forecasting using r - rob j. hyndman · fit3

Forecasting using R 1

Forecasting using R

Rob J Hyndman

1.4 Exponential smoothing

Outline

1 Simple exponential smoothing

2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7

6 Taxonomy of exponential smoothing methods

Forecasting using R Simple exponential smoothing 2

Simple methodsTime series y1, y2, . . . , yT.Random walk forecasts

yT+h|T = yT

Average forecasts

yT+h|T =1T

T∑t=1

yt

Want something in between that weights most recentdata more highly.Simple exponential smoothing uses a weightedmoving average with weights that decreaseexponentially.


Simple Exponential Smoothing

Forecast equationyT+1|T = αyT + α(1− α)yT−1 + α(1− α)2yT−2 + · · ·

where 0 ≤ α ≤ 1.

Weights assigned to observations for:Observation α = 0.2 α = 0.4 α = 0.6 α = 0.8

yT 0.2 0.4 0.6 0.8yT−1 0.16 0.24 0.24 0.16yT−2 0.128 0.144 0.096 0.032yT−3 0.1024 0.0864 0.0384 0.0064yT−4 (0.2)(0.8)4 (0.4)(0.6)4 (0.6)(0.4)4 (0.8)(0.2)4yT−5 (0.2)(0.8)5 (0.4)(0.6)5 (0.6)(0.4)5 (0.8)(0.2)5


Simple Exponential SmoothingComponent form

Forecast equation yt+h|t = `tSmoothing equation `t = αyt + (1− α)`t−1

`t is the level (or the smoothed value) of the series attime t.yt+1|t = αyt + (1− α)yt|t−1Iterate to get exponentially weighted moving averageform.

Weighted average form

yT+1|T =T−1∑j=0α(1− α)jyT−j + (1− α)T`0


Optimisation

Need to choose value for α and `0Similarly to regression — we choose α and `0 byminimising SSE:

SSE =T∑t=1(yt − yt|t−1)2.

Unlike regression there is no closed form solution —use numerical optimization.


Example: Oil production

oil %>% ses(PI=FALSE) %>% autoplot

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1980 2000 2020Time

y

Forecasts from Simple exponential smoothing


Outline


2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7


Forecasting using R Trend methods 8

Holt’s linear trendComponent form

Forecast yt+h|t = `t + hbtLevel `t = αyt + (1− α)(`t−1 + bt−1)Trend bt = β∗(`t − `t−1) + (1− β∗)bt−1,

Two smoothing parameters α and β∗ (0 ≤ α, β∗ ≤ 1).`t level: weighted average between yt one-step aheadforecast for time t, (`t−1 + bt−1 = yt|t−1)bt slope: weighted average of (`t − `t−1) and bt−1,current and previous estimate of slope.Choose α, β∗, `0, b0 to minimise SSE.


Holt’s method in R

window(ausair, start=1990, end=2004) %>%holt(h=5, PI=FALSE) %>% autoplot

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50

1990 1995 2000 2005Time

y

Forecasts from Holt's method


Damped trend method

Component formyt+h|t = `t + (φ + φ2 + · · · + φh)bt`t = αyt + (1− α)(`t−1 + φbt−1)bt = β∗(`t − `t−1) + (1− β∗)φbt−1.

Damping parameter 0 < φ < 1.If φ = 1, identical to Holt’s linear trend.As h→∞, yT+h|T → `T + φbT/(1− φ).Short-run forecasts trended, long-run forecastsconstant.


Example: Air passengers

window(ausair, start=1990, end=2004) %>%holt(damped=TRUE, h=5, PI=FALSE) %>% autoplot

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1990 1995 2000 2005Time

y

Forecasts from Damped Holt's method


Example: Sheep in Asia

livestock2 <- window(livestock, start=1970,end=2000)

fit1 <- ses(livestock2)fit2 <- holt(livestock2)fit3 <- holt(livestock2, damped = TRUE)

accuracy(fit1, livestock)accuracy(fit2, livestock)accuracy(fit3, livestock)


Example: Sheep in AsiaSES Linear trend Damped trend

α 1.00 0.98 0.98β∗ 0.00 0.00φ 0.98`0 263.92 258.88 253.69b0 5.03 5.70Training RMSE 14.77 13.92 14.00Test RMSE 25.46 11.88 15.50Test MAE 20.38 10.67 13.95Test MAPE 4.60 2.53 3.21Test MASE 2.26 1.18 1.55

