NEURAL NETWORK AND TIME SERIES AS TOOLS FOR

SALES FORECASTING

Maria Emilia Camargo, Walter Priesnitz Filho, Marcelo Mahado Barbosa Pinto

University of Caxias do Sul, CAMVA, Vacaria, RS, Brasil

Angela Isabel dos Santos

Department of Statistical, University of Santa Maria, RS, Brasil

Keywords: Neural Network, Time series

Abstract: This paper presents the use of times series AutoRegressive Integrated Moving Average ARIMA(p,d,q)

model with interventions, and neural network back-propagation model in analyzing the behavior of sales in

a medium size enterprise located in Rio Grande do Sul Brazil for the period January 1984 – December 2000.

The forecasts obtained using the neural network back-propagation model were found to be more accurate

than those of ARIMA model with interventions.

1 INTRODUCTION

In various areas of research, the models obtained

using neural networks are found to be better

compared to other ways of modeling. Lapedes and

Farber (1987), published an article where they

trained the network to generate a line series using a

specific equation. They obtained agreeable results

which provided accurate forecasting. In this paper,

the sales data collected from a medium size

enterprise located in Rio Grande do Sul, Brazil for

the period January 1984 – December 2000 have

been analyzed using the time series ARIMA model

with interventions and the back – propagation neural

network model.

Initially the exploratory analysis, autocorrelation

and parcial autocorrelation functions have been used

to analyze the time series sales data to verify the

existence of seasonal components, non - stationarity,

and the randomness in the data. The number of

intermediate layers for the back-propagation model

has been determined by trial and error approach.

Error analysis of the time series has been carried out

accurate forecasting. The results obtained are

presents and discussed.

2 METHODOLOGIES

2.1 AutoRegressive Integrated

Moving Average (ARIMA) Model

with Interventions

The intervention analysis is a transfer function

stochastic model from which it is possible to

interpret and incorporate its effects to time series

model (Box and Jenkins, 1970). The major effects of

intervention can be noted observing the changes in

inclination level of time series. Based on these

changes, the error variables can be altered and

components that were not present in the model can

be introduced. Let a time series for which you have

verified and estimated an ARIMA model with you

were making forecasts for certain time. At a given

instant, an independent phenomenon occurred that

originated a time series, but whose effects can be

manifested on the time series. This external event,

whose effects will influence the time series being

studied, should be incorporated in the model as an

additional. This addition is called intervention (Box

& Tiao, 1973). The intervention model can be

represented by:

476

Emilia Camargo M., Priesnitz Filho W., Mahado Barbosa Pinto M. and Isabel dos Santos A. (2004).

NEURAL NETWORK AND TIME SERIES AS TOOLS FOR SALES FORECASTING.

In Proceedings of the Sixth International Conference on Enterprise Information Systems, pages 476-478

DOI: 10.5220/0002648304760478

 SciTePress

ttj

(B)XVt)x,f(k, +

∑

(1)

ttj

(B)X

(B)

t)x,f(k, +

∑

(2)

where: X

j=1.2, ........ k are exogenous variables

(interventions). Eventually.

−

can be used where

b is the time space occurred between the input and

the output series (discrepancy between the time

series); k is a set of unknown parameters which

appears in v

(B) or in ω

(B) and σ

(B); where: v

(B)

= ω

(B)/σ

(B), j=1,2,........ k is the transfer function

of j-th exogenous variable, such that V

(B), ω

(B)

and σ

(B)are polynomials in B and

η is the noise

that can be represented by an ARIMA model with

seasonal components. Each series

X is an

indicator that assumes the values 0 or 1,

representing, respectively, the absence or the

presence of j-th intervention , that is, the non-

occurrence or the occurrence of j-th event. The time

series indicators of the intervention can be

represented by three types of binary variables:

Impulse function (type 1); step function (type 2);

and impulse seasonal function (type 3).

2.2 Neural Network

The back-propagation model is a paradigm

commonly used in the areas of signal recognition,

and principally in forecasting of time series ( Beale

and Jackson, 1991). The back-propagation model

uses a topology of 3 or more layears. The

connections between the units are interlayers that are

directed from the input layer to the output layer. The

neural network adjusts itself as a time series

forecasting model in the following form:

1. The input units are made up of information

relevant to the forecasting:

2. The weights are the model parameters and be

estimated through learning by the network that takes

into account the pair of inputs to the respective

output targets (real values of the time series);

3. The hidden layers are the links between the input

and output layers. Its role is important for the

selection of better sets of weights, since the non-

linearity of the model can be located in the

activating function of the hidden units;

4. The output layer is made up of only one unit and

carries the needed information for forecasting;

5. The network is trained for estimating the model

parameters. After training, forecasting for the

periods 1,2... etc.. in the can be generated on the

output layer;

6. The hyper parameters are the values provided by

the user that in general are constants. They are: rate

of learning, momentum term, and the varying

interval size of weights.

