George Athanasopoulos
(with Rob J. Hyndman, Nikos Kourentzes and
Fotis Petropoulos)
Forecasting hierarchical (and
grouped) time series
Total
A
AA
AB
C
B
AC
BA
BB
BC
CA
CB
CC
1
Outline
1
Hierarchical and grouped time series
2
Optimal forecasts
3
Approximately optimal forecasts
4
Temporal hierarchies
5
References
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
2
Summary
3 key developments:
1
2
3
We generalise the forecasting process in
"properly" accounting for grouped data.
(Empirical Application 1).
We advance the “optimal combination”
approach by proposing two new estimators
based on WLS.
Both now implemented in the hts package.
We introduce temporal hierarchies.
(Empirical Application 2).
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
3
Summary
3 key developments:
1
2
3
We generalise the forecasting process in
"properly" accounting for grouped data.
(Empirical Application 1).
We advance the “optimal combination”
approach by proposing two new estimators
based on WLS.
Both now implemented in the hts package.
We introduce temporal hierarchies.
(Empirical Application 2).
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
3
Summary
3 key developments:
1
2
3
We generalise the forecasting process in
"properly" accounting for grouped data.
(Empirical Application 1).
We advance the “optimal combination”
approach by proposing two new estimators
based on WLS.
Both now implemented in the hts package.
We introduce temporal hierarchies.
(Empirical Application 2).
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
3
Summary
3 key developments:
1
2
3
We generalise the forecasting process in
"properly" accounting for grouped data.
(Empirical Application 1).
We advance the “optimal combination”
approach by proposing two new estimators
based on WLS.
Both now implemented in the hts package.
We introduce temporal hierarchies.
(Empirical Application 2).
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
3
Hierarchical versus grouped
Table: Geographical Hierarchy
Level
Australia
States and Territories
Total series per level
1
7
VIC, NSW, QLD, SA, WA, NT, TAS;
Zones
27
VIC (5): Metro, West Coast, East Coast, Nth East, Nth West;
NSW (6): Metro, Nth Coast, Sth Coast, Sth, Nth, ACT;
QLD (4): Metro, Central Coast, Nth Coast, Inland;
Regions
76
Metro VIC: Melbourne, Peninsula, Geelong;
Metro NSW: Sydney, Illawarra, Central Coast;
Metro QLD: Brisbane, Gold Coast, Sunshine Coast;
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
4
Australian domestic tourism
Hierarchical:
Australia (1)
States (7)
Zones (27)
Regions (76)
Total: 111 series
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
5
Australian domestic tourism
Purpose of travel (PoT):
Holiday
Visiting Friends and Relatives
Business
Other
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
5
Australian domestic tourism
Hierarchical:
Australia (1)
PoT (×4):
AustraliaPoT (4)
States (7)
StatesPoT (28)
Zones (27)
ZonesPoT (108)
Regions (76)
RegionsPoT (304)
Total: 111 series
Forecasting hierarchical (and grouped) time series
Total: 444 series
Hierarchical and grouped time series
5
Australian domestic tourism
Hierarchical:
Australia (1)
PoT (×4):
AustraliaPoT (4)
States (7)
StatesPoT (28)
Zones (27)
ZonesPoT (108)
Regions (76)
RegionsPoT (304)
Total: 111 series
Total: 444 series
Grouped
Grand total: 555 series
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
5
Forecasting such structures
Existing methods:
ã Bottom-up
ã Top-down
ã Middle-out
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
6
Forecasting such structures
Existing methods:
ã Bottom-up
ã Top-down
ã Middle-out
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
6
Hierarchical data
Total
A
B
C
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
7
Hierarchical data
Total
A
B
C
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
7
Hierarchical data
Total
A
B
Yt
=
C
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Yt
YA,t
YB,t
YC,t
Forecasting hierarchical (and grouped) time series
Hierarchical and grouped time series
7
Hierarchical data
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Total
A
B
Yt
=
C
Yt
YA,t
YB,t
YC,t
=
1
1
0
0
1
0
1
0
|
{z
S
Forecasting hierarchical (and grouped) time series
1
Y
A
,
t
0
YB,t
0
YC,t
1
}
Hierarchical and grouped time series
7
Hierarchical data
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Total
A
B
Yt
=
C
Yt
YA,t
YB,t
YC,t
=
1
1
0
0
1
0
1
0
|
{z
S
Forecasting hierarchical (and grouped) time series
1
Y
A
,
t
0
YB,t
0
YC,t
1 | {z }
}
Bt
Hierarchical and grouped time series
7
Hierarchical data
Yt : observed aggregate of all
series at time t.
