The document explores the challenges and methodologies of forecasting in various contexts, highlighting the inherent difficulties and limitations of predictive models. It examines specific case studies, including stock market trends, drug sales under the Australian Pharmaceutical Benefits Scheme, and peak electricity demand predictions in South Australia. The discussion emphasizes the need for robust forecasting models that account for volatility and changing environments while noting advancements in forecasting techniques, such as exponential smoothing state space models.

1. Exploring the
boundaries of
predictability
What can we forecast
and when should we give up?
Rob J Hyndman
Exploring the boundaries of predictability 1

2. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability Forecasting is difficult 2

3. Forecasting is difficult
Exploring the boundaries of predictability Forecasting is difficult 3

4. Reputations can be made and lost
“I think there is a world market for maybe
five computers.” (Chairman of IBM, 1943)
“Computers in the future may weigh no
more than 1.5 tons.” (Popular Mechanics, 1949)
“There is no reason anyone would want a
computer in their home.” (President, DEC, 1977)
Exploring the boundaries of predictability Forecasting is difficult 4

5. Reputations can be made and lost
“I think there is a world market for maybe
five computers.” (Chairman of IBM, 1943)
“Computers in the future may weigh no
more than 1.5 tons.” (Popular Mechanics, 1949)
“There is no reason anyone would want a
computer in their home.” (President, DEC, 1977)
Exploring the boundaries of predictability Forecasting is difficult 4

6. Reputations can be made and lost
“I think there is a world market for maybe
five computers.” (Chairman of IBM, 1943)
“Computers in the future may weigh no
more than 1.5 tons.” (Popular Mechanics, 1949)
“There is no reason anyone would want a
computer in their home.” (President, DEC, 1977)
Exploring the boundaries of predictability Forecasting is difficult 4

7. Forecasting in a changing environment
A good forecasting model captures the way things
move, not just where things are.
a highly volatile environment will continue to
be highly volatile;
a business with fluctuating sales will continue
to have fluctuating sales;
an economy that has gone through booms and
busts will continue to go through booms and
busts.
“If we could first know where we are and whither we
are tending, we could better judge what to do and
how to do it.” Abraham Lincoln
Exploring the boundaries of predictability Forecasting is difficult 5

8. Forecasting in a changing environment
A good forecasting model captures the way things
move, not just where things are.
a highly volatile environment will continue to
be highly volatile;
a business with fluctuating sales will continue
to have fluctuating sales;
an economy that has gone through booms and
busts will continue to go through booms and
busts.
“If we could first know where we are and whither we
are tending, we could better judge what to do and
how to do it.” Abraham Lincoln
Exploring the boundaries of predictability Forecasting is difficult 5

9. Forecasting in a changing environment
A good forecasting model captures the way things
move, not just where things are.
a highly volatile environment will continue to
be highly volatile;
a business with fluctuating sales will continue
to have fluctuating sales;
an economy that has gone through booms and
busts will continue to go through booms and
busts.
“If we could first know where we are and whither we
are tending, we could better judge what to do and
how to do it.” Abraham Lincoln
Exploring the boundaries of predictability Forecasting is difficult 5

10. Forecasting in a changing environment
A good forecasting model captures the way things
move, not just where things are.
a highly volatile environment will continue to
be highly volatile;
a business with fluctuating sales will continue
to have fluctuating sales;
an economy that has gone through booms and
busts will continue to go through booms and
busts.
“If we could first know where we are and whither we
are tending, we could better judge what to do and
how to do it.” Abraham Lincoln
Exploring the boundaries of predictability Forecasting is difficult 5

11. Forecasting in a changing environment
A good forecasting model captures the way things
move, not just where things are.
a highly volatile environment will continue to
be highly volatile;
a business with fluctuating sales will continue
to have fluctuating sales;
an economy that has gone through booms and
busts will continue to go through booms and
busts.
“If we could first know where we are and whither we
are tending, we could better judge what to do and
how to do it.” Abraham Lincoln
Exploring the boundaries of predictability Forecasting is difficult 5

12. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability Stock market data 6

13. Stock market data
Exploring the boundaries of predictability Stock market data 7
Daily closing prices: Yahoo
Date
$
20304050
2012 2013 2014 2015

14. Stock market data
Exploring the boundaries of predictability Stock market data 7
Forecasts from ETS(M,N,N)
Date
$
20304050
2012 2013 2014 2015

15. Stock market data
Exploring the boundaries of predictability Stock market data 8
Forecasts from ETS(M,N,N)
Date
$
20304050
2012 2013 2014 2015
fit <- ets(yahoo)
plot(forecast(fit, h=100))