3 EXPERIMENTAL RESULTS

3.1 ARIMA Model with Intervention

The adjusted model is with interventions ARIMA

(1,1,3)x(0,1,0) The estimated parameters and

statistics of the model are prented in Table1.

Table 1: Estimated parameters and statistics “t” for the

univariate model with interventions for the time series

sales data

Parameter Estimated Value Statistic”t”

-0.2553 -2.02

-0.3987 -3.38

-0.1256 -3.78

-0.3765 -4.10

0.1857 3.97

0.1074 2.18

0.6876 1.99

The adjusted statistic and noise statistics for the

model are:

R² = 0.9647: Mean ≅ 0.0000; and

Variance = 0.00138

The identified interventions for the sales data

during the period analyzed are presented in Table 2

considering a level of significance of 5%.

Table 2: Types of detected interventions

Type of intervention Instant Period

→ 1

99 MAR/87

→ 1

131 NOV/89

→ 2

123 MAR/89

→ 3

115, 127 JUN/88

→ 2

108 DEC/87

NEURAL NETWORK AND TIME SERIES AS TOOLS FOR SALES FORECASTING

477

The types of interventions occurred are: impulse,

step, and seasonal impulse. It observed that the

estimated coefficients of the intervention variables

have their expected signals. That is ξ

and ξ

have negative signals, when

and ξ

have

positive signals:

1. The first interventions represents the reflections

due to Cruzado Plan that imposed freezing of prices,

which was in vigor from March to November of

1986:

2. The intervention of November 1989, is the

reflection of heterodox shock of Summer Plan.

3. The intervention occurred in March 1989 is due to

the price increase in consequence to inflationary

memory;

4. The increase in sales in June 1988 is characterized

by the seasonal effect, since every year starting from

1988 there has been increase in sales (more

emphasize in the year 1988); and a harmonic

observed with reference to the month June is highly

significant.

5. Finally X

represents the intervention due to

Bresser Plan a new tentative of freezing the prices,

this time for a very short period of time, from July to

October 1987.

3.2 Neural Network

Architecture: The chosen architecture (after testing

various architectures by evaluating.

1. 14units in the input layer, in the following form: 2

past lags: X

and X

t-1

; 12 seasonal units.

2. two units in the hidden layer;

3. one unit in the output layer X

t-1

Training

: The sales series was trained 1400 times,

updating the weight for every 30 repetitions. The

learning constant was maintained at 0.12 and in the

last 300 repetitions, a memory loss term of 0.4 was

used. This term was used to provide more weight for

the most recent observations. The momentum term

used was 0.7. The varying interval size of the weight

was 4.

3.3 Comparison of forecasts

The performance of two approaches for sales we

calculate the Absolute Percentage Error (MAPE).

The two methods (ARIMA and Neural Network)

provide the following MAPE: 7.63% and 5.38%,

respectively.

The results show that the neural network model

adjust well to the sales data is the preferred model

for forecast on the basis of MAPE.

4 CONCLUSION

We presented in this paper two approaches for the

study the sales data collected from medium size

entriprise located in Rio Grande do Sul, Brazil for

periodo January 1984 – December 2000.

The ARIMA model interventions presented a

residual variation of 0.0014 where as the neural

network model presented a residual variation of

0.0001. The MAPE for the neural network model

was 5.38% and for the ARIMA model with

interventions was 7.63%. The sales time series

presented a marked seasonality for which it was

necessary to use 12 binary units (0 or 1) for

determining the relative weight for weight for each

month. The results show that the neural network

model is the preferred model for forecast on the

basis of MAPE

The model obtained by the neural network was

superior to ARIMA model, in adjustment as well as

in forecasting for the data analyzed.

REFERENCES

Beale, R. and T. Jackson, T., 1991, Neural Computing –

an introduction. Adam Higler Publ.

Box, G.E.P. and Jenkins, G.M., 1970, Time Series

Analysis: Forecasting and Control. San Francisco.

Holden-Day.

Box, G.E.P. and Tiao,G.C.,1970. Intervention Analysis

with Applications to economic and Environmental

Problems. Journal of The American Statistical

Association.

Lapedes, A and Farber, R, 1987, Nonlinear Signal

Processing Using Neural Network: Prediction and

System Modeling, Proceeding of the IEEE – Los

Alamos National Laboratory report LA-UR-87-2662

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