1. To evaluate the seasonal nature of price movement of TOP vegetables for major market in Gujarat.
2. To identify the temporal business opportunity in TOP vegetables for major market in Gujarat.
3. To forecast price of TOP vegetables for major markets of Gujarat using neural network model.

Market:

The market will be selected based on the largest amount of average yearly arrivals of the TOP vegetables in the market as well as the consistency with which the relevant time series data have been regularly available for a long time.

Harvest and time frame: On the basis of area, production, and the significance of the vegetable crop for the area, the TOP vegetables crop is chosen for the study. The research will be carried by using data from wholesale price time series collected on a monthly basis between January, 2001 and October, 2024.

Data: The Agricultural Produce Market Committee and AGMARKNET's secondary time series data on monthly TOP vegetables market prices will be used in the study.

Analytical Framework

The following statistical tools will be used to assess the various objectives listed above. The seasonal character of price variations can be measured using a variety of techniques. The most popular technique for determining seasonal indexes is the ratio to moving average method (Meera & Sharma, 2016). The average seasonal price variation of commodities over a period of years, as indicated by a seasonal price index, can be used as a benchmark for comparison (Rahn, 1968). This approach reduces differences caused by causes other than seasonality by taking into account the effects of price inflation, cyclical production shifts, and technological advancement (Rahn, 1968; Flaskerud & Johnson, 2000). It aids in better purchasing, selling, and storage decisions and can be used to calculate agricultural storage profitability (Rahn, 1968; Jayaramu, 2015).

• To evaluate the seasonal nature of price movement of TOP vegetables for major market in Gujarat.

For the estimation of seasonal index, a 12-month moving average will be calculated as follows:

And the Centre Moving Average (CMA) was calculated as:

Seasonal index is the ratio of observed value of price (Yt) to centred moving average.

Where,

E = Seasonal Factor or index

Yt = Observed price in period t

CMA = Centred Moving average.

Seasonal Price Variation will be examined by calculating average seasonal index using monthly data where each month’s price/seasonal index (EA) will computed as the average of the same month’s seasonal index for all years included in the moving average time series. This will be calculated by arranging the seasonal indices month-wise for each year and calculating the average for each month.

The indices are expressed as a percentage of the moving average price and adjusted to the base of 100 and is called Adjusted Seasonal Index or Grand Seasonal index (GSI). This will be calculated by multiplying each month’s average seasonal index by a correction factor as:

Where,  is the correction factor and, EA = Average Seasonal index for a month

The magnitude of price variability will be calculated as percentage of difference between highest and lowest de-seasonalized price in each year, as shown in following equation:

Where,

Vt = magnitude of price variability in year t.

The de-seasonalized price will be calculated by taking the ratio of actual price (Yt) to the average seasonal index (EA) for that month. The variability of price will be analyzed by calculating the average of price variability of each year.

• To identify the temporal business opportunity in TOP vegetables for major market in Gujarat.

Gross storage return (GSR) can be used to evaluate the feasibility of storage. Higher value of GSR indicates that the return to storage is higher in market and hence farmers can store commodity for future sale (Ngare, Simtowe, & Massingue, 2014). The GSR can be calculated by computing the percentage increase from seasonal low to seasonal high of Gross seasonal Index (Ngare et al. 2014; Mani et al. 2018). Mathematically, it can be computed as:

The higher percentage of GSR meant the return to storage for major vegetables will be higher in market and vice versa.

• To forecast price of TOP vegetables for major markets of Gujarat using neural network model.

Model for a neural network by supplying the implicit functional representation of time, which allows a static neural network like the multilayer perceptron to be endowed with dynamic qualities, time series data can be described using ANN (Haykin, 1999). By incorporating either long-term or short-term memory, depending on the retention period, into the architecture of a static network, a neural network can be made dynamic. Time delay, which can be applied at the input layer of the neural network, is one straightforward method of incorporating short-term memory into the structure of a neural network. A Time-Delay Neural Network (TDNN) (Figure 1), used in the current study, is an illustration of such an architecture.

The number of layers and total number of nodes in each layer are both factors in the ANN construction for a specific time series prediction issue. As there is no theoretical foundation for calculating these characteristics, it is often discovered by experimentation. It has been demonstrated that neural networks with a single hidden layer are capable of approximating any non-linear function with the right amount of hidden layer nodes and training data. We employed a neural network with one hidden layer for this study. The number of input nodes that are lagged observations of the same variable is crucially important when performing time series analysis since it aids in modelling the autocorrelation structure of the data. The number of output nodes can be calculated quite easily. One output node has been employed in this study. Using an iterative process, multi-step forecasting is carried out using the Box-Jenkins ARIMA Time Series modelling approach. In order to forecast future value, this includes using the forecast value as an input. The model with fewer nodes in the hidden layer is always preferable because it performs better for out-of-sample forecasting and doesn't suffer from over-fitting. Equation (1) provides the general expression for the final output value yt+1 in a multi layer feed forward time delay neural network.