16. Stock market data
Exploring the boundaries of predictability Stock market data 9
Daily Log Returns: Yahoo
Date
%
−10−50510
2012 2013 2014 2015

17. Stock market data
Exploring the boundaries of predictability Stock market data 9
Forecasts from ARIMA(0,0,3) with non−zero mean
Date
%
−10−50510
2012 2013 2014 2015

18. Stock market data
Exploring the boundaries of predictability Stock market data 10
Forecasts from ARIMA(0,0,3) with non−zero mean
Date
%
−10−50510
2012 2013 2014 2015
fit <- auto.arima(logreturns)
plot(forecast(fit, h=100, bootstrap=TRUE))

19. Efficient Market Hypothesis
Market efficiency causes existing values to
always incorporate and reflect all relevant
information. Thus, current values are their
own forecasts.
Consequences
No such thing as an undervalued stock or
inflated stock.
Insider information or waiting a long time
are the only ways to win.
In reality, slight inefficiencies exist but are
usually insufficient to beat transaction costs.
Exploring the boundaries of predictability Stock market data 11

20. Efficient Market Hypothesis
Market efficiency causes existing values to
always incorporate and reflect all relevant
information. Thus, current values are their
own forecasts.
Consequences
No such thing as an undervalued stock or
inflated stock.
Insider information or waiting a long time
are the only ways to win.
In reality, slight inefficiencies exist but are
usually insufficient to beat transaction costs.
Exploring the boundaries of predictability Stock market data 11

21. Efficient Market Hypothesis
Market efficiency causes existing values to
always incorporate and reflect all relevant
information. Thus, current values are their
own forecasts.
Consequences
No such thing as an undervalued stock or
inflated stock.
Insider information or waiting a long time
are the only ways to win.
In reality, slight inefficiencies exist but are
usually insufficient to beat transaction costs.
Exploring the boundaries of predictability Stock market data 11

22. Efficient Market Hypothesis
Market efficiency causes existing values to
always incorporate and reflect all relevant
information. Thus, current values are their
own forecasts.
Consequences
No such thing as an undervalued stock or
inflated stock.
Insider information or waiting a long time
are the only ways to win.
In reality, slight inefficiencies exist but are
usually insufficient to beat transaction costs.
Exploring the boundaries of predictability Stock market data 11

23. Efficient Market Hypothesis
Market efficiency causes existing values to
always incorporate and reflect all relevant
information. Thus, current values are their
own forecasts.
Consequences
No such thing as an undervalued stock or
inflated stock.
Insider information or waiting a long time
are the only ways to win.
In reality, slight inefficiencies exist but are
usually insufficient to beat transaction costs.
Exploring the boundaries of predictability Stock market data 11

24. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability Drug sales 12

25. Australian drug sales
The Pharmaceutical Benefits Scheme (PBS) is
the Australian government drugs subsidy scheme.
Many drugs bought from pharmacies (“drug
stores” in the US) are subsidised to allow more
equitable access to modern drugs.
The cost to government is determined by the
number and types of drugs purchased.
Currently nearly 1% of GDP.
The total cost is budgeted based on forecasts
of drug usage.
Exploring the boundaries of predictability Drug sales 13

26. Australian drug sales
The Pharmaceutical Benefits Scheme (PBS) is
the Australian government drugs subsidy scheme.
Many drugs bought from pharmacies (“drug
stores” in the US) are subsidised to allow more
equitable access to modern drugs.
The cost to government is determined by the
number and types of drugs purchased.
Currently nearly 1% of GDP.
The total cost is budgeted based on forecasts
of drug usage.
Exploring the boundaries of predictability Drug sales 13

27. Australian drug sales
The Pharmaceutical Benefits Scheme (PBS) is
the Australian government drugs subsidy scheme.
Many drugs bought from pharmacies (“drug
stores” in the US) are subsidised to allow more
equitable access to modern drugs.
The cost to government is determined by the
number and types of drugs purchased.
Currently nearly 1% of GDP.
The total cost is budgeted based on forecasts
of drug usage.
Exploring the boundaries of predictability Drug sales 13

28. Australian drug sales
The Pharmaceutical Benefits Scheme (PBS) is
the Australian government drugs subsidy scheme.
Many drugs bought from pharmacies (“drug
stores” in the US) are subsidised to allow more
equitable access to modern drugs.
The cost to government is determined by the
number and types of drugs purchased.
Currently nearly 1% of GDP.
The total cost is budgeted based on forecasts
of drug usage.
Exploring the boundaries of predictability Drug sales 13

