Forecasting using - Rob J Hyndman exponential smoothing Forecasting using R Simple exponential...
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Forecasting using
5. Exponential smoothing methods
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Forecasting using R 1
Rob J Hyndman
Outline
1 Simple exponential smoothing
2 Non-seasonal trend methods
Forecasting using R Simple exponential smoothing 2
Simple methods
Random walk forecasts
yT+1|T = yT
Average forecasts
yT+1|T =1
T
T∑t=1
yt
Want something in between that weights mostrecent data more highly.Simple exponential smoothing uses a weightedmoving average with weights that decreaseexponentially.
Forecasting using R Simple exponential smoothing 3
Simple methods
Random walk forecasts
yT+1|T = yT
Average forecasts
yT+1|T =1
T
T∑t=1
yt
Want something in between that weights mostrecent data more highly.Simple exponential smoothing uses a weightedmoving average with weights that decreaseexponentially.
Forecasting using R Simple exponential smoothing 3
Simple methods
Random walk forecasts
yT+1|T = yT
Average forecasts
yT+1|T =1
T
T∑t=1
yt
Want something in between that weights mostrecent data more highly.Simple exponential smoothing uses a weightedmoving average with weights that decreaseexponentially.
Forecasting using R Simple exponential smoothing 3
Simple methods
Random walk forecasts
yT+1|T = yT
Average forecasts
yT+1|T =1
T
T∑t=1
yt
Want something in between that weights mostrecent data more highly.Simple exponential smoothing uses a weightedmoving average with weights that decreaseexponentially.
Forecasting using R Simple exponential smoothing 3
Simple Exponential Smoothing
Forecast equation
yT+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)4
yT−5 (0.2)(0.8)5 (0.4)(0.6)5 (0.6)(0.4)5 (0.8)(0.2)5
Forecasting using R Simple exponential smoothing 4
Simple Exponential Smoothing
Forecast equation
yT+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)4
yT−5 (0.2)(0.8)5 (0.4)(0.6)5 (0.6)(0.4)5 (0.8)(0.2)5
Forecasting using R Simple exponential smoothing 4
Simple Exponential Smoothing
Weighted average form
yt+1|t = αyt + (1− α)yt|t−1
for t = 1, . . . , T, where 0 ≤ α ≤ 1 is the smoothingparameter.
The process has to start somewhere, so we let thefirst forecast of y1 be denoted by `0. Then
y2|1 = αy1 + (1− α)`0
y3|2 = αy2 + (1− α)y2|1
y4|3 = αy3 + (1− α)y3|2...
yT+1|T = αyT + (1− α)yT|T−1Forecasting using R Simple exponential smoothing 5
Simple Exponential Smoothing
Weighted average form
yt+1|t = αyt + (1− α)yt|t−1
for t = 1, . . . , T, where 0 ≤ α ≤ 1 is the smoothingparameter.
The process has to start somewhere, so we let thefirst forecast of y1 be denoted by `0. Then
y2|1 = αy1 + (1− α)`0
y3|2 = αy2 + (1− α)y2|1
y4|3 = αy3 + (1− α)y3|2...
yT+1|T = αyT + (1− α)yT|T−1Forecasting using R Simple exponential smoothing 5
Simple Exponential Smoothingyt+1|t = αyt + (1− α)yt|t−1
Substituting each equation into the following equation:
y3|2 = αy2 + (1− α)y2|1
= αy2 + (1− α) [αy1 + (1− α)`0]
= αy2 + α(1− α)y1 + (1− α)2`0
y4|3 = αy3 + (1− α)[αy2 + α(1− α)y1 + (1− α)2`0]
= αy3 + α(1− α)y2 + α(1− α)2y1 + (1− α)3`0
...
yT+1|T = αyT + α(1− α)yT−1 + α(1− α)2yT−2 + · · ·+ (1− α)T`0
Exponentially weighted average
yT+1|T =T−1∑j=0
α(1− α)jyT−j + (1− α)T`0
Forecasting using R Simple exponential smoothing 6
Simple Exponential Smoothingyt+1|t = αyt + (1− α)yt|t−1
Substituting each equation into the following equation:
y3|2 = αy2 + (1− α)y2|1
= αy2 + (1− α) [αy1 + (1− α)`0]
= αy2 + α(1− α)y1 + (1− α)2`0
y4|3 = αy3 + (1− α)[αy2 + α(1− α)y1 + (1− α)2`0]
= αy3 + α(1− α)y2 + α(1− α)2y1 + (1− α)3`0
...
yT+1|T = αyT + α(1− α)yT−1 + α(1− α)2yT−2 + · · ·+ (1− α)T`0
Exponentially weighted average
yT+1|T =T−1∑j=0
α(1− α)jyT−j + (1− α)T`0
Forecasting using R Simple exponential smoothing 6
Simple exponential smoothing
Initialization
Last term in weighted moving average is(1− α)T ˆ0.
