Comparisons of forecasting for hepatitis in Guangxi Province, China by using three neural networks models

Materials

The incidence of hepatitis data, including hepatitis A, B, C, and E, were collected on a monthly base from the Chinese National Surveillance System (http://www.gxhfpc.gov.cn/xxgks/yqxx/yqyb/list_459_2.html) and the Guangxi Health Information Network (http://www.phsciencedata.cn/Share/ky_sjml.jsp?id=8defcfc2-b9a4-4225-b92c-ebb002321cea&show=0) from January 2004 to December 2014. These data composed the time series X = {x(0), x(1), …, x(131)}. The information belongs to the government statistical data and is available to the public. Hepatitis D has not been considered because the data cannot be obtained from the Chinese National Surveillance System and the Guangxi Health Information Network. The incidence dataset between 2004 and 2013 was used as the training sample to fit the model, and the dataset in 2014 was used as the testing sample.

Methods

Three ANN methods: BPNN-GA, WNN, and GRNN, were used for prediction and their performances were compared.

BPNN-GA model

BPNN is a multi-layered feed-forward neural network; the main features are that the signal transports forward, and the error transports backward. The input signal will be processed layer-by-layer from the input layer to the output layer. The next state of the neuron is only affected by the front state of the neuron in the layer. If the expected output was not received, the weights and the thresholds of the network will be adjusted by the error that transports backward. Therefore, the desired output will be achieved in an iterative manner (Ramesh Babu & Arulmozhivarman, 2013).

If the model of BPNN has i input nodes, j hidden nodes, and k output nodes, there will be weight variables of dimensionality N = i × j between the input layer and the hidden layer, j × 1 threshold variables in hidden layer, weight variable of dimensionality M = j × k between the hidden layer and the output layer, and k × 1 threshold variables in the output layer (Huang, Gong & Gong, 2015). The topology structure is shown in Fig. 1.

Genetic algorithm (GA) is a search heuristic that mimics the process of natural selection. This heuristic is routinely used to generate useful solutions to optimization and search problems (Mitchell, 1996). The initial weights and thresholds of BPNN are optimized by GA, which is called the BPNN-GA method. The algorithm of the BPNN-GA flow chart is shown in Fig. 2.

WNN model

WNN is a kind of neural network with a structure that is established on the basis of BPNN, and the wavelet basis function is taken as the transfer function is in hidden layer nodes. The signal also transports forward and the error transports backward. The topology structure is shown in Fig. 3. WNN includes two new variables, a scale factor, a displacement factor, which give it excellent functional approximation. The WNN method is composed of relatively less expensive terms that often has fast functional approximation abilities and good predicting precision (due to its ability to sift out the parameters). Compared to BPNN, the weight coefficient of WNN has the characteristics of linearity, and the objective function of learning has the feature of convexity. These properties will avoid being nonlinear in local optimization when the network is trained (Antonios & Zapranis, 2014; Alexandridis & Zapranis, 2013).

The formula for calculating the hidden layer for an input signal sequence is x_i(i = 1, 2, …, k) is as follows: $h(j)=h_j\left[\frac{{\displaystyle \sum _{i=1}^k{\omega }_{ij}{x}_i-b_j}}{a_j}\right]j=1,2,\mathrm{...},l$ where k is the number of input signal; l is the number of nodes in the hidden layer; h(j) is the output of the jth node in the hidden layer; h_j is the wavelet basis function; ω_ij is the weights between the input layer and hidden layer; a_j is the scale factor of h_j and b_j is the displacement factor of h_j.

The formula to calculate the output layer is as follows. $y(k)={\displaystyle \sum _{i=1}^l{\omega }_{ik}