# forecasting using - rob j hyndman 1exponential smoothing methods so far...

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Forecasting using

6. ETS models

OTexts.com/fpp/7/

Forecasting using R 1

Rob J Hyndman

Outline

1 Exponential smoothing methods so far

2 Holt-Winters seasonal method

3 Taxonomy of exponential smoothingmethods

4 Exponential smoothing state space models

Forecasting using R Exponential smoothing methods so far 2

Exponential smoothing methods

Simple exponential smoothing: no trend.ses(x)

Holts method: linear trend.holt(x)

Exponential trend method.holt(x, exponential=TRUE)

Damped trend method.holt(x, damped=TRUE)

Damped exponential trend method.holt(x, damped=TRUE, exponential=TRUE)

Forecasting using R Exponential smoothing methods so far 3

Exponential smoothing methods

Simple exponential smoothing: no trend.ses(x)

Holts method: linear trend.holt(x)

Exponential trend method.holt(x, exponential=TRUE)

Damped trend method.holt(x, damped=TRUE)

Damped exponential trend method.holt(x, damped=TRUE, exponential=TRUE)

Forecasting using R Exponential smoothing methods so far 3

Exponential smoothing methods

Simple exponential smoothing: no trend.ses(x)

Holts method: linear trend.holt(x)

Exponential trend method.holt(x, exponential=TRUE)

Damped trend method.holt(x, damped=TRUE)

Damped exponential trend method.holt(x, damped=TRUE, exponential=TRUE)

Forecasting using R Exponential smoothing methods so far 3

Exponential smoothing methods

Simple exponential smoothing: no trend.ses(x)

Holts method: linear trend.holt(x)

Exponential trend method.holt(x, exponential=TRUE)

Damped trend method.holt(x, damped=TRUE)

Damped exponential trend method.holt(x, damped=TRUE, exponential=TRUE)

Forecasting using R Exponential smoothing methods so far 3

Exponential smoothing methods

Simple exponential smoothing: no trend.ses(x)

Holts method: linear trend.holt(x)

Exponential trend method.holt(x, exponential=TRUE)

Damped trend method.holt(x, damped=TRUE)

Damped exponential trend method.holt(x, damped=TRUE, exponential=TRUE)

Forecasting using R Exponential smoothing methods so far 3

Outline

1 Exponential smoothing methods so far

2 Holt-Winters seasonal method

3 Taxonomy of exponential smoothingmethods

4 Exponential smoothing state space models

Forecasting using R Holt-Winters seasonal method 4

Holt-Winters additive methodHolt and Winters extended Holts method tocapture seasonality.Three smoothing equationsone for the level,one for trend, and one for seasonality.Parameters: 0 1, 0 1,0 1 and m = period of seasonality.

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

Forecasting using R Holt-Winters seasonal method 5

Holt-Winters additive methodHolt and Winters extended Holts method tocapture seasonality.Three smoothing equationsone for the level,one for trend, and one for seasonality.Parameters: 0 1, 0 1,0 1 and m = period of seasonality.

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

Forecasting using R Holt-Winters seasonal method 5

Holt-Winters additive methodHolt and Winters extended Holts method tocapture seasonality.Three smoothing equationsone for the level,one for trend, and one for seasonality.Parameters: 0 1, 0 1,0 1 and m = period of seasonality.

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

Forecasting using R Holt-Winters seasonal method 5

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

Forecasting using R Holt-Winters seasonal method 5

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

Forecasting using R Holt-Winters seasonal method 5

Holt-Winters additive method

yt+h|t = `t + hbt + stm+h+m`t = (yt stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt `t1 bt1) + (1 )stm

h+m = b(h 1) mod mc+ 1Forecasting using R Holt-Winters seasonal method 5

Holt-Winters multiplicative method

Holt-Winters multiplicative method

yt+h|t = (`t + hbt)stm+h+m`t = (yt/stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt/(`t1 + bt1)) + (1 )stm

Most textbooks use st = (yt/`t) + (1 )stmWe optimize for , , , `0, b0, s0, s1, . . . , s1m.

