Forecasting intermittent data with complex patterns

Transcription

1 Forecasting intermittent data with complex patterns Ivan Svetunkov, John Boylan, Patricia Ramos and Jose Manuel Oliveira ISF th June 2018 Marketing Analytics and Forecasting

2 Introduction In the previous episodes (ISFs)... (Pictures are from Rick and Morty by Roiland and Harmon, )

3 Introduction Svetunkov and Boylan (2017) proposed an intermittent multiplicative state-space model. We showed that this model underlies Croston (1972) and Teunter et al. (2011) methods. We extended that model, presenting at ISF2017 the idea of using ETS(M,N,N), ETS(M,M,N) or ETS(M,Md,N) for demand sizes.

4 Introduction We showed how to select between different ETS models in this context. The approach worked well on a WF wholesale data from Johnston et al. (1999). The conclusion was: you can use one model for both intermittent and non-intermittent data.

5 Motivation Now we want more!

6 Motivation The reason is this: Real time series example Demand Time

7 Motivation How do you deal with this type of data? There is a trend, but the demand is intermittent. We can predict the increase in demand sizes with iets(m,m,n) p...

8 Motivation iets(mmn) Series Fitted values Point forecast Forecast origin

9 Motivation...Which underforecasts. Because we deal with the following: iss, Probability based

10 Motivation We need to capture complex patterns in the occurrence part of demand...

11 The model

12 iets model y t = o t z t, (1) where o t Bernoulli(p t ), z t is a statistical model of our choice and p t is another statistical model.

13 Intermittent state-space model Example. iets(m,n,n) p (with probability-based occurrence): y t = o t z t z t = l t 1 (1 + ɛ t ) l t = l t 1 (1 + αɛ t ) o t Bernoulli(p t ) p t = l p,t 1 (1 + ɛ p,t ) l p,t = l p,t 1 (1 + α p ɛ p,t ) (2)

14 Intermittent state-space model Example. iets(m,n,n) p (with probability-based occurrence): y t = o t z t } z t = l t 1 (1 + ɛ t ) Demand sizes l t = l t 1 (1 + αɛ t ) o t Bernoulli(p t ) } p t = l p,t 1 (1 + ɛ p,t ) Demand occurrence l p,t = l p,t 1 (1 + α p ɛ p,t ) (3) 1 + ɛ t logn(0, σ 2 ), which means that z t is always positive. 1 + ɛ p,t logn(0, σ 2 p). So far, so good?

15 The problem But there is a tiny problem...

16 The problem The problems appear, when we start introducing additional components and variables in the model of p t : p t = l p,t 1 b p,t 1 (1 + ɛ p,t ) l p,t = l p,t 1 b p,t 1 (1 + α p ɛ p,t ), (4) b p,t = b p,t 1 (1 + β p ɛ p,t ) p t should be in [0, 1] But if trend is positive, p t might become greater than one. Cutting off values is inhumane...

17 Logistic transform The solution - use a different model for p t. If we knew the true p t, then we could use logit transform: ( ) pt q t = log 1 p t (5) q t is defined on (, ). We can use any model for q t.

18 Logistic transform For example we can use ETS(A,A,N): q t = l q,t 1 + b q,t 1 + ɛ q,t l q,t = l q,t 1 + b q,t 1 + α q ɛ q,t, (6) b q,t = b q,t 1 + β q ɛ q,t where ɛ q,t N (0, σ 2 q). We can extend this model with exogenous variables or seasonal components. This would mean that in some cases the probability of occurrence increases / decreases.

19 Logistic transform In fact, we don t need to know either p t or q t, we only need to know ɛ q,t, initial values of l q,0 and b q,0, and smoothing parameters values. The latter four can be estimated... IF we have ɛ q,t But it is unobservable, so...

20 Logistic model One can only dream... right?

21 Logistic transform. The rise of errors There is a solution... Use the inverse transform if the value ˆq t is known: ˆp t = exp(ˆq t) 1 + exp(ˆq t ) (7) Compare the predicted probability ˆp t with the outcome o t : u t = o t ˆp t (8) The problem now is to translate this error into ɛ q,t.