YX,t : observation on series X at
time t.
Bt : vector of all series at
bottom level in time t.
Total
A
B
Yt
Yt
=
C
Yt
YA,t
YB,t
YC,t
= SBt
=
1
1
0
0
1
0
1
0
|
{z
S
Forecasting hierarchical (and grouped) time series
1
Y
A
,
t
0
YB,t
0
YC,t
1 | {z }
}
Bt
Hierarchical and grouped time series
7
Hierarchical data
Total
A
AX
Yt =
AY
Yt
YA,t
YB,t
YC,t
YAX,t
YAY ,t
YAZ,t
YBX,t
YBY ,t
YCX,t
YCY ,t
C
B
AZ
BX
1
1
0
0
1
0
0
0
0
0
0
=
|
1
1
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0
{z
S
Forecasting hierarchical (and grouped) time series
CX
BY
1
0
0
1
0
0
0
0
0
1
0
CY
1
0
0
1
0
0
0
0
0
0
1
YAX,t
Y
AY,t
YAZ,t
YBX,t
YBY,t
Y
CX,t
YCY,t
| {z }
Bt
}
Hierarchical and grouped time series
8
Hierarchical data
Total
A
AX
Yt =
AY
Yt
YA,t
YB,t
YC,t
YAX,t
YAY ,t
YAZ,t
YBX,t
YBY ,t
YCX,t
YCY ,t
C
B
AZ
BX
1
1
0
0
1
0
0
0
0
0
0
=
|
1
1
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0
{z
S
Forecasting hierarchical (and grouped) time series
CX
BY
1
0
0
1
0
0
0
0
0
1
0
CY
1
0
0
1
0
0
0
0
0
0
1
YAX,t
Y
AY,t
YAZ,t
YBX,t
YBY,t
Y
CX,t
YCY,t
| {z }
Bt
}
Hierarchical and grouped time series
8
Hierarchical data
Total
A
AX
Yt =
AY
Yt
YA,t
YB,t
YC,t
YAX,t
YAY ,t
YAZ,t
YBX,t
YBY ,t
YCX,t
YCY ,t
C
B
AZ
BX
1
1
0
0
1
0
0
0
0
0
0
=
|
1
1
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
0
0
0
1
0
0
{z
S
Forecasting hierarchical (and grouped) time series
CX
BY
1
0
0
1
0
0
0
0
0
1
0
CY
1
0
0
1
0
0
0
0
0
0
1
YAX,t
Y
AY,t
YAZ,t
YBX,t
YBY,t
Y
CX,t
YCY,t
| {z }
Bt
}
Yt = SBt
Hierarchical and grouped time series
8
Grouped data
Total
Total
A
AX
B
AY
BX
Yt
YA,t
YB,t
YX,t
Yt = YY ,t =
Y
AX,t
Y
AY ,t
YBX,t
YBY ,t
1
1
0
1
0
1
0
0
0
|
X
BY
1
1
0
0
1
0
1
0
0
AX
1
0
1
1
0
0
0
1
0
Y
BX
AY
BY
1
0
1
YAX,t
0
YAY ,t
1
YBX,t
0
YBY ,t
0
| {z }
Bt
0
1
{z
S
Forecasting hierarchical (and grouped) time series
}
Hierarchical and grouped time series
9
Grouped data
Total
Total
A
AX
B
AY
BX
Yt
YA,t
YB,t
YX,t
Yt = YY ,t =
Y
AX,t
Y
AY ,t
YBX,t
YBY ,t
1
1
0
1
0
1
0
0
0
|
X
BY
1
1
0
0
1
0
1
0
0
AX
1
0
1
1
0
0
0
1
0
Y
BX
AY
BY
1
0
1
YAX,t
0
YAY ,t
1
YBX,t
0
YBY ,t
0
| {z }
Bt
0
1
{z
S
Forecasting hierarchical (and grouped) time series
}
Hierarchical and grouped time series
9
Grouped data
Total
Total
A
AX
B
AY
BX
Yt
YA,t
YB,t
YX,t
Yt = YY ,t =
Y
AX,t
Y
AY ,t
YBX,t
YBY ,t
1
1
0
1
0
1
0
0
0
|
X
BY
1
1
0
0
1
0
1
0
0
AX
1
0
1
1
0
0
0
1
0
S
