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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 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

    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

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