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450

1970 1980 1990 2000 2010Year

Live

stoc

k, s

heep

in A

sia

(mill

ions

)

Data

SES

Holt's

Damped trend


Outline


2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7


Forecasting using R Lab session 6 15

Lab Session 6


Outline


2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7


Forecasting using R Seasonal methods 17

Holt-Winters additive methodHolt and Winters extended Holt’s method to captureseasonality.Component form

yt+h|t = `t + hbt + st−m+h+m`t = α(yt − st−m) + (1− α)(`t−1 + bt−1)bt = β∗(`t − `t−1) + (1− β∗)bt−1st = γ(yt − `t−1 − bt−1) + (1− γ)st−m,

h+m = b(h− 1) mod mc + 1 = largest integer not greater than(h− 1) mod m. Ensures estimates from the final year are usedfor forecasting.Parameters: 0 ≤ α ≤ 1, 0 ≤ β∗ ≤ 1, 0 ≤ γ ≤ 1− α andm =period of seasonality (e.g. m = 4 for quarterly data).


Holt-Winters additive method

Seasonal component is usually expressed asst = γ∗(yt − `t) + (1− γ∗)st−m.Substitute in for `t:st = γ∗(1− α)(yt − `t−1 − bt−1) + [1− γ∗(1− α)]st−mWe set γ = γ∗(1− α).The usual parameter restriction is 0 ≤ γ∗ ≤ 1, whichtranslates to 0 ≤ γ ≤ (1− α).


Holt-Winters multiplicativeFor when seasonal variations are changing proportional to thelevel of the series.Component form

yt+h|t = (`t + hbt)st−m+h+m.

`t = αyt

st−m+ (1− α)(`t−1 + bt−1)

bt = β∗(`t − `t−1) + (1− β∗)bt−1st = γ

yt(`t−1 + bt−1)

+ (1− γ)st−m

With additive method st is in absolute terms:within each year

∑i si ≈ 0.

With multiplicative method st is in relative terms:within each year

∑i si ≈ m.


Example: Visitor Nights

aust <- window(austourists,start=2005)fit1 <- hw(aust,seasonal="additive")fit2 <- hw(aust,seasonal="multiplicative")

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2005 2007 2009 2011 2013Year

Inte

rnat

iona

l vis

itor

nigh

t in

Aus

tral

ia (

mill

ions

)

Data

HW additive forecasts

HW multiplicative forecasts


Estimated components

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40

45

0.6372

0.6374

0.6376

0.6378

−10

−5

0

5

10

levelslope

season

2006 2008 2010Year

Additive states

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36

40

44

0.6

0.7

0.8

0.9

1.0

1.1

0.8

0.9

1.0

1.1

1.2

levelslope

season

2006 2008 2010Year

Multiplicative states


Holt-Winters damped method

Often the single most accurate forecasting method forseasonal data:

yt+h|t = [`t + (φ + φ2 + · · · + φh)bt]st−m+h+m`t = α(yt/st−m) + (1− α)(`t−1 + φbt−1)bt = β∗(`t − `t−1) + (1− β∗)φbt−1st = γ

yt(`t−1 + φbt−1)

+ (1− γ)st−m


Outline


2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7



Lab Session 7


Outline


2 Trend methods

3 Lab session 6

4 Seasonal methods

5 Lab session 7


Forecasting using R Taxonomy of exponential smoothing methods 26

Exponential smoothing methodsSeasonal Component

Trend N A MComponent (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)

(N,N): Simple exponential smoothing(A,N): Holt’s linear method(Ad,N): Additive damped trend method(A,A): Additive Holt-Winters’ method(A,M): Multiplicative Holt-Winters’ method(Ad,M): Damped multiplicative Holt-Winters’ method

There are also multiplicative trend methods (not recommended).Forecasting using R Taxonomy of exponential smoothing methods 27