Forecasting hierarchical (and grouped) time series
BX
AY
BY
1
0
1
YAX,t
0
YAY ,t
1
YBX,t
0
YBY ,t
0
| {z }
Bt
0
1
{z
Y
Yt = SBt
}
Hierarchical and grouped time series
9
Outline
1
Hierarchical and grouped time series
2
Optimal forecasts
3
Approximately optimal forecasts
4
Temporal hierarchies
5
References
Forecasting hierarchical (and grouped) time series
Optimal forecasts
10
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal forecasts
Key idea: forecast reconciliation
å Ignore structural constraints and forecast
every series of interest independently.
å Adjust forecasts to impose constraints.
ˆ n(h) be vector of initial h-step forecasts,
Let Y
made at time n, stacked in same order as Yt .
Yt
= SBt .
ˆ n(h)
So Y
= Sβn(h) + εh .
βn(h) = E[Bn+h | Y1, . . . , Yn].
εh has zero mean and covariance Σh.
Estimate βn (h) using GLS?
Forecasting hierarchical (and grouped) time series
Optimal forecasts
11
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Revised forecasts
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Revised forecasts
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Revised forecasts
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Revised forecasts
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Optimal combination forecasts
˜ n(h)
Y
ˆ n (h )
= Sβˆn(h) = S(S0Σ†hS)−1S0Σ†hY
Revised forecasts
Initial forecasts
= (S0Σ†hS)−1S0Σ†h.
Σ†h is generalized inverse of Σh.
Optimal P
Revised forecasts unbiased: SPS
= S.
Revised forecasts minimum variance:
˜ n(h)|Y1, . . . , Yn]
Var[Y
Problem:
= S(S0Σ†hS)−1S0.
Σh hard to estimate.
Forecasting hierarchical (and grouped) time series
Optimal forecasts
12
Outline
1
Hierarchical and grouped time series
2
Optimal forecasts
3
Approximately optimal forecasts
4
Temporal hierarchies
5
References
Forecasting hierarchical (and grouped) time series
Approximately optimal forecasts
13
Approx. optimal forecasts
˜ n(h)
Y
ˆ n(h)
= S(S0Σ†hS)−1S0Σ†hY
Solution 1: OLS
Assume εh ≈ SεB,h where εB,h is the forecast
error at bottom level.
If Moore-Penrose generalized inverse used,
then (S0 Σ† S)−1 S0 Σ† = (S0 S)−1 S0 .
˜ n(h) = S(S0S)−1S0Y
ˆ n ( h)
Y
Forecasting hierarchical (and grouped) time series
Approximately optimal forecasts
14
Approx. optimal forecasts
˜ n(h)
Y
ˆ n(h)
= S(S0Σ†hS)−1S0Σ†hY
Solution 2: Rescaling
Suppose we approximate Σh by its diagonal.
