Rob J Hyndman Automatic algorithms for time … algorithms for time series forecasting 3 Motivation...

Rob J Hyndman

Automatic algorithmsfor time seriesforecasting

Follow along using R

Requirements

Install the fpp package and its dependencies.

Automatic algorithms for time series forecasting 2

Motivation

1 Common in business to have over 1000products that need forecasting at least monthly.

2 Forecasts are often required by people who areuntrained in time series analysis.

Specifications

Automatic forecasting algorithms must:

å determine an appropriate time series model;

å estimate the parameters;

å compute the forecasts with prediction intervals.


Example: Asian sheep


Numbers of sheep in Asia

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

250

300

350

400

450

500

550



Automatic ETS forecasts

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

250

300

350

400

450

500

550


library(fpp)fit <- ets(livestock)fcast <- forecast(fit)plot(fcast)


Automatic ETS forecasts

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

250

300

350

400

450

500

550

Example: Cortecosteroid sales


Monthly cortecosteroid drug sales in Australia

Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

Example: Cortecosteroid sales


Forecasts from ARIMA(3,1,3)(0,1,1)[12]

Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

Auto ARIMA

fit <- auto.arima(h02)fcast <- forecast(fit)plot(fcast)



Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

Outline

1 Forecasting competitions

2 Exponential smoothing

3 ARIMA modelling

4 Automatic nonlinear forecasting?

5 Time series with complex seasonality

6 Recent developments

Automatic algorithms for time series forecasting Forecasting competitions 8

Makridakis and Hibon (1979)



This was the first large-scale empirical evaluation oftime series forecasting methods.

Highly controversial at the time.

Difficulties:8 How to measure forecast accuracy?8 How to apply methods consistently and objectively?8 How to explain unexpected results?

Common thinking was that the moresophisticated mathematical models (ARIMAmodels at the time) were necessarily better.If results showed ARIMA models not best, itmust be because analyst was unskilled.


Consequences of M&H (1979)

As a result of this paper, researchers started to:

å consider how to automate forecasting methods;

å study what methods give the best forecasts;

å be aware of the dangers of over-fitting;

å treat forecasting as a different problem fromtime series analysis.

Makridakis & Hibon followed up with a newcompetition in 1982:

1001 seriesAnyone could submit forecasts (avoiding thecharge of incompetence)Multiple forecast measures used.


M-competition


M-competition

Main findings1 Statistically sophisticated or complex methods do

not necessarily provide more accurate forecaststhan simpler ones.

2 The relative ranking of the performance of thevarious methods varies according to the accuracymeasure being used.

3 The accuracy when various methods are beingcombined outperforms, on average, the individualmethods being combined and does very well incomparison to other methods.

4 The accuracy of the various methods depends uponthe length of the forecasting horizon involved.


M3 competition



“The M3-Competition is a final attempt by the authors tosettle the accuracy issue of various time series methods. . .The extension involves the inclusion of more methods/researchers (in particular in the areas of neural networksand expert systems) and more series.”

3003 seriesAll data from business, demography, finance andeconomics.Series length between 14 and 126.Either non-seasonal, monthly or quarterly.All time series positive.M&H claimed that the M3-competition supported thefindings of their earlier work.However, best performing methods far from “simple”.


Makridakis and Hibon (2000)Best methods:

Theta

A very confusing explanation.

Shown by Hyndman and Billah (2003) to be average oflinear regression and simple exponential smoothingwith drift, applied to seasonally adjusted data.

Later, the original authors claimed that theirexplanation was incorrect.

Forecast Pro

A commercial software package with an unknownalgorithm.

Known to fit either exponential smoothing or ARIMAmodels using BIC.


M3 results (recalculated)

Method MAPE sMAPE MASE

Theta 17.42 12.76 1.39

ForecastPro 18.00 13.06 1.47

ForecastX 17.35 13.09 1.42

Automatic ANN 17.18 13.98 1.53

B-J automatic 19.13 13.72 1.54


M3 results (recalculated)


Theta 17.42 12.76 1.39

ForecastPro 18.00 13.06 1.47

ForecastX 17.35 13.09 1.42


B-J automatic 19.13 13.72 1.54


ä Calculations do not match

published paper.

ä Some contestants apparently

submitted multiple entries but only

best ones published.

Outline



3 ARIMA modelling




Automatic algorithms for time series forecasting Exponential smoothing 18

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










N,N: Simple exponential smoothingA,N: Holt’s linear method










N,N: Simple exponential smoothingA,N: Holt’s linear methodAd,N: Additive damped trend method










N,N: Simple exponential smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend method










N,N: Simple exponential smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend methodMd,N: Multiplicative damped trend method










N,N: Simple exponential smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend methodMd,N: Multiplicative damped trend methodA,A: Additive Holt-Winters’ method










N,N: Simple exponential smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend methodMd,N: Multiplicative damped trend methodA,A: Additive Holt-Winters’ methodA,M: Multiplicative Holt-Winters’ method










There are 15 separate exp. smoothing methods.










