[PPT] - Improving forecasting by estimating time series structural PowerPoint Presentation

SLIDE 1

Improving forecasting by estimating time series structural components across multiple frequencies

Nikolaos Kourentzes Fotios Petropoulos Juan R. Trapero

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1. Motivation
2. The idea behind the algorithm
3. Multiple Aggregation Prediction Algorithm
4. Empirical evaluation
5. Conclusions

Agenda

Multiple Aggregation Prediction Algorithm

SLIDE 3

Motivation

Forecasting Forecasting is crucial for several operations of organisations

Short- and long-term objectives
Demand and inventory planning
Capacity planning
Pricing and marketing strategy planning
Budgeting
etc.

Requirement for large number of forecasts → Automa,on Key issues in forecasting automation:

Model selection
Model parameterisation
Forecast reconciliation

SLIDE 4

Motivation

Exponential Smoothing Let us consider the example of Exponential Smoothing method (ETS)

Considered one of the most reliable and robust methods for automatic univariate

forecasting [Makridakis & Hibbon, 2000, Hyndman et al., 2002, Gardner, 2006]

It is a family of methods: ETS (error type, trend type, seasonality type)
Error: Additive or Multiplicative
Trend: None or Additive or Multiplicative, Linear or Damped/Exponential
Seasonality: None or Additive or Multiplicative
Adequate for a most types of time series

SLIDE 5

Motivation

Optimisation and model selection We have an optimisation problem of estimating the smoothing parameters , , , ′ and the initial state This is done by maximising the likelihood of the model: For automatic forecasting we can consider up to 30 different models. This introduces a model selection problem. Hyndman et al., 2002 suggested to solve this via the Akaike Information Criterion (AIC) and provided supporting empirical evidence We select the model with the best AIC, which we use to forecast ∗ ,

2
∗

!, " 2# Number of smoothing parameters and initial states … for well-behaved data

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Motivation

Issues What can go wrong in parameter and model selection:

Business time series are often short Limited data
Estimation of parameters can fail miserably (for monthly data optimise up to 18

parameters, with often no more than 36 observations)

Model selection can fail as well (30 models over-fitting?)
Both optimisation and model selection are myopic Focus on data fitting in the

past, rather than ‘forecastability’

Special cases:

10 20 30 40 50 60 70 80 100 120 140 160 180 200 220 Sales Month Demand Fit Forecast

True model: Additive trend, additive seasonality Identified model: No trend, additive seasonality Why? In-sample variance explained mostly by seasonality Reliable automatic forecasting requires robust parameter estimation and model selection

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Idea

Time/Frequency domains Given a monthly time series:

20 40 60 80 100 120 1000 2000 3000 4000 5000 6000 7000 Month Demand 0.1 0.2 0.3 0.4 0.5 5 10 15 x 10

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Frequency Power

Low frequency components = Level + Trend Seasonality and its harmonics Time series plot Power spectrum We can look at a time series in the classical way, or in the frequency domain Differences, in frequency domain:

Components are separated
ETS is a filter, with smoothing parameters deciding its shape
Initial states cannot be retrieved

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Idea

Temporal Aggregation Given a monthly time series we can do temporal non-overlapping aggregating

20 40 60 80100 120 1000 2000 3000 4000 5000 6000 7000 Period Aggregation Level 1 0.1 0.2 0.3 0.4 5 10 15 x 10

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Frequency Power 10 20 30 40 1000 2000 3000 4000 5000 6000 7000 Period Aggregation Level 3 0.1 0.2 0.3 0.4 1 2 3 4 5 x 10

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Frequency Power 5 10 15 20 1000 2000 3000 4000 5000 6000 7000 Period Aggregation Level 6 0.1 0.2 0.3 0.4 0.5 1 1.5 2 2.5 x 10

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Frequency Power 2 4 6 8 10 12 14 1000 2000 3000 4000 5000 6000 7000 Period Aggregation Level 9 0.1 0.2 0.3 0.4 5 10 15 x 10

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Frequency Power 2 4 6 8 10 1000 2000 3000 4000 5000 6000 7000 Period Aggregation Level 12 0.1 0.2 0.3 0.4 2 4 6 8 10 12 x 10

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Frequency Power

Monthly Quarterly Half-annually 9-monthly Annually Seasonality Weak seasonality No seasonality No seasonality Strong seasonality

SLIDE 9

Idea

Temporal Aggregation Temporal non-overlapping aggregation:

Show to be beneficial for forecasting accuracy ADIDA algorithm [Nikolopoulos et al.,

2011]

Step 1: Aggregate time series Step 2: Forecast time series (motivated by intermittent data) Step 3: Disaggregate time series

Good performance for slow and fast moving goods [Nikolopoulos et al., 2011, Babai et al., 2012]
Reduces noise as aggregation level increases, but removes component information

[Spithourakis et al., 2012]

Consider aggregating monthly time series and disaggregating, seasonality is lost.

Reconstruction would limit only to deterministic forms.

