A GA-BP neural network for nonlinear time-series forecasting and its application in cigarette sales forecast

3.1 The training and prediction principle of BP neural network

The basic steps of using the BP neural network for time-series modeling and forecasting are mainly divided into three steps:

1. Determine the dimensionality of the input layer: First, divide the time series into two parts. The first part is roughly twice the size of the other part, the size of the starting window can be chosen arbitrarily; that is, the number of input neurons can be set to any initial value, use the first part to train the network, and the resulting network is used to predict the second part and calculate the prediction error. Change the window from small to large, until as the window size increases, the prediction accuracy is no longer significantly improved, and the window size at this time is the dimension of the input layer.

2. Train the network: Use all sequences as training samples to train the network to obtain the neural network prediction model of the time sequence [14].

3. Prediction: Use the obtained model to make predictions. The main problem faced in the practical application of time-series modeling and forecasting using the BP neural network, the first is the learning efficiency of BP network. Since the BP algorithm uses the gradient descent method to adjust the connection weights, it is inevitable that the network learning speed will be slower, and it is easy to sink into a local minimum or enter a flat area, resulting in failure to converge; At the same time, BP network, by its nature, is just a nonlinear mapping system, not a nonlinear dynamic system; without dynamic adaptability, it is difficult to meet the requirements of real-time systems; In addition, in order to obtain satisfactory prediction accuracy, the accuracy of the sample data is required to be high.

Second, it can be seen from the above modeling steps that the main factors that affect the modeling speed are as follows:

1. When determining the order, use multiple sets of samples for training, prediction, and comparison of prediction accuracy, and then get the number of network input layer units, which is bound to consume a lot of time;

2. Use all samples for training, and then make predictions, the increase in the number of samples can easily lead to the expansion of calculations, especially when a new sample is added, all previous samples must be added to the retraining, which will not only extend the training time [15]; moreover, because the information contained in the BP network is limited, it is easy to cause the network to fail to converge. In response to these problems, a dynamic modeling and prediction method is proposed below. Now introduce this method with one-step prediction, and the multi-step prediction method can be analogized.

3.2 The key process of optimizing neural network with GA calculation

The process of optimizing the connection weight and threshold of the neural network with the GA algorithm has the following three main processes: (i) The expression of the gene (i.e., the code that determines the weight and threshold). (ii) Estimation of individual fitness. (iii) Use evolutionary operators (including selection, crossover, and mutation). On the basis of the above three steps, the algorithm iteratively optimizes until the conditions are met.

1. Coding

   First build a BP neural network, the ownership value and threshold of the network (including the weight matrix from the input layer to the hidden layer, the weight matrix from the hidden layer to the output layer, the hidden layer threshold, and the output layer threshold) are regarded as a set of ordered chromosomes; according to the number of weights and thresholds, it is represented by a real variable of the corresponding dimension. The direct use of real number coding is because: (i) The number of patterns in the population is only related to the population size and chromosome length [16]. (ii) It is a direct natural description of the continuous parameter optimization problem, there is no encoding and decoding process between decimal and binary. (iii) It can improve the accuracy and speed of calculation, reduce the complexity of calculation, and improve the efficiency of calculation.

   (1) X = [ w 11 , w 12 , ⋯ w min , v 11 , v 12 , ⋯ v p m , θ 1 , θ 2 , ⋯ θ m , t 1 , t 2 , ⋯ t p ] .

2. Adaptability

   In the evolution of GAs, the evaluation of chromosomes is done by fitness function, the calculation of the fitness function value is very important, and it is the basis for selecting the operation [17]. The search goal of the GA is to obtain network weights and thresholds that minimize the sum of squared errors of the network in all evolutionary generations; the GA is evolving in the direction of increasing the value of the fitness function. According to the neural network corresponding to each individual (weight and threshold), the sum of squared errors of the BP network is calculated, and the fitness function uses the reciprocal of the sum of squares of network errors:

   (2) E ( X i ) = 1 2 ∑ k = 1 I ∑ j = 1 p ( d k j i − o k j i ) 2 ,

   (3) f ( X i ) = E − 1 ( X i ) ,

   where o k j i represents the ith individual (chromosome string) (i.e., the ordered vector of weights and thresholds), the output value of the kth training sample at the jth output node; d k j i is the expected output value; l is the number of training samples; p is the number of neurons in the output layer, and i = 1,2,…,N (N is the population size). Obviously, f(X _i ) represents the fitness of the ith gene chain.

3. Evolutionary operation

1. Gene selection: According to formula (8), the fitness value of each individual in the population can be obtained. Sort their sizes in descending order, and then use the fitness ratio selection method (roulette selection) to get the probability of their appearance in the offspring individuals, parental evolution is selected and regenerated in this way [18].

2. Keegan Crossover: The use of crossover operators enables the algorithm to find a better individual coding structure from a global perspective. Suppose the two gene chains to be involved in the crossover operation are X _i and X _j , (X _i ’s fitness is greater than X _j ’s fitness), the chromosomes at corresponding positions on the chain are A and B, respectively, and the following two intermediate variables are defined:

   (4) Δ j i = min 1 + p c 2 ( x i − x j ) , x max max 1 + p c 2 ( x i − x j ) , x min ,

   (5) Δ j i = max 1 + p c 2 ( x j − x i ) + x j , x min max 1 + p c 2 ( x j − x i ) + x j , x min .