There are 15 separate exp. smoothing methods.Each can have an additive or multiplicative error,giving 30 separate models.










There are 15 separate exp. smoothing methods.Each can have an additive or multiplicative error,giving 30 separate models.Only 19 models are numerically stable.










There are 15 separate exp. smoothing methods.Each can have an additive or multiplicative error,giving 30 separate models.Only 19 models are numerically stable.Multiplicative trend models give poor forecastsleaving 15 models.










General notation E T S : ExponenTial Smoothing

Examples:A,N,N: Simple exponential smoothing with additive errorsA,A,N: Holt’s linear method with additive errorsM,A,M: Multiplicative Holt-Winters’ method with multiplicative errors










General notation E T S : ExponenTial Smoothing↑

TrendExamples:

A,N,N: Simple exponential smoothing with additive errorsA,A,N: Holt’s linear method with additive errorsM,A,M: Multiplicative Holt-Winters’ method with multiplicative errors










General notation E T S : ExponenTial Smoothing↑ ↖

Trend SeasonalExamples:











General notation E T S : ExponenTial Smoothing↗ ↑ ↖

Error Trend SeasonalExamples:











General notation E T S : ExponenTial Smoothing↗ ↑ ↖

Error Trend SeasonalExamples:



Innovations state space models

å All ETS models can be written in innovationsstate space form (IJF, 2002).

å Additive and multiplicative versions give thesame point forecasts but different predictionintervals.

ETS state space model

xt−1

εt

yt


State space modelxt = (level, slope, seasonal)


xt−1

εt

yt

xt




xt−1

εt

yt

xt yt+1

εt+1




xt−1

εt

yt

xt yt+1

εt+1 xt+1




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2 yt+3

εt+3




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2 yt+3

εt+3 xt+3




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2 yt+3

εt+3 xt+3 yt+4

εt+4




xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2 yt+3

εt+3 xt+3 yt+4

εt+4



EstimationCompute likelihood L fromε1, ε2, . . . , εT.Optimize L wrt modelparameters.


xt−1

εt

yt

xt yt+1

εt+1 xt+1 yt+2

εt+2 xt+2 yt+3

εt+3 xt+3 yt+4

εt+4



EstimationCompute likelihood L fromε1, ε2, . . . , εT.Optimize L wrt modelparameters.

Q: How to choosebetween the 15useful ETS models?

Cross-validationTraditional evaluation


● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● timeTraining data Test data


Standard cross-validation



● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●



Time series cross-validation



● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●



Time series cross-validation



● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●●● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●

Also known as “Evaluation ona rolling forecast origin”

Akaike’s Information Criterion

AIC = −2 log(L) + 2k

where L is the likelihood and k is the number ofestimated parameters in the model.

This is a penalized likelihood approach.If L is Gaussian, then AIC ≈ c + T log MSE + 2kwhere c is a constant, MSE is from one-stepforecasts on training set, and T is the length ofthe series.

Minimizing the Gaussian AIC is asymptoticallyequivalent (as T →∞) to minimizing MSE fromone-step forecasts on test set via time seriescross-validation.Automatic algorithms for time series forecasting Exponential smoothing 23

Akaike’s Information Criterion

AIC = −2 log(L) + 2k

Corrected AICFor small T, AIC tends to over-fit. Bias-correctedversion:

AICC = AIC + 2(k+1)(k+2)T−k

CV-MSE too time consuming for most automaticforecasting purposes. Also requires large T.AICc asymptotically equivalent, can be used onsmall samples and is very fast to compute.


ets algorithm in R


Based on Hyndman, Koehler,Snyder & Grose (IJF 2002):

Apply each of 15 models that areappropriate to the data. Optimizeparameters and initial valuesusing MLE.

Select best method using AICc.

Produce forecasts using bestmethod.

Obtain prediction intervals usingunderlying state space model.