−1
ˆ1
Let Λ = diagonal Σ
contain inverse
one-step ahead in-sample forecast error
variances.
˜ n(h) = S(S0ΛS)−1S0ΛY
ˆ n ( h)
Y
Forecasting hierarchical (and grouped) time series
Approximately optimal forecasts
15
Approx. optimal forecasts
˜ n(h)
Y
ˆ n(h)
= S(S0ΛS)−1S0ΛY
Solution 3: Averaging
If the bottom level error series are
approximately uncorrelated and have similar
variances, then Λ is inversely proportional to
the number of series contributing to each
node.
So set Λ to be the inverse row sums of S:
Λ = diag(S × 1)−1
where 1 = (1, 1, . . . , 1)0 .
Forecasting hierarchical (and grouped) time series
Approximately optimal forecasts
16
Outline
1
Hierarchical and grouped time series
2
Optimal forecasts
3
Approximately optimal forecasts
4
Temporal hierarchies
5
References
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
17
Temporal hierarchies: quarterly
Annual
Semi-Anual1
Q1
Semi-Anual2
Q2
Q3
Q4
Basic idea:
å Forecast series at each available
frequency.
å Optimally combine forecasts within the
same year.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
18
Temporal hierarchies: quarterly
Annual
Semi-Anual1
Q1
Semi-Anual2
Q2
Q3
Q4
Basic idea:
å Forecast series at each available
frequency.
å Optimally combine forecasts within the
same year.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
18
Temporal hierarchies: monthly
Annual
Semi-Anual1
Q1
M1
M2
Semi-Anual2
Q3
Q2
M3
M4
M5
M6
M7
M8
Q4
M9
M10 M11 M12
Aggregate: 3, 6, 12
Alternatively: 2, 4, 12.
How about: 2, 3, 4, 6, 12?
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
19
Temporal hierarchies: monthly
Annual
FourM1
BiM1
M1
M2
BiM2
M3
FourM3
FourM2
M4
BiM3
M5
M6
BiM5
BiM4
M7
M8
M9
BiM6
M10 M11 M12
Aggregate: 3, 6, 12
Alternatively: 2, 4, 12.
How about: 2, 3, 4, 6, 12?
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
19
Temporal hierarchies: monthly
Annual
FourM1
BiM1
M1
M2
BiM2
M3
FourM3
FourM2
M4
BiM3
M5
M6
BiM5
BiM4
M7
M8
M9
BiM6
M10 M11 M12
Aggregate: 3, 6, 12
Alternatively: 2, 4, 12.
How about: 2, 3, 4, 6, 12?
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
19
Monthly data
= 0
1
0
A
SemiA1
SemiA2
FourM1
FourM2
FourM3
Q1
..
.
Q4
BiM1
..
.
BiM6
M1
..
.
M12
|
{z
(28×1)
}
1
1
0
1
0
0
1
1
1
0
1
0
0
1
1
1
0
1
0
0
1
1
1
0
1
0
0
0
1
1
0
0
1
0
0
1
1
0
0
1
0
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
0 0 0 0
0 0 0 0
..
.
0 0 0 0
0
0
1
0
1
0
0
0
1
..
.
0
1
0
0
0
0
0
0
0
|
Forecasting hierarchical (and grouped) time series
I12
{z
S
M1
M2
M3
M
4
M5
M6
1 M7
0 M8
M9
1M10
M11
M
12
| {z }
} Bt
1
0
1
0
0
1
0
Temporal hierarchies
20
Experimental setup:
M3 forecasting competition (Makridakis
and Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of
12 observations each.
756 quarterly series with a test sample of
8 observations each.
Forecast each series with ETS (ARIMA)
models. Methods performed well in the
actual competition.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
21
Experimental setup:
M3 forecasting competition (Makridakis
and Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of
12 observations each.
