Sales Forecasting using Dynamic Bayesian Networks Steve Djajasaputra SNN Nijmegen The Netherlands.

Sales Forecasting using

Dynamic Bayesian Networks

Steve Djajasaputra

SNN Nijmegen

The Netherlands

STW 6/2003 Steve Djajasaputra (SNN Nijmegen) 2

Table of Content

1. Why Sales Forecasting?

2. Method

3. Results & Discussions

4. Conclusions

5. Further Research

6. Acknowledgements


1. Why Sales Forecasting?

• Sales Forecasting bring advantage for your business:– reducing logistic cost– improving your services

• targeted marketing• lower backorder

But in practice… is this really happening?


The Answer is… YES! An Example of Success Story:

• Bayesian statistical technology for predicting newspaper sales

• 1 to 4% more sales with same deliveries • 3 to 12 % less deliveries to achieve same total

amount of sales


But time-series forecasting is not always easy!

So….

• Searching better forecasting technology

• Aggregation of different group of products can be helpful

• Clustering methodology for aggregation

• Bayesian Methodology: generative model


2. Method

• Dynamic Bayesian Networks

• Forecasting

• The Inputs



• Y is our observation– e.g. sales of different products:

beer y1, beer y2,…– X-axis: the time t (e.g.weeks)

t

Y1

Y2

Y3

Y4

Y



• In our model, we assume that our observation Y is generated with this dynamic:

• X are inputs, for example: sales of bier last week, weather information, prices, day labeling

are “hidden variables”, which are unobserved/unknown

is noise N(0,2)



• Hierarchical model:• Our hidden variables depend on

other unobserved/unknown hidden variables M.

• Several from different product share the same M.

–A is a transition matrix for –G is a transition matrix for M is noise N(0, ) is noise N(0, M )



• Inference & Learning:– We have Y & X data in our model– But we don’t know the values of

hidden variables: , M and their initial values

– We also don’t know the correct value of parameters: , M ,A,G and their initial values

– We solve these problems in Bayesian paradigm, using EM Algorithm.


Forecasting

Steps:• Training step: find the model

parameters and hidden variables 1:T given the data from observation window X1:T,Y1:T, using EM algorithm and Kalman smoothing.

• Forecasting step: predict T+h and YT+h

h is the horizon of forecasting• Updating step: update the hidden

variables 1:T+h given the real value YT+h

• Repeat the forecasting & updating steps above in iterations.

t

Y O b se rved Y

Fo re c a ste d Y

O b se rva tio n wind o w fo r tra ining ste p

t= T t= T+ h


The Inputs

• By Autocorrelation & FFT Spectrum analysis, for inputs (Xi,t) I decided to use: – Seasonality markers– Recent sales (1 week ago)– Last month sales (4 weeks ago)

• We need to keep the number of inputs as small as possible to avoid over-fitting.

• Since I consider seasonality & recent sales, my model is somewhat comparable with “SC” model which is used by Pim Ouwehand.


3. Results & Discussions

• An Example of Result• Residual Analysis• Nonlinear Transformation• The Offset Problem• Removing Outliers• Our Bayesian Approach vs Conventional

Econometric Methods• Need More Informative Inputs


An Example of Result

• Mean Absolute Deviation (MAD) is 2346 beers

Y-axis:O is the real valueX is the prediction

X-axis: weeksTraining steps: week 5..204Forecasting steps: week 205..260, 1 week horizon

• This result is about the range of Winter method used by Pim Ouwehand.

MAD=timetotalproducttotal

YYpredictediproductttime

titi

_*__ _

,,


Residual Analysis

• To validate our model.• It’s showed that the residues

(error) are noise as we assumed.– Y predicted vs Error (figure on top)– Error vs time (figure on bottom)– Autocorrelation and FFT of Error– Cross correlation Error vs Inputs


Nonlinear Transformation

• To make data more linear & gaussian since we assume our model is linear and the data is assumed to be gaussian distributed.

e.g. Log, Sigmoid


The Offset Problem• Due to the stationary assumption, the software gives

over(under)estimated forecasting if the trend is exist.• Solutions:

– Removing trend (e.g. taking difference)– Updating the parameters after forecasting step.

Legend:Left: moving averaged Beer-2 vs weeksRight: Predicted Beer-2 vs weeks


Removing Outliers

• The plot shows that the data is very noisy.

• Most of the outliers are below the mean, perhaps due to “out of stock” problem. Thus it will be helpful if we can get “out of stock” label for input in our forecasting model.

Sales of 10 beers (normalized) vs weeks


Our Bayesian Approach vs Conventional Econometric methods

• Econometric regression methods (e.g. Winter Method used by Pim Ouwehand) works well to fit the data.

t

Y 1Y


• However, we don’t want just do fitting the data. We want to understand the process behind the data that we observed (i.e. hidden/unobserved variables).

• We want to have a generative model of the beer buyers.

• This generative model helps you to understand the “hidden process” in the market. This is a valuable insight for business decision, e.g. by simulation.


We need More Informative Inputs

Pric e ? D isc o unt/Bo nus?

Ad ve rtise m e nt?

Ava ila b ility (vs o ut o f sto c k) ?N ic e we a the r?

Ho t we a the r?Thirsty?

G o ing o ut?

Whic h b e e r I b o ug ht ye ste d a y?

Be e r Buye rs M o d e l


4. Conclusions

• This preliminary research (only with the sales data without other informative inputs) showed that the result is about in the range of Winter method.

• We need more informative input data for a better model.

• Hacking data (e.g. removing trend, nonlinear transformation) slightly improves the result. But this is not the main purpose of this research.


Conclusions… continued

• We are not only just fitting the data but constructing a generative model, which is useful for understanding business process behind the sales.

• This understanding help you to shape your strategy to achieve more profit.


5. Further Research

• Clustering and Structural Learning• Non stationary process• Non linear model• Approximations

– Variational– Factorial– Monte Carlo (MCMC)


6. Acknowledgements

• Our sponsor: STW

• Tom Heskes (KUN)

• Pim Ouwehand (TUE)

• Bart Bakker (Phillips, was in KUN)

• Data providers/ Business Partners : Schuitema, Technie Unie, OPG.


Appendix: Clustering Insights

• On Observed data Y


• On hidden variables:– 1,2:seasonality– 3:last month– 4:last week

Sales Forecasting using Dynamic Bayesian Networks Steve Djajasaputra SNN Nijmegen The Netherlands.

Documents

sales slide

sales forecasting

forecasting model

netherlands slide

sigmoid slide

generative model slide

horizon of forecasting

overunderestimated forecasting