Introduction Intermittent state-space model Model selection Experiments Finale References One for all: forecasting intermittent and non-intermittent demand using one model Ivan Svetunkov and John Boylan Presentation at University of Bath 30th March 2017 Lancaster Centre for Forecasting Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction Typical forecasting task in supply chain is to produce forecasts for many products. Demand on each of the products may have its own characteristics and in general can be: • non-intermittent; • intermittent. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction 12 10 8 Demand 6 4 2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Time Figure: Example of intermittent data. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction ETS(ANN) 10 8 6 4 2 0 1980.0 1980.5 1981.0 1981.5 1982.0 1982.5 Series Point forecast Fitted values Forecast origin Figure: Simple Exponential Smoothing applied to the intermittent data. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction Intermittent data is considered as a separate case. It is identified and then forecasted, usually using Croston (1972): y t = 1 ˆ q t ˆ z t ˆ , (1) z t = α z z t − 1 + (1 − α z )ˆ ˆ z t − 1 q t = α q q t − 1 + (1 − α q )ˆ ˆ q t − 1 where z t are the demand sizes, q t are the demand intervals, α z and α q are the smoothing parameters. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction 12 10 8 6 4 2 0 5 10 15 20 25 30 Period Figure: Intermittent data and Croston’s forecast. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction We also have SBA (Syntetos and Boylan, 2005), TSB (Teunter et al., 2011), HES (Prestwich et al., 2014), INARMA etc. All of them are separated from ETS / ARIMA / regression / etc. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction ETS(MAN) 10000 9500 9000 8500 8000 7500 7000 1980 1982 1984 1986 1988 1990 1992 1994 Series Point forecast Fitted values Forecast origin Figure: Non-intermittent data and a forecast. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction How to categorise the data? Johnston and Boylan (1996), Syntetos et al. (2005), Petropoulos and Kourentzes (2015) BUT! Products can change their characteristics... LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction Demand on a fast moving product may become obsolete... 800 600 Demand 400 200 0 2 4 6 8 10 Time LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Introduction ...or demand is just building up. 300 250 200 Demand 150 100 50 0 1 2 3 4 5 6 7 8 Time LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Problems • Products can change their characteristics; • Croston / TSB are based on SES. Overall: 1. We need a model that could switch between intermittent / non-intermittent regimes; 2. We may need trend and / or seasonality; 3. We need to apply that model to a wide variety of data. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Intermittent state-space model (iSS)
Introduction Intermittent state-space model Model selection Experiments Finale References Intermittent state-space model The model is based on the original idea of Croston (1972): y t = o t z t , (2) where o t ∼ Bernoulli ( p t ) and z t is a statistical model of our choice. z t can be ETS, ARIMA, regression, diffusion model, etc. o t = 1 means that there is a sale. o t = 0 means no sale today. If o t = 1 , for all t , then this is non-intermittent model. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References General state-space (based on Hyndman et al. (2008)) General state-space model for iETS: y t = o t ( w ( v t − 1 ) + r ( v t − 1 , ǫ t )) (3) . v t = f ( v t − 1 ) + g ( v t − 1 , ǫ t ) v t is the vector of states, w is the measurement function, f is the transition function, g is the persistence function, where o t ∼ Bernoulli ( p t ) and ǫ t is the error term. LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Intermittent state-space model Multiplicative model is preferred (paper submitted to IJF). Example. iETS(M,N,N) with time varying probability: y t = o t z t z t = l t − 1 (1 + ǫ t ) (4) , l t = l t − 1 (1 + αǫ t ) 1 + ǫ t ∼ logN (0 , σ 2 ) , which means that z t is always positive. States are updated on every observation (potential demand). But sales happen only when o t = 1 . Underlies SES, when o t = 1 . LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References How to model the probability? p t has a statistical model of its own. So far we have developed three models for p t : • Fixed probability model; p t = p for all t . • Croston’s model; 1 p t = 1+ q t , where q t is ETS(M,N,N). • TSB model. p t ∼ Beta ( a t , b t ) , where a t and b t are ETS(M,N,N). LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Examples iETS(M,N,N) with fixed probability... iETS(MNN) 80 60 40 20 0 2 4 6 8 10 Series Point forecast Forecast origin Fitted values 95% prediction interval LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Examples Fixed probability. iSS, Fixed probability 1.0 0.8 0.6 0.4 0.2 0.0 2 4 6 8 10 Series Point forecast Fitted values Forecast origin LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Examples iETS(M,Md,N) with TSB and demand becoming obsolete iETS(MMdN) 800 600 400 200 0 2 4 6 8 10 12 Series Point forecast Forecast origin Fitted values 95% prediction interval LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Examples Time varying probability, TSB style. iSS, TSB 1.0 0.8 0.6 0.4 0.2 0.0 2 4 6 8 10 12 Series Point forecast Fitted values Forecast origin LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
Introduction Intermittent state-space model Model selection Experiments Finale References Examples iETS(M,N,N) with Croston and building up level of demand... iETS(MNN) 600 500 400 300 200 100 0 2 4 6 8 Series Point forecast Forecast origin Fitted values 95% prediction interval LC F Ivan Svetunkov and John Boylan LCF One for all: forecasting intermittent and non-intermittent demand using one model
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