Post on 11-May-2015
description
Advances inautomatictime series forecasting
Advances in automatic time series forecasting 1
Rob J Hyndman
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Motivation 2
Motivation
Advances in automatic time series forecasting Motivation 3
Motivation
Advances in automatic time series forecasting Motivation 3
Motivation
Advances in automatic time series forecasting Motivation 3
Motivation
Advances in automatic time series forecasting Motivation 3
Motivation
Advances in automatic time series forecasting Motivation 3
Motivation
Advances in automatic time series forecasting Motivation 3
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.
3 Some types of data can be decomposed into alarge number of univariate time series thatneed to be forecast.
Specifications
Automatic forecasting algorithms must:
å determine an appropriate time series model;
å estimate the parameters;
å compute the forecasts with prediction intervals.Advances in automatic time series forecasting Motivation 4
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.
3 Some types of data can be decomposed into alarge number of univariate time series thatneed to be forecast.
Specifications
Automatic forecasting algorithms must:
å determine an appropriate time series model;
å estimate the parameters;
å compute the forecasts with prediction intervals.Advances in automatic time series forecasting Motivation 4
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.
3 Some types of data can be decomposed into alarge number of univariate time series thatneed to be forecast.
Specifications
Automatic forecasting algorithms must:
å determine an appropriate time series model;
å estimate the parameters;
å compute the forecasts with prediction intervals.Advances in automatic time series forecasting Motivation 4
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.
3 Some types of data can be decomposed into alarge number of univariate time series thatneed to be forecast.
Specifications
Automatic forecasting algorithms must:
å determine an appropriate time series model;
å estimate the parameters;
å compute the forecasts with prediction intervals.Advances in automatic time series forecasting Motivation 4
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.
3 Some types of data can be decomposed into alarge number of univariate time series thatneed to be forecast.
Specifications
Automatic forecasting algorithms must:
å determine an appropriate time series model;
å estimate the parameters;
å compute the forecasts with prediction intervals.Advances in automatic time series forecasting Motivation 4
Example: Cortecosteroid sales
Advances in automatic time series forecasting Motivation 5
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
1.6
Example: Cortecosteroid sales
Advances in automatic time series forecasting Motivation 5
Automatic ARIMA forecasts
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
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Exponential smoothing 6
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
Advances in automatic time series forecasting Exponential smoothing 7
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
Advances in automatic time series forecasting Exponential smoothing 7
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 smoothingA,N: Holt’s linear method
Advances in automatic time series forecasting Exponential smoothing 7
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 smoothingA,N: Holt’s linear methodAd,N: Additive damped trend method
Advances in automatic time series forecasting Exponential smoothing 7
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 smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend method
Advances in automatic time series forecasting Exponential smoothing 7
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 smoothingA,N: Holt’s linear methodAd,N: Additive damped trend methodM,N: Exponential trend methodMd,N: Multiplicative damped trend method
Advances in automatic time series forecasting Exponential smoothing 7
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 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
Advances in automatic time series forecasting Exponential smoothing 7
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 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
Advances in automatic time series forecasting Exponential smoothing 7
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
There are 15 separate exponential smoothingmethods.
Advances in automatic time series forecasting Exponential smoothing 7
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
There are 15 separate exponential smoothingmethods.Each can have an additive or multiplicativeerror, giving 30 separate models.
Advances in automatic time series forecasting Exponential smoothing 7
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
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
General notation E T S : ExponenTial Smoothing↑
Trend SeasonalExamples:
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
General notation E T S : ExponenTial Smoothing ↑
Error Trend SeasonalExamples:
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
General notation E T S : ExponenTial Smoothing ↑
Error Trend SeasonalExamples:
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
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
General notation E T S : ExponenTial Smoothing ↑
Error Trend SeasonalExamples:
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 errorsAdvances in automatic time series forecasting Exponential smoothing 8
Innovations state space models
å All ETS models can be written in innovationsstate space form.
å Additive and multiplicative versions give thesame point forecasts but different predictionintervals.
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate tothe data. Optimize parameters and initialvalues using MLE (or some other criterion).
Select best method using AIC:
AIC = −2 log(Likelihood) + 2p
where p = # parameters.
Produce forecasts using best method.
Obtain prediction intervals using underlyingstate space model.