So value of `0 plays a role in all subsequentforecasts.
Weight is small unless α close to zero or Tsmall.
Common to set `0 = y1. Better to treat it as aparameter, along with α.
Forecasting using R Simple exponential smoothing 7
Simple exponential smoothing
Initialization
Last term in weighted moving average is(1− α)T ˆ0.
So value of `0 plays a role in all subsequentforecasts.
Weight is small unless α close to zero or Tsmall.
Common to set `0 = y1. Better to treat it as aparameter, along with α.
Forecasting using R Simple exponential smoothing 7
Simple exponential smoothing
Initialization
Last term in weighted moving average is(1− α)T ˆ0.
So value of `0 plays a role in all subsequentforecasts.
Weight is small unless α close to zero or Tsmall.
Common to set `0 = y1. Better to treat it as aparameter, along with α.
Forecasting using R Simple exponential smoothing 7
Simple exponential smoothing
Initialization
Last term in weighted moving average is(1− α)T ˆ0.
So value of `0 plays a role in all subsequentforecasts.
Weight is small unless α close to zero or Tsmall.
Common to set `0 = y1. Better to treat it as aparameter, along with α.
Forecasting using R Simple exponential smoothing 7
Simple exponential smoothing
Year
No.
str
ikes
in U
S
1950 1960 1970 1980 1990
3500
4000
4500
5000
5500
6000
Forecasting using R Simple exponential smoothing 8
Simple exponential smoothing
Optimization
We can choose α and `0 by minimizing MSE:
MSE =1
T − 1
T∑t=2
(yt − yt|t−1)2
Unlike regression there is no closed formsolution — use numerical optimization.
Forecasting using R Simple exponential smoothing 10
Simple exponential smoothing
Optimization
We can choose α and `0 by minimizing MSE:
MSE =1
T − 1
T∑t=2
(yt − yt|t−1)2
Unlike regression there is no closed formsolution — use numerical optimization.
Forecasting using R Simple exponential smoothing 10
Simple exponential smoothing
Forecasting using R Simple exponential smoothing 12
0.0 0.2 0.4 0.6 0.8 1.0
0.8
1.0
1.2
1.4
1.6
1.8
2.0
alpha
MS
E (
'000
000
)
α = 0.68
Simple exponential smoothing
Multi-step forecasts
yT+h|T = yT+1|T, h = 2,3, . . .
A “flat” forecast function.
Remember, a forecast is an estimated mean ofa future value.
So with no trend, no seasonality, and no otherpatterns, the forecasts are constant.
Forecasting using R Simple exponential smoothing 13
Simple exponential smoothing
Multi-step forecasts
yT+h|T = yT+1|T, h = 2,3, . . .
A “flat” forecast function.
Remember, a forecast is an estimated mean ofa future value.
So with no trend, no seasonality, and no otherpatterns, the forecasts are constant.
Forecasting using R Simple exponential smoothing 13
Simple exponential smoothing
Multi-step forecasts
yT+h|T = yT+1|T, h = 2,3, . . .
A “flat” forecast function.
Remember, a forecast is an estimated mean ofa future value.
So with no trend, no seasonality, and no otherpatterns, the forecasts are constant.
Forecasting using R Simple exponential smoothing 13
Simple exponential smoothing
Multi-step forecasts
yT+h|T = yT+1|T, h = 2,3, . . .
A “flat” forecast function.
Remember, a forecast is an estimated mean ofa future value.
So with no trend, no seasonality, and no otherpatterns, the forecasts are constant.