Selection of aggregation level No theoretical grounding [Nikolopoulos et al., 2011,

Spithourakis et al., 2011]

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What if we do not select an aggregation level?

use multiple

10 20 30 40 100 120 140 160 180 200 Period Demand Aggregation level 1 ETS(A,N,A)

ETS(A,M,A) ETS(A,M,N) ETS(A,A,N)

Issues:

Different model
Different length
Combination

2 4 6 8 10 12 14 16 100 120 140 160 180 200 Period Demand Aggregation level 3 1 2 3 4 5 6 7 100 120 140 160 180 200 Period Demand Aggregation level 7 1 1.5 2 2.5 3 3.5 4 100 120 140 160 180 200 Period Demand Aggregation level 12

Idea

Temporal Aggregation

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Idea

Combination Forecast combination:

Forecast combination is widely considered as beneficial for forecasting accuracy

and forecast error variance [Bates & Granger, 1969, Makridakis & Winkler, 1983, Clemen, 1989, Hibon &

Evgeniou, 2005]

Simple combination methods (average, median) considered robust, relatively

accurate to more complex methods [Clemen, 1989, Timmermann, 2006, Jose & Winkler, 2008] Issues:

If there are different model types to be combined then the resulting forecast does

not fit well with any component!

10 20 30 40 100 120 140 160 180 200 Period Demand 10 20 30 40 100 120 140 160 180 200 Period Demand

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The MAPA algorithm

Part 1

10 20 30 40 50 60 5000 10000 15000 5 10 15 20 25 30 5000 10000 15000 5 10 15 20 5000 10000 15000 1 2 3 4 5 6 5000 10000 15000 1 2 3 4 5 6 5000 10000 15000 1 2 3 4 5 5000 10000 15000

…

$

%1'

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%2'

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%3'

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%10'

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Aggregate

10 20 30 40 50 60 70 5000 10000 15000

Fit state space ETS

10 20 30 40 50 60

44.26
44.24
44.22

10 20 30 40 50 60 2000 4000 6000 10 20 30 40 50 60 0.5 1 1.5 2

Save states

Level Trend Season

Fit state space ETS

5 10 15 20 25 30 35 2000 4000 6000 8000 10000

5 10 15 20 25 30 1 1.2 1.4 5 10 15 20 25 30

96.27
96.26
96.25

5 10 15 20 25 30 2000 4000 6000

Save states

Level Trend Season

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The MAPA algorithm

Part 2 Transform states to additive and to original sampling frequency Combine states (components) Produce forecasts

SLIDE 14

Empirical Evaluation

Assess the performance of MAPA on four datasets:

645 annual time series from the M3 competition [Makridakis & Hibbon, 2000]
1483 semi-annual time series from the FRED database
756 quarterly time series from the M3 competition
1428 monthly time series from the M3 competition

Setup identical to M3 competition to allow comparison with published results. FRED semi-annual setup same as M3 quarterly.

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Empirical Evaluation

Better than benchmark ETS The longer the horizon the better the relative performance

Annual data: 2 aggregation levels Semi-annual data: 2 aggregation levels

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Empirical Evaluation

With seasonality present MAPA outperforms Comb The longer the horizon the better the relative performance

Quarterly data: 4 aggregation levels Monthly data: 12 aggregation levels

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Empirical Evaluation

Summary On average better performance than exponential smoothing

Significant for practice, most systems and organisations use exponential smoothing
Switching from ETS to MAPA requires small and transparent changes
Particularly good for long term forecasts Both high- and low-frequency time

series components captured:

Same forecast useful for operational, tactical and strategic horizons
Reconciles short-term forecasting with long-term forecasting
Operational forecasts naturally aggregate to predictions for capacity planning,
etc. Implications for supply chain and operations management

Can we improve further on the short term forecasts?

Standard time series modelling approach: combine MAPA with ETS using simple

average.

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Empirical Evaluation

Combined ETS-MAPA Combined ETS with MAPA(Mean) and (MAPA(Median)) MAPA better MAPA-ETS better ETS better In the MAPA-ETS combination we can show that each state is eventually calculated as: (level) w1=(K+1)/2K and w2 to wK equal to 0.5K, K is the maximum temporal aggregation level.

Temporal hierarchies! Grounding for theoretically identifying optimum aggregation

combination and variable combination weights conditional on the forecast horizon Best literature result: 13.83

SLIDE 19

Conclusions

MAPA provides a framework to better identify and estimate the different time series

components better forecasts

On average outperforms ETS, one of the most widely used, robust and accurate

univariate forecasting methods

Evidence of hierarchies in time may lead to theoretical optimum aggregation levels

and variable combination weights

Implications for short- and long-term forecasts

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Nikos Kourentzes

Lancaster University Management School Centre for Forecasting - Lancaster, LA1 4YX email: n.kourentzes@lancaster.ac.uk

Questions?

SLIDE 21

Algorithm

Part 3 Multiple Aggregation Prediction Algorithm

Step 1: Aggregation . . . . . .

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Y

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Y 2 = k 3 = k K k =

Step 2: Forecasting

ETS Model Selection ETS Model Selection ETS Model Selection ETS Model Selection

. . . . . .

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b

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Step 3: Combination

+

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K &

Strengthens and attenuates components Estimation of parameters at multiple levels Robustness on model selection and parameterisation