756 quarterly series with a test sample of
8 observations each.
Forecast each series with ETS (ARIMA)
models. Methods performed well in the
actual competition.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
21
Experimental setup:
M3 forecasting competition (Makridakis
and Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of
12 observations each.
756 quarterly series with a test sample of
8 observations each.
Forecast each series with ETS (ARIMA)
models. Methods performed well in the
actual competition.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
21
Experimental setup:
M3 forecasting competition (Makridakis
and Hibon, 2000, IJF). In total 3003 series.
1,428 monthly series with a test sample of
12 observations each.
756 quarterly series with a test sample of
8 observations each.
Forecast each series with ETS (ARIMA)
models. Methods performed well in the
actual competition.
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
21
Results: Monthly
sMAPE
(obs)
Forecast Horizon (h)
Annual SemiA FourM
(1)
(2)
(3)
Q
(4)
BiM
(6)
M
(12)
Average
ETS
Initial
9.66
9.18
9.76
10.14
10.82
12.85
10.40
Bottom-up
8.38
9.14
9.78
10.06
11.04
12.85
10.21
OLS
Scaling
Averaging
7.80
7.64
7.51
8.64
8.44
8.31
9.39
9.15
9.05
9.72 10.68 12.68
9.49 10.45 12.40
9.38 10.34 12.30
9.82
9.60
9.48
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
22
Results: Quarterly
sMAPE
(obs)
Forecast Horizon (h)
Annual Semi-Ann Quart
(2)
(4)
(8)
Average
ETS
Initial
10.50
9.97
9.84
10.10
Bottom-up
8.87
9.35
9.84
9.35
OLS
Scaling
Averaging
9.31
8.75
8.81
9.78
9.19
9.26
10.28
9.70
9.78
9.79
9.21
9.28
Forecasting hierarchical (and grouped) time series
Temporal hierarchies
23
Outline
1
Hierarchical and grouped time series
2
Optimal forecasts
3
Approximately optimal forecasts
4
Temporal hierarchies
5
References
Forecasting hierarchical (and grouped) time series
References
24
More information
Vignette on CRAN
Forecasting hierarchical (and grouped) time series
References
25
References
RJ Hyndman, RA Ahmed, G Athanasopoulos, and HL
Shang (2011). “Optimal combination forecasts for
hierarchical time series”. Computational Statistics and
Data Analysis 55(9), 2579-2589.
G Athanasopoulos, RA Ahmed, RJ Hyndman,(2009).
“Hierarchical forecasts for Australian domestic tourism”.
International Journal of Forecasting 25, 146-166.
RJ Hyndman, et al., (2014). hts: Hierarchical time series.
cran.r-project.org/package=hts.
RJ Hyndman and G Athanasopoulos (2014). Forecasting:
principles and practice. OTexts. www.otexts.org/fpp/.
Forecasting hierarchical (and grouped) time series
References
26
References
RJ Hyndman, RA Ahmed, G Athanasopoulos, and HL
Shang (2011). “Optimal combination forecasts for
hierarchical time series”. Computational Statistics and
Data Analysis 55(9), 2579-2589.
å GEmail:
Athanasopoulos, RA Ahmed, RJ Hyndman,(2009).
“Hierarchical
forecasts for Australian domestic tourism”.
[email protected]
International Journal of Forecasting 25, 146-166.
Acknowledgments: Rob J Hyndman, Roman
RJ
Hyndman,
al., (2014).
hts: Hierarchical
time series.
Ahmed,
HanetShang,
Shanika
Wickramasuriya,
cran.r-project.org/package=hts.
Nikolaos Kourentzes, Fotios Petropoulos.
RJ
Hyndman and G Athanasopoulos
(2014). Forecasting:
Acknowledgments:
Excellent research
principles and practice. OTexts. www.otexts.org/fpp/.
assistance by Earo Wang.
Forecasting hierarchical (and grouped) time series
References
26
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