Forecasting using R Simple exponential smoothing 13
Example: Oil production
Time Observed α = 0.2 α = 0.6 α = 0.89∗
Year period t values yt Level `t
– 0 – 446.7 446.7 447.5∗
1996 1 446.7 446.7 446.7 446.71997 2 454.5 448.2 450.6 453.61998 3 455.7 449.7 453.1 455.41999 4 423.6 444.5 438.4 427.12000 5 456.3 446.8 447.3 453.12001 6 440.6 445.6 444.0 441.92002 7 425.3 441.5 434.6 427.12003 8 485.1 450.3 459.9 478.92004 9 506.0 461.4 483.0 503.12005 10 526.8 474.5 504.9 524.22006 11 514.3 482.5 509.6 515.32007 12 494.2 484.8 501.9 496.5
h Forecasts yT+h|T2008 1 – 484.8 501.9 496.52009 2 – 484.8 501.9 496.52010 3 – 484.8 501.9 496.5
∗α = 0.89 and `0 = 447.5 are obtained by minimising SSE over periods t = 1, 2, . . . , 12.
Forecasting using R Simple exponential smoothing 14
Example: Oil production
Forecasting using R Simple exponential smoothing 15
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1996 1998 2000 2002 2004 2006 2008 2010
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dataα = 0.2α = 0.6α = 0.89
SES in R
fit1 <- ses(oildata, alpha=0.2,initial="simple", h=3)
fit2 <- ses(oildata, alpha=0.6,initial="simple", h=3)
fit3 <- ses(oildata, h=3)
accuracy(fit1)accuracy(fit2)accuracy(fit3)
Forecasting using R Simple exponential smoothing 16
Equivalent forms
Weighted average form
yt+1|t = αyt + (1− α)yt|t−1
Error correction form
yt+1|t = yt|t−1 + α(yt − yt|t−1)
Component form
yt+h|t = `t
`t = αyt + (1− α)`t−1
`t = estimate of level of series.
Forecasting using R Simple exponential smoothing 17
Equivalent forms
Weighted average form
yt+1|t = αyt + (1− α)yt|t−1
Error correction form
yt+1|t = yt|t−1 + α(yt − yt|t−1)
Component form
yt+h|t = `t
`t = αyt + (1− α)`t−1
`t = estimate of level of series.
Forecasting using R Simple exponential smoothing 17
Equivalent forms
Weighted average form
yt+1|t = αyt + (1− α)yt|t−1
Error correction form
yt+1|t = yt|t−1 + α(yt − yt|t−1)
Component form
yt+h|t = `t
`t = αyt + (1− α)`t−1
`t = estimate of level of series.
Forecasting using R Simple exponential smoothing 17
Outline
1 Simple exponential smoothing
2 Non-seasonal trend methods
Forecasting using R Non-seasonal trend methods 18
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Holt (1957) extended SES to allow forecastingof data with trends.Two smoothing parameters: α and β∗ (withvalues between 0 and 1).
yt+h|t = `t + hbt`t = αyt + (1− α)(`t−1 + bt−1)
bt = β∗(`t − `t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the seriesat time tbt denotes an estimate of the slope of theseries at time t.
Forecasting using R Non-seasonal trend methods 19
Holt’s local trend method
Optimization
Need to find α and β∗ by minimizing the valueof MSE.We also optimize MSE for `0 and b0.Optimizing in four dimensions is getting tricky!
Forecasting using R Non-seasonal trend methods 20
Holt’s method in R
fit1 <- holt(strikes)plot(fit1$model)plot(fit1, plot.conf=FALSE)lines(fitted(fit1), col="red")fit1$model
fit2 <- ses(strikes)plot(fit2$model)plot(fit2, plot.conf=FALSE)lines(fit1$mean, col="red")
accuracy(fit1)accuracy(fit2)
Forecasting using R Non-seasonal trend methods 21
Comparing Holt and SES
Holt’s method will almost always have betterin-sample RMSE because it is optimized overone additional parameter.
It may not be better on other measures.
You need to compare out-of-sample RMSE(using a test set) for the comparison to beuseful.
But we don’t have enough data.
A better method for comparison will be in thenext session!
Forecasting using R Non-seasonal trend methods 22
Comparing Holt and SES
Holt’s method will almost always have betterin-sample RMSE because it is optimized overone additional parameter.
It may not be better on other measures.
You need to compare out-of-sample RMSE(using a test set) for the comparison to beuseful.