Method performed very well in M3 competition.Advances in automatic time series forecasting Exponential smoothing 9
Exponential smoothing
Advances in automatic time series forecasting Exponential smoothing 10
Automatic ETS forecasts
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
Exponential smoothing
library(forecast)fit <- ets(h02)fcast <- forecast(fit)plot(fcast)
Advances in automatic time series forecasting Exponential smoothing 11
Automatic ETS forecasts
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
Exponential smoothing> fitETS(M,Md,M)
Smoothing parameters:alpha = 0.3318beta = 4e-04gamma = 1e-04phi = 0.9695
Initial states:l = 0.4003b = 1.0233s = 0.8575 0.8183 0.7559 0.7627 0.6873 1.2884
1.3456 1.1867 1.1653 1.1033 1.0398 0.9893
sigma: 0.0651
AIC AICc BIC-121.97999 -118.68967 -65.57195
Advances in automatic time series forecasting Exponential smoothing 12
References
Hyndman, Koehler, Snyder, Grose (2002). “Astate space framework for automaticforecasting using exponential smoothingmethods”. International Journal of Forecasting18(3), 439–454
Hyndman, Koehler, Ord, Snyder (2008).Forecasting with exponential smoothing: thestate space approach. Berlin: Springer-Verlag.www.exponentialsmoothing.net
Hyndman (2012). forecast: Forecastingfunctions for time series.cran.r-project.org/package=forecast
Advances in automatic time series forecasting Exponential smoothing 13
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting ARIMA modelling 14
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.
Advances in automatic time series forecasting ARIMA modelling 15
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.
Hyndman & Khandakar (JSS, 2008) algorithm:Select no. differences d via KPSS unit root test.Select p,q, c by minimising AIC.Use stepwise search to traverse model space,starting with a simple model and consideringnearby variants.
Advances in automatic time series forecasting ARIMA modelling 15
How does auto.arima() work?
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 AIC.Use stepwise search to traverse model space,starting with a simple model and consideringnearby variants.
Advances in automatic time series forecasting ARIMA modelling 16
Auto ARIMA
Advances in automatic time series forecasting ARIMA modelling 17
Automatic ARIMA forecasts
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
Auto ARIMA
fit <- auto.arima(h02)fcast <- forecast(fit)plot(fcast)
Advances in automatic time series forecasting ARIMA modelling 18
Automatic ARIMA forecasts
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
Auto ARIMA
> fitSeries: h02ARIMA(3,1,3)(0,1,1)[12]
Coefficients:ar1 ar2 ar3 ma1 ma2 ma3
-0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353s.e. 0.2198 0.3293 0.1268 0.2227 0.2755 0.212
sma1-0.5931
s.e. 0.0651
sigma^2 estimated as 0.002706: log likelihood=290.25AIC=-564.5 AICc=-563.71 BIC=-538.48
Advances in automatic time series forecasting ARIMA modelling 19
References
Hyndman, Khandakar (2008). “Automatic timeseries forecasting : the forecast package for R”.Journal of Statistical Software 26(3)
Hyndman (2011). “Major changes to theforecast package”.robjhyndman.com/researchtips/forecast3/.
Hyndman (2012). forecast: Forecastingfunctions for time series.cran.r-project.org/package=forecast
Advances in automatic time series forecasting ARIMA modelling 20
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Time series with complex seasonality 21
Examples
Advances in automatic time series forecasting Time series with complex seasonality 22
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
Advances in automatic time series forecasting Time series with complex seasonality 22
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
Advances in automatic time series forecasting Time series with complex seasonality 22
Turkish electricity demand
Days
Ele
ctric
ity d
eman
d (G
W)
2000 2002 2004 2006 2008
1015
2025
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
M seasonal periods
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
M seasonal periods
global and local trend
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
M seasonal periods
global and local trend
ARMA error
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
M seasonal periods
global and local trend
ARMA error
Fourier-like seasonalterms
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
Advances in automatic time series forecasting Time series with complex seasonality 23
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
Box-Cox transformation
M seasonal periods
global and local trend
ARMA error
Fourier-like seasonalterms
TBATSTrigonometric
Box-Cox
ARMA
Trend
Seasonal
Examples
fit <- tbats(gasoline)fcast <- forecast(fit)plot(fcast)
Advances in automatic time series forecasting Time series with complex seasonality 24
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)
Advances in automatic time series forecasting Time series with complex seasonality 25
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)
Advances in automatic time series forecasting Time series with complex seasonality 26
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
References
Automatic algorithm described inDe Livera, Hyndman, Snyder (2011).“Forecasting time series with complex seasonalpatterns using exponential smoothing”. Journalof the American Statistical Association106(496), 1513–1527.
Slightly improved algorithm implemented inHyndman (2012). forecast: Forecastingfunctions for time series.cran.r-project.org/package=forecast.
More work required!