22 Logistic transform. The rise of errors u t lies in ( 1, 1). We transform u t, so that it lies in (0, 1): u t = 1+ut 2. and then use logit transform to obtain an estimate of error: ( ) 1 + ot ˆp t e q,t = log. (9) 1 o t + ˆp t If o t = ˆp t, then e q,t = 0 (because o t is binary).

23 iets l model So the final model is: y t = o t z t z t ETS(Y,Y,Y) o t Bernoulli(p t ) exp(qt) 1+exp(q t) p t = q t ETS(X,X,X) ( ) e q,t = log 1+ot ˆp t 1 o t+ˆp t, (10) where ETS(Y,Y,Y) is a multiplicative ETS, and ETS(X,X,X) is an additive one.

24 iets l model How does iets l work? iets(mmn)

25 iets l model The occurrence part of iets l iss, Logistic probability Series Point forecast Fitted values Forecast origin

26 iets l model So now we have: an extendable model......that captures complex patterns for demand sizes......and demand occurrence part.

27 Experiment Yes! Now it is time for experiments...

28 Experiment Intermittent data of a Portuguese retailer SKUs with at least 4 non-zero demands each. Weekly data, 173 observations. h = {1, 2, 3, 4}. Rolling with 52 origins. smse, sme, sapis (Petropoulos and Kourentzes, 2015).

29 Experiment Croston, TSB and imapa from tsintermittent package es function from smooth package with: ETS(A,N,N), iets(m,y,n) f - fixed probability, selection of trend, iets(m,y,n) i - Croston style, iets(m,y,n) p - TSB style, iets(m,n,n) l (A,N,N) - logistic with ETS(A,N,N) for occurrence, iets(m,y,n) l (A,N,N) - similar + selection of trend, iets(m,y,n) l (A,X,N) - similar + selection for occurrence.

30 Results Model sme smse sapis iets(myn) l (AXN) iets(mnn) l (ANN) iets(myn) l (ANN) iets(mnn) p ETS(AAN) iets(mnn) i imapa iets(mnn) f TSB Croston

31 Results Based on the iets l : Demand occurrence Demand sizes No trend Trend Overall No trend 31.9% 35.7% 67.6% Trend 19.0% 13.4% 32.4% Overall 50.9% 49.1% 100%

32 Conclusions

33 Conclusions We now have a more general modelling framework; We have a new type of model for intermittent data; We can capture complex patterns in intermittent data; The approach seems to work well in practice.

34 What s next? More thorough analysis of results driven by the data; Go multivariate vector intermittent models: 1. Extend the iets l to ives l ; 2. Group time series based on the characteristics; 3. Capture seasonality across similar time series; 4. Introduce exogenous variables; 5. Forecast groups of time series. In the end we should have a universal time series forecasting approach...

35 And that s how it s done!

36 Thank you for your attention! Ivan Svetunkov Marketing Analytics and Forecasting

37 Croston, J. D., sep Forecasting and Stock Control for Intermittent Demands. Operational Research Quarterly ( ) 23 (3), 289. URL Johnston, F. R., Boylan, J. E., Meadows, M., Shale, E., dec Some Properties of a Simple Moving Average when Applied to Forecasting a Time Series. The Journal of the Operational Research Society 50 (12), URL // Petropoulos, F., Kourentzes, N., jun Forecast combinations for intermittent demand. Journal of the Operational Research Society 66 (6), URL 62{%}7B{%}25{%}7D5Cn /jors http:

38 //link.springer.com/ /jors Roiland, J., Harmon, D., Rick and morty. Adult Swim. Svetunkov, I., Boylan, J. E., Multiplicative State-Space Models for Intermittent Time Series. Teunter, R. H., Syntetos, A. A., Babai, M. Z., nov Intermittent demand: Linking forecasting to inventory obsolescence. European Journal of Operational Research 214 (3), URL S