Invited talk, Australian Statistical Conference, Adelaide, July 2012.

1. Rob J Hyndman
Advances in
automatic
time series forecasting
Advances in automatic time series forecasting 1

2. 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

3. Motivation
Advances in automatic time series forecasting Motivation 3

4. Motivation
Advances in automatic time series forecasting Motivation 3

5. Motivation
Advances in automatic time series forecasting Motivation 3

6. Motivation
Advances in automatic time series forecasting Motivation 3

7. Motivation
Advances in automatic time series forecasting Motivation 3

8. Motivation
Advances in automatic time series forecasting Motivation 3

9. Motivation
1 Common in business to have over 1000
products that need forecasting at least monthly.
2 Forecasts are often required by people who are
untrained in time series analysis.
3 Some types of data can be decomposed into a
large number of univariate time series that
need to be forecast.
Specifications
Automatic forecasting algorithms must:
¯ determine an appropriate time series model;
¯ estimate the parameters;
¯ compute the forecasts with prediction intervals.
Advances in automatic time series forecasting Motivation 4

10. Motivation
1 Common in business to have over 1000
products that need forecasting at least monthly.
2 Forecasts are often required by people who are
untrained in time series analysis.
3 Some types of data can be decomposed into a
large number of univariate time series that
need to be forecast.
Specifications
Automatic forecasting algorithms must:
¯ determine an appropriate time series model;
¯ estimate the parameters;
¯ compute the forecasts with prediction intervals.
Advances in automatic time series forecasting Motivation 4

11. Motivation
1 Common in business to have over 1000
products that need forecasting at least monthly.
2 Forecasts are often required by people who are
untrained in time series analysis.
3 Some types of data can be decomposed into a
large number of univariate time series that
need to be forecast.
Specifications
Automatic forecasting algorithms must:
¯ determine an appropriate time series model;
¯ estimate the parameters;
¯ compute the forecasts with prediction intervals.
Advances in automatic time series forecasting Motivation 4

12. Motivation
1 Common in business to have over 1000
products that need forecasting at least monthly.
2 Forecasts are often required by people who are
untrained in time series analysis.
3 Some types of data can be decomposed into a
large number of univariate time series that
need to be forecast.
Specifications
Automatic forecasting algorithms must:
¯ determine an appropriate time series model;
¯ estimate the parameters;
¯ compute the forecasts with prediction intervals.
Advances in automatic time series forecasting Motivation 4

13. Motivation
1 Common in business to have over 1000
products that need forecasting at least monthly.
2 Forecasts are often required by people who are
untrained in time series analysis.
3 Some types of data can be decomposed into a
large number of univariate time series that
need to be forecast.
Specifications
Automatic forecasting algorithms must:
¯ determine an appropriate time series model;
¯ estimate the parameters;
¯ compute the forecasts with prediction intervals.
Advances in automatic time series forecasting Motivation 4

14. Example: Cortecosteroid sales
Monthly cortecosteroid drug sales in Australia
1.6
1.4
Total scripts (millions)
1.2
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting Motivation 5

15. Example: Cortecosteroid sales
Automatic ARIMA forecasts
1.6
1.4
Total scripts (millions)
1.2
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting Motivation 5

16. 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

17. Exponential smoothing methods
Seasonal Component
Trend 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

18. Exponential smoothing methods
Seasonal Component
Trend 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

19. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Advances in automatic time series forecasting Exponential smoothing 7

20. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Ad ,N: Additive damped trend method
Advances in automatic time series forecasting Exponential smoothing 7

21. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Ad ,N: Additive damped trend method
M,N: Exponential trend method
Advances in automatic time series forecasting Exponential smoothing 7

22. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Ad ,N: Additive damped trend method
M,N: Exponential trend method
Md ,N: Multiplicative damped trend method
Advances in automatic time series forecasting Exponential smoothing 7

23. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Ad ,N: Additive damped trend method
M,N: Exponential trend method
Md ,N: Multiplicative damped trend method
A,A: Additive Holt-Winters’ method
Advances in automatic time series forecasting Exponential smoothing 7

24. Exponential smoothing methods
Seasonal Component
Trend 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
A,N: Holt’s linear method
Ad ,N: Additive damped trend method
M,N: Exponential trend method
Md ,N: Multiplicative damped trend method
A,A: Additive Holt-Winters’ method
A,M: Multiplicative Holt-Winters’ method
Advances in automatic time series forecasting Exponential smoothing 7

