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Volume No 04, Special Issue No. 01, April 2015 ISSN (online): 2394-1537

APPLICATION OF ARIMA MODEL USING SPSS

SOFTWARE - A CASE STUDY IN SUPPLY CHAIN
MANAGEMENT
Dr.N.Chitra 1 , Ra.Shanmathi 2 , Dr.R.Rajesh 3
1
Assistant Professor, 2 Under-Graduate Student, Department of Mathematics,
Thiagarajar College of Engineering, Madurai, Tamil Nadu,(India)
3
Professor, Department of Social Science, Horticultural College and Research Institute,
TNAU, Periyakulam, Tamil Nadu, (India)

ABSTRACT
This paper was an attempt to apply Auto-Regressive Integrated Moving Average (ARIMA) model in Supply
Chain management (SCM) of fresh vegetables analysis using SPSS Software.The ARIMA methodology
developed by Box and Jenkins was used in this paper. Time Series refers an ordered sequence of values of a
variable at equally spaced time intervals. Time series occur frequently when looking at agricultural data
applications. The analysis was carried out using time series data on the supply of fresh vegetables during the
period from 2002 to 2011 which was collected from the office of the Deputy Directorate of Agri-business
situated at Madurai, India. The analysis of monthly supply and price of vegetables data was used to find out
seasonal pattern. The seasonal index for vegetable supply was highest in March and April and was lower in the
November and December. The seasonal variation in vegetables price was high in the period of November to
January and price low in March. It was further inferred that forecast value for the supply of fresh brinjal ,
Bhendi and Green Chilly was low in January’13 and it was high in December for these vegetables whereas in
the forecasted value for the supply of fresh tomato and green chilly variations were found. The study suggested
that cold storage capacities may be developed in the needy places to increase the benefits of consumers and
farmers to increase the efficiency of Supply Chain Management.

Keywords: Auto-Regressive Integrated Moving Average model, Seasonal Index, SPSS software,
Supply Chain Management, Time series analysis.

I. INTRODUCTION
Time series analysis accounts for the fact that data points taken over time may have an internal structure (such
as autocorrelation, trend or seasonal variation) that should be accounted for. The usage of time series models are
(i) obtaining an understanding of the underlying forces and structure that produced the observed data and (ii)
fitting a model and proceed to forecasting, monitoring or even feedback and feed forward control.The modeling
and forecasting procedures discussed in Identifying Patterns in Time Series Data involved knowledge about the
mathematical model of the process. However, in real-life research and practice, patterns of the data are unclear,
individual observations involve considerable error, and we still need not only to uncover the hidden patterns in

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the data but also generate forecasts. The ARIMA methodology developed by Box and Jenkins (1976) allows to
do enormous popularity in many areas and research practice confirms its power and flexibility.
The Box-Jenkins ARMA model is a combination of the AR and MA models.
Xt=δ+ϕ1Xt−1+ϕ2Xt−2+⋯+ϕpXt−p+At−θ1At−1−θ2At−2−⋯−θqAt−q (1)
where the terms in equation (1) have the same meaning as given for the AR and MA model.
A couple of notes on this model,
1. The Box-Jenkins model assumes that the time series is stationary. Box and Jenkins recommend differencing
non-stationary series one or more times to achieve stationarity. Doing so produces an ARIMA model, with the
"I" standing for "Integrated".
2. Some formulations transform the series by subtracting the mean of the series from each data point. This
yields a series with a mean of zero. Whether you need to do this or not is dependent on the software you use to
estimate the model.
3. Box-Jenkins models can be extended to include seasonal autoregressive and seasonal moving average terms.
Although this complicates the notation and mathematics of the model, the underlying concepts for seasonal
autoregressive and seasonal moving average terms are similar to the non-seasonal autoregressive and moving
average terms.
4. The most general Box-Jenkins model includes difference operators, autoregressive terms, moving average
terms, seasonal difference operators, seasonal autoregressive terms, and seasonal moving average terms. As with
modelling in general, however, only necessary terms should be included in the model.