where, f and g denote the activation function at the hidden and output layers, respectively; p is the number of input nodes (tapped delay); q is the number of hidden nodes; βij is the weight attached to the connection between ith input node to the jth node of hidden layer; αj is the weight attached to the connection from the jth hidden node to the output node; and yt-i is the ith input (lag) of the model. Each node of the hidden layer receives the weighted sum of all the inputs, including a bias term for which the value of input variable will always be one. This weighted sum of input variables is then transformed by each hidden node using the activation function f which is usually a non-linear sigmoid function. In a similar manner, the output node also receives the weighted sum of the output of all the hidden nodes and produces an output by transforming the weighted sum using its activation function g. In the time series analysis, f is often chosen as the Logistic Sigmoid function and g, as an identity function. The logistic function is expressed as Equation (2):

For p tapped delay nodes, q hidden nodes, one output node and biases at both hidden and output layers, the total number of parameters (weights) in a three layer feed forward neural network is q(p + 2) + 1. For a univariate time series forecasting problem, the past observations of a given variable serve as input variables. The TDNN model attempts to map the following function:

where, yt+1 pertains to the observation at time t+1, p is the number of lagged observation, w is the vector of network weights, and εt+1 is the error-term at time t+1. Hence, TDNN acts like a non-linear autoregressive model.

The ARIMA Model

In an Auto-Regressive Integrated Moving Average (ARIMA) model, time series variable is assumed to be a linear function of the previous actual values and random shocks. In general, an ARIMA model is characterized by the notation ARIMA (p, d, q), where p, d and q denote orders of Auto-Regression (AR), Integration (differencing) and Moving Average (MA), respectively. ARIMA is a parsimonious approach which can represent both stationary and non-stationary processes. An ARMA (p, q) process is defined by Equation (4):

where, yt and εt are the actual value and random error at time period t, respectively, Φi (i=1, 2,……,p) and φi (j=1, 2,……,q) are the model parameters. The random errors, εt are assumed to be independently and identically distributed with a mean of zero and a constant variance of σ2.

The first step in the process of ARIMA modelling is to check for the stationarity of the series as the estimation procedure is available only for a stationary series. A series is regarded stationary if its statistical characteristics such as the mean and the autocorrelation structures are constant over time. The stochastic trend of the series is removed by differencing, while logarithmic transformation is employed to stabilize the variance. After appropriate transformation and differencing, multiple ARMA models are chosen on the basis of Auto-Correlation Function (ACF) and Partial Auto- Correlation Function (PACF) that closely fit the data. Then, the parameters of the tentative models are estimated through any non-linear optimization procedure such that the overall measure of errors is minimized or the likelihood function is maximized. Lastly, diagnostic checking for model adequacy is performed for all the estimated models through the plot of residual ACF and using Portmonteau test. The most suitable ARIMA model is selected using the smallest Akaike Information Criterion (AIC) or SchwarzBayesian Criterion (SBC) value and the lowest root mean square error (RMSE).

The Hybrid ARIMA - TDNN model

In this section, the time series decomposition is proposed in which ARIMA and TDNN models are combined in order to obtain a robust and efficient methodology for time series forecasting. Accordingly, we postulate that our time series data can be decomposed into a linear and a nonlinear component (Rojas et al., 2008), viz.

where, yt is the observed time series data, Lt is the linear auto-regressive component, and Nt is the non-linear component. In this approach, we apply an ARIMA model to the data series to fit the linear part and the residuals are modelled using neural network model only if there is an evidence of non-linearity for the series. Figure 2 shows a schematic diagram of this method. Let rt be the residual at time t of the linear component, then

where, L^ t is the estimate of the linear auto-regressive component. For non-linear components, we apply neural network model, i.e.

where, p is the number of input delays and f is the nonlinear function. So the combined forecast is given by Equation (8):

 where, εt is the error-term of the combined model at time t. Here, it is assumed that since ARIMA model cannot capture the nonlinear structure of the data, the residual of linear model will contain information about nonlinearity. Hence, the hybrid architecture is expected to exploit the feature and strength of both the models in order to improve the overall forecasting performance. The residuals which are obtained from fitted ARIMA model are utilized to test non linearity. The test statistic is given by Equation (9):

where, r(i) is the autocorrelation of the squared residuals, and h is the number of autocorrelations.

Forecast Evaluation Methods

The forecasting ability of different models will be assessed with respect to two common performance measures, viz. the root mean squared error (RMSE) and the mean absolute deviation (MAD). The RMSE measures the overall performance of a model and is given by Equation (10):

where, yt is the actual value for time t, y^ t is the predicted value for time t, and n is the number of predictions. The second criterion, the mean absolute deviation is a measure of average error for each point forecast and is given by Equation (11):

where the symbols have the same meaning as above.