29. Forecasting the PBS
In 2001: $4.5 billion budget, under-forecasted
by $800 million.
Subject to covert marketing, volatile products,
uncontrollable expenditure.
Monthly data on thousands of drug groups and
4 concession types available from 1991.
Data were aggregated to annual values, and
only the first three years were being used in
estimating the forecasts.
All forecasts being done with the FORECAST
function in MS-Excel!
Exploring the boundaries of predictability Drug sales 14

30. Forecasting the PBS
In 2001: $4.5 billion budget, under-forecasted
by $800 million.
Subject to covert marketing, volatile products,
uncontrollable expenditure.
Monthly data on thousands of drug groups and
4 concession types available from 1991.
Data were aggregated to annual values, and
only the first three years were being used in
estimating the forecasts.
All forecasts being done with the FORECAST
function in MS-Excel!
Exploring the boundaries of predictability Drug sales 14

31. Forecasting the PBS
In 2001: $4.5 billion budget, under-forecasted
by $800 million.
Subject to covert marketing, volatile products,
uncontrollable expenditure.
Monthly data on thousands of drug groups and
4 concession types available from 1991.
Data were aggregated to annual values, and
only the first three years were being used in
estimating the forecasts.
All forecasts being done with the FORECAST
function in MS-Excel!
Exploring the boundaries of predictability Drug sales 14

32. Forecasting the PBS
In 2001: $4.5 billion budget, under-forecasted
by $800 million.
Subject to covert marketing, volatile products,
uncontrollable expenditure.
Monthly data on thousands of drug groups and
4 concession types available from 1991.
Data were aggregated to annual values, and
only the first three years were being used in
estimating the forecasts.
All forecasts being done with the FORECAST
function in MS-Excel!
Exploring the boundaries of predictability Drug sales 14

33. Forecasting the PBS
In 2001: $4.5 billion budget, under-forecasted
by $800 million.
Subject to covert marketing, volatile products,
uncontrollable expenditure.
Monthly data on thousands of drug groups and
4 concession types available from 1991.
Data were aggregated to annual values, and
only the first three years were being used in
estimating the forecasts.
All forecasts being done with the FORECAST
function in MS-Excel!
Exploring the boundaries of predictability Drug sales 14

34. ATC drug classification
A Alimentary tract and metabolism14 classes
A10 Drugs used in diabetes84 classes
A10B Blood glucose lowering drugs
A10BA Biguanides
A10BA02 Metformin
Exploring the boundaries of predictability Drug sales 15

35. ETS forecasts of PBS data
Exploring the boundaries of predictability Drug sales 16
Total cost: A03 concession safety net group
$thousands
1995 2000 2005 2010
020040060080010001200

36. ETS forecasts of PBS data
Exploring the boundaries of predictability Drug sales 16
Total cost: A05 general copayments group
$thousands
1995 2000 2005 2010
050100150200250

37. ETS forecasts of PBS data
Exploring the boundaries of predictability Drug sales 16
Total cost: D01 general copayments group
$thousands
1995 2000 2005 2010
0100200300400500600700

38. ETS forecasts of PBS data
Exploring the boundaries of predictability Drug sales 16
Total cost: S01 general copayments group
$thousands
1995 2000 2005 2010
0100020003000400050006000

39. ETS forecasts of PBS data
Exploring the boundaries of predictability Drug sales 16
Total cost: R03 general copayments group
$thousands
1995 2000 2005 2010
1000200030004000500060007000

40. Forecasting the PBS
As part of this project, we developed an
automatic forecasting algorithm for exponential
smoothing state space models based on the
AIC.
Exponential smoothing models allowed for
time-changing trend and seasonal patterns.
Forecast MAPE reduced from 15–20% to 0.6%.
State space models provide prediction intervals
which give a sense of uncertainty.
Algorithm now implemented as ets function
in forecast package in R.
Exploring the boundaries of predictability Drug sales 17

41. Forecasting the PBS
As part of this project, we developed an
automatic forecasting algorithm for exponential
smoothing state space models based on the
AIC.
Exponential smoothing models allowed for
time-changing trend and seasonal patterns.
Forecast MAPE reduced from 15–20% to 0.6%.
State space models provide prediction intervals
which give a sense of uncertainty.
Algorithm now implemented as ets function
in forecast package in R.
Exploring the boundaries of predictability Drug sales 17

42. Forecasting the PBS
As part of this project, we developed an
automatic forecasting algorithm for exponential
smoothing state space models based on the
AIC.
Exponential smoothing models allowed for
time-changing trend and seasonal patterns.
Forecast MAPE reduced from 15–20% to 0.6%.
State space models provide prediction intervals
which give a sense of uncertainty.
Algorithm now implemented as ets function
in forecast package in R.
Exploring the boundaries of predictability Drug sales 17