1.1. Two Common Processes

1.1.1. Autoregressive process
Most time series consist of elements that are serially dependent in the sense that you can estimate a coefficient
or a set of coefficients that describe consecutive elements of the series from specific, time-lagged (previous)
elements. This can be summarized in the equation:

Xt = + 1*x(t-1) + 2*x(t-2) + 3*x(t-3) + ... + (2)

Where in equation (2) is a constant (intercept), and 1, 2, 3 are the autoregressive model parameters.
Put into words, each observation is made up of a random error component (random shock,) and a linear
combination of prior observations.
1.1.2. Stationarity Requirement
Note that an autoregressive process will only be stable if the parameters are within a certain range; for example,

if there is only one autoregressive parameter then is must fall within the interval of -1 < < 1. Otherwise, past
effects would accumulate and the values of successive xt' s would move towards infinity, that is, the series
would not be stationary. If there is more than one autoregressive parameter, similar (general) restrictions on the
parameter values can be defined.
1.1.3. Moving Average Process
Independent from the autoregressive process, each element in the series can also be affected by the past error (or
random shock) that cannot be accounted for by the autoregressive component, that is:
xt = µ + t - 1t - 1*(t-1) - t - 2*(t-2) - t - 3*(t-3) (3)

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Where in equation (3) µ is a constant, and 1, 2, 3 are the moving average model parameters.
Put into words, each observation is made up of a random error component (random shock, ) and a linear
combination of prior random shocks.
1.1.4. Invertibility Requirement
Without going into too much detail, there is a "duality" between the moving average process and the
autoregressive process (e.g., see Box & Jenkins, 1976; Montgomery, Johnson, & Gardiner, 1990), that is, the
moving average equation above can be rewritten (inverted) into an autoregressive form (of infinite order).
However, analogous to the stationarity condition described above, this can only be done if the moving average
parameters follow certain conditions, that is, if the model is invertible. Otherwise, the series will not
be stationary.

II. ARIMA METHODOLOGY

2.1. Auto- Regressive Moving Average Model
The general model introduced by Box and Jenkins (1976) includes autoregressive as well as moving average
parameters, and explicitly includes differencing in the formulation of the model. Specifically, the three types of
parameters in the model are: the autoregressive parameters (p), the number of differencing passes (d), and
moving average parameters (q). In the notation introduced by Box and Jenkins, models are summarized as
ARIMA (p, d, q); so, for example, a model described as (0, 1, 2) means that it contains 0 (zero) autoregressive
(p) parameters and 2 moving average (q) parameters which were computed for the series after it was differenced
once.

2.2. Identification
As mentioned earlier, the input series for ARIMA needs to be stationary, that is, it should have a constant
mean, variance, and autocorrelation through time. Therefore, usually the series first needs to be differenced until
it is stationary (this also often requires log transforming the data to stabilize the variance). The number of times
the series needs to be differenced to achieve stationarity is reflected in the d parameter (see the previous
paragraph). In order to determine the necessary level of differencing, you should examine the plot of the data
and autocorrelogram. Significant changes in level (strong upward or downward changes) usually require first
order non seasonal (lag=1) differencing; strong changes of slope usually require second order non seasonal
differencing. Seasonal patterns require respective seasonal differencing (see below). If the estimated
autocorrelation coefficients decline slowly at longer lags, first order differencing is usually needed.
However, you should keep in mind that some time series may require little or no differencing, and that over
differenced series produce less stable coefficient estimates.
At this stage (which is usually called Identification phase, see below) we also need to decide how many
autoregressive (p) and moving average (q) parameters are necessary to yield an effective but
still parsimonious model of the process (parsimonious means that it has the fewest parameters and greatest
number of degrees of freedom among all models that fit the data). In practice, the numbers of
the p or q parameters very rarely need to be greater than 2 (see below for more specific recommendations).

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2.3. Estimation and Forecasting

At the next step (Estimation), the parameters are estimated (using function minimization procedures, see below;
for more information on minimization procedures see also Nonlinear Estimation), so that the sum of squared
residuals is minimized. The estimates of the parameters are used in the last stage (Forecasting) to calculate new
values of the series (beyond those included in the input data set) and confidence intervals for those predicted
values. The estimation process is performed on transformed (differenced) data; before the forecasts are
generated, the series needs to be integrated (integration is the inverse of differencing) so that the forecasts are
expressed in values compatible with the input data. This automatic integration feature is represented by the letter
I in the name of the methodology (ARIMA = Auto-Regressive Integrated Moving Average).

