Forecasting: principles and practice · Outline 1Variance stabilization 2Box-Cox transformations...
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Forecasting: principles and practice 1 Forecasting: principles and practice Rob J Hyndman 2.2 Transformations
Transcript of Forecasting: principles and practice · Outline 1Variance stabilization 2Box-Cox transformations...
Forecasting: principles and practice 1
Forecasting: principlesand practice
Rob J Hyndman
2.2 Transformations
Outline
1 Variance stabilization
2 Box-Cox transformations
3 Back-transformation
4 Lab session 9
Forecasting: principles and practice Variance stabilization 2
Variance stabilizationIf the data show different variation at different levels ofthe series, then a transformation can be useful.Denote original observations as y1, . . . , yn andtransformed observations as w1, . . . ,wn.Mathematical transformations for stabilizing variation
Square root wt =√yt ↓
Cube root wt = 3√yt Increasing
Logarithm wt = log(yt) strength
Logarithms, in particular, are useful because they are moreinterpretable: changes in a log value are relative (percent)changes on the original scale.
Forecasting: principles and practice Variance stabilization 3
Variance stabilizationIf the data show different variation at different levels ofthe series, then a transformation can be useful.Denote original observations as y1, . . . , yn andtransformed observations as w1, . . . ,wn.Mathematical transformations for stabilizing variation
Square root wt =√yt ↓
Cube root wt = 3√yt Increasing
Logarithm wt = log(yt) strength
Logarithms, in particular, are useful because they are moreinterpretable: changes in a log value are relative (percent)changes on the original scale.
Forecasting: principles and practice Variance stabilization 3
Variance stabilizationIf the data show different variation at different levels ofthe series, then a transformation can be useful.Denote original observations as y1, . . . , yn andtransformed observations as w1, . . . ,wn.Mathematical transformations for stabilizing variation
Square root wt =√yt ↓
Cube root wt = 3√yt Increasing
Logarithm wt = log(yt) strength
Logarithms, in particular, are useful because they are moreinterpretable: changes in a log value are relative (percent)changes on the original scale.
Forecasting: principles and practice Variance stabilization 3
Variance stabilizationIf the data show different variation at different levels ofthe series, then a transformation can be useful.Denote original observations as y1, . . . , yn andtransformed observations as w1, . . . ,wn.Mathematical transformations for stabilizing variation
Square root wt =√yt ↓
Cube root wt = 3√yt Increasing
Logarithm wt = log(yt) strength
Logarithms, in particular, are useful because they are moreinterpretable: changes in a log value are relative (percent)changes on the original scale.
Forecasting: principles and practice Variance stabilization 3
Variance stabilization
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Electricity production
Forecasting: principles and practice Variance stabilization 4
Variance stabilization
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Square root electricity production
Forecasting: principles and practice Variance stabilization 5
Variance stabilization
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Cube root electricity production
Forecasting: principles and practice Variance stabilization 6
Variance stabilization
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Log electricity production
Forecasting: principles and practice Variance stabilization 7
Variance stabilization
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Inverse electricity production
Forecasting: principles and practice Variance stabilization 8
Outline
1 Variance stabilization
2 Box-Cox transformations
3 Back-transformation
4 Lab session 9
Forecasting: principles and practice Box-Cox transformations 9
Box-Cox transformations
Each of these transformations is close to a member of thefamily of Box-Cox transformations:
wt = log(yt), λ = 0;(yλt − 1)/λ, λ 6= 0.
λ = 1: (No substantive transformation)λ = 1
2 : (Square root plus linear transformation)λ = 0: (Natural logarithm)λ = −1: (Inverse plus 1)
Forecasting: principles and practice Box-Cox transformations 10
Box-Cox transformations
Each of these transformations is close to a member of thefamily of Box-Cox transformations:
wt = log(yt), λ = 0;(yλt − 1)/λ, λ 6= 0.
λ = 1: (No substantive transformation)λ = 1
2 : (Square root plus linear transformation)λ = 0: (Natural logarithm)λ = −1: (Inverse plus 1)
Forecasting: principles and practice Box-Cox transformations 10
Box-Cox transformations
Forecasting: principles and practice Box-Cox transformations 11
Box-Cox transformations
autoplot(BoxCox(elec,lambda=1/3))
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Time
Box
Cox
(ele
c, la
mbd
a =
1/3
)
Forecasting: principles and practice Box-Cox transformations 12
Box-Cox transformationsyλt for λ close to zero behaves like logs.If some yt = 0, then must have λ > 0if some yt < 0, no power transformation is possibleunless all yt adjusted by adding a constant to allvalues.Choose a simple value of λ. It makes explanationeasier.Results are relatively insensitive to value of λOften no transformation (λ = 1) needed.Transformation often makes little difference toforecasts but has large effect on PI.Choosing λ = 0 is a simple way to force forecasts tobe positiveForecasting: principles and practice Box-Cox transformations 13
Automated Box-Cox transformations
(BoxCox.lambda(elec))
## [1] 0.2654076
This attempts to balance the seasonal fluctuationsand random variation across the series.Always check the results.A low value of λ can give extremely large predictionintervals.