43. Forecasting the PBS
As part of this project, we developed an
automatic forecasting algorithm for exponential
smoothing state space models based on the
AIC.
Exponential smoothing models allowed for
time-changing trend and seasonal patterns.
Forecast MAPE reduced from 15–20% to 0.6%.
State space models provide prediction intervals
which give a sense of uncertainty.
Algorithm now implemented as ets function
in forecast package in R.
Exploring the boundaries of predictability Drug sales 17

44. Forecasting the PBS
As part of this project, we developed an
automatic forecasting algorithm for exponential
smoothing state space models based on the
AIC.
Exponential smoothing models allowed for
time-changing trend and seasonal patterns.
Forecast MAPE reduced from 15–20% to 0.6%.
State space models provide prediction intervals
which give a sense of uncertainty.
Algorithm now implemented as ets function
in forecast package in R.
Exploring the boundaries of predictability Drug sales 17

45. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability Extreme electricity demand 18

46. The problem
We want to forecast the peak electricity
demand in a half-hour period in twenty years
time.
We have fifteen years of half-hourly electricity
data, temperature data and some economic
and demographic data.
The location is South Australia: home to the
most volatile electricity demand in the world.
Sounds impossible?
Exploring the boundaries of predictability Extreme electricity demand 19

47. The problem
We want to forecast the peak electricity
demand in a half-hour period in twenty years
time.
We have fifteen years of half-hourly electricity
data, temperature data and some economic
and demographic data.
The location is South Australia: home to the
most volatile electricity demand in the world.
Sounds impossible?
Exploring the boundaries of predictability Extreme electricity demand 19

48. South Australian demand data
Exploring the boundaries of predictability Extreme electricity demand 20

49. South Australian demand data
Exploring the boundaries of predictability Extreme electricity demand 20
Black Saturday →

50. The heatwave
Exploring the boundaries of predictability Extreme electricity demand 21

51. South Australian demand data
Exploring the boundaries of predictability Extreme electricity demand 22
SA State wide demand (summer 2015)
SAStatewidedemand(GW)
1.01.52.02.53.0
Oct Nov Dec Jan Feb Mar

52. South Australian demand data
Exploring the boundaries of predictability Extreme electricity demand 22

53. Temperature data (Sth Aust)
Exploring the boundaries of predictability Extreme electricity demand 23

54. Temperature data (Sth Aust)
Exploring the boundaries of predictability Extreme electricity demand 24
10 20 30 40
1.01.52.02.53.03.5
Time: 12 midnight
Temperature (deg C)
Demand(GW)
Workday
Non−workday

55. Predictors
calendar effects
prevailing and recent weather conditions
climate changes
economic and demographic changes
changing technology
Modelling framework
Semi-parametric additive models
with correlated errors.
Each half-hour period modelled separately for:
each season (x2)
work-days/non-work-days (x2)
morning/afternoon/evening (x3)
Total: 48 × 2 × 2 × 3 = 576 models.
Variables selected to provide best out-of-sample
predictions using cross-validation on last two years.
Exploring the boundaries of predictability Extreme electricity demand 25

56. Predictors
calendar effects
prevailing and recent weather conditions
climate changes
economic and demographic changes
changing technology
Modelling framework
Semi-parametric additive models
with correlated errors.
Each half-hour period modelled separately for:
each season (x2)
work-days/non-work-days (x2)
morning/afternoon/evening (x3)
Total: 48 × 2 × 2 × 3 = 576 models.
Variables selected to provide best out-of-sample
predictions using cross-validation on last two years.
Exploring the boundaries of predictability Extreme electricity demand 25

57. Half-hourly models
Exploring the boundaries of predictability Extreme electricity demand 26
Demand (January 2015)
Date in January
SAdemand(GW)
012345
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Actual
Predicted
Temperatures (January 2015)
)
40
temp_23090
temp_23083

58. Peak demand distribution
Exploring the boundaries of predictability Extreme electricity demand 27
PoE (seasonal interpretation, summer)
Year
PoEDemand(non−industrial)
2.53.03.5
2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015
10 %
50 %
90 %
q
q
q
q
q
q
q
q
q q
q
q
q
q
Summer

59. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability What can we forecast? 28

60. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

61. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

62. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

63. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

64. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

65. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

66. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

67. What can we forecast?
Exploring the boundaries of predictability What can we forecast? 29

68. Predictability factors
1 We understand and can measure the causal factors.
2 There is a lot of historical data available.
3 The forecasts do not affect the thing we are trying to
forecast.
4 The future is somewhat similar to the past.
Exploring the boundaries of predictability What can we forecast? 30