2.4. Identification Phase

2.4.1 Number of Parameters to be Estimated
Before the estimation can begin, we need to decide on (identify) the specific number and type of ARIMA
parameters to be estimated. The major tools used in the identification phase are plots of the series, correlograms
of auto correlation (ACF), and partial autocorrelation (PACF). The decision is not straightforward and in less
typical cases requires not only experience but also a good deal of experimentation with alternative models (as
well as the technical parameters of ARIMA). However, a majority of empirical time series patterns can be
sufficiently approximated using one of the 5 basic models that can be identified based on the shape of the
autocorrelogram (ACF) and partial auto correlogram (PACF). Also, note that since the number of parameters (to
be estimated) of each kind is almost never greater than 2, it is often practical to try alternative models on the
same data.
1. One autoregressive (p) parameter: ACF - exponential decay; PACF - spike at lag 1, no correlation for other
lags.
2. Two autoregressive (p) parameters: ACF - a sine-wave shape pattern or a set of exponential decays; PACF -
spikes at lags 1 and 2, no correlation for other lags.
3. One moving average (q) parameter: ACF - spike at lag 1, no correlation for other lags; PACF - damps out
exponentially.
4. Two moving average (q) parameters: ACF - spikes at lags 1 and 2, no correlation for other lags; PACF - a
sine-wave shape pattern or a set of exponential decays.
5. One autoregressive (p) and one moving average (q) parameter: ACF - exponential decay starting at lag 1;
PACF - exponential decay starting at lag 1.

2.5. Seasonal Models

Multiplicative seasonal ARIMA is a generalization and extension of the method introduced in the previous
paragraphs to series in which a pattern repeats seasonally over time. In addition to the non-seasonal parameters,
seasonal parameters for a specified lag (established in the identification phase) need to be estimated. Analogous
to the simple ARIMA parameters, these are: seasonal autoregressive (ps), seasonal differencing (ds), and
seasonal moving average parameters (qs). For example, the model (0,1,2)(0,1,1) describes a model that includes
no autoregressive parameters, 2 regular moving average parameters and 1 seasonal moving average parameter,
and these parameters were computed for the series after it was differenced once with lag 1, and once seasonally
differenced. The seasonal lag used for the seasonal parameters is usually determined during the identification
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phase and must be explicitly specified. The general recommendations concerning the selection of parameters to
be estimated (based on ACF and PACF) also apply to seasonal models. The main difference is that in seasonal
series, ACF and PACF will show sizable coefficients at multiples of the seasonal lag (in addition to their overall
patterns reflecting the non seasonal components of the series).

III. SUPPLY CHAIN MANAGEMENT (SCM)

Marketing of agricultural produce is different and more challenging than many industrial products because of
the perishability, seasonality and bulkiness. The very nature of small size of land holdings by the farmers, varied
climatic conditions, production spread over wide geographical area, mainly in remote villages, diversified
consumption pattern habits of the Indian consumers and poor Supply Chain (SC) infrastructure makes marketing
for vegetables more complicated. At the same time, Indian consumers demand fresh vegetables. Thus SCM
plays a crucial role in marketing vegetables. Supply Chain efficiency not only helps in increased production and
per capita consumption, but also contributes to economic development of the country. As a result, SCM throws
both challenges and opportunities in marketing of vegetables. Efficient SCM in marketing, not only increases
the profitability and efficiency of retailers, but also adds value to different stakeholders like cultivators
(farmers), consolidators and consumers. Consumption of vegetables is still far below the recommended level for
effective health promotion and disease prevention in developed and developing countries

IV. METHODOLOGY
ARIMA model was applied using SPSS software in the time Series data on the supply of fresh vegetables during
the period from 2002 to 2011.The data was collected from the office of the Deputy Directorate of Agri-business
situated at Madurai, India. The particulars on the constraints in supply of fresh vegetables to the study area were
collected from the farmers by conducting personal interview with 30 farmers. Research has shown the need to
increase the supply of vegetables to support demand-generating interventions aimed at improving people‟s
eating habits and reducing the risk of chronic disease. With this background, a pioneering attempt was made to
assess the supply of fresh vegetables in the leading farmer market in Madurai district in India with the following
specific objectives:
 To assess the price and supply of fresh vegetables in the study area
 To workout the seasonal index for major vegetables sold in the study area
 To forecast the price and supply of fresh vegetables in the study area and
 To identify the constraints in supply of fresh vegetables to the study area and to suggest the ways to
overcome the problems in the supply of fresh vegetables.