Advances in automatic time series forecasting Time series with complex seasonality 27
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Hierarchical and grouped time series 28
Introduction
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29
Introduction
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29
Introduction
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29
Introduction
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29
Hierarchical/grouped time series
A hierarchical time series is a collection ofseveral time series that are linked together in ahierarchical structure.
A grouped time series is a collection of timeseries that are aggregated in a number ofnon-hierarchical ways.
Example: daily numbers of calls to HP call centresare grouped by product type and location of callcentre.
Forecasts should be “aggregate consistent”,unbiased, minimum variance.
How to compute forecast intervals?Advances in automatic time series forecasting Hierarchical and grouped time series 30
Hierarchical/grouped time series
A hierarchical time series is a collection ofseveral time series that are linked together in ahierarchical structure.
A grouped time series is a collection of timeseries that are aggregated in a number ofnon-hierarchical ways.
Example: daily numbers of calls to HP call centresare grouped by product type and location of callcentre.
Forecasts should be “aggregate consistent”,unbiased, minimum variance.
How to compute forecast intervals?Advances in automatic time series forecasting Hierarchical and grouped time series 30
Hierarchical/grouped time series
A hierarchical time series is a collection ofseveral time series that are linked together in ahierarchical structure.
A grouped time series is a collection of timeseries that are aggregated in a number ofnon-hierarchical ways.
Example: daily numbers of calls to HP call centresare grouped by product type and location of callcentre.
Forecasts should be “aggregate consistent”,unbiased, minimum variance.
How to compute forecast intervals?Advances in automatic time series forecasting Hierarchical and grouped time series 30
Hierarchical/grouped time series
A hierarchical time series is a collection ofseveral time series that are linked together in ahierarchical structure.
A grouped time series is a collection of timeseries that are aggregated in a number ofnon-hierarchical ways.
Example: daily numbers of calls to HP call centresare grouped by product type and location of callcentre.
Forecasts should be “aggregate consistent”,unbiased, minimum variance.
How to compute forecast intervals?Advances in automatic time series forecasting Hierarchical and grouped time series 30
Hierarchical/grouped time series
A hierarchical time series is a collection ofseveral time series that are linked together in ahierarchical structure.
A grouped time series is a collection of timeseries that are aggregated in a number ofnon-hierarchical ways.
Example: daily numbers of calls to HP call centresare grouped by product type and location of callcentre.
Forecasts should be “aggregate consistent”,unbiased, minimum variance.
How to compute forecast intervals?Advances in automatic time series forecasting Hierarchical and grouped time series 30
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Y t = [Yt, YA,t, YB,t, YC,t]′ =
1 1 11 0 00 1 00 0 1
YA,tYB,tYC,t
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Y t = [Yt, YA,t, YB,t, YC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Y t = [Yt, YA,t, YB,t, YC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
︸ ︷︷ ︸
YK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Notation
Total
A B C
K : number of levels inhierarchy (excl. Total).
Y t = [Yt, YA,t, YB,t, YC,t]′ =
1 1 11 0 00 1 00 0 1
︸ ︷︷ ︸
S
YA,tYB,tYC,t
︸ ︷︷ ︸
YK,tY t = SYK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 31
Yt : observed aggregate of allseries at time t.
YX,t : observation on series X attime t.
Y i,t : vector of all series at level iin time t.
Y t = [Yt,Y1,t, . . . ,YK,t]′
Hierarchical dataTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
Y t =
YtYA,tYB,tYC,tYAX,tYAY,tYAZ,tYBX,tYBY,tYBZ,tYCX,tYCY,tYCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYAZ,tYBX,tYBY,tYBZ,tYCX,tYCY,tYCZ,t
︸ ︷︷ ︸
YK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 32
Hierarchical dataTotal
A
AX AY AZ
B
BX BY BZ
C
CX CY CZ
Y t =
YtYA,tYB,tYC,tYAX,tYAY,tYAZ,tYBX,tYBY,tYBZ,tYCX,tYCY,tYCZ,t
=
1 1 1 1 1 1 1 1 11 1 1 0 0 0 0 0 00 0 0 1 1 1 0 0 00 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 00 1 0 0 0 0 0 0 00 0 1 0 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 0 0 0 00 0 0 0 0 1 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYAZ,tYBX,tYBY,tYBZ,tYCX,tYCY,tYCZ,t
︸ ︷︷ ︸
YK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 32
Grouped data
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Y t =
YtYA,tYB,tYX,tYY,tYAX,tYAY,tYBX,tYBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYBX,tYBY,t
︸ ︷︷ ︸
YK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 33
Grouped data
Total
A
AX AY
B
BX BY
Total
X
AX BX
Y
AY BY
Y t =
YtYA,tYB,tYX,tYY,tYAX,tYAY,tYBX,tYBY,t
=
1 1 1 11 1 0 00 0 1 11 0 1 00 1 0 11 0 0 00 1 0 00 0 1 00 0 0 1
︸ ︷︷ ︸
S
YAX,tYAY,tYBX,tYBY,t
︸ ︷︷ ︸
YK,t
Advances in automatic time series forecasting Hierarchical and grouped time series 33
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast everyseries of interest independently.