69. Predictability factors
1 We understand and can measure the causal factors.
2 There is a lot of historical data available.
3 The forecasts do not affect the thing we are trying to
forecast.
4 The future is somewhat similar to the past.
Exploring the boundaries of predictability What can we forecast? 30

70. Predictability factors
Measure Big No Future
Causal Data Feedback ∼ Past
Finance N Y N short-term
Economics N N N short-term
Drugs partly Y Y short-term
Electricity short-term Y Y short-term
Weather short-term Y Y short-term
Astronomy Y Y Y Y
Exploring the boundaries of predictability What can we forecast? 31

71. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability M3 competition data 32

72. M3 forecasting competition
Exploring the boundaries of predictability M3 competition data 33

73. M3 forecasting competition
Exploring the boundaries of predictability M3 competition data 33

74. M3 forecasting competition
“The M3-Competition is a final attempt by the authors to
settle the accuracy issue of various time series methods. . .
The extension involves the inclusion of more methods/
researchers (in particular in the areas of neural networks
and expert systems) and more series.”
Makridakis & Hibon, IJF 2000
3003 series
All data from business, demography, finance and
economics.
Series length between 14 and 126.
Either non-seasonal, monthly or quarterly.
All time series positive.
Exploring the boundaries of predictability M3 competition data 34

75. M3 forecasting competition
Exploring the boundaries of predictability M3 competition data 35
q

76. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Exploring the boundaries of predictability M3 competition data 36
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).

77. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Exploring the boundaries of predictability M3 competition data 36
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).

78. Key idea
Examples for time series
lag correlation
size and direction of trend
strength of seasonality
timing of peak seasonality
spectral entropy
Called “features” or “characteristics” in the
machine learning literature.
Exploring the boundaries of predictability M3 competition data 36
John W Tukey
Cognostics
Computer-produced diagnostics
(Tukey and Tukey, 1985).

79. An STL decomposition
Exploring the boundaries of predictability M3 competition data 37
Time
1984 1986 1988 1990 1992
6000650070007500

80. An STL decomposition
Yt = St + Tt + Rt
60007000
data
−1500
seasonal
60007000
trend
−60−2020
1984 1986 1988 1990 1992
remainder
time
Exploring the boundaries of predictability M3 competition data 37

81. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

82. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

83. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

84. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

85. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

86. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

87. Candidate features
STL decomposition
Yt = St + Tt + Rt
Seasonal period
Strength of seasonality: 1 − Var(Rt)
Var(Yt−Tt)
Strength of trend: 1 − Var(Rt)
Var(Yt−St)
Spectral entropy: H = −
π
−π fy(λ) log fy(λ)dλ,
where fy(λ) is spectral density of Yt.
Low values of H suggest a time series that is
easier to forecast (more signal).
Autocorrelations: r1, r2, r3, . . .
Optimal Box-Cox transformation parameter λ
Exploring the boundaries of predictability M3 competition data 38

88. Candidate features
Exploring the boundaries of predictability M3 competition data 39
Seasonality
N0001
1976 1978 1980 1982 1984 1986 1988
100030005000
N2602
1978 1980 1982 1984 1986
01000020000
N1906
1984 1986 1988 1990 1992
2000600010000

89. Candidate features
Exploring the boundaries of predictability M3 competition data 39
Trend
N0125
1976 1978 1980 1982 1984 1986 1988
200040006000
N1978
1982 1984 1986 1988 1990 1992
30005000
N0546
1975 1980 1985
100040007000

90. Candidate features
Exploring the boundaries of predictability M3 competition data 39
ACF1
N0001
1987 1988 1989 1990
580060006200
N2658
1987 1988 1989 1990 1991
300050007000
N2409
1984 1986 1988 1990 1992
700080009000

91. Candidate features
Exploring the boundaries of predictability M3 competition data 39
Spectral entropy
N2487
1964 1966 1968 1970 1972 1974
250040005500
N0794
1986 1988 1990 1992
30004500
N0121
1976 1978 1980 1982 1984 1986 1988
200024002800

92. Candidate features
Exploring the boundaries of predictability M3 competition data 39
Box Cox
N0002
1976 1978 1980 1982 1984 1986 1988
200040006000
N0468
1960 1965 1970 1975 1980 1985
500070009000
N0354
1960 1965 1970 1975 1980 1985
20006000

93. Candidate features
Exploring the boundaries of predictability M3 competition data 40
SpecEntr
0.0 0.4 0.8 2 6 10 0.0 0.4 0.8
0.50.9
0.00.6
Trend
Season
0.00.6
28
Freq
ACF
−0.40.6
0.5 0.7 0.9
0.00.6
0.0 0.4 0.8 −0.4 0.2 0.8
Lambda