25. Exponential smoothing methods
Seasonal Component
Trend 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 smoothing
methods.
Advances in automatic time series forecasting Exponential smoothing 7

26. Exponential smoothing methods
Seasonal Component
Trend 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 smoothing
methods.
Each can have an additive or multiplicative
error, giving 30 separate models.
Advances in automatic time series forecasting Exponential smoothing 7

27. Exponential smoothing methods
Seasonal Component
Trend 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 errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

28. Exponential smoothing methods
Seasonal Component
Trend 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 errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

29. Exponential smoothing methods
Seasonal Component
Trend 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
Examples:
A,N,N: Simple exponential smoothing with additive errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

30. Exponential smoothing methods
Seasonal Component
Trend 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 Seasonal
Examples:
A,N,N: Simple exponential smoothing with additive errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

31. Exponential smoothing methods
Seasonal Component
Trend 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 Seasonal
Examples:
A,N,N: Simple exponential smoothing with additive errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

32. Exponential smoothing methods
Seasonal Component
Trend 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 Seasonal
Examples:
A,N,N: Simple exponential smoothing with additive errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

33. Exponential smoothing methods
Innovations state space modelsComponent
Seasonal
Trend N A M
¯ AllComponent
ETS models can (None) (Additive) (Multiplicative)
be written in innovations
N state space form.
(None) N,N N,A N,M
A (Additive) A,N A,A A,M
¯ Additive and multiplicative versions give,M
A (Additive damped) A ,N A ,A A
the
d d d d
M
same point forecasts but different prediction
(Multiplicative) M,N M,A M,M
Md intervals. damped) M ,N
(Multiplicative M ,A M ,M d d d
General notation E T S : ExponenTial Smoothing
↑
Error Trend Seasonal
Examples:
A,N,N: Simple exponential smoothing with additive errors
A,A,N: Holt’s linear method with additive errors
M,A,M: Multiplicative Holt-Winters’ method with multiplicative errors
Advances in automatic time series forecasting Exponential smoothing 8

34. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

35. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

36. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

37. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

38. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

39. Automatic forecasting
From Hyndman et al. (IJF, 2002):
Apply each of 30 models that are appropriate to
the data. Optimize parameters and initial
values 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 underlying
state space model.
Method performed very well in M3 competition.
Advances in automatic time series forecasting Exponential smoothing 9

40. Exponential smoothing
Automatic ETS forecasts
1.6
1.4
Total scripts (millions)
1.2
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting Exponential smoothing 10

41. Exponential smoothing
Automatic ETS forecasts
1.6
library(forecast)
1.4
fit <- ets(h02)
fcast <- forecast(fit)
Total scripts (millions)
1.2
plot(fcast)
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting Exponential smoothing 11

42. Exponential smoothing
> fit
ETS(M,Md,M)
Smoothing parameters:
alpha = 0.3318
beta = 4e-04
gamma = 1e-04
phi = 0.9695
Initial states:
l = 0.4003
b = 1.0233
s = 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

43. References
Hyndman, Koehler, Snyder, Grose (2002). “A
state space framework for automatic
forecasting using exponential smoothing
methods”. International Journal of Forecasting
18(3), 439–454
Hyndman, Koehler, Ord, Snyder (2008).
Forecasting with exponential smoothing: the
state space approach. Berlin: Springer-Verlag.
www.exponentialsmoothing.net
Hyndman (2012). forecast: Forecasting
functions for time series.
cran.r-project.org/package=forecast
Advances in automatic time series forecasting Exponential smoothing 13

44. 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

45. How does auto.arima() work?
A non-seasonal ARIMA process
φ(B)(1 − B)d yt = c + θ(B)εt
Need to select appropriate orders p, q, d, and
whether to include c.
Advances in automatic time series forecasting ARIMA modelling 15

46. How does auto.arima() work?
A non-seasonal ARIMA process
φ(B)(1 − B)d yt = c + θ(B)εt
Need to select appropriate orders p, q, d, and
whether 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 considering
nearby variants.
Advances in automatic time series forecasting ARIMA modelling 15

47. How does auto.arima() work?
A seasonal ARIMA process
Φ(Bm )φ(B)(1 − B)d (1 − Bm )D yt = c + Θ(Bm )θ(B)εt
Need to select appropriate orders p, q, d, P, Q, D, and
whether 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 considering
nearby variants.
Advances in automatic time series forecasting ARIMA modelling 16