Forecasting: principles and practice Box-Cox transformations 14
Automated Box-Cox transformations
(BoxCox.lambda(elec))
## [1] 0.2654076
This attempts to balance the seasonal fluctuationsand random variation across the series.Always check the results.A low value of λ can give extremely large predictionintervals.
Forecasting: principles and practice Box-Cox transformations 14
Outline
1 Variance stabilization
2 Box-Cox transformations
3 Back-transformation
4 Lab session 9
Forecasting: principles and practice Back-transformation 15
Back-transformation
Wemust reverse the transformation (or back-transform) toobtain forecasts on the original scale. The reverse Box-Coxtransformations are given by
yt = exp(wt), λ = 0;(λWt + 1)1/λ, λ 6= 0.
Forecasting: principles and practice Back-transformation 16
Back-transformation
fit <- snaive(elec, lambda=1/3)autoplot(fit)
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elec
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Forecasts from Seasonal naive method
Forecasting: principles and practice Back-transformation 17
Back-transformation
autoplot(fit, include=120)
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elec
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Forecasts from Seasonal naive method
Forecasting: principles and practice Back-transformation 18
Back transformation
Back-transformed point forecasts are medians.Back-transformed PI have the correct coverage.
Back-transformed meansLet X be have mean µ and variance σ2.
Let f(x) be back-transformation function, and Y = f(X).
E[Y] = E[f(X)] = f(µ) + 12σ
2[f′′(µ)]2.
Forecasting: principles and practice Back-transformation 19
Back transformation
Back-transformed point forecasts are medians.Back-transformed PI have the correct coverage.
Back-transformed meansLet X be have mean µ and variance σ2.
Let f(x) be back-transformation function, and Y = f(X).
E[Y] = E[f(X)] = f(µ) + 12σ
2[f′′(µ)]2.
Forecasting: principles and practice Back-transformation 19
Back transformationBox-Cox back-transformation:
yt = exp(wt) λ = 0;(λWt + 1)1/λ λ 6= 0.
f(x) =
ex λ = 0;(λx + 1)1/λ λ 6= 0.
f′′(x) =
ex λ = 0;(1− λ)(λx + 1)1/λ−2 λ 6= 0.
E[Y] =
eµ[1 + σ2
2
]λ = 0;
(λµ + 1)1/λ[1 + σ2(1−λ)
2(λµ+1)2
]λ 6= 0.
Forecasting: principles and practice Back-transformation 20
Back transformationBox-Cox back-transformation:
yt = exp(wt) λ = 0;(λWt + 1)1/λ λ 6= 0.
f(x) =
ex λ = 0;(λx + 1)1/λ λ 6= 0.
f′′(x) =
ex λ = 0;(1− λ)(λx + 1)1/λ−2 λ 6= 0.
E[Y] =
eµ[1 + σ2
2
]λ = 0;
(λµ + 1)1/λ[1 + σ2(1−λ)
2(λµ+1)2
]λ 6= 0.
Forecasting: principles and practice Back-transformation 20
Back-transformationelec %>% snaive(lambda=1/3, biasadj=FALSE) %>%
autoplot(include=120)
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Forecasting: principles and practice Back-transformation 21
Back-transformationelec %>% snaive(lambda=1/3, biasadj=TRUE) %>%
autoplot(include=120)
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Forecasts from Seasonal naive method
Forecasting: principles and practice Back-transformation 22
Back-transformationeggs %>% ses(lambda=1/3, biasadj=FALSE) %>%
autoplot
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Forecasts from Simple exponential smoothing
Forecasting: principles and practice Back-transformation 23
Back-transformationeggs %>% ses(lambda=1/3, biasadj=TRUE) %>%
autoplot
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Forecasts from Simple exponential smoothing
Forecasting: principles and practice Back-transformation 24
Outline
1 Variance stabilization
2 Box-Cox transformations
3 Back-transformation
4 Lab session 9
Forecasting: principles and practice Lab session 9 25
Lab Session 9
Forecasting: principles and practice Lab session 9 26
