Recursive Chain Forecasting: A Hybrid Time Series Model | Blueocean MI

www.blueoceanmi.com

Bishal Neogi, Senior Data ScientistSoumajyoti Mazumder, Data ScientistReetika Choudhary, Data Scientistwith inputs and guidance from Eron Kar, AVP, Advanced Analytics

Recursive Chain Forecasting:

A Hybrid Time Series Model

Trend/Comparative Analysis for Product A and Product B

Seasonality Analysis and Normalization

Model Validation Forecasting Analysis

ABSTRACT

A technology major had a few pre-existing brands in the market which they wanted to replace with a single easy-to-carry, less space-occupying product across the major U.S. electronic/technology retailers for all the states. In due course of the analysis, we came across the parent product (Product A), which had 3 different pre-existing variants. The company plans to replace all those with a single branded product (Product B). Our analysis methodology assesses the time-series sales data for a stipulated time period, with all possible conventional tool kits. What really cuts through is our hybrid methodology, which magnifies the model precision over and above the standalone known techniques.

OBJECTIVE

• To understand the sales trend of Product A and Product B over a period of time• To determine the effect of seasonality which acts as a key influencer of Product A (and consequently

Product B)• Data harmonization for reducing seasonal impact and carrying out an unbiased analysis over a period of

time• To study the effect on a cross-sell unit volume of a product due to the change in sales volume of Product A

and/or Product B• To develop a final time series model to establish the unit sales volume for Product A and B

DATA

Our data set comprises of a product and its sub-product referred to as Product A and Product B respectively throughout the analysis. We have considered bivariate monthly series for our analysis, starting from July 2010 til March 2013.

As we can see in the figure 1 above, both Product A and Product B sales are following a similar pattern throughout the observed time period.

CHOICE OF MODEL

For monthly data, an additive model incorporates an underlying assumption that the difference between the values of any two distinct months will approximately remain same each year. In other words, the amplitude of

Comparison of Product A and Product B sales data

0 5 10 15 20 25 30

0

1000

00

2000

00

300

000

Prod

uct A

(Bla

ck) a

nd P

rodu

ct B

(Red

)

Time

TABLE 1Variable

Product A

Product B

Description

Sales

Sales

Figure 1: Data Plot of Product A and Product B Sales

2

the seasonal effect will be same every year. Based on this assumption, the graphical results and the logical understanding of the extreme possible hypothetical situation where the sales can reach a point of zero, we have agreed on to make use of the Additive model.

For a clear picture, consider the two graphs below (Figure 2). On observation, you will notice that the Additive model best fits the given data. Hence, we conclude that choice of the Additive model is the most appropriate of all.

Figure 2 – Multiplicative and Additive Model Head-to-Head

APPROACH 1:

DECOMPOSITION

To proceed, we first decompose the data into its respective components, including trend (Tt), seasonality (St) and irregularity (It). The equation below gives a concise view of our analytical modelling technique, and the graphs further substantiate it with pictorial representation.

Data = Tt + St + It

Figure 3.1: Decomposition for Product A Sales Figure 3.2: Decomposition for Product B