Exponential smoothing


Forecasts from ETS(M,A,N)

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

300

400

500

600


fit <- ets(livestock)fcast <- forecast(fit)plot(fcast)


Forecasts from ETS(M,A,N)

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

300

400

500

600



Forecasts from ETS(M,N,M)

Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

1.6


fit <- ets(h02)fcast <- forecast(fit)plot(fcast)


Forecasts from ETS(M,N,M)

Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

1.6


> fitETS(M,N,M)

Smoothing parameters:alpha = 0.4597gamma = 1e-04

Initial states:l = 0.4501s = 0.8628 0.8193 0.7648 0.7675 0.6946 1.2921

1.3327 1.1833 1.1617 1.0899 1.0377 0.9937

sigma: 0.0675

AIC AICc BIC-115.69960 -113.47738 -69.24592


M3 comparisons


Theta 17.42 12.76 1.39

ForecastPro 18.00 13.06 1.47

ForecastX 17.35 13.09 1.42


B-J automatic 19.13 13.72 1.54

ETS 17.38 13.13 1.43




www.OTexts.org/fpp

Exercise

1 Use ets to find the best ETS models for thefollowing series: ibmclose, eggs, bricksq,hsales.

2 Try ets with cangas and lynx. What do youlearn?

3 Can you find another series for which ets givesbad forecasts?


Outline



3 ARIMA modelling




Automatic algorithms for time series forecasting ARIMA modelling 34

ARIMA models

yt−1

yt−2

yt−3

yt

Inputs Output


ARIMA models

yt−1

yt−2

yt−3

εt

yt

Inputs Output


Autoregression (AR)model

ARIMA models

yt−1

yt−2

yt−3

εt

εt−1

εt−2

yt

Inputs Output


Autoregression movingaverage (ARMA) model

ARIMA models

yt−1

yt−2

yt−3

εt

εt−1

εt−2

yt

Inputs Output



EstimationCompute likelihood L fromε1, ε2, . . . , εT.Use optimizationalgorithm to maximize L.

ARIMA models

yt−1

yt−2

yt−3

εt

εt−1

εt−2

yt

Inputs Output



EstimationCompute likelihood L fromε1, ε2, . . . , εT.Use optimizationalgorithm to maximize L.

ARIMA modelAutoregression movingaverage (ARMA) modelapplied to differences.

ARIMA modelling


Auto ARIMA


Forecasts from ARIMA(0,1,0) with drift

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

250

300

350

400

450

500

550

Auto ARIMA

fit <- auto.arima(livestock)fcast <- forecast(fit)plot(fcast)


Forecasts from ARIMA(0,1,0) with drift

Year

mill

ions

of s

heep

1960 1970 1980 1990 2000 2010

250

300

350

400

450

500

550

Auto ARIMA



Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

Auto ARIMA

fit <- auto.arima(h02)fcast <- forecast(fit)plot(fcast)



Year

Tota

l scr

ipts

(m

illio

ns)

1995 2000 2005 2010

0.4

0.6

0.8

1.0

1.2

1.4

Auto ARIMA

> fitSeries: h02ARIMA(3,1,3)(0,1,1)[12]

Coefficients:ar1 ar2 ar3 ma1 ma2 ma3 sma1

-0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353 -0.5931s.e. 0.2198 0.3293 0.1268 0.2227 0.2755 0.212 0.0651

sigma^2 estimated as 0.002706: log likelihood=290.25AIC=-564.5 AICc=-563.71 BIC=-538.48


How does auto.arima() work?

A non-seasonal ARIMA process

φ(B)(1− B)dyt = c + θ(B)εt

Need to select appropriate orders p,q,d, andwhether to include c.


Algorithm choices driven by forecast accuracy.


A non-seasonal ARIMA process

φ(B)(1− B)dyt = c + θ(B)εt

Need to select appropriate orders p,q,d, andwhether to include c.

Hyndman & Khandakar (JSS, 2008) algorithm:Select no. differences d via KPSS unit root test.Select p,q, c by minimising AICc.Use stepwise search to traverse model space,starting with a simple model and consideringnearby variants.


Algorithm choices driven by forecast accuracy.


A seasonal ARIMA process

Φ(Bm)φ(B)(1− B)d(1− Bm)Dyt = c + Θ(Bm)θ(B)εt

Need to select appropriate orders p,q,d, P,Q,D, andwhether to include c.

Hyndman & Khandakar (JSS, 2008) algorithm:Select no. differences d via KPSS unit root test.Select D using OCSB unit root test.Select p,q, P,Q, c by minimising AICc.Use stepwise search to traverse model space,starting with a simple model and consideringnearby variants.


M3 comparisons


Theta 17.42 12.76 1.39

ForecastPro 18.00 13.06 1.47

B-J automatic 19.13 13.72 1.54

ETS 17.38 13.13 1.43

AutoARIMA 19.12 13.85 1.47


Exercise

1 Use auto.arima to find the best ARIMA modelsfor the following series: ibmclose, eggs,bricksq, hsales.

2 Try auto.arima with cangas and lynx. What doyou learn?

3 Can you find a series for which auto.arimagives bad forecasts?

4 How would you compare the ETS and ARIMAresults?


Outline



3 ARIMA modelling




Automatic algorithms for time series forecasting Automatic nonlinear forecasting? 46

Automatic nonlinear forecasting

Automatic ANN in M3 competition did poorly.