å Adjust forecasts to impose constraints.
Let Yn(h) be vector of initial forecasts for horizon h,made at time n, stacked in same order as Y t.
Y t = SYK,t . So Yn(h) = Sβn(h) + εh .
βn(h) = E[YK,n+h | Y1, . . . ,Yn].εh has zero mean and covariance matrix Σh.Estimate βn(h) using GLS?
Revised forecasts: Yn(h) = Sβn(h)Advances in automatic time series forecasting Hierarchical and grouped time series 34
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecastsYn(h) = Sβn(h) = S(S′Σ†hS)−1S′Σ†hYn(h)
Revised forecasts Base forecasts
Σ†h is generalized inverse of Σh.
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK,h where εK,h is theforecast error at bottom level.Then Σh ≈ SΩhS′ where Ωh = Var(εK,h).If Moore-Penrose generalized inverse used,then
(S′Σ†S)−1S′Σ† = (S′S)−1S′.
Yn(h) = S(S′S)−1S′Yn(h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Optimal combination forecasts
Yn(h) = S(S′S)−1S′Yn(h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies dueto size of S matrix.
Optimal weights are S(S′S)−1S′
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36
Weights: S(S′S)−1S′ =0.75 0.25 0.25 0.250.25 0.75 −0.25 −0.250.25 −0.25 0.75 −0.250.25 −0.25 −0.25 0.75
Optimal combination forecasts
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Weights: S(S′S)−1S′ =
0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.080.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.060.23 −0.17 0.58 −0.17 −0.06 −0.06 −0.06 0.19 0.19 0.19 −0.06 −0.06 −0.060.23 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.190.08 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.020.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.270.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.270.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 −0.27 0.73
Advances in automatic time series forecasting Hierarchical and grouped time series 37
Optimal combination forecasts
Total
A
AA AB AC
B
BA BB BC
C
CA CB CC
Weights: S(S′S)−1S′ =
0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.080.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.060.23 −0.17 0.58 −0.17 −0.06 −0.06 −0.06 0.19 0.19 0.19 −0.06 −0.06 −0.060.23 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.190.08 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.020.08 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.020.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.270.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.270.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 −0.27 0.73
Advances in automatic time series forecasting Hierarchical and grouped time series 37
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Features and problemsYn(h) = S(S′S)−1S′Yn(h)
Var[Yn(h)] = SΩhS′ = Σh.
Advances in automatic time series forecasting Hierarchical and grouped time series 38
Covariates can be included in base forecasts.Point forecasts are always consistent.Need to estimate Ωh to produce prediction intervals.Very simple and flexible method. Can work with anyhierarchical or grouped time series.
Hyndman, Ahmed, Athanasopoulos, Shang (2011).“Optimal combination forecasts for hierarchicaltime series”. Computational Statistics and DataAnalysis 55(9), 2579–2589Hyndman, Ahmed, Shang (2011). hts: Hierarchicaltime series. cran.r-project.org/package=hts
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Functional time series 39
Fertility rates
Advances in automatic time series forecasting Functional time series 40
Some notationLet yt,x be the observed data in period t at age x,t = 1, . . . ,n.
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 41
Estimate ft(x) using penalized regression splines.
Estimate µ(x) as mean ft(x) across years.
Estimate βt,k and φk(x) using functional (weighted)principal components.
εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Some notationLet yt,x be the observed data in period t at age x,t = 1, . . . ,n.
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 41
Estimate ft(x) using penalized regression splines.
Estimate µ(x) as mean ft(x) across years.
Estimate βt,k and φk(x) using functional (weighted)principal components.
εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Some notationLet yt,x be the observed data in period t at age x,t = 1, . . . ,n.
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 41
Estimate ft(x) using penalized regression splines.
Estimate µ(x) as mean ft(x) across years.
Estimate βt,k and φk(x) using functional (weighted)principal components.
εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Some notationLet yt,x be the observed data in period t at age x,t = 1, . . . ,n.
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 41
Estimate ft(x) using penalized regression splines.
Estimate µ(x) as mean ft(x) across years.
Estimate βt,k and φk(x) using functional (weighted)principal components.
εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Some notationLet yt,x be the observed data in period t at age x,t = 1, . . . ,n.
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 41
Estimate ft(x) using penalized regression splines.
Estimate µ(x) as mean ft(x) across years.
Estimate βt,k and φk(x) using functional (weighted)principal components.
εt,xiid∼ N(0,1) and et(x)
iid∼ N(0, v(x)).
Fertility application
Advances in automatic time series forecasting Functional time series 42
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2006)
Age
Fer
tility
rat
e
Fertility model
Advances in automatic time series forecasting Functional time series 43
15 20 25 30 35 40 45 50
05
1015
Age
Mu
15 20 25 30 35 40 45 50
0.00
0.10
0.20
0.30
AgeP
hi 1
Year
Bet
a 1
1920 1960 2000
−20
−10
05
10
15 20 25 30 35 40 45 50
−0.
2−
0.1
0.0
0.1
0.2
Age
Phi
2Year
Bet
a 2
1920 1960 2000
−10
010
2030
Functional time series model
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
The eigenfunctions φk(x) show the mainregions of variation.The scores βt,k are uncorrelated byconstruction. So we can forecast each βt,k usinga univariate time series model.Univariate ARIMA models used for automaticforecasting of scores.
Advances in automatic time series forecasting Functional time series 44
Functional time series model
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
The eigenfunctions φk(x) show the mainregions of variation.The scores βt,k are uncorrelated byconstruction. So we can forecast each βt,k usinga univariate time series model.Univariate ARIMA models used for automaticforecasting of scores.
Advances in automatic time series forecasting Functional time series 44
Functional time series model
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
The eigenfunctions φk(x) show the mainregions of variation.The scores βt,k are uncorrelated byconstruction. So we can forecast each βt,k usinga univariate time series model.Univariate ARIMA models used for automaticforecasting of scores.
Advances in automatic time series forecasting Functional time series 44
Functional time series model
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
The eigenfunctions φk(x) show the mainregions of variation.The scores βt,k are uncorrelated byconstruction. So we can forecast each βt,k usinga univariate time series model.Univariate ARIMA models used for automaticforecasting of scores.
Advances in automatic time series forecasting Functional time series 44
Forecasts
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
Advances in automatic time series forecasting Functional time series 45
Forecasts
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
where vn+h,k = Var(βn+h,k | β1,k, . . . , βn,k)and y = [y1,x, . . . , yn,x].
Advances in automatic time series forecasting Functional time series 45
E[yn+h,x | y] = µ(x) +K∑
k=1
βn+h,k φk(x)
Var[yn+h,x | y] = σ2µ(x) +
K∑k=1
vn+h,k φ2k(x) + σ2
t (x) + v(x)
Forecasts of ft(x)
Advances in automatic time series forecasting Functional time series 46
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2006)
Age
Fer
tility
rat
e
Forecasts of ft(x)
Advances in automatic time series forecasting Functional time series 46
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2006)
Age
Fer
tility
rat
e
Forecasts of ft(x)
Advances in automatic time series forecasting Functional time series 46
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2006)
Age
Fer
tility
rat
e
Forecasts of ft(x)
Advances in automatic time series forecasting Functional time series 46
15 20 25 30 35 40 45 50
050
100
150
200
250
Australia fertility rates (1921−2006)
Age
Fer
tility
rat
e
80% prediction intervals
References
Hyndman, Ullah (2007). “Robust forecasting ofmortality and fertility rates: A functional dataapproach”. Computational Statistics and DataAnalysis 51(10), 4942–4956
Hyndman, Shang (2009). “Forecastingfunctional time series (with discussion)”.Journal of the Korean Statistical Society 38(3),199–221
Hyndman (2012). demography: Forecastingmortality, fertility, migration and populationdata.cran.r-project.org/package=demography
Advances in automatic time series forecasting Functional time series 47
Outline
1 Motivation
2 Exponential smoothing
3 ARIMA modelling
4 Time series with complex seasonality
5 Hierarchical and grouped time series
6 Functional time series
7 Grouped functional time series
Advances in automatic time series forecasting Grouped functional time series 48
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting groups
Let ft,j(x) be the smoothed mortality rate for age xin group j in year t.
Groups may be males and females.
Groups may be states within a country.
Expected that groups will behave similarly.
Coherent forecasts do not diverge over time.
Existing functional models do not imposecoherence.