48. Auto ARIMA
Automatic ARIMA forecasts
1.6
1.4
Total scripts (millions)
1.2
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting ARIMA modelling 17

49. Auto ARIMA
Automatic ARIMA forecasts
1.6
fit <- auto.arima(h02)
fcast <- forecast(fit)
1.4
plot(fcast)
Total scripts (millions)
1.2
1.0
0.8
0.6
0.4
1995 2000 2005 2010
Year
Advances in automatic time series forecasting ARIMA modelling 18

50. Auto ARIMA
> fit
Series: h02
ARIMA(3,1,3)(0,1,1)[12]
Coefficients:
ar1 ar2 ar3 ma1 ma2 ma3
-0.3648 -0.0636 0.3568 -0.4850 0.0479 -0.353
s.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=290.25
AIC=-564.5 AICc=-563.71 BIC=-538.48
Advances in automatic time series forecasting ARIMA modelling 19

51. References
Hyndman, Khandakar (2008). “Automatic time
series forecasting : the forecast package for R”.
Journal of Statistical Software 26(3)
Hyndman (2011). “Major changes to the
forecast package”.
robjhyndman.com/researchtips/forecast3/.
Hyndman (2012). forecast: Forecasting
functions for time series.
cran.r-project.org/package=forecast
Advances in automatic time series forecasting ARIMA modelling 20

52. 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

53. Examples
US finished motor gasoline products
6500 7000 7500 8000 8500 9000 9500
Thousands of barrels per day
1992 1994 1996 1998 2000 2002 2004
Weeks
Advances in automatic time series forecasting Time series with complex seasonality 22

54. Examples
Number of calls to large American bank (7am−9pm)
400
300
Number of call arrivals
200
100
3 March 17 March 31 March 14 April 28 April 12 May
5 minute intervals
Advances in automatic time series forecasting Time series with complex seasonality 22

55. Examples
Turkish electricity demand
25
Electricity demand (GW)
20
15
10
2000 2002 2004 2006 2008
Days
Advances in automatic time series forecasting Time series with complex seasonality 22

56. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0;
yt =
log yt if ω = 0.
M
(ω) (i)
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt
i=1 j=1
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

57. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
log yt if ω = 0.
M
(ω) (i)
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt
i=1 j=1
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

58. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
log yt if ω = 0.
M
(ω) (i) M seasonal periods
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt
i=1 j=1
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

59. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
log yt if ω = 0.
M
(ω) (i) M seasonal periods
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t global and local trend
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt
i=1 j=1
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

60. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
log yt if ω = 0.
M
(ω) (i) M seasonal periods
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t global and local trend
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt ARMA error
i=1 j=1
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

61. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
log yt if ω = 0.
M
(ω) (i) M seasonal periods
yt = t −1 + φbt −1 + st−mi + dt
i=1
= t−1 + φbt−1 + αdt
t global and local trend
bt = (1 − φ)b + φbt−1 + β dt
p q
dt = φi dt−i + θj εt−j + εt ARMA error
i=1 j=1
Fourier-like seasonal
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
terms
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

62. TBATS model
yt = observation at time t
ω
(ω) (yt − 1)/ω if ω = 0; Box-Cox transformation
yt =
TBATSif ω = 0.
log yt
M
(ω) Trigonometric d
=
(i) M seasonal periods
yt t −1 + φbt −1 +
st − m + t i
Box-Cox1 i=
t = t −1 + φbt −1 + αdt global and local trend
ARMA
bt = (1 − φ)b + φbt−1 + β dt
p Trendq
dt = φi dt−i + θj ε + ε ARMA error
i=1
Seasonalt−j t
j=1
Fourier-like seasonal
(i) (i) (i) ∗(i) (i) (i)
ki sj,t = sj,t−1 cos λj + sj,t−1 sin λj + γ1 dt
terms
(i) (i)
st = sj,t (i) (i) (i) ∗(i) (i) (i)
sj,t = −sj,t−1 sin λj + sj,t−1 cos λj + γ2 dt
j=1
Advances in automatic time series forecasting Time series with complex seasonality 23

63. Examples
Forecasts from TBATS(0.999, {2,2}, 1, {<52.1785714285714,8>})
10000
fit <- tbats(gasoline)
fcast <- forecast(fit)
Thousands of barrels per day
plot(fcast)
9000
8000
7000
1995 2000 2005
Weeks
Advances in automatic time series forecasting Time series with complex seasonality 24