Linear methods did best in the NN3competition!

Very few machine learning methods getpublished in the IJF because authors cannotdemonstrate their methods give betterforecasts than linear benchmark methods,even on supposedly nonlinear data.

Some good recent work by Kourentzes andCrone on automated ANN for time series.

Watch this space!

Automatic algorithms for time series forecasting Automatic nonlinear forecasting? 47

Outline



3 ARIMA modelling




Automatic algorithms for time series forecasting Time series with complex seasonality 48

Examples


US finished motor gasoline products

Weeks

Tho

usan

ds o

f bar

rels

per

day

1992 1994 1996 1998 2000 2002 2004

6500

7000

7500

8000

8500

9000

9500

Examples


Number of calls to large American bank (7am−9pm)

5 minute intervals

Num

ber

of c

all a

rriv

als

100

200

300

400

3 March 17 March 31 March 14 April 28 April 12 May

Examples


Turkish electricity demand

Days

Ele

ctric

ity d

eman

d (G

W)

2000 2002 2004 2006 2008

1015

2025

TBATS model

TBATSTrigonometric terms for seasonality

Box-Cox transformations for heterogeneity

ARMA errors for short-term dynamics

Trend (possibly damped)

Seasonal (including multiple and non-integer periods)

Automatic algorithm described in AM De Livera,RJ Hyndman, and RD Snyder (2011). “Forecastingtime series with complex seasonal patterns usingexponential smoothing”. Journal of the AmericanStatistical Association 106(496), 1513–1527.


TBATS model

yt = observation at time t

y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t


s(i)j,t = s(i)j,t−1 cosλ(i)j + s∗(i)j,t−1 sinλ(i)j + γ(i)1 dt

s(i)j,t = −s(i)j,t−1 sinλ(i)j + s∗(i)j,t−1 cosλ(i)j + γ(i)2 dt

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t




Box-Cox transformation

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t





M seasonal periods

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t





M seasonal periods

global and local trend

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t





M seasonal periods


ARMA error

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t





M seasonal periods


ARMA error

Fourier-like seasonalterms

TBATS model


y(ω)t =

{(yωt − 1)/ω if ω 6= 0;

log yt if ω = 0.

y(ω)t = `t−1 + φbt−1 +M∑i=1

s(i)t−mi+ dt

`t = `t−1 + φbt−1 + αdt

bt = (1− φ)b + φbt−1 + βdt

dt =

p∑i=1

φidt−i +

q∑j=1

θjεt−j + εt

s(i)t =

ki∑j=1

s(i)j,t





M seasonal periods


ARMA error

Fourier-like seasonalterms

TBATSTrigonometric

Box-Cox

ARMA

Trend

Seasonal

Examples

fit <- tbats(gasoline)fcast <- forecast(fit)plot(fcast)


Forecasts from TBATS(0.999, {2,2}, 1, {<52.1785714285714,8>})

Weeks

Tho

usan

ds o

f bar

rels

per

day

1995 2000 2005

7000

8000

9000

1000

0

Examples

fit <- tbats(callcentre)fcast <- forecast(fit)plot(fcast)


Forecasts from TBATS(1, {3,1}, 0.987, {<169,5>, <845,3>})

5 minute intervals

Num

ber

of c

all a

rriv

als

010

020

030

040

050

0

3 March 17 March 31 March 14 April 28 April 12 May 26 May 9 June

Examples

fit <- tbats(turk)fcast <- forecast(fit)plot(fcast)


Forecasts from TBATS(0, {5,3}, 0.997, {<7,3>, <354.37,12>, <365.25,4>})

Days

Ele

ctric

ity d

eman

d (G

W)

2000 2002 2004 2006 2008 2010

1015

2025

Outline



3 ARIMA modelling




Automatic algorithms for time series forecasting Recent developments 55

Further competitions

1 2011 tourism forecasting competition.

2 Kaggle and other forecasting platforms.

3 GEFCom 2012: Point forecasting of

electricity load and wind power.

4 GEFCom 2014: Probabilistic forecasting

of electricity load, electricity price,

wind energy and solar energy.


Forecasts about forecasting

1 Automatic algorithms will become moregeneral — handling a wide variety of timeseries.

2 Model selection methods will take accountof multi-step forecast accuracy as well asone-step forecast accuracy.

3 Automatic forecasting algorithms formultivariate time series will be developed.

4 Automatic forecasting algorithms thatinclude covariate information will bedeveloped.


Rob J Hyndman Automatic algorithms for time … algorithms for time series forecasting 3 Motivation...

Documents

Transcript of Rob J Hyndman Automatic algorithms for time … algorithms for time series forecasting 3 Motivation...