94. Dimension reduction for time series
Exploring the boundaries of predictability M3 competition data 41
q

95. Dimension reduction for time series
Exploring the boundaries of predictability M3 competition data 41
q
SpecEntr
0.0 0.4 0.8 2 6 10 0.0 0.4 0.8
0.50.9
0.00.6
Trend
Season
0.00.6
28
Freq
ACF
−0.40.6
0.5 0.7 0.9
0.00.6
0.0 0.4 0.8 −0.4 0.2 0.8
Lambda
Feature
calculation

96. Dimension reduction for time series
Exploring the boundaries of predictability M3 competition data 41
q
SpecEntr
0.0 0.4 0.8 2 6 10 0.0 0.4 0.8
0.50.9
0.00.6
Trend
Season
0.00.6
28
Freq
ACF
−0.40.6
0.5 0.7 0.9
0.00.6
0.0 0.4 0.8 −0.4 0.2 0.8
Lambda
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−3
−2
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0
1
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−2 0 2 4
PC1
PC2
Feature
calculation
Principal
component
decomposition

97. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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−1
0
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3
−2 0 2 4
PC1
PC2
First two PCs explain 68% of variation.

98. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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−3
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PC1
PC2
3
6
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12
value
Freq

99. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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−2 0 2 4
PC1
PC2
0.00
0.25
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value
Season

100. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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−2
−1
0
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2
3
−2 0 2 4
PC1
PC2
0.25
0.50
0.75
value
Trend

101. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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PC1
PC2
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0.5
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ACF

102. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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SpecEntr

103. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 42
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PC2
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Lambda

104. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 43
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−2 0 2 4
−3−2−10123
PC1
PC2
q
q
q
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105. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 43
N1627
1990 1991 1992 1993 1994
N0636
1920 1925 1930 1935 1940 1945
N2981
0 10 20 30 40 50 60
N2699
1984 1986 1988 1990 1992

106. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 43
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−2 0 2 4
−3−2−10123
PC1
PC2
Yahoo log returns series

107. Feature space of M3 data
Exploring the boundaries of predictability M3 competition data 43
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PC2
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108. Evolving new time series
Exploring the boundaries of predictability M3 competition data 44
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−2 0 2 4
−3−2−10123
PC1
PC2
We randomly
generate new time
series with specified
feature vectors using
a genetic algorithm.

109. Evolving new time series
Exploring the boundaries of predictability M3 competition data 44
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−4
−2
0
2
4
−2 0 2 4 6
PC1
PC2
Targets
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−4
−2
0
2
4
−2
PC2

110. Evolving new time series
Exploring the boundaries of predictability M3 competition data 44
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PC1
PC2
Evolved yearly data

111. Evolving new time series
Exploring the boundaries of predictability M3 competition data 44
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PC1
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E

112. Evolving new time series
Exploring the boundaries of predictability M3 competition data 44
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−4
−2
0
2
4
−2 0 2 4 6
PC1
PC2
Evolved monthly data

113. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability Yahoo web traffic 45

114. Yahoo web-traffic
Tens of thousands of time series collected at
one-hour intervals over one month.
Consisting of several server metrics (e.g. CPU usage
and paging views) from many server farms globally.
Aim: find unusual (anomalous) time series.
Exploring the boundaries of predictability Yahoo web traffic 46

115. Yahoo web-traffic
busy233busy271busy50busy200busy369
2014−11−09
2014−11−10
2014−11−11
2014−11−12
2014−11−13
2014−11−14
2014−11−15
2014−11−16
2014−11−17
2014−11−18
2014−11−19
2014−11−20
2014−11−21
2014−11−22
2014−11−23
2014−11−24
2014−11−25
2014−11−26
2014−11−27
2014−11−28
2014−11−29
2014−11−30
2014−12−01
2014−12−02
2014−12−03
2014−12−04
2014−12−05
2014−12−06
2014−12−07
2014−12−08
2014−12−09
2014−12−10
2014−12−11
2014−12−12
date
25
30
35
40
45
20
25
30
35
40
50
60
70
10
15
20
25
50
60
memory460memory429memory147memory413memory484
2014−11−09
2014−11−10
2014−11−11
2014−11−12
2014−11−13
2014−11−14
2014−11−15
2014−11−16
2014−11−17
2014−11−18
2014−11−19
2014−11−20
2014−11−21
2014−11−22
2014−11−23
2014−11−24
2014−11−25
2014−11−26
2014−11−27
2014−11−28
2014−11−29
2014−11−30
2014−12−01
2014−12−02
2014−12−03
2014−12−04
2014−12−05
2014−12−06
2014−12−07
2014−12−08
2014−12−09
2014−12−10
2014−12−11
2014−12−12
date
value
0
5000
10000
15000
20000
200
400
600
0
5000
10000
15000
20000
500
1000
0
5000
10000
15000
20000
25000
2014−11−09
2014−11−10
2014−11−11
2014−11−12
2014−11−13
2014−11−14
2014−11−15
2014−11−16
2014−11−17
2014−11−18
2014−11−19
2014−11−20
2014−11−21
2014−11−22
2014−11−23
2014−11−24
2014−11−25
2014−11−26
2014−11−27
2014−11−28
2014−11−29
2014−11−30
2014−12−01
2014−12−02
2014−12−03
2014−12−04
2014−12−05
2014−12−06
2014−12−07
2014−12−08
2014−12−09
2014−12−10
date
value
Exploring the boundaries of predictability Yahoo web traffic 47

