Description of data
In order to carry out our analysis, historical weekly jute prices data for Coochbehar market, Baxirhat market and Tufanganj market of Coochbehar district have been taken from the Agricultural Marketing Information Network (https://agmarknet.gov.in) portal. The data for the Coochbehar market has been collected from January, 2009 to December, 2022 (672 weeks) and for Baxirhat and Tufanganj markets from January, 2010 to December, 2022 (624 weeks). The first 80% observations are used for the model building purpose and the rest 20% observations are used for model validation. In the present study, statistical analyses have been carried out using the powerful software “RStudio” (https://www.rstudio.com).
Non-linearity test
In the present study, we have applied a nonlinearity test to determine the existence of nonlinear dynamics given by
Brock et al. (1996). The BDS test is a non-parametric test having the null hypothesis (H
0): Data is independently and identically distributed (
i.i.d.).
i.e., data is linear against the alternative hypothesis (H
A): Data is not independently and identically distributed indicating presence of nonlinearity in the data.
Time- delay neural network (TDNN)
Artificial neural network (ANN) is a nonlinear data driven self-adaptive approach and is powerful tools for performing nonlinear modelling without a priori knowledge about the relationships between input and output variables. Artificial Neural Network is an information processing system which is inspired by the models of biological neural network (
Sivanandam et al., 2008). It is an adaptive system that changes its structure or internal information that flows through the network during the training time (
Sheela and Deepa, 2013). One of the most dominant advantages of ANN models over other nonlinear statistical models is that ANN is universal approximator that is able to approximate a large class of functions with a high degree of accuracy (
Zhang and Qi, 2005). There are several types of neural networks: Feed forward NN, Radial Basis Function (RBF) NNand Recurrent neural network. A single hidden layer feedforward ANN with one output node is most prominent network used in time series modelling and forecasting (
Zhang et al., 1998;
Zhang, 2003;
Anjoy et al., 2017;
Ray et al., 2023). Time-Delay Neural Network (TDNN) has gained significant traction in modelling non-linear patterns and generating non-linear forecasts in the arena of time series forecasting (
Coban and Tezcan, 2022;
Zhong et al., 2022). The major advantage of this model is that it does not require any presumption about the considered time series data; rather, the pattern of the data is quite important to it, usually referred as the data-driven approach. The structure between output (Y
t) and the inputs (Y
t-1Y
t-2,...,Y
t-p) by multilayer feed-forward time-delay neural network (p × q × 1) is given with the following mathematical representation:
Where,
⌽
j(j = 0,1,2,...,q)= Weights between hidden layer and output layer.
⊝
ij(i = 0,1,2,...,p ; j = 1,2,...,q )= Weights connect input layer and hidden layer.
⌽
0 and ⊝
0j= Bias terms.
p= Number of input nodes.
q= Number of hidden nodes.
ε
t= White noise.
g(.)= Hidden layer activation function.
A graphical presentation of Time-delay neural network (TDNN) with one hidden layer is given in Fig 1.
For a univariate time series forecasting problem, lagged values of the time series can be used as inputs to a neural network. 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 take a weight one. This weighted sum of input variables is then transformed by each hidden node using the activation function g(.) which is a non-linear 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 its activation function
f(.).
The activation function (transfer function) determines the relationship between inputs and outputs of a node and a network. In general, the activation function introduces a degree of nonlinearity that is valuable for most ANN applications (
Zhang et al., 1998). In time series analysis, hidden layer activation function is often chosen as the Logistic (Sigmoid) function and output nodes, as an Identity function (
Jha and Sinha,2012). Sigmoid or Logistic activation function for hidden layer is written as an equation (
Kumar et al., 2020;
Priyadarshi et al., 2023):
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. The neural network structure for a particular problem in time series prediction includes determination of the number of layers and total number of nodes in each layer. This is usually determined through experimentation of the given data as there is no theoretical basis for determining these parameters. Hence, a fashionable approach is trial and error until obtaining the appropriate parameters (
Kaastra and Boyd, 1996;
Zhang, 1998).
Support vector regression (SVR)
Support vector machine (SVM) is a machine learning algorithm introduced by
Vapnik (1995) for classification problems. It was originally developed for pattern recognition thereafter promoted to support vector regression (
Vapnik et al., 1997) for regression problems by incorporating -insensitive loss function to penalize data when they are greater than . The support vector regression (SVR) is a method that can handle overfitting so that it produces good performance in regression and prediction of time series (
Setiawan et al., 2021). The SVR technique provides a nonlinear mapping function to map the training dataset into a high dimensional feature space (
Yeh et al., 2011).
Considering a data set of
N elements
;
Where
X
i ∈ R
n input vector, Y
i∈ R is scalar output corresponding to x
i.
The general formula for linear support vector regression is given as:
Where,
w= Weight vector N dimension,
φ (x)= Function that maps to the feature space with dimensions.
b= Biased.
The solution of W and b in above equation can be obtained by solving the following minimization problem (
Sermpinis et al., 2014):
With the constrains:
Where,
y
i= Actual value of
i period.
φ (x
i )= Estimated value of period.
This regularized risk function minimizes both empirical error and regularized term simultaneously and implements structural risk minimization (SRM) principle to avoid under and over fitting of training data. The first term ll w ll
2 in the objective function, employing the concept of maximizing the distance of two separated training data. It is used to regularize weight sizes to penalize large weightsand to maintain regression function flatness. The second term
penalizes training errors of
f (
x) and
y by using the ε- insensitive loss function. C > 0 is a constant known as the penalty parameter determines how much the error deviation is from the tolerable limit ε. Training errors above ε are denoted as 𝄽
i , whereas training errors below are denoted as 𝄽
i* The SVR illustration is given in Fig 2.
The formula above is a Convex Linear Programming. NLP Optimization Problem which functions to minimize the quadratic function to be converted into a constraint. This limitation can be solved by using the Lagrange Multiplier function. The process of deriving formulas is very long and complicated. After going through mathematical stages, a new equation is obtained with the function (
Muthiah et al., 2021):
Where,
α
i*, α
i = Lagrangian multipliers.
x
i= Support vector.
x
j = Test vector.
The above functions can be used to solve linear problems. The function K ( x
i , x
j ) is called the kernel function and the value of the kernel equals the inner product of two vectors, x
i and x
j, in the feature space respectively; that is:
The radial basis functions (RBF) kernel function is often used for forecasting (
Lin et al., 2007;
Cho, 2024). A nonlinear RBF kernel function defined as follows (
Ghanbari and Goldani, 2022):
Where,
ℽ = RBF width parameter.
The value of this parameter with the trade-off between the minimum fitting error and estimated function is determined. The SVR model with RBF contains three tuning parameters in the K (x
i, x
j ) : 1. The parameter of loss function ε 2. Penalty factor C and 3. The parameter of kernel function ℽ.
2.2.5 Accuracy measures =
Root mean square error (RMSE) =
Mean absolute error (MAE)
Mean absolute percentage error (MAPE)
Where,
y
i and y
i= Actual value and predicted value of response variable.
