m + _h ⁺ _m

__t = α (y_t / s_t−m) + (1 − α)(__{t− 1} + b_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) b_{t− 1}

s_t = γ (y_t /(__{t− 1}+ b_{t− 1})) + (1 − γ) s_t−m

= (^_t + ^hbt) ^s_t−m + _h ⁺ _m

__t = α (y_t / s_t−m) + (1 − α)(__{t− 1} + φb_{t− 1}) b_t = β^∗ (__t − __{t− 1}) + (1 − β^∗) φb_{t− 1}

s_t = γ (y_t /(__{t− 1}+ φb_{t− 1})) + (1 − γ) s_t−m

= (^_t + ^φh ^bt) ^s_t−m + _h ⁺ _m

__t = α (y_t / s_t−m) + (1 − α) __{t− 1} b_{t− 1}

b_t = β^∗ (__t / __{t− 1}) + (1 − β^∗) b_{t− 1}

s_t = γ (y_t /(__{t− 1} b_{t− 1})) + (1 − γ) s_t−m

^{y ˆ} t + h|t = ^_t ^{bh s} ₋ +⁺ t t m h_m

2 Getting Started

_t = αyt + (1 − α) _t− 1 bt^φ− 1

_t = α (yt − st−m) + (1 − α) _t− 1 bt^φ− 1

_t = α (yt / st−m) + (1 −α) _t− 1 bt^φ− 1

b

t

= β^∗ (_

/ _

t− 1

) + (1 − β^∗) b^φ

b

t

= β^∗ (_

/ _

t− 1

) + (1 − β^∗) b^φ

b

t

= β^∗ (_

/ _

t− 1

) + (1 − β^∗) b^φ

Md

t

t− 1

t

t− 1

t

t− 1

st = γ (yt − _t− 1 bt^φ− 1) + (1 − γ) st−m

st = γ (yt /(_t− 1 bt^φ− 1)) + (1 − γ) st−m

y ˆ t + h|t = _t bt^φ^h

^y ^ˆ t + h|t = ^_t ^bt^φ^h + ^st−m + hm ^+

^y ^ˆ t + h|t = ^_t ^bt^φ^h ^st−m + hm ^+

In each case, __t denotes the series level at time t, b_t denotes the slope at time t, s_t denotes the seasonal component of the series at time t, and m denotes the number of seasons in a year; α, β^∗, γ and φ are constants, φ_h = φ + φ ² + · · · + φ^h and h ⁺ _m = ^_(h − 1) mod m ^_ + 1.

^µt + h|t

2.5 State Space Models

of a future observation given knowledge of the past. We denote this as

= E(y_{t + h} | x_t), where x_t contains the unobserved components such

as __t, b_t and s_t. For h = 1 we use µ_{t +1} ≡ µ_{t +1 |t} as a shorthand notation. For most models, these conditional means will be identical to the point forecasts given earlier, so that µ_{t + h|t} = y ˆ _{t + h|t}. However, for other models (those with multiplicative trend or multiplicative seasonality), the conditional mean and the point forecast will differ slightly for h ≥ 2.

2.5.1 State Space Models for Holt’s Linear Method

We now illustrate the ideas using Holt’s linear method.

Additive Error Model: ETS(A,A,N)

Let µ_t = y ˆ _t = __{t− 1} + b_{t− 1} denote the one-step forecast of y_t assuming we know the values of all parameters. Also let ε _t = y_t − µ_t denote the one-step forecast error at time t. From (2.4c), we find that

	yt = _t− 1+ bt− 1+ ε t,	(2.8)
and using (2.4a) and (2.4b) we can write
_t	= _t− 1 + bt− 1 + αε t,	(2.9)
bt	= bt− 1 + β^∗ (_t − _t− 1 − bt− 1) = bt− 1 + αβ^∗ε t.	(2.10)

We simplify the last expression by setting β = αβ^∗. The three equations above constitute a state space model underlying Holt’s method. We can write it in standard state space notation by defining the state vector as x_t = (__t, b_t) and expressing (2.8)–(2.10) as

yt = [1 1] xt− 1+ ε t,		(2.11a)
xt =^_	1 1	^_ xt− 1	+ ^_	α	^_ ε t.	(2.11b)
0 1	β

The model is fully specified once we state the distribution of the error term ε _t. Usually we assume that these are independent and identically distributed, following a Gaussian distribution with mean 0 and variance σ ², which we write as ε _t ∼ NID(0, σ ²).