Advances in automatic time series forecasting Grouped functional time series 49
Forecasting the coefficients
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
We use ARIMA models for each coefficientβ1,j,k, . . . , βn,j,k.The ARIMA models are non-stationary for thefirst few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Advances in automatic time series forecasting Grouped functional time series 50
Forecasting the coefficients
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
We use ARIMA models for each coefficientβ1,j,k, . . . , βn,j,k.The ARIMA models are non-stationary for thefirst few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Advances in automatic time series forecasting Grouped functional time series 50
Forecasting the coefficients
yt,x = ft(x) + σt(x)εt,x
ft(x) = µ(x) +K∑
k=1
βt,k φk(x) + et(x)
We use ARIMA models for each coefficientβ1,j,k, . . . , βn,j,k.The ARIMA models are non-stationary for thefirst few coefficients (k = 1,2)
Non-stationary ARIMA forecasts will diverge.Hence the mortality forecasts are not coherent.
Advances in automatic time series forecasting Grouped functional time series 50
Male fts model
Advances in automatic time series forecasting Grouped functional time series 51
0 20 40 60 80
−8
−6
−4
−2
Age
µ M(x
)
0 20 40 60 80
0.1
0.2
0.3
0.4
Age
φ 1(x
)
Year
β t1
1960 2000
−6
−4
−2
02
0 20 40 60 80
−0.
6−
0.2
0.2
0.4
Age
φ 2(x
)
Year
β t2
1960 2000
−0.
6−
0.2
0.0
0.2
0 20 40 60 80
−0.
4−
0.2
0.0
0.2
0.4
Age
φ 3(x
)
Year
β t3
1960 2000
−0.
4−
0.2
0.0
0.2
Australian male death rates
Female fts model
Advances in automatic time series forecasting Grouped functional time series 52
0 20 40 60 80
−8
−6
−4
−2
Age
µ F(x
)
0 20 40 60 80
0.1
0.2
0.3
0.4
Age
φ 1(x
)
Year
β t1
1960 2000
−4
−2
02
0 20 40 60 80
−0.
4−
0.2
0.0
0.2
0.4
Age
φ 2(x
)
Year
β t2
1960 2000
−0.
5−
0.3
−0.
10.
1
0 20 40 60 80
−0.
4−
0.2
0.0
0.2
Age
φ 3(x
)
Year
β t3
1960 2000
−0.
4−
0.2
0.0
0.2
Australian female death rates
Australian mortality forecasts
Advances in automatic time series forecasting Grouped functional time series 53
0 20 40 60 80 100
−10
−8
−6
−4
−2
(a) Males
Age
Log
deat
h ra
te
0 20 40 60 80 100
−10
−8
−6
−4
−2
(b) Females
Age
Log
deat
h ra
te
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.
Advances in automatic time series forecasting Grouped functional time series 54
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.
Advances in automatic time series forecasting Grouped functional time series 54
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.
Advances in automatic time series forecasting Grouped functional time series 54
Mortality product and ratios
Key idea
Model the geometric mean and the mortality ratioinstead of the individual rates for each sexseparately.
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
Product and ratio are approximatelyindependent
Ratio should be stationary (for coherence) butproduct can be non-stationary.
Advances in automatic time series forecasting Grouped functional time series 54
Mortality rates
Advances in automatic time series forecasting Grouped functional time series 55
Mortality rates
Advances in automatic time series forecasting Grouped functional time series 56
Model product and ratios
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt(x)] = µr(x) +L∑`=1
γt,`ψ`(x) + wt(x).
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).No restrictions for βt,1, . . . , βt,K.Forecasts: fn+h|n,M(x) = pn+h|n(x)rn+h|n(x)
fn+h|n,F(x) = pn+h|n(x)/rn+h|n(x).
Advances in automatic time series forecasting Grouped functional time series 57
Model product and ratios
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt(x)] = µr(x) +L∑`=1
γt,`ψ`(x) + wt(x).
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).No restrictions for βt,1, . . . , βt,K.Forecasts: fn+h|n,M(x) = pn+h|n(x)rn+h|n(x)
fn+h|n,F(x) = pn+h|n(x)/rn+h|n(x).
Advances in automatic time series forecasting Grouped functional time series 57
Model product and ratios
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt(x)] = µr(x) +L∑`=1
γt,`ψ`(x) + wt(x).
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).No restrictions for βt,1, . . . , βt,K.Forecasts: fn+h|n,M(x) = pn+h|n(x)rn+h|n(x)
fn+h|n,F(x) = pn+h|n(x)/rn+h|n(x).
Advances in automatic time series forecasting Grouped functional time series 57
Model product and ratios
pt(x) =√ft,M(x)ft,F(x) and rt(x) =
√ft,M(x)
/ft,F(x).
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt(x)] = µr(x) +L∑`=1
γt,`ψ`(x) + wt(x).