3. Genetic mutation: In order to make the individual closer to the optimal solution from a local perspective, and to accelerate the convergence of the algorithm when approaching the optimal solution neighborhood, use a uniformly distributed random number to replace the original gene, so that the individual can move freely in the search space; that is, the mutation point k is randomly selected among the parent individuals, and the new gene value of the mutation point is:

where X _min and X _max are the minimum and maximum values of the initial individual target variables, respectively; β is a random number uniformly distributed in [O, 1]. If the crossover operator finds better individuals globally, the mutation operator adjusts and optimizes the coding structure in the details of the search space. The use of gene mutation can improve the local search ability of the algorithm and maintain the diversity of the group [19]. The combined flowchart of the BP network and GA algorithm is shown in Figure 3.

Figure 3

GA-BP network flowchart.

3.3 Forecasting methods of nonlinear time series

For one-dimensional nonlinear time series x(t), t ∈ [0,1], if you want to predict the value of x(t + 1) time, first, we must construct the structural form of the BP network, that is, determine the number of input nodes and output nodes, better reveal the relevance of non-time series in the time-delayed state space, for a single time series, and use the overlap and partial overlap methods of the training part and the test part to predict. The specific method is as follows:

x ₁, x ₂, x ₃,… x ₀ is a single time series, to predict the value of x _n+1, x _k , x _k+1, …x _k+s can be taken as k input samples, use − 1 2 + 1 1 + exp ( − x k + s ) as the kth corresponding teacher, and the corresponding value x _k+s+1 is predicted.

3.4 Cigarette sales forecast

Compared with annual sales and quarterly sales forecasts, monthly cigarette sales forecasts are more difficult. The monthly data of the influencing factors used in the annual and quarterly sales forecasts are difficult to collect, and it is inconvenient to do multiple regression analysis. Therefore, academia usually uses the time-series method to predict the monthly sales of cigarettes and summarizes the monthly sales forecasts to predict quarterly or annual sales.

Time-series analysis is the theory and method of establishing mathematical models through curve fitting and parameter estimation based on the time-series data obtained from system observations. Time-series forecasting is simple and practical, the existing literature has proposed a variety of methods for double trend time-series forecasting, and the most common one is the autoregressive moving average model, which requires time-series data to be stationary after differentiation, in addition, when doing multi-step forecasting. It is easy to bias toward the average value just like the exponential model and the threshold regression model, resulting in large errors; BP neural network has also been widely used, but it often ignores some huge noise or non-stationary data, ignores the overall growth trend of the sequence, and makes its prediction results generally lower than actual observations [20]. Although the call blocking probability model is structurally isotropic, the BP neural network has been partially improved, however, the cyclical volatility of the dual trend time series is ignored, and its forecasting effect is relatively poor. The gray G(1, 1) model can only fit the trend part of the time series well, but for periodic volatility, its prediction accuracy is significantly reduced. The traditional moving average method and exponential smoothing method often have lag errors.

4.1 Prediction by time-series decomposition method

According to the steps of the time-series decomposition method in the second section, the original data from 2015.01 to 2020.12 are analyzed and forecasted. Step 1: Calculate the long-term trend T, draw the data after moving average and centralization processing into a TC scatter diagram (Figure 5), and perform curve estimation.

Figure 5

TC scatter plot of monthly cigarette sales.

Figure 5 shows that the entire image is a straight line with a negative slope, therefore, consider the time t① as the independent variable, use TC as the dependent variable to do a linear fitting of sales T, according to the processed cigarette sales data, the regression equation of the model fitted with Eviews is:

The regression coefficients are statistically significant at the 95% confidence level. The coefficient of determination R ² = 0.983582 of the regression equation indicates that the regression model has a high degree of fit. The P value of the F test is close to 0, which means that the model is statistically significant [27,28].

Step 2: Calculate the seasonal index, as shown in Table 2.

Step 3: Get the predicted value. Calculate T according to formula (4), and then calculate the final predicted value according to formula (6). In the monthly forecast results, the average relative error ② is 4.451%, and the range is between 0.134 and 29.499%; among them, the relative error of 89.286% falls within the 10% interval. The comparison between monthly sales and forecast results is shown in Figure 6.

Figure 6

Comparison of the effect of the monthly sales forecast.

As can be seen from Figure 6, the time-series decomposition method can effectively simulate the seasonal and periodic characteristics of cigarette sales based on historical sales data, but the error of some series points is too large. In the prediction results, there are 28 sequence points with an error greater than 5%; among them, points greater than 10% basically occur in December, January, and February (marked by underline), and the rest are distributed in April, May, August, and September. The sales data still include the influence of certain fixed factors, causing the forecast error to be too large in a fixed period; these sequence points are similar to traditional Chinese festivals, so it can be inferred that the forecast errors in the previous few months are relatively large, mainly because of the influence of traditional lunar festivals (such as Spring Festival, Dragon Boat Festival, and Mid-Autumn Festival) [29,30]. Traditional festivals are different from the solar calendar, and the data are summarized based on the solar calendar; therefore, there is an abnormality during traditional festivals that is related to this, but this factor interferes with the cyclical law, and the forecast error will fluctuate randomly with the distribution of the lunar calendar, which reduces the reliability of the forecast.

By comparing the above experiment analysis: (i) based on BP neural network, prediction model is indeed a considerable prediction method, simply increasing a certain number of hidden layers is conducive to the model prediction accuracy, but also increasing the complexity of the model, reducing the model training efficiency, and by introducing the dynamic learning rate in the model training process can improve the model training efficiency to a certain extent. (ii) After using the GA optimization, the training speed of the neural network model has been greatly improved, the convergence speed of the model is accelerated, and the prediction accuracy of the model is also improved. (iii) In the experiment, the GRNN neural network was used to establish a time-series prediction model to predict the cigarette sales volume. Compared with the GRNN model and the GA-BP neural network prediction model, according to the two model evaluation indicators of the average absolute error and the mean square error, the accuracy of the GRNN prediction model is slightly lacking. Therefore, the GA-BP neural network prediction model studied in this article has a high feasibility.