116. Feature space
ACF1: first order autocorrelation = Corr(Yt, Yt−1)
Strength of trend and seasonality based on STL
Trend linearity and curvature
Size of seasonal peak and trough
Spectral entropy
Lumpiness: variance of block variances (block size 24).
Spikiness: variances of leave-one-out variances of STL remainders.
Level shift: Maximum difference in trimmed means of consecutive
moving windows of size 24.
Variance change: Max difference in variances of consecutive
moving windows of size 24.
Flat spots: Discretize sample space into 10 equal-sized intervals.
Find max run length in any interval.
Number of crossing points of mean line.
Kullback-Leibler score: Maximum of
DKL(P Q) = P(x) ln P(x)/Q(x)dx where P and Q are estimated by
kernel density estimators applied to consecutive windows of size 48.
Change index: Time of maximum KL score
Exploring the boundaries of predictability Yahoo web traffic 48

117. Principal component analysis
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ACF1
lumpiness
entropy
lshiftvchange
cpoints
fspots
trend
linearity
curvature
spikiness
season
peaktrough
klscore
change.idx
−4
−2
0
2
−2.5 0.0 2.5
standardized PC1 (28.7% explained var.)
standardizedPC2(17.3%explainedvar.)
Exploring the boundaries of predictability Yahoo web traffic 49

118. What is “anomalous”
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ACF1
lumpiness
entropy
lshiftvchange
cpoints
fspots
trend
linearity
curvature
spikiness
season
peaktrough
klscore
change.idx
−4
−2
0
2
−2.5 0.0 2.5
standardized PC1 (28.7% explained var.)
standardizedPC2(17.3%explainedvar.)
We need a measure of the “anomalousness” of a time
series.
1 Rank points based on their local density.
2 Rank points based on whether they are within
α-convex hulls of different radius.
Exploring the boundaries of predictability Yahoo web traffic 50

119. What is “anomalous”
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ACF1
lumpiness
entropy
lshiftvchange
cpoints
fspots
trend
linearity
curvature
spikiness
season
peaktrough
klscore
change.idx
−4
−2
0
2
−2.5 0.0 2.5
standardized PC1 (28.7% explained var.)
standardizedPC2(17.3%explainedvar.)
We need a measure of the “anomalousness” of a time
series.
1 Rank points based on their local density.
2 Rank points based on whether they are within
α-convex hulls of different radius.
Exploring the boundaries of predictability Yahoo web traffic 50

120. What is “anomalous”
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ACF1
lumpiness
entropy
lshiftvchange
cpoints
fspots
trend
linearity
curvature
spikiness
season
peaktrough
klscore
change.idx
−4
−2
0
2
−2.5 0.0 2.5
standardized PC1 (28.7% explained var.)
standardizedPC2(17.3%explainedvar.)
We need a measure of the “anomalousness” of a time
series.
1 Rank points based on their local density.
2 Rank points based on whether they are within
α-convex hulls of different radius.
Exploring the boundaries of predictability Yahoo web traffic 50

121. Bivariate density ranking
Exploring the boundaries of predictability Yahoo web traffic 51
−5 0 5
−8−6−4−20246
pc1
pc2
1
2
3
4
5

122. Bivariate density ranking
Exploring the boundaries of predictability Yahoo web traffic 51
0
10000
20000
30000
40000
0
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0
10000
20000
30000
40000
50000
0
10000
20000
30000
40000
0
1000
2000
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4000
5000
S7793S8494S10464S7833S1715
2015−02−28
2015−03−01
2015−03−02
2015−03−03
2015−03−04
2015−03−05
2015−03−06
2015−03−07
2015−03−08
2015−03−09
2015−03−10
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2015−03−12
2015−03−13
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2015−03−26
2015−03−27
2015−03−28
2015−03−29
2015−03−30
2015−03−31
2015−04−01
date
value