Additive model implemental Multiplicative model implemental

5 10 15 20 2011.0 2012.0 2013.0

1000

00

1400

00

1800

00

1000

00

1400

00

1800

00

Prod

uct B

Dec

ompo

sed

Addi

tve

Index Time5 10 15 20 2011.0 2012.0 2013.0

1000

00

1400

00

1800

00

-4e+

13

0

e+00

2e

+13

4

e+13

Prod

uct B

Dec

ompo

sed

Addi

tve

Index Time

Decomposition Trend AnalysisSeasonalityIndexing and Normalization

StationarityCheck and Granger Causality

Regression

2010.5 2011.0 2011.5 2012.0 2012.5 2013.0

Time

=300

0

0 2

000-

2000

0 4

0000

16

5000

18

0000

15

0000

2

5000

0

ranb

o s

easo

nal

tren

d o

bser

ved

2010.5 2011.0 2011.5 2012.0 2012.5 2013.0

Time

=300

0

0 2

000-

2000

0 4

0000

16

5000

18

0000

15

0000

2

5000

0

ranb

o s

easo

nal

tren

d o

bser

ved

Decomposition of additive time series Decomposition of additive time series

3

80000 100000 120000 140000 160000 180000

1500

00

200

000

25

0000

Prod

uct A

Sal

es

Product B Sales

Trend Plot of A and Product B Sales

2010.5 2011.0 2011.5 2012.0 2012.5 2013.0

-200

00

200

00

600

00

Product A seasonal Component plot

Prod

uct A

$ s

easo

nal

Time

Plot of Adjusted Seasonal for Product A

Prod

uct A

$ s

easo

nal

Time2010.5 2011.0 2011.5 2012.0 2012.5 2013.0

-200

00

200

00

600

00

4

On observing the graphs above (Figure 3.1 and Figure 3.2), we notice that the sales trend of Product A & Product B up to 2012, has a linear increasing pattern, post which, their behaviour becomes constant. However, as for the seasonality, both the products portray almost the same behaviour.

TREND

Next, we plot the data of both the products together to capture permanent changes in the given time period. The trend plot in Figure 4 helps us look for an overall pattern hidden in the data and in the long run, even helps us forecast future values.

Figure 4: Product A and Product B Trend Plot

SEASONALITY IDENTIFICATION AND NORMALIZATION

Further, the periodic movement depicted by the seasonality index helps identify the general monthly behaviour masked in the data set (Figure 5). Considering a period of 12 months, we use the method of “Seasonal means” to remove seasonality from data (Figure 6).

Figure 5: Product A Sales Seasonality index Figure 6: Product A: Adjusted for Seasonality

STATIONARITY CHECK

The stationarity check tells us that both Product A and Product B are non-stationary in nature, up to a legitimate level. Hence, to make the series stationary, we do the 2nd order differencing. The referred test gives us a fair-enough evidence to claim that the seasonally adjusted and twice-differenced data is stationary at 5% level of significance.

After the removal of Non-stationarity in both Product A and Product B, we fit them into a linear regression framework.

MARS on complete data Stationarity check Fitting MARS on the

normalized data

-1e+05 -5e+04 0e+00 5e+04 1e+05

-1e+

05

0e+0

0

1e+0

5

∇2 Product B

∇2 Pro

duct

A

5

CAUSALITY

Since we are modelling a bivariate time series data, we have used Granger Causality test to check the causality dependence between the variables. The test gives us following insights:• On checking causality between Product A and Product B, we see that Product A is granger causal to

Product B up to order 3. The results stand valid up to 47% level of significance.• However, the point to be noted here is that for the above statement, the reverse does not hold true. On

checking causality between Product B and Product A, we see that Product B does not granger cause Product A at 5% level of significance.

LINEAR REGRESSION

To see if our model best fits the linear regression framework, we plot the adjusted data and obtain the below mentioned graph in Figure 7:

Figure 7: Linear Regression ∇2 Product A and ∇2 ProductB

At a 0.93 level of significance, we conclude that the data best fits the linear model and hence the derived model can be as represented below:

∇2 Product A= α+β* ∇2 ProductB

APPROACH 2:

MULTIPLE ADAPTIVE REGRESSION SPLINES (MARS)

First we use MARS on the raw data (Figure 8) where the results share an R-square of 70%, which is quite good but surely not better than the LM fit. Next we improve on data by stationarizing it and then implementing MARS on it (Figure 8). On doing this, we derive a better fit, with an accuracy level of almost 95%.

The model that we derive on using MARS as our methodology is as given below:

∇1 Product A= α1-β1*h(C1-∇1 Product B)+γ1*h(∇1 Product B-C2)

0 5 10 15 20 25 30

1500

00

PRO

DU

CT

A

2000

0025

0000

Plot of original data and MARS upon it

INDEX

0 5 10 15 20 25 30

-1e+

05

∇^1

Prod

_A d

ata

0e+0

01e

+05

Plot of stationarized data and MARS upon it

INDEX

-100000

-200

000

2000

0

0 50000 150000

Predicted Values

Resid

ual V

alue

s

-200

0020

000

0

-2 -1 0 1 2

Sim

ple

Qua

ntile

s

Theoretical Quantiles

6

Figure 8: MARS Implementation

CHOICE OF FINAL MODEL

It can clearly be observed that MARS gives us better results but simultaneously introduces more complexity into the process. Had the improvement in the result been significantly large, usage of MARS was justified but in our case, since the improvement is just by 1% when compared to the LM fit, MARS cannot be claimed to be an appropriate method. Hence, we conclude that approach 1 i.e. LM is a better choice when weighed on the cost-complexity-efficiency criteria. Thus our final model is:

∇2 Product A= α+β*∇2 Product B

MODEL VALIDATION

To validate our model selection process, we proceed to model validation. One of the reasons why model validation is necessary is that the high R-square is not sufficient to conclude the goodness of fit. Hence, we conduct few tests like Residual Analysis and plot the normal Q-Q plot (Figure 10) to analyze the goodness of fit of the regression. We check if the residuals are random and if the model's predictive performance weakens substantially when we infuse new variables into the estimation process. Apart from this, we even adopt standard methods to detect the various outliers among residuals.

If you observe the residual plot below (Figure 9), you would see that the residuals are behaving in a random fashion along the line y=0 and this suggests that our model fits the data well. Further, autocorrelation tests have yielded 87% proof that no auto correlation exists between residual and time.

Figure 9: Residual Plot Figure 10: Normal Q-Q plot

FORECASTINGUsing our final model as mentioned below, we now move ahead with forecasting sales for product A assuming that the sales for product B is a given, and that the seasonal effect is known to us.

∇2 Product A= α+β*∇2 Product B

we fit ARIMA on the residuals, to obtain the forecasted value.For example, if we have to forecast for t=n+2,period to be T (say) and the known value of Product B is X1, then we will use the below mentioned equation:

Product An+2 =α+β(Product Bn+2 - 2Product Bn+1+Product Bn )+2Product An+1 - Product An+Product A seasonal component for the period T + εn+2

We have named the above-built hybrid methodology as “Recursive Chain Forecast.” One limitation with this method is that it cannot be used for point estimation as the chain can simply not be broken. Hence, it is necessary to conduct all the previous forecast calculations beforehand.

TECHNICAL ASPECTS & ANOMALIES

If we perform a linear regression analysis on the basic variables (i.e. Product A regressed over Product B) we obtain quite standard R2; a deep dive into the data, evidently shows signs of non-stationarity. A resultant spurious relationship between the variables may be observed. To overcome the spuriousness, we find both the series to be I(2) series and execute Cointegration.

Furthermore, our initial regression analysis showed a Durbin-Watson (DW) statistic of 1.0491<1.37=dL , on

account of which we reject the hypothesis that the series is not positively auto-correlated, which we eliminated after stationarizing both the basic variables & obtain DW= 2.2614>1.501=d

U. We infer hence that

there is no positive autocorrelation in the latter.

Post cointegrating from I(2) to I(0), we run linear regression for our final model. The underlying anomaly here is that, the DW statistic for the final model shows a very little amount of negative autocorrelation if both the series are cointegrated.

KEY BUSINESS INSIGHTS

Seasonality analysis for Product A and Product B:• Seasonality affects most of the commodity sale patterns within a given geography in a uniform way.

Forecasting Model for Product A Unit Sales:• It includes how Product A sales affect Product B unit sales in an aggregative manner in the future.

Product A – Product B association:• Granger Causality, Recursive Chain Forecasting, and Residual Analysis inferentially determine that

Product B drives Product A sales.• It is evident that in some periods or quarters Product A & Product B behave quite alike.

7

www.facebook.com/blueoceanmi www.twitter.com/blueoceanmi

© Copyright 2013 blueocean market intelligence, All Rights Reserved.

For more information, visit www.blueoceanmi.com.

4835 E Cactus Road, #300Scottsdale, AZ 85254 USAp: 602.441.2474

11280 Northup Way, #E200Bellevue, WA 98005 USA

Vrindavan Tech VillageBuilding 2A, Ground Floor East TowerSarjapur Outer Ring RoadBangalore, 560 037 INDIAp: 91.80.41785800

Contact:

DURJOY PATRANABISHSenior Vice President - Analyticse: [email protected]