64. Examples
Forecasts from TBATS(1, {3,1}, 0.987, {<169,5>, <845,3>})
500
fit <- tbats(callcentre)
fcast <- forecast(fit)
400
plot(fcast)
Number of call arrivals
300
200
100
0
3 March 17 March 31 March 14 April 28 April 12 May 26 May 9 June
5 minute intervals
Advances in automatic time series forecasting Time series with complex seasonality 25

65. Examples
Forecasts from TBATS(0, {5,3}, 0.997, {<7,3>, <354.37,12>, <365.25,4>})
fit <- tbats(turk)
25
fcast <- forecast(fit)
Electricity demand (GW)
plot(fcast)
20
15
10
2000 2002 2004 2006 2008 2010
Days
Advances in automatic time series forecasting Time series with complex seasonality 26

66. References
Automatic algorithm described in
De Livera, Hyndman, Snyder (2011).
“Forecasting time series with complex seasonal
patterns using exponential smoothing”. Journal
of the American Statistical Association
106(496), 1513–1527.
Slightly improved algorithm implemented in
Hyndman (2012). forecast: Forecasting
functions for time series.
cran.r-project.org/package=forecast.
More work required!
Advances in automatic time series forecasting Time series with complex seasonality 27

67. 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

68. Introduction
Total
A B C
AA AB AC BA BB BC CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29

69. Introduction
Total
A B C
AA AB AC BA BB BC CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29

70. Introduction
Total
A B C
AA AB AC BA BB BC CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29

71. Introduction
Total
A B C
AA AB AC BA BB BC CA CB CC
Examples
Manufacturing product hierarchies
Pharmaceutical sales
Net labour turnover
Advances in automatic time series forecasting Hierarchical and grouped time series 29

72. Hierarchical/grouped time series
A hierarchical time series is a collection of
several time series that are linked together in a
hierarchical structure.
A grouped time series is a collection of time
series that are aggregated in a number of
non-hierarchical ways.
Example: daily numbers of calls to HP call centres
are grouped by product type and location of call
centre.
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

73. Hierarchical/grouped time series
A hierarchical time series is a collection of
several time series that are linked together in a
hierarchical structure.
A grouped time series is a collection of time
series that are aggregated in a number of
non-hierarchical ways.
Example: daily numbers of calls to HP call centres
are grouped by product type and location of call
centre.
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

74. Hierarchical/grouped time series
A hierarchical time series is a collection of
several time series that are linked together in a
hierarchical structure.
A grouped time series is a collection of time
series that are aggregated in a number of
non-hierarchical ways.
Example: daily numbers of calls to HP call centres
are grouped by product type and location of call
centre.
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

75. Hierarchical/grouped time series
A hierarchical time series is a collection of
several time series that are linked together in a
hierarchical structure.
A grouped time series is a collection of time
series that are aggregated in a number of
non-hierarchical ways.
Example: daily numbers of calls to HP call centres
are grouped by product type and location of call
centre.
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

76. Hierarchical/grouped time series
A hierarchical time series is a collection of
several time series that are linked together in a
hierarchical structure.
A grouped time series is a collection of time
series that are aggregated in a number of
non-hierarchical ways.
Example: daily numbers of calls to HP call centres
are grouped by product type and location of call
centre.
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

77. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
Advances in automatic time series forecasting Hierarchical and grouped time series 31

78. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
Advances in automatic time series forecasting Hierarchical and grouped time series 31

79. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
 
1 1 1  
1 YA,t
0 0 
Y t = [Yt , YA,t , YB,t , YC,t ] = 
0
 YB,t 
1 0
YC,t
0 0 1
Advances in automatic time series forecasting Hierarchical and grouped time series 31

80. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
 
1 1 1  
1 YA,t
0 0 
Y t = [Yt , YA,t , YB,t , YC,t ] = 
0
 YB,t 
1 0
YC,t
0 0 1
S
Advances in automatic time series forecasting Hierarchical and grouped time series 31

81. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
 
1 1 1  
1 Y A, t
0 0 
Y t = [Yt , YA,t , YB,t , YC,t ] = 
0
 YB , t 
1 0
YC,t
0 0 1
Y K ,t
S
Advances in automatic time series forecasting Hierarchical and grouped time series 31