4.1. Time Series Analysis Method used in the Estimation of Seasonal Index
The seasonality in the arrival of the major vegetables was examined with the help of monthly seasonal index.
Monthly arrival pattern expressed in terms of monthly seasonal indices for Brinjal, Tomato, Bhendi, Small
Onion and Green chilly showed variation over the period of months from 2002 to 2011. Seasonal Index is
calculated by Moving Average method. The analysis of monthly supply and price of vegetables data was used to
find out seasonal pattern..

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The monthly data on vegetable supply and price for the period from 2002 to 2011 were used for forecasting the
supply of fresh vegetables by ARIMA models using Box-Jenkins methodology.. The Box-Jenkins procedure is
concerned with fitting a mixed Auto Regressive Integrated Moving Average (ARIMA) model to a given set of
data. The main objective in fitting this ARIMA model is to identify the stochastic process of the time series and
predict the future values accurately.
4.1.1. Identification
Appropriate values of p, d and q are found first. The tools used for identification are the Autocorrelation
Function (ACF), the Partial Autocorrelation Function (PACF) and the resulting correlograms and partial
correlograms.
The general ARIMA (p, d, q) model is presented in simple form as:
 (B) d Xt =  (B) Ut (4)
Where B is the backshift operator defined by:
BmXt = Xt – m (m = 0,1,2 . . . p) (5)
 (B) is autoregressive operator of order „ p‟ defined by:
 ( B ) = 1 – 1 B1 – 2 B2 – . . . – p Bp (6)
 is the backward difference operator defined by:
 Xt = Xt – Xt–1 = (1–B) Xt (7)
d means the dth difference of the series values Xt,  (B) is the moving average operator of order „q‟ defined by:
 (B) = 1 – 1B1 – 2B2 – . . . – q Bq (8)
Ut is white noise process having a normal probability distribution with mean zero and variance  u.
2

An example of ARIMA model is given below to clarify the general representation of the ARIMA (1,1,1) in
explaining some features of the general ARIMA (p,d,q) model.
As could be seen in ARIMA(1,1,1) model where p=1, d=1, q=1
 (B) = 1 – 1 B1
1 = (1–B)1 = 1 – B and
 (B) = 1 –  1B1 = 1 –  1B1
Thus the model becomes,
(1–1B) (1–B) Xt = (1–1B1) Ut
i.e. (1 – 1B1) (Xt – Xt–1) = Ut – 1 Ut–1
i.e. Xt – t Xt–1 – Xt–1 + t Xt–2 = Ut – 1 Ut–1
i.e. (Xt – Xt–1) –  (Xt – Xt–1) = Ut – 1 Ut–1
i.e. Wt – 1 Wt–1 = Ut – 1 Ut–1
1 and 1 are the parameters of the model ARIMA (1, 1, 1).
Similarly i ( i = 1,2,….p) and j (j = 1,2…q) are the parameters of the general ARIMA (p, d, q) model. i (i
= 1,2…p) are the Autoregressive (AR) parameters and j (j = 1,2…q) are the moving average (MA) parameters.

V. RESULTS AND DISCUSSION

The results of the study are furnished below.

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5.1. Seasonal Index

Seasonal supply and price indices of fresh vegetables are presented in TABLE 1and TABLE 2 respectively.
Table 1: Seasonal Index for Supply of Major Fresh Vegetables

Table 2: Seasonal Index-Price for Major Fresh Vegetables

It could be seen from the table that the seasonal index for vegetable supply was highest in March and April and
was lower in the November, December. The seasonal variation in vegetables price was high in the period of

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November to January and price low in March. When supply was more in season period, the price was low and
vice versa. The surplus of fresh vegetables could be for value addition. The farmers could get remunerative
prices for their value added products.