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).No restrictions for βt,1, . . . , βt,K.Forecasts: fn+h|n,M(x) = pn+h|n(x)rn+h|n(x)
fn+h|n,F(x) = pn+h|n(x)/rn+h|n(x).
Advances in automatic time series forecasting Grouped functional time series 57
Product model
Advances in automatic time series forecasting Grouped functional time series 58
0 20 40 60 80
−8
−6
−4
−2
Age
µ P(x
)
0 20 40 60 80
0.1
0.2
0.3
0.4
Age
φ 1(x
)
Year
β t1
1960 2000
−4
−2
02
0 20 40 60 80
−0.
4−
0.2
0.0
0.2
0.4
Age
φ 2(x
)
Year
β t2
1960 2000
−0.
6−
0.2
0.0
0.2
0 20 40 60 80
−0.
4−
0.2
0.0
0.2
Age
φ 3(x
)
Year
β t3
1960 2000
−0.
20.
00.
10.
20.
3
Ratio model
Advances in automatic time series forecasting Grouped functional time series 59
0 20 40 60 80
0.1
0.2
0.3
0.4
0.5
Age
µ R(x
)
0 20 40 60 80
−0.
20.
00.
20.
4
Age
φ 1(x
)
Year
β t1
1960 2000
−0.
4−
0.2
0.0
0.2
0 20 40 60 80
0.0
0.2
0.4
0.6
Age
φ 2(x
)
Year
β t2
1960 2000
−0.
2−
0.1
0.0
0.1
0.2
0 20 40 60 80
−0.
40.
00.
40.
8
Age
φ 3(x
)
Year
β t3
1960 2000
−0.
100.
000.
10
Product forecasts
Advances in automatic time series forecasting Grouped functional time series 60
0 20 40 60 80 100
−10
−8
−6
−4
−2
Age
Log
of g
eom
etric
mea
n de
ath
rate
Ratio forecasts
Advances in automatic time series forecasting Grouped functional time series 61
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Age
Sex
rat
io: M
/F
Coherent forecasts
Advances in automatic time series forecasting Grouped functional time series 62
0 20 40 60 80 100
−10
−8
−6
−4
−2
(a) Males
Age
Log
deat
h ra
te
0 20 40 60 80 100
−10
−8
−6
−4
−2
(b) Females
Age
Log
deat
h ra
te
Ratio forecasts
Advances in automatic time series forecasting Grouped functional time series 63
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Independent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
0 20 40 60 80 100
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Coherent forecasts
Age
Sex
rat
io o
f rat
es: M
/F
Life expectancy forecasts
Advances in automatic time series forecasting Grouped functional time series 64
Life expectancy forecasts
Year
Age
1960 1980 2000 2020
7075
8085
1960 1980 2000 2020
7075
8085Life expectancy difference: F−M
Year
Num
ber
of y
ears
1960 1980 2000 2020
24
68
Coherent forecasts for J groups
pt(x) = [ft,1(x)ft,2(x) · · · ft,J(x)]1/J
and rt,j(x) = ft,j(x)/pt(x),
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt,j(x)] = µr,j(x) +L∑
l=1
γt,l,jψl,j(x) + wt,j(x).
Advances in automatic time series forecasting Grouped functional time series 65
pt(x) and all rt,j(x)are approximatelyindependent.
Ratios satisfy constraintrt,1(x)rt,2(x) · · · rt,J(x) = 1.
log[ft,j(x)] = log[pt(x)rt,j(x)]
Coherent forecasts for J groups
pt(x) = [ft,1(x)ft,2(x) · · · ft,J(x)]1/J
and rt,j(x) = ft,j(x)/pt(x),
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt,j(x)] = µr,j(x) +L∑
l=1
γt,l,jψl,j(x) + wt,j(x).
Advances in automatic time series forecasting Grouped functional time series 65
pt(x) and all rt,j(x)are approximatelyindependent.
Ratios satisfy constraintrt,1(x)rt,2(x) · · · rt,J(x) = 1.
log[ft,j(x)] = log[pt(x)rt,j(x)]
Coherent forecasts for J groups
pt(x) = [ft,1(x)ft,2(x) · · · ft,J(x)]1/J
and rt,j(x) = ft,j(x)/pt(x),
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt,j(x)] = µr,j(x) +L∑
l=1
γt,l,jψl,j(x) + wt,j(x).
Advances in automatic time series forecasting Grouped functional time series 65
pt(x) and all rt,j(x)are approximatelyindependent.