123. α-convex hulls
The space generated by point pairs that can be
touched by an empty disc of radius α.
α → ∞ gives a convex hull.
Points can become isolated when α is small.
We rank points based on the value of α when
they become isolated.
Exploring the boundaries of predictability Yahoo web traffic 52

124. α-convex hulls
The space generated by point pairs that can be
touched by an empty disc of radius α.
α → ∞ gives a convex hull.
Points can become isolated when α is small.
We rank points based on the value of α when
they become isolated.
Exploring the boundaries of predictability Yahoo web traffic 52

125. α-convex hulls
The space generated by point pairs that can be
touched by an empty disc of radius α.
α → ∞ gives a convex hull.
Points can become isolated when α is small.
We rank points based on the value of α when
they become isolated.
Exploring the boundaries of predictability Yahoo web traffic 52

126. α-convex hulls
The space generated by point pairs that can be
touched by an empty disc of radius α.
α → ∞ gives a convex hull.
Points can become isolated when α is small.
We rank points based on the value of α when
they become isolated.
Exploring the boundaries of predictability Yahoo web traffic 52

127. α-convex hull
Exploring the boundaries of predictability Yahoo web traffic 53
−5 0 5
−8−6−4−20246
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128. α-convex hull ranking
Exploring the boundaries of predictability Yahoo web traffic 54
−5 0 5
−8−6−4−20246
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5

129. α-convex hull ranking
Exploring the boundaries of predictability Yahoo web traffic 54
0
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S10464S7793S8494S1715S7826
2015−02−28
2015−03−01
2015−03−02
2015−03−03
2015−03−04
2015−03−05
2015−03−06
2015−03−07
2015−03−08
2015−03−09
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2015−03−28
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2015−03−30
2015−03−31
2015−04−01
date
value

130. HDR versus α-convex hull
HDR boxplot
−5 0 5
−8−6−4−20246
pc1
pc2
1
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α-convex hull
−5 0 5
−8−6−4−20246
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Exploring the boundaries of predictability Yahoo web traffic 55

131. Top 5 anomalous time series
HDR
0
10000
20000
30000
40000
0
2000
4000
6000
0
10000
20000
30000
40000
50000
0
10000
20000
30000
40000
0
1000
2000
3000
4000
5000
S7793S8494S10464S7833S1715
2015−02−28
2015−03−01
2015−03−02
2015−03−03
2015−03−04
2015−03−05
2015−03−06
2015−03−07
2015−03−08
2015−03−09
2015−03−10
2015−03−11
2015−03−12
2015−03−13
2015−03−14
2015−03−15
2015−03−16
2015−03−17
2015−03−18
2015−03−19
2015−03−20
2015−03−21
2015−03−22
2015−03−23
2015−03−24
2015−03−25
2015−03−26
2015−03−27
2015−03−28
2015−03−29
2015−03−30
2015−03−31
2015−04−01
date
value
α-convex hull
0
10000
20000
30000
40000
50000
0
10000
20000
30000
40000
0
2000
4000
6000
0
1000
2000
3000
4000
5000
0
10000
20000
30000
S10464S7793S8494S1715S7826
2015−02−28
2015−03−01
2015−03−02
2015−03−03
2015−03−04
2015−03−05
2015−03−06
2015−03−07
2015−03−08
2015−03−09
2015−03−10
2015−03−11
2015−03−12
2015−03−13
2015−03−14
2015−03−15
2015−03−16
2015−03−17
2015−03−18
2015−03−19
2015−03−20
2015−03−21
2015−03−22
2015−03−23
2015−03−24
2015−03−25
2015−03−26
2015−03−27
2015−03−28
2015−03−29
2015−03−30
2015−03−31
2015−04−01
date
value
Exploring the boundaries of predictability Yahoo web traffic 56

132. Outline
1 Forecasting is difficult
2 Stock market data
3 Drug sales
4 Extreme electricity demand
5 What can we forecast?
6 M3 competition data
7 Yahoo web traffic
8 What next?
Exploring the boundaries of predictability What next? 57

133. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

134. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

135. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

136. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

137. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

138. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

139. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58

140. What next?
Use feature space to select forecasting models.
Develop a more comprehensive set of features
that are reliable measures and fast to compute.
e.g., for finance data.
Consider other dimension reduction methods
and more than 2 dimensions.
Develop dynamic & interactive visualization tools.
Make methods available in an R package.
Some of the methods are already available in the
anomalous package for R on github.
¯ Papers: robjhyndman.com
¯ Code: github.com/robjhyndman
¯ Email: Rob.Hyndman@monash.edu
Exploring the boundaries of predictability What next? 58