5.2. Forecasting with ARIMA Model

For forecasting fresh vegetables supply and price, ARIMA model was used only after transforming the variable
under forecasting into a stationary series. ARIMA models are developed basically to forecast the corresponding
variable. To judges the forecasting ability of the fitted ARIMA model, important measure of the sample period
forecasts accuracy was computed. The forecasts for supply and price of fresh vegetable during 2013 are
presented in TABLE 3 and TABLE 4 respectively.
Table 3: Forecast for Supply of Fresh Vegetables (Kgs)

It was inferred that forecast value for the supply of fresh brinjal , Bhendi and Green Chilly was low in
January‟13 and it was high in December for these vegetables. But a variation was found in the forecast value
for the supply of fresh tomato and green chilly.
Table 4: Forecast for Price of Fresh Vegetables (Rs/Qtl)

It was inferred that increasing trend in the forecast value for the price of fresh vegetables were found in the
study area.
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5.3. Constraints in Supply of Fresh vegetables

Garette‟s ranking technique was used to analyze the problem faced by the farmers in supplying fresh vegetables
to the study area. A perusal of the table showed that most widely reported problem was the insufficient weighing
machine, sudden price fluctuation, insufficient stall facility and insufficient cold storage facility were reported
by the farmers. The results were presented in the TABLE 5.
Table 5: Constraint Identification Using Garette’s Ranking Method

S.No Constraint Mean Rank

Score
1 Insufficient of weighing 58.30 I
machine
2 Sudden price fluctuation 39.87 II

3 Insufficient stall facility 22.87 III

4 Insufficient cold storage 17.93 IV

facility
5 No standard grading 5.77 V
procedure

VI. CONCLUSION
The inferences from the study are furnished below:
In Supply Chain Management of fresh vegetables, availability of fresh vegetables to consumers in good quality
and correct quantity and in reasonable prices and insisted. Further the farmers should gain reasonable profit in
selling their products in the markets. Here the study helped to identify surplus and deficit periods in the supply
of fresh vegetables and the favorable price period for their crops. Accordingly the farmers can plan their
vegetables production pattern.
The seasonal index for vegetable supply was highest in March and April and was lower in the November and
December. The seasonal variation in vegetables price was high in the period of November to January and price
low in March. It was further inferred that forecast value for the supply of fresh brinjal ,Bhendi and Green
Chilly was low in January‟13 and it was high in December for these vegetables whereas in the forecasted value
for the supply of fresh tomato and green chilly variations were found.
The results paved the way for the following policy implications.
1) To increase the efficiency of Supply Chain Management, cold storage capacities may be developed in the
needy places to increase the benefits of consumers and farmers.
2) Value addition techniques for fresh vegetables may be educated to the farmers to increase their income .
Further availability value added vegetable products will satisfy the specific requirements of consumers as
well. For that, necessary infrastructural facilities can be generated based on the intensity of vegetables
cultivation area.
3) In the supply Chain management of fresh vegetables, good transport facilities and financial schemes and
good marketing environment can be provided to the farming community to minimise the loss and to
provide fresh vegetables throughout year to the consumers.

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REFERENCES
[1] Box, G.E.P. and G.M. Jenkin, Time Series of Analysis, Forecasting and Control, San Franscico, Holden-
Day, California, USA, 1976.
[2] Rahulamin, MD, and Razzaque, MA, “Autoregressive Integrated Moving Average Modeling for Monthly
Potato Pulses in Bangladesh”, Journal of Financial Management and Analysis, 13(1), pp.74-80, 2000.
[3] Raj Pravin.T., SWOT Analysis on Uzhavar Santhai, M.Sc(Ag) Thesis, TNAU, Coimbatore, 2001.
[4] Subhadra. M.R. , Suresh. K.A. and Reeja George. P., Constraint analysis of farmers operating mixed
farming in kerala, Department of Veterinary and A.H Extension, College of Veterinary and Animal
Sciences, Mannuthy, Thrissur- 680651, India.
[5] Ramasubramaniam.V, IASRI, Library Avenue, New Delhi.

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