Ratios satisfy constraintrt,1(x)rt,2(x) · · · rt,J(x) = 1.
log[ft,j(x)] = log[pt(x)rt,j(x)]
Coherent forecasts for J groups
pt(x) = [ft,1(x)ft,2(x) · · · ft,J(x)]1/J
and rt,j(x) = ft,j(x)/pt(x),
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt,j(x)] = µr,j(x) +L∑
l=1
γt,l,jψl,j(x) + wt,j(x).
Advances in automatic time series forecasting Grouped functional time series 65
pt(x) and all rt,j(x)are approximatelyindependent.
Ratios satisfy constraintrt,1(x)rt,2(x) · · · rt,J(x) = 1.
log[ft,j(x)] = log[pt(x)rt,j(x)]
Coherent forecasts for J groups
pt(x) = [ft,1(x)ft,2(x) · · · ft,J(x)]1/J
and rt,j(x) = ft,j(x)/pt(x),
log[pt(x)] = µp(x) +K∑
k=1
βt,kφk(x) + et(x)
log[rt,j(x)] = µr,j(x) +L∑
l=1
γt,l,jψl,j(x) + wt,j(x).
Advances in automatic time series forecasting Grouped functional time series 65
pt(x) and all rt,j(x)are approximatelyindependent.
Ratios satisfy constraintrt,1(x)rt,2(x) · · · rt,J(x) = 1.
log[ft,j(x)] = log[pt(x)rt,j(x)]
Coherent forecasts for J groups
µj(x) = µp(x) + µr,j(x) is group mean
zt,j(x) = et(x) + wt,j(x) is error term.
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).
No restrictions for βt,1, . . . , βt,K.
Advances in automatic time series forecasting Grouped functional time series 66
log[ft,j(x)] = log[pt(x)rt,j(x)] = log[pt(x)] + log[rt,j]
= µj(x) +K∑
k=1
βt,kφk(x) +L∑`=1
γt,`,jψ`,j(x) + zt,j(x)
Coherent forecasts for J groups
µj(x) = µp(x) + µr,j(x) is group mean
zt,j(x) = et(x) + wt,j(x) is error term.
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).
No restrictions for βt,1, . . . , βt,K.
Advances in automatic time series forecasting Grouped functional time series 66
log[ft,j(x)] = log[pt(x)rt,j(x)] = log[pt(x)] + log[rt,j]
= µj(x) +K∑
k=1
βt,kφk(x) +L∑`=1
γt,`,jψ`,j(x) + zt,j(x)
Coherent forecasts for J groups
µj(x) = µp(x) + µr,j(x) is group mean
zt,j(x) = et(x) + wt,j(x) is error term.
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).
No restrictions for βt,1, . . . , βt,K.
Advances in automatic time series forecasting Grouped functional time series 66
log[ft,j(x)] = log[pt(x)rt,j(x)] = log[pt(x)] + log[rt,j]
= µj(x) +K∑
k=1
βt,kφk(x) +L∑`=1
γt,`,jψ`,j(x) + zt,j(x)
Coherent forecasts for J groups
µj(x) = µp(x) + µr,j(x) is group mean
zt,j(x) = et(x) + wt,j(x) is error term.
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).
No restrictions for βt,1, . . . , βt,K.
Advances in automatic time series forecasting Grouped functional time series 66
log[ft,j(x)] = log[pt(x)rt,j(x)] = log[pt(x)] + log[rt,j]
= µj(x) +K∑
k=1
βt,kφk(x) +L∑`=1
γt,`,jψ`,j(x) + zt,j(x)
Coherent forecasts for J groups
µj(x) = µp(x) + µr,j(x) is group mean
zt,j(x) = et(x) + wt,j(x) is error term.
γt,` restricted to be stationary processes:either ARMA(p,q) or ARFIMA(p,d,q).
No restrictions for βt,1, . . . , βt,K.
Advances in automatic time series forecasting Grouped functional time series 66
log[ft,j(x)] = log[pt(x)rt,j(x)] = log[pt(x)] + log[rt,j]
= µj(x) +K∑
k=1
βt,kφk(x) +L∑`=1
γt,`,jψ`,j(x) + zt,j(x)
References
Hyndman, Booth, Yasmeen (2012). “Coherentmortality forecasting: the product-ratio methodwith functional time series models”.Demography, to appear
Hyndman (2012). demography: Forecastingmortality, fertility, migration and populationdata.cran.r-project.org/package=demography
Advances in automatic time series forecasting Grouped functional time series 67
For further information
robjhyndman.com
Slides and references for this talk.
Links to all papers and books.
Links to R packages.
A blog about forecasting research.
Advances in automatic time series forecasting Grouped functional time series 68