Introduction Motivation The model The problem Logistic model Experiment Finale References Forecasting intermittent data with complex patterns Ivan Svetunkov, John Boylan, Patricia Ramos and Jose Manuel Oliveira ISF 2018 19th June 2018 Marketing Analytics and Forecasting Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction In the previous episodes (ISFs)... (Pictures are from ‘Rick and Morty’ by Roiland and Harmon, 2013–2017)
Introduction Motivation The model The problem Logistic model Experiment Finale References 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. CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References 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. CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Motivation Now we want more!
Introduction Motivation The model The problem Logistic model Experiment Finale References Motivation The reason is this: Real time series example 30 20 Demand 10 5 0 0 20 40 60 80 100 120 CMAF Time Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References 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 ... CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Motivation iETS(MMN) 30 25 20 15 10 5 0 0 20 40 60 80 100 120 Series Point forecast ● CMAF Fitted values Forecast origin Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Motivation ...Which underforecasts. Because we deal with the following: iSS, Probability−based 0.8 0.4 0.0 CMAF 0 20 40 60 80 100 120 Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Motivation We need to capture complex patterns in the occurrence part of demand... CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
The model
Introduction Motivation The model The problem Logistic model Experiment Finale References 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. CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References 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 + ǫ t ) l t = l t − 1 (1 + αǫ t ) (2) o t ∼ Bernoulli ( p t ) p t = l p,t − 1 (1 + ǫ p,t ) l p,t = l p,t − 1 (1 + α p ǫ p,t ) CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References 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 + ǫ t ) Demand sizes l t = l t − 1 (1 + αǫ t ) (3) o t ∼ Bernoulli ( p t ) � p t = l p,t − 1 (1 + ǫ p,t ) Demand occurrence l p,t = l p,t − 1 (1 + α p ǫ p,t ) 1 + ǫ t ∼ logN (0 , σ 2 ) , which means that z t is always positive. 1 + ǫ p,t ∼ logN (0 , σ 2 p ) . So far, so good? CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
The problem But there is a tiny problem...
Introduction Motivation The model The problem Logistic model Experiment Finale References 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 + ǫ p,t ) l p,t = l p,t − 1 b p,t − 1 (1 + α p ǫ p,t ) , (4) b p,t = b p,t − 1 (1 + β p ǫ 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... CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Logistic transform The solution - use a different model for p t . If we knew the true p t , then we could use logit transform: � � p t q t = log (5) 1 − p t q t is defined on ( −∞ , ∞ ) . We can use any model for q t . CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Logistic transform For example we can use ETS(A,A,N): q t = l q,t − 1 + b q,t − 1 + ǫ q,t l q,t = l q,t − 1 + b q,t − 1 + α q ǫ q,t , (6) b q,t = b q,t − 1 + β q ǫ q,t where ǫ 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. CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Logistic transform In fact, we don’t need to know either p t or q t , we only need to know ǫ 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 ǫ q,t But it is unobservable, so... CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Logistic model One can only dream... right?
Introduction Motivation The model The problem Logistic model Experiment Finale References Logistic transform. The rise of errors There is a solution... Use the inverse transform if the value ˆ q t is known: exp(ˆ q t ) p t = ˆ (7) 1 + exp(ˆ q t ) Compare the predicted probability ˆ p t with the outcome o t : u t = o t − ˆ (8) p t The problem now is to translate this error into ǫ q,t . CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References Logistic transform. The rise of errors u t lies in ( − 1 , 1) . We transform u t , so that it lies in (0 , 1) : t = 1+ u t u ′ 2 . and then use logit transform to obtain an estimate of error: � 1 + o t − ˆ � p t e q,t = log (9) . 1 − o t + ˆ p t If o t = ˆ p t , then e q,t = 0 (because o t is binary). CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References 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 ) (10) exp( q t ) , p t = 1+exp( q t ) q t ∼ ETS(X,X,X) � � 1+ o t − ˆ p t e q,t = log 1 − o t +ˆ p t where ETS(Y,Y,Y) is a multiplicative ETS, and ETS(X,X,X) is an additive one. CMAF Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References iETS l model How does iETS l work? iETS(MMN) 25 15 5 0 CMAF 0 20 40 60 80 100 120 Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
Introduction Motivation The model The problem Logistic model Experiment Finale References iETS l model The occurrence part of iETS l iSS, Logistic probability 1.0 0.8 0.6 0.4 0.2 0.0 0 20 40 60 80 100 120 CMAF Series Point forecast ● Fitted values Forecast origin Ivan Svetunkov, John Boylan, Patricia Ramosand Jose Manuel Oliveira LCF Forecasting intermittent data with complex patterns
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