82. Notation
Yt : observed aggregate of all
Total series at time t.
YX,t : observation on series X at
time t.
A B C
Y i,t : vector of all series at level i
K : number of levels in in time t.
hierarchy (excl. Total). Y t = [Yt , Y 1,t , . . . , Y K ,t ]
 
1 1 1  
1 Y A, t
0 0 
Y t = [Yt , YA,t , YB,t , YC,t ] = 
0
 YB , t 
1 0
YC,t
0 0 1
Y K ,t
Y t = SY K ,t S
Advances in automatic time series forecasting Hierarchical and grouped time series 31

83. Hierarchical data
Total
A B C
AX AY AZ BX BY BZ CX CY CZ
 Yt
 1 1 1 1 1 1 1 1 1

YA,t 1 1 1 0 0 0 0 0 0
YB,t  0 0 0 1 1 1 0 0 0 Y
 AX,t

 YC,t  0 0 0 0 0 0 1 1 1 Y AY ,t
YAX,t
  1 0 0 0 0 0 0 0 0
 Y AZ,t 
 YAY ,t
 0 1 0 0 0 0 0 0 0
 Y BX,t 
Yt =  =
 YAZ,t
 0 0 1 0 0 0 0 0 0
 Y
,
BY t
 
 YBX,t  0 0 0 1 0 0 0 0 0 Y ,
BZ t

YBY ,t  0 0 0 0 1 0 0 0 0 Y
 ,
CX t

 YBZ,t  0 0 0 0 0 1 0 0 0 Y CY ,t
 YCX,t  0 0 0 0 0 0 1 0 0 Y CZ,t
YCY ,t 0 0 0 0 0 0 0 1 0
YCZ,t 0 0 0 0 0 0 0 0 1 Y K ,t
S
Advances in automatic time series forecasting Hierarchical and grouped time series 32

84. Hierarchical data
Total
A B C
AX AY AZ BX BY BZ CX CY CZ
 Yt
 1 1 1 1 1 1 1 1 1

YA,t 1 1 1 0 0 0 0 0 0
YB,t  0 0 0 1 1 1 0 0 0 Y
 AX,t

 YC,t  0 0 0 0 0 0 1 1 1 Y AY ,t
YAX,t
  1 0 0 0 0 0 0 0 0
 Y AZ,t 
 YAY ,t
 0 1 0 0 0 0 0 0 0
 Y BX,t 
Yt =  =
 YAZ,t
 0 0 1 0 0 0 0 0 0
 Y
,
BY t
 
 YBX,t  0 0 0 1 0 0 0 0 0 Y ,
BZ t

YBY ,t  0 0 0 0 1 0 0 0 0 Y
 ,
CX t

 YBZ,t  0 0 0 0 0 1 0 0 0 Y CY ,t
 YCX,t  0 0 0 0 0 0 1 0 0 Y CZ,t
YCY ,t 0 0 0 0 0 0 0 1 0
YCZ,t 0 0 0 0 0 0 0 0 1 Y K ,t
S
Advances in automatic time series forecasting Hierarchical and grouped time series 32

85. Grouped data
Total Total
A B X Y
AX AY BX BY AX BX AY BY
   
Yt 1 1 1 1
 YA,t  1 1 0 0
 YB,t  0 0 1 1  
    YAX,t
 YX,t  1 0 1 0
 YAY ,t 
Yt =  YY ,t  = 0 1 0 1 
  
YAX,t  1 YBX,t 
   0 0 0
 YBY ,t
YAY ,t  0 1 0 0
   
YBX,t 0 0 1 0 Y K ,t
YBY ,t 0 0 0 1
S
Advances in automatic time series forecasting Hierarchical and grouped time series 33

86. Grouped data
Total Total
A B X Y
AX AY BX BY AX BX AY BY
   
Yt 1 1 1 1
 YA,t  1 1 0 0
 YB,t  0 0 1 1  
    YAX,t
 YX,t  1 0 1 0
 YAY ,t 
Yt =  YY ,t  = 0 1 0 1 
  
YAX,t  1 YBX,t 
   0 0 0
 YBY ,t
YAY ,t  0 1 0 0
   
YBX,t 0 0 1 0 Y K ,t
YBY ,t 0 0 0 1
S
Advances in automatic time series forecasting Hierarchical and grouped time series 33

87. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

88. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

89. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

90. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

91. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

92. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

93. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

94. Forecasts
Key idea: forecast reconciliation
¯ Ignore structural constraints and forecast every
series of interest independently.
¯ Adjust forecasts to impose constraints.
ˆ
Let Y n (h) be vector of initial forecasts for horizon h,
made at time n, stacked in same order as Y t .
Y t = SY K ,t . ˆ
So Y n (h) = Sβn (h) + εh .
βn (h) = E[Y K ,n+h | Y 1 , . . . , Y n ].
εh has zero mean and covariance matrix Σh .
Estimate βn (h) using GLS?
˜
Revised forecasts: Y n (h) = Sβn (h) ˆ
Advances in automatic time series forecasting Hierarchical and grouped time series 34

95. Optimal combination forecasts
˜ ˆ ˆ † †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Σ† is generalized inverse of Σh .
h
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

96. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

97. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

98. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

99. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

100. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

101. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

102. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

103. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

104. Optimal combination forecasts
˜ ˆ ˆ† †
Y n (h) = Sβn (h) = S(S Σh S)−1 S Σh Y n (h)
Revised forecasts Base forecasts
†
Σh is generalized inverse of Σh .
Problem: Don’t know Σh and hard to estimate.
Solution: Assume εh ≈ SεK ,h where εK ,h is the
forecast error at bottom level.
Then Σh ≈ SΩh S where Ωh = Var(εK ,h ).
If Moore-Penrose generalized inverse used,
then
(S Σ† S)−1 S Σ† = (S S)−1 S .
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
Advances in automatic time series forecasting Hierarchical and grouped time series 35

105. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

106. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

107. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

108. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

109. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

110. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Total
A B C
Advances in automatic time series forecasting Hierarchical and grouped time series 36

111. Optimal combination forecasts
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
GLS =OLS.
Optimal weighted average of base forecasts.
Computational difficulties in big hierarchies due
to size of S matrix.
Optimal weights are S(S S)−1 S
Weights are independent of the data!
Weights: S(S S)−1 S =
Total 
0.75 0.25 0.25 0.25

 0.25
 0.75 −0.25 −0.25 

 0.25 −0.25 0.75 −0.25 
A B C
0.25 −0.25 −0.25 0.75
Advances in automatic time series forecasting Hierarchical and grouped time series 36

112. Optimal combination forecasts
Total
A B C
AA AB AC BA BB BC CA CB CC
Weights: S(S S)−1 S =
0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
 
 0.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 
−0.17 −0.17 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 
 
 0.23 0.58 0.19 0.19 0.19
 
 0.23
 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.19 

 0.08
 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 

 0.08
 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 

 0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 
0.08 −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

113. Optimal combination forecasts
Total
A B C
AA AB AC BA BB BC CA CB CC
Weights: S(S S)−1 S =
0.69 0.23 0.23 0.23 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08 0.08
 
 0.23 0.58 −0.17 −0.17 0.19 0.19 0.19 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 
−0.17 −0.17 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 
 
 0.23 0.58 0.19 0.19 0.19
 
 0.23
 −0.17 −0.17 0.58 −0.06 −0.06 −0.06 −0.06 −0.06 −0.06 0.19 0.19 0.19 

 0.08
 0.19 −0.06 −0.06 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 0.19 −0.06 −0.06 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 0.19 −0.06 −0.06 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 −0.02 −0.02 −0.02 

 0.08
 −0.06 0.19 −0.06 −0.02 −0.02 −0.02 −0.27 −0.27 0.73 −0.02 −0.02 −0.02 

 0.08
 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 0.73 −0.27 −0.27 

 0.08 −0.06 −0.06 0.19 −0.02 −0.02 −0.02 −0.02 −0.02 −0.02 −0.27 0.73 −0.27 
0.08 −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

114. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

115. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

116. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

117. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

118. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

119. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Advances in automatic time series forecasting Hierarchical and grouped time series 38

120. Features and problems
˜ ˆ
Y n (h) = S(S S)−1 S Y n (h)
˜
Var[Y n (h)] = SΩh S = Σh .
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 any
hierarchical or grouped time series.
Hyndman, Ahmed, Athanasopoulos, Shang (2011).
“Optimal combination forecasts for hierarchical
time series”. Computational Statistics and Data
Analysis 55(9), 2579–2589
Hyndman, Ahmed, Shang (2011). hts: Hierarchical
time series. cran.r-project.org/package=hts
Advances in automatic time series forecasting Hierarchical and grouped time series 38