Multiplicative Error Model: ETS(M,A,N)

A model with multiplicative error can be derived similarly, by first setting ε _t = (y_t − µ_t)/ µ_t, so that ε _t is a relative error. Then, following a similarapproach to that for additive errors, we find

y_t = (__{t− 1}+ b_{t− 1})(1+ ε _t), __t = (__{t− 1} + b_{t− 1})(1 + αε _t),

bt = bt− 1+ β (_t− 1+ bt− 1) ε t,

	2 Getting Started
or
	yt	= [1 1] xt− 1(1 + ε t),
	xt	= ^_	1 1	^_ xt− 1	+ [1 1] xt− 1	_	α	^_ ε t.
	0 1	β

Again we assume that ε _t ∼ NID(0, σ ²).

Of course, this is a nonlinear state space model, which is usually consid-ered difficult to handle in estimating and forecasting. However, that is one of the many advantages of the innovations form of state space models—we can still compute forecasts, the likelihood and prediction intervals for this nonlinear model with no more effort than is required for the additive error model.

2.5.2 State Space Models for All Exponential Smoothing Methods

We now give the state space models for all 30 exponential smoothing variations. The general model involves a state vector x_t = (__t, b_t, s_t, s_{t− 1},..., s_{t−m +1})and state space equations of the form

yt = w (xt− 1) + r (xt− 1) ε t,	(2.12a)
xt = f (xt− 1) + g (xt− 1) ε t,	(2.12b)

where {ε _t } is a Gaussian white noise process with variance σ ², and µ_t = w (x_{t− 1}). The model with additive errors has r (x_{t− 1}) =1, so that y_t = µ_t + ε _t. The model with multiplicative errors has r (x_{t− 1}) = µ_t, so that y_t = µ_t (1 + ε _t). Thus, ε _t = (y_t − µ_t)/ µ_t is the relative error for the multiplicative model.The models are not unique. Clearly, any value of r (x_{t− 1}) will lead to identical point forecasts for y_t.

Each of the methods in Table 2.1 can be written in the form given in (2.12a) and (2.12b). The underlying equations for the additive error models are given in Table 2.2. We use β = αβ^∗ to simplify the notation. Multiplicative error models are obtained by replacing ε _t with µ_t ε _t in the equations of Table 2.2. The resulting multiplicative error equations are given in Table 2.3.

Some of the combinations of trend, seasonality and error can occasionally lead to numerical difficulties; specifically, any model equation that requires division by a state component could involve division by zero. This is a problem for models with additive errors and either multiplicative trend or multiplicative seasonality, as well as the model with multiplicative errors, multiplicative trend and additive seasonality. These models should there-fore be used with caution. The properties of these models are discussed in Chap. 15.

The multiplicative error models are useful when the data are strictly pos-itive, but are not numerically stable when the data contain zeros or negative

Table 2.2. State space equations for each additive error model in the classification.

Trend Seasonal

N

^µt = ^_t− 1

N __t = __{t− 1} + αε _t

^µt = ^_t− 1+ ^bt− 1

^{A _}t = ^_t− 1+ ^bt− 1+ ^αε t ^bt = ^bt− 1+ ^βε t

	^µt = ^_t− 1+ ^φ^bt− 1
Ad	^_t = ^_t− 1+ ^φ^bt− 1+ ^αε t
	bt = φbt− 1+ βε t
	^µt = ^_t− 1 ^bt− 1
M	^_t = ^_t− 1 ^bt− 1+ ^αε t
	^bt = ^bt− 1+ ^βε t ^/ ^_t− 1
	^µt = ^_t− 1 ^bt^φ− 1
Md	^_t = ^_t− 1 ^bt^φ− 1+ ^αε t
	^bt = ^bt^φ− 1+ ^βε t ^/ ^_t− 1

A

^µt = ^_t− 1+ ^st−m ^_t = ^_t− 1+ ^αε t s_t = s_t−m + γε _t

^µt = ^_t− 1+ ^bt− 1+ ^st−m ^_t = ^_t− 1+ ^bt− 1+ ^αε t ^bt = ^bt− 1+ ^βε t

s_t = s_t−m + γε _t

^µt = ^_t− 1+ ^φbt− 1+ ^st−m ^_t = ^_t− 1+ ^φbt− 1+ ^αε t b_t = φb_{t− 1}+ βε _t

s_t = s_t−m + γε _t

^µt = ^_t− 1 ^bt− 1+ ^st−m ^_t = ^_t− 1 ^bt− 1+ ^αε t ^bt = ^bt− 1+ ^βε t ^{/ _}t− 1 s_t = s_t−m + γε _t

^µt = ^_t− 1 ^b_t^φ_{− 1}+ ^st−m ^_t = ^_t− 1 ^b_t^φ_{− 1}+ ^αε t ^bt = ^b_t^φ_{− 1}+ ^βε t ^{/ _}t− 1

Дата добавления: 2015-10-24