Improving forecasting by estimating time series structural components across multiple frequencies

Transcription

1 Improving forecasting by estimating time series structural components across multiple frequencies Nikolaos Kourentzes Fotios Petropoulos Juan R Trapero

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

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

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, 2, Hyndman et al, 22, Gardner, 26] 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

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:, 2 For automatic forecasting we can consider up to 3 different models This introduces a model selection problem Hyndman et al, 22 suggested to solve this via the Akaike Information Criterion (AIC) and provided supporting empirical evidence!," 2# Number of smoothing parameters and initial states We select the model with the best AIC, which we use to forecast for well-behaved data

6 What can go wrong in parameter and model selection: Business time series are often short Limited data Motivation Issues 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 (3 models over-fitting?) Both optimisation and model selection are myopic Focus on data fitting in the past, rather than forecastability Special cases: 22 2 Demand Fit Forecast True model: Additive trend, additive seasonality Sales Month Identified model: No trend, additive seasonality Why? In-sample variance explained mostly by seasonality Reliable automatic forecasting requires robust parameter estimation and model selection

7 Idea Time/Frequency domains Given a monthly time series: 7 Time series plot 15 x 16 Power spectrum 6 Demand 4 3 Power Month Frequency Low frequency components = Level + Trend Seasonality and its harmonics 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

8 Given a monthly time series we can do temporal non-overlapping aggregating 7 Aggregation Level 1 Idea Temporal Aggregation Monthly Quarterly Half-annually 9-monthly Annually 7 Aggregation Level 3 7 Aggregation Level 6 7 Aggregation Level 9 Aggregation Level x 16 5 x x x x Power 1 5 Power 3 2 Power 15 1 Power 1 5 Power Frequency Strong seasonality Frequency Frequency Frequency Frequency Seasonality Weak seasonality No seasonality No seasonality

9 Idea Temporal Aggregation Temporal non-overlapping aggregation: Show to be beneficial for forecasting accuracy ADIDA algorithm [Nikolopouloset al, 211] 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, 211, Babai et al, 212] Reduces noise as aggregation level increases, but removes component information [Spithourakis et al, 212] Consider aggregating monthly time series and disaggregating, seasonality is lost Reconstruction would limit only to deterministic forms Selection of aggregation level No theoretical grounding [Nikolopouloset al, 211, Spithourakis et al, 211]

10 What if we do not select an aggregation level? use multiple Idea Temporal Aggregation 2 Aggregation level 1 ETS(A,N,A) 2 Aggregation level 3 ETS(A,M,A) Demand Demand Aggregation level 7 ETS(A,M,N) Aggregation level 12 ETS(A,A,N) Issues: Different model Different length Combination Demand Demand

11 Forecast combination: Idea 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, 25] Simple combination methods (average, median) considered robust, relatively accurate to more complex methods [Clemen, 1989, Timmermann, 26, Jose & Winkler, 28] Issues: If there are different model types to be combined then the resulting forecast does not fit well with any component! Demand Demand

12 $ %1' $ %2' $ %3' Aggregate The MAPA algorithm Fit state space ETS Save states Level Part 1 $ %1' $ %11' $ %12' Trend Season

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

14 Empirical Evaluation Assess the performance of MAPA on four datasets: 645 annual time series from the M3 competition [Makridakis& Hibbon, 2] 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

15 Empirical Evaluation Annual data: 2 aggregation levels Semi-annual data: 2 aggregation levels Better than benchmark ETS The longer the horizon the better the relative performance

16 Empirical Evaluation Quarterly data: 4 aggregation levels Monthly data: 12 aggregation levels With seasonality present MAPA outperforms Comb The longer the horizon the better the relative performance

17 On average better performance than exponential smoothing Empirical Evaluation Summary 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

18 Empirical Evaluation Combined ETS-MAPA Combined ETS with MAPA(Mean) and (MAPA(Median)) MAPA better MAPA-ETS better ETS better Best literature result: 1383 In the MAPA-ETS combination we can show that each state is eventually calculated as: (level) w 1 =(K+1)/2K and w 2 to w K equal to 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

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

20 Questions? Nikos Kourentzes Lancaster University Management School Centre for Forecasting -Lancaster, LA1 4YX

21 Multiple Aggregation Prediction Algorithm Algorithm Part 3 Step 1: Aggregation Step 2: Forecasting Step 3: Combination [ 1] Y ETS Model Selection l [ 1] [ 1] b [ 1] s + [ 1] Yˆ k = 2 k = 3 k = K [ 2] Y [ 3] Y [ K ] Y ETS Model Selection ETS Model Selection ETS Model Selection l [ 2] b [ 2] s l [ 3] b s [ 2] [ 3] [ 3] l [ K ] [ K ] b [ K ] s 1 K 1 K 1 K & l b s Strengthens and attenuates components Estimation of parameters at multiple levels Robustness on model selection and parameterisation