Recursive formulae


7/ exponential smoothing 145

Trend SeasonalN A M

yt+h|t = `t yt+h|t = `t + st−m+h+m

yt+h|t = `tst−m+h+m

N `t = αyt + (1−α)`t−1 `t = α(yt − st−m) + (1−α)`t−1 `t = α(yt/st−m) + (1−α)`t−1st = γ(yt − `t−1) + (1−γ)st−m st = γ(yt/`t−1) + (1−γ)st−m

yt+h|t = `t + hbt yt+h|t = `t + hbt + st−m+h+m

yt+h|t = (`t + hbt)st−m+h+m

A `t = αyt + (1−α)(`t−1 + bt−1) `t = α(yt − st−m) + (1−α)(`t−1 + bt−1) `t = α(yt/st−m) + (1−α)(`t−1 + bt−1)bt = β∗(`t − `t−1) + (1− β∗)bt−1 bt = β∗(`t − `t−1) + (1− β∗)bt−1 bt = β∗(`t − `t−1) + (1− β∗)bt−1

st = γ(yt − `t−1 − bt−1) + (1−γ)st−m st = γ(yt/(`t−1 − bt−1)) + (1−γ)st−myt+h|t = `t +φhbt yt+h|t = `t +φhbt + st−m+h+

myt+h|t = (`t +φhbt)st−m+h+

m

Ad `t = αyt + (1−α)(`t−1 +φbt−1) `t = α(yt − st−m) + (1−α)(`t−1 +φbt−1) `t = α(yt/st−m) + (1−α)(`t−1 +φbt−1)bt = β∗(`t − `t−1) + (1− β∗)φbt−1 bt = β∗(`t − `t−1) + (1− β∗)φbt−1 bt = β∗(`t − `t−1) + (1− β∗)φbt−1

st = γ(yt − `t−1 −φbt−1) + (1−γ)st−m st = γ(yt/(`t−1 −φbt−1)) + (1−γ)st−myt+h|t = `tb

ht yt+h|t = `tb

ht + st−m+h+

myt+h|t = `tb

ht st−m+h+

m

M `t = αyt + (1−α)`t−1bt−1 `t = α(yt − st−m) + (1−α)`t−1bt−1 `t = α(yt/st−m) + (1−α)`t−1bt−1bt = β∗(`t/`t−1) + (1− β∗)bt−1 bt = β∗(`t/`t−1) + (1− β∗)bt−1 bt = β∗(`t/`t−1) + (1− β∗)bt−1

st = γ(yt − `t−1bt−1) + (1−γ)st−m st = γ(yt/(`t−1bt−1)) + (1−γ)st−m

yt+h|t = `tbφht yt+h|t = `tb

φht + st−m+h+

myt+h|t = `tb

φht st−m+h+

m

Md `t = αyt + (1−α)`t−1bφt−1 `t = α(yt − st−m) + (1−α)`t−1b

φt−1 `t = α(yt/st−m) + (1−α)`t−1b

φt−1

bt = β∗(`t/`t−1) + (1− β∗)bφt−1 bt = β∗(`t/`t−1) + (1− β∗)bφt−1 bt = β∗(`t/`t−1) + (1− β∗)bφt−1st = γ(yt − `t−1b

φt−1) + (1−γ)st−m st = γ(yt/(`t−1b

φt−1)) + (1−γ)st−m

Table 7.8: Formulae for recursivecalculations and point forecasts. Ineach case, `t denotes the series levelat time t, bt denotes the slope at timet, st denotes the seasonal componentof the series at time t, and m denotesthe number of seasons in a year; α, β∗,γ and φ are smoothing parameters,φh = φ + φ2 + · · · + φh and h+

m =b(h− 1) mod mc+ 1.

Method Initial values

(N,N) l0 = y1

(A,N) (Ad,N) l0 = y1, b0 = y2 − y1

(M,N) (Md,N) l0 = y1, b0 = y2/y1

(A,A) (Ad,A) l0 = 1m (y1 + · · ·+ ym)

b0 = 1m

[ym+1−y1

m + · · ·+ ym+m−ymm

]

s0 = y1 − `0, s−1 = y2 − `0, . . . , s−m+1 = ym − `0

(A,M) (Ad,M) l0 = 1m (y1 + · · ·+ ym)

b0 = 1m

[ym+1−y1

m + · · ·+ ym+m−ymm

]

s0 = y1/`0, s−1 = y2/`0, . . . , s−m+1 = ym/`0

Table 7.9: Initialisation strategies forsome of the more commonly usedexponential smoothing methods.

R functions

Simple exponential smoothing: no trend.ses(y)Holt’s method: linear trend.holt(y)Damped trend method.holt(y, damped=TRUE)Holt-Winters methodshw(y, damped=TRUE, seasonal="additive")hw(y, damped=FALSE, seasonal="additive")hw(y, damped=TRUE, seasonal="multiplicative")hw(y, damped=FALSE, seasonal="multiplicative")


forecasting using r - rob j. hyndman · fit3

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