But we don’t have enough data.
A better method for comparison will be in thenext session!
Forecasting using R Non-seasonal trend methods 22
Comparing Holt and SES
Holt’s method will almost always have betterin-sample RMSE because it is optimized overone additional parameter.
It may not be better on other measures.
You need to compare out-of-sample RMSE(using a test set) for the comparison to beuseful.
But we don’t have enough data.
A better method for comparison will be in thenext session!
Forecasting using R Non-seasonal trend methods 22
Comparing Holt and SES
Holt’s method will almost always have betterin-sample RMSE because it is optimized overone additional parameter.
It may not be better on other measures.
You need to compare out-of-sample RMSE(using a test set) for the comparison to beuseful.
But we don’t have enough data.
A better method for comparison will be in thenext session!
Forecasting using R Non-seasonal trend methods 22
Comparing Holt and SES
Holt’s method will almost always have betterin-sample RMSE because it is optimized overone additional parameter.
It may not be better on other measures.
You need to compare out-of-sample RMSE(using a test set) for the comparison to beuseful.
But we don’t have enough data.
A better method for comparison will be in thenext session!
Forecasting using R Non-seasonal trend methods 22
Exponential trend method
Multiplicative version of Holt’s method
yt+h|t = `tbht
`t = αyt + (1− α)(`t−1bt−1)
bt = β∗(`t/`t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the series attime tbt denotes an estimate of the relative growth of theseries at time t.In R: holt(x, exponential=TRUE)Comparing additive and multiplicative trendmethods in-sample is ok because they have thesame number of parameters to optimize.
Forecasting using R Non-seasonal trend methods 23
Exponential trend method
Multiplicative version of Holt’s method
yt+h|t = `tbht
`t = αyt + (1− α)(`t−1bt−1)
bt = β∗(`t/`t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the series attime tbt denotes an estimate of the relative growth of theseries at time t.In R: holt(x, exponential=TRUE)Comparing additive and multiplicative trendmethods in-sample is ok because they have thesame number of parameters to optimize.
Forecasting using R Non-seasonal trend methods 23
Exponential trend method
Multiplicative version of Holt’s method
yt+h|t = `tbht
`t = αyt + (1− α)(`t−1bt−1)
bt = β∗(`t/`t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the series attime tbt denotes an estimate of the relative growth of theseries at time t.In R: holt(x, exponential=TRUE)Comparing additive and multiplicative trendmethods in-sample is ok because they have thesame number of parameters to optimize.
Forecasting using R Non-seasonal trend methods 23
Exponential trend method
Multiplicative version of Holt’s method
yt+h|t = `tbht
`t = αyt + (1− α)(`t−1bt−1)
bt = β∗(`t/`t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the series attime tbt denotes an estimate of the relative growth of theseries at time t.In R: holt(x, exponential=TRUE)Comparing additive and multiplicative trendmethods in-sample is ok because they have thesame number of parameters to optimize.
Forecasting using R Non-seasonal trend methods 23
Exponential trend method
Multiplicative version of Holt’s method
yt+h|t = `tbht
`t = αyt + (1− α)(`t−1bt−1)
bt = β∗(`t/`t−1) + (1− β∗)bt−1
`t denotes an estimate of the level of the series attime tbt denotes an estimate of the relative growth of theseries at time t.In R: holt(x, exponential=TRUE)Comparing additive and multiplicative trendmethods in-sample is ok because they have thesame number of parameters to optimize.
Forecasting using R Non-seasonal trend methods 23
Damped trend method
Gardner and McKenzie (1985) suggested thatthe trends should be “damped” to be moreconservative for longer forecast horizons.Two smoothing parameters: α and β∗ (withvalues between 0 and 1), and one dampingparameter 0 < φ < 1.
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
φ dampens the trend so it approaches aconstant.
Forecasting using R Non-seasonal trend methods 24
Damped trend method
Gardner and McKenzie (1985) suggested thatthe trends should be “damped” to be moreconservative for longer forecast horizons.Two smoothing parameters: α and β∗ (withvalues between 0 and 1), and one dampingparameter 0 < φ < 1.
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
φ dampens the trend so it approaches aconstant.
