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.

Variance stabilization

1960 1970 1980 1990

Electricity production

1960 1970 1980 1990

Square root electricity production

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Cube root electricity production

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Log electricity production

−8e−04

−6e−04

−4e−04

−2e−04

1960 1970 1980 1990

Inverse electricity production

Outline

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)

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)

autoplot(BoxCox(elec,lambda=1/3))

1960 1970 1980 1990

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.

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.

Outline

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.

Back-transformation

fit <- snaive(elec, lambda=1/3)autoplot(fit)

1960 1970 1980 1990

Forecasts from Seasonal naive method

Back-transformation

autoplot(fit, include=120)

1987.5 1990.0 1992.5 1995.0 1997.5

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.

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.

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

]λ = 0;

(λµ + 1)1/λ[1 + σ2(1−λ)

2(λµ+1)2

]λ 6= 0.

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

]λ = 0;

(λµ + 1)1/λ[1 + σ2(1−λ)

2(λµ+1)2

]λ 6= 0.

Back-transformationelec %>% snaive(lambda=1/3, biasadj=FALSE) %>%

autoplot(include=120)

1987.5 1990.0 1992.5 1995.0 1997.5

Back-transformationelec %>% snaive(lambda=1/3, biasadj=TRUE) %>%

autoplot(include=120)

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Back-transformationeggs %>% ses(lambda=1/3, biasadj=FALSE) %>%

autoplot

1920 1950 1980

Forecasts from Simple exponential smoothing

Back-transformationeggs %>% ses(lambda=1/3, biasadj=TRUE) %>%

autoplot

1920 1950 1980

Forecasts from Simple exponential smoothing

Outline

4 Lab session 9

Forecasting: principles and practice Lab session 9 25

Lab Session 9

Forecasting: principles and practice Lab session 9 26

Forecasting: principles and practice · Outline 1Variance stabilization 2Box-Cox transformations...

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