Forecasting using R Holt-Winters seasonal method 6

Holt-Winters multiplicative method

Holt-Winters multiplicative method

yt+h|t = (`t + hbt)stm+h+m`t = (yt/stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = (yt/(`t1 + bt1)) + (1 )stm

Most textbooks use st = (yt/`t) + (1 )stmWe optimize for , , , `0, b0, s0, s1, . . . , s1m.

Forecasting using R Holt-Winters seasonal method 6

Damped Holt-Winters method

Damped Holt-Winters multiplicative method

yt+h|t = [`t + (1 + + 2 + + h1)bt]stm+h+m

`t = (yt/stm) + (1 )(`t1 + bt1)bt =

(`t `t1) + (1 )bt1st = [yt/(`t1 + bt1)] + (1 )stm

This is often the single most accurateforecasting method for seasonal data.

Forecasting using R Holt-Winters seasonal method 7

A confusing array of methods?

All these methods can be confusing!

How to choose between them?

The ETS framework provides an

automatic way of selecting the best

method.

It was developed to solve the problem

of automatically forecasting

pharmaceutical sales across thousands

of products.Forecasting using R Holt-Winters seasonal method 8

A confusing array of methods?

All these methods can be confusing!

How to choose between them?

The ETS framework provides an

automatic way of selecting the best

method.

It was developed to solve the problem

of automatically forecasting

pharmaceutical sales across thousands

of products.Forecasting using R Holt-Winters seasonal method 8

A confusing array of methods?

All these methods can be confusing!

How to choose between them?

The ETS framework provides an

automatic way of selecting the best

method.

It was developed to solve the problem

of automatically forecasting

pharmaceutical sales across thousands

of products.Forecasting using R Holt-Winters seasonal method 8

A confusing array of methods?

All these methods can be confusing!

How to choose between them?

The ETS framework provides an

automatic way of selecting the best

method.

It was developed to solve the problem

of automatically forecasting

pharmaceutical sales across thousands

of products.Forecasting using R Holt-Winters seasonal method 8

Outline

1 Exponential smoothing methods so far

2 Holt-Winters seasonal method

3 Taxonomy of exponential smoothingmethods

4 Exponential smoothing state space models

Forecasting using R Taxonomy of exponential smoothing methods 9

Exponential smoothing methods

Seasonal ComponentTrend N A M

Component (None) (Additive) (Multiplicative)

N (None) N,N N,A N,M

A (Additive) A,N A,A A,M

Ad (Additive damped) Ad,N Ad,A Ad,M

M (Multiplicative) M,N M,A M,M

Md (Multiplicative damped) Md,N Md,A Md,M

Forecasting using R Taxonomy of exponential smoothing methods 10

Exponential smoothing methods

Seasonal ComponentTrend N A M

Component (None) (Additive) (Multiplicative)

N (None) N,N N,A N,M

A (Additive) A,N A,A A,M

Ad (Additive damped) Ad,N Ad,A Ad,M

M (Multiplicative) M,N M,A M,M

Md (Multiplicative damped) Md,N Md,A Md,M

(N,N): Simple exponential smoothing

Forecasting using R Taxonomy of exponential smoothing methods 10

Exponential smoothing methods

Seasonal ComponentTrend N A M

Component (None) (Additive) (Multiplicative)

N (None) N,N N,A N,M

A (Additive) A,N A,A A,M

Ad (Additive damped) Ad,N Ad,A Ad,M

M (Multiplicative) M,N M,A M,M

Md (Multiplicative damped) Md,N Md,A Md,M

(N,N): Simple exponential smoothing(A,N): Holts linear method

Forecasting using R Taxonomy of exponential smoothing methods 10

Exponential smoothing methods

Seasonal ComponentTrend N A M

Component (None) (Additive) (Multiplicative)

N (None) N,N N,A N,M

A (Additive) A,N A,A A,M

Ad (Additive damped) Ad,N Ad,A Ad,M

M (Multiplicative) M,N M,A M,M

Md (Multiplicative dampe