Forecasting using R Non-seasonal trend methods 24
Damped trend method
Gardner and McKenzie (1985) suggested thatthe trends should be “damped” to be moreconservative for longer forecast horizons.Two smoothing parameters: α and β∗ (withvalues between 0 and 1), and one dampingparameter 0 < φ < 1.
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
φ dampens the trend so it approaches aconstant.
Forecasting using R Non-seasonal trend methods 24
Damped trend method
Gardner and McKenzie (1985) suggested thatthe trends should be “damped” to be moreconservative for longer forecast horizons.Two smoothing parameters: α and β∗ (withvalues between 0 and 1), and one dampingparameter 0 < φ < 1.
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
φ dampens the trend so it approaches aconstant.
Forecasting using R Non-seasonal trend methods 24
Damped trend method
Gardner and McKenzie (1985) suggested thatthe trends should be “damped” to be moreconservative for longer forecast horizons.Two smoothing parameters: α and β∗ (withvalues between 0 and 1), and one dampingparameter 0 < φ < 1.
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
φ dampens the trend so it approaches aconstant.
Forecasting using R Non-seasonal trend methods 24
Damped trend method
Forecasting using R Non-seasonal trend methods 25
Forecasts from damped Holt's method
1950 1960 1970 1980 1990
2500
3500
4500
5500
Damped trend method
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
If φ = 1, this is the same as Holt’s method.
φ can be estimated along with α and β∗ byminimizing the MSE.
Damped trend method often gives betterforecasts than linear trend.
Forecasts converge to `T + φbT/(1− φ) ash→∞.
Forecasting using R Non-seasonal trend methods 26
Damped trend method
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
If φ = 1, this is the same as Holt’s method.
φ can be estimated along with α and β∗ byminimizing the MSE.
Damped trend method often gives betterforecasts than linear trend.
Forecasts converge to `T + φbT/(1− φ) ash→∞.
Forecasting using R Non-seasonal trend methods 26
Damped trend method
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
If φ = 1, this is the same as Holt’s method.
φ can be estimated along with α and β∗ byminimizing the MSE.
Damped trend method often gives betterforecasts than linear trend.
Forecasts converge to `T + φbT/(1− φ) ash→∞.
Forecasting using R Non-seasonal trend methods 26
Damped trend method
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
If φ = 1, this is the same as Holt’s method.
φ can be estimated along with α and β∗ byminimizing the MSE.
Damped trend method often gives betterforecasts than linear trend.
Forecasts converge to `T + φbT/(1− φ) ash→∞.
Forecasting using R Non-seasonal trend methods 26
Damped trend method
yt+h|t = `t + (φ+ φ2 + · · ·+ φh−1)bt`t = αyt + (1− α)(`t−1 + φbt−1)
bt = β∗(`t − `t−1) + (1− β∗)φbt−1
If φ = 1, this is the same as Holt’s method.
φ can be estimated along with α and β∗ byminimizing the MSE.
Damped trend method often gives betterforecasts than linear trend.
Forecasts converge to `T + φbT/(1− φ) ash→∞.
Forecasting using R Non-seasonal trend methods 26
Multiplicative damped trend method
Taylor (2003) introduced multiplicative damping.
yt+h|t = `tb(φ+φ2+···+φh)t
`t = αyt + (1− α)(`t−1bφt−1)
bt = β∗(`t/`t−1) + (1− β∗)bφt−1
φ = 1 gives exponential trend method
Forecasts converge to `T + bφ/(1−φ)T as h→∞.
Forecasting using R Non-seasonal trend methods 27
Multiplicative damped trend method
Taylor (2003) introduced multiplicative damping.
yt+h|t = `tb(φ+φ2+···+φh)t
`t = αyt + (1− α)(`t−1bφt−1)
bt = β∗(`t/`t−1) + (1− β∗)bφt−1
φ = 1 gives exponential trend method
Forecasts converge to `T + bφ/(1−φ)T as h→∞.
Forecasting using R Non-seasonal trend methods 27
Multiplicative damped trend method
Taylor (2003) introduced multiplicative damping.
yt+h|t = `tb(φ+φ2+···+φh)t
`t = αyt + (1− α)(`t−1bφt−1)
bt = β∗(`t/`t−1) + (1− β∗)bφt−1
φ = 1 gives exponential trend method
Forecasts converge to `T + bφ/(1−φ)T as h→∞.
Forecasting using R Non-seasonal trend methods 27