Using Temporal Hierarchies to Predict Tourism Demand

Transcription

1 Using Temporal Hierarchies to Predict Tourism Demand International Symposium on Forecasting 29 th June 2015 Nikolaos Kourentzes Lancaster University George Athanasopoulos Monash University Lancaster Centre for Forecasting

2 O u t l i n e Part I. 1. Tourism demand forecasting background 2. Revisiting univariate methods 3. Modelling special events Part II. 4. Temporal hierarchies 5. Cross-temporal hierarchies

3 T o u r i s m D e m a n d F o r e c a s t i n g Statistical forecasting has shown promising performance over expert judgement [Athanasopoulos & Hyndman, 2009] Univariate time series methods are inexpensive, simple and relatively accurate [Athanasopoulos & Hyndman, 2009; Athanasopoulos et al., 2011] However, predicting for different objectives (horizons) typically requires different statistical methods [Chatfield, 2000]: Requiring to build, evaluate and select amongst multiple models Raising the complexity difficult for non-expert practitioners Lack of good univariate long-term forecasting methods

4 T o u r i s m D e m a n d F o r e c a s t i n g Athanasopoulos et al. (2009) looked at the following univariate approaches: Seasonal Naïve ARIMA Exponential Smoothing (ETS) & Damped Trend Theta ForecastPro and methods with explanatory Autoregressive distributed lag model Time varying parameter model Vector autoregressive model considering various monetary and macroeconomic indicators.

5 T o u r i s m D e m a n d F o r e c a s t i n g The findings were: Time series methods were best Methods with additional variables did not perform well Additional modelling complexity Variable selection (initial set and included in the model) Requirement for out-of-sample variable knowledge The literature is inconclusive: Allen & Fildes (2001); Song et al. (2003) find otherwise Witt & Witt (1995); Kuledran & King (1997) support time series methods Fildes et al. (2011) find benefits from additional variables, but suggest that univariate methods are equally good and simpler. Nonetheless, structural irregularities still require treatment (e.g. outliers)

6 O b j e c t i v e s This work has two objectives: 1. Revisit univariate methods for Tourism demand Understand poor performance of Theta in previous works. Tackle the effect of outliers in model specification and parameterisation. 2. Consider the issue of different forecasting objectives and horizons We will employ the novel temporal hierarchies modelling approach with the aim of producing forecasts that satisfy different planning objectives. Particular we aim at longer term predictions, while using simple methods.

7 T i m e S e r i e s M e t h o d s : T h e t a The Theta method was first introduced at the M3 competition outperforming all other methods and approaches some cases substantially. However its performance was not validated for by Athanasopoulos et al. (2011), where Theta was outperformed by both ETS and ARIMA. We argue that this is because the method as introduced by Assimakopoulos and Nikolopoulos (2000) was very restrictive does not suit characteristics of tourism demand.

8 H o w d o e s T h e t a w o r k? First a time series is decomposed using classical multiplicative decomposition: Deterministic decomposition Stochastic decomposition We propose to allow for the seasonal pattern to evolve by using a pure seasonal model instead: Obviously when γ 0 then it is the deterministic case.

9 H o w d o e s T h e t a w o r k? Then the deseasonalised time series is broken down in two lines: a linear trend long term trend 2 x (deseasonalised data - linear trend) inflate variability Each series is forecasted separately using linear regression and single exponential smoothing and their forecast is then combined: Hyndman and Billah (2003) showed that this `arcane method is simply a single exponential smoothing with a drift term long term linear trend.

10 H o w d o e s T h e t a w o r k? Finally the forecast of the deseasonalised time series is re-seasonalised with the indices calculated previously to give the final forecast: So Theta is simply a two-stage method, where seasonality is first estimated and subsequently a single exponential smoothing with drift is fitted.

11 F u r t h e r r e f i n e m e n t s We can test whether there is a trend in the time series and remove the drift term if unnecessary simpler model, fit better data process If we keep relaxing the type and presence of trend then Theta ETS. In practice some of the observed performance of Theta is due to its lack of the flexibilty of ETS smaller chance for modelling mistakes restrict modelling uncertainty over ETS We will see that this effort to mitigate modelling uncertainty is what gives the biggest accuracy gains.

12 E m p i r i c a l E v a l u a t i o n A u s t r a l i a D e m a n d Data from Athanasopoulos et al. (2011): 56 quarterly regional tourism demand time series 9 years long, last 3 years used for test set Predict up to 2 years ahead.

13 E m p i r i c a l E v a l u a t i o n A u s t r a l i a D e m a n d Literature Theta agree with previous results Refined Theta The refined Theta performs best, while still being a very simple method. It captures: Stochastic seasonality, when present Linear trend, when present Smooths remainder irregularities ETS is more flexible more difficult to specify cost in accuracy

14 A f u r t h e r r e f i n e m e n t E f f e c t o f s p e c i a l e v e n t s Tourism series are sensitive to special events that may cause spikes in tourism demand. Such events may be: well known, recorded and tracked unknown, but can be revealed by time series analysis residual analysis Theta with events Events are tracked with binary dummies (one for each event) Linear trend estimation includes time and dummies Single exponential smoothing: Pure seasonal: Trend and smoothing parameters estimation robust to events Error drops to 32.58% from 34.01%

15 O b j e c t i v e 1 : R e v i s i t u n i v a r i a t e f o r e c a s t s Producing univariate forecasts is simple and cheap. Literature has shown that they can be very accurate. We revisited some inconsistent results (Theta performance) and showed that simple refinements to fit better tourism demand process can increase accuracy substantially We argued that although model flexibility is good, too much may harm accuracy as it becomes difficult to specify to appropriate model. We will take the last point further and introduce an approach to mitigate this uncertainty for any forecasting method.

16 O u t l i n e Part I. 1. Tourism demand forecasting background 2. Revisiting univariate methods 3. Modelling special events Part II. 4. Temporal hierarchies 5. Cross-temporal hierarchies

17 L o n g t e r m f o r e c a s t i n g Organisations have to often produce forecasts for: Short term operational horizons Medium term tactical horizons Long term strategic horizons We know that different forecasting models are better for different forecast horizons We also know that it helps to forecast long horizons using aggregate data ETS(M,A d,a) - AIC: x ETS(A,A,N) - AIC: d Months Years These forecasts often do not agree, which one is `correct?

18 A u t o m a t i c t i m e s e r i e s f o r e c a s t i n g How automatic forecasting is done? 1. Build, parameterise and apply a reasonable pool of forecasting models 2. Model selection 3. Model application Issues with automatic modelling: Model selection How good is the best fit model? How reliable? Sampling uncertainty Identified model/parameters stable as new data appear? Transparency/Trust Practitioners do not trust systems that change substantially ETS(M,A d,a) - AIC: ETS(A,A d,m) - AIC: Months Issues for resource/capacity planning, procurement, budgeting, etc

19 Sales What can go wrong: W h a t c a n g o w r o n g? A n i l l u s t r a t i o n 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: Demand Fit Forecast True model: Additive trend, additive seasonality Month Identified model: No trend, additive seasonality Why? In-sample variance explained mostly by seasonality

20 Te m p o r a l A g g r e g a t i o n a n d F o r e c a s t i n g Kourentzes et al. (2014) argued that it is important to model the time series at different temporal aggregation levels in parallel reveals different types of information. At lower levels of aggregation we can observe better high frequency components such as seasonality, special events, etc. At higher levels of aggregation we can observe better low frequency components such as trends and cycles. By looking at different temporal aggregation levels, in contrast to conventional model building, we can mitigate modelling uncertainty. Using multiple levels Multiple Aggregation Prediction Algorithm (MAPA) Robust against model misspecification Superior long term accuracy

21 Power Power Power Power Power Te m p o r a l A g g r e g a t i o n a n d F o r e c a s t i n g Given a monthly time series we can do temporal non-overlapping aggregating Monthly Quarterly Half-annually 9-monthly Annually Aggregation Level 1 Aggregation Level 3 Aggregation Level 6 Aggregation Level 9 Aggregation Level Period Period Period Period Period 15 x x x x x Frequency Strong seasonality Frequency Frequency Frequency Frequency Seasonality Weak seasonality No seasonality No seasonality

22 I n t r o d u c i n g Te m p o r a l H i e r a r c h i e s Analogous to geographical hierarchies we can construct temporal hierarchies, where we consider for a single time series multiple levels of temporal aggregation. The idea is that similarly to cross-sectional hierarchies, we can take advantage of the structure to increase accuracy at short and long term and align forecasts of different horizons. MAPA suggests that we also mitigate modelling uncertainty.

23 I n t r o d u c i n g Te m p o r a l H i e r a r c h i e s The process: 1. Temporally aggregate time series up to the annual level (filter high frequency components) Create a temporal hierarchy 2. At each level fit a forecasting model (it can be the same or different) 3. Reconcile forecasts similar to cross sectional hierarchies optimal combination theory [Athanasopoulos et al., 2009; Hyndman et al., 2011] 4. Use reconciled forecast as final appropriate for all short, medium and long term predictions.

24 A n e x a m p l e Weekly Bi-Weekly Monthly Quarterly Semi-annual Annual Red is the prediction of the base model (ARIMA) Blue is the temporal hierarchy reconciled forecasts (based on ARIMA) Observe how information is `borrowed between temporal levels. Base models for instance provide very poor weekly and annual forecasts

25 F o r e c a s t i n g w i t h Te m p o r a l H i e r a r c h i e s We set aggregation levels k to be a factor of m, the highest sampling frequency per year; e.g. for quarterly series m = 4. We consider all aggregation levels that retain the frequency integer:

26 F o r e c a s t i n g w i t h Te m p o r a l H i e r a r c h i e s Let us collect these in one column vector: Where for our example with m=4:

27 F o r e c a s t i n g w i t h Te m p o r a l H i e r a r c h i e s Let h be the required forecast horizon at the annual level. For each aggregation level k we generate m/k x h base forecasts and stack them the same way as before: From optimal cross-sectional reconciliation we already have that: The only question is how to approximate Σ h. Everything else is known and the reconciled forecast combines information from all levels.

28 F o r e c a s t i n g w i t h Te m p o r a l H i e r a r c h i e s We have proposed two approximations: 1. Variance scaling: where we make use of the one-step ahead forecast error variances: 2. Structural scaling: where we make use of the imposed hierarchical structure and assume approximately uncorrelated errors at the bottom level. We have found that both work very well with similar performance.

29 Consider the previous dataset: E m p i r i c a l E v a l u a t i o n The improvement comes from: Further mitigating model uncertainty combine multiple views of the data

30 D e c i s i o n m a k i n g a n d h i e r a r c h i c a l f o r e c a s t i n g Hierarchical (or grouped) forecasting can improve accuracy, but their true strength lies in the reconciliation of the forecasts aligning forecasts is crucial for decision making. Is the reconciliation achieved useful for decision making? Cross-sectional Reconcile across different items. Units may change at different levels of hierarchy. Suppose an electricity demand hierarchy: lower and higher levels have same units. All levels relevant for decision making. Suppose a supply chain hierarchy. Weekly sales of SKU are useful. Weekly sales of organisation are not! Needed at different time scale. Temporal Reconcile across time units/horizons. Units of items do not change. Consider hospital admissions short and long term are useful for decision making. Suppose a supply chain hierarchy. Weekly sales of SKU is useful for operations. Yearly sales of a single SKU may be useful, but often not! Operational Tactical Strategic forecasts.

31 C r o s s - t e m p o r a l h i e r a r c h i e s Temporal hierarchies permit aligning operational, tactical and strategic planning, while offering accuracy gains useful for decision making BUT there can be cases that strategic level forecasts are not required for each item, but at an aggregate level. Let us consider tourism demand for Australia as an example. Local authorities can make use of detailed forecasts (temporal/spatial) but at a country level weekly forecasts are of limited use. Temporal: tactical strategic Cross-sectional: local country Cross temporal can support decisions at both dimensions: Tactical/local; strategic/local; tactical/country; strategic/country

32 C r o s s - t e m p o r a l h i e r a r c h i e s : T o u r i s m d e m a n d

33 C r o s s - t e m p o r a l h i e r a r c h i e s : T o u r i s m d e m a n d The gains in accuracy are geared towards long term and are more pronounced when modelling uncertainty is higher

34 C o n c l u s i o n s Temporal hierarchies provide a new class of hierarchical forecasts that can be produced for any time series. Applicable to forecasts produced by any means theoretically elegant hierarchical combination of forecasts. Joins operational, tactical and strategic decision making by reconciling forecasts satisfies a business need that has remained unmet Potential to increase forecasting accuracy and mitigate modelling uncertainty Combining cross-sectional and temporal hierarchies: forecasts reconciled across conventional hierarchy and forecast horizons one number forecast superior decision making.

35 R e s o u r c e s R was used for the experiments Forecasting methods: ETS, and `traditional Theta forecast package, available on CRAN Revised theta TStools packages: available in Github. Temporal hierarchies: Code will be released shortly, but MAPA (precursor method) is available on CRAN Cross-section hierarchies: hts package, available on CRAN For code and updates visit my research blog:

36 Thank you for your attention! Questions? Nikolaos Kourentzes Lancaster University Management School Lancaster Centre for Forecasting - Lancaster, LA1 4YX n.kourentzes@lancaster.ac.uk Forecasting blog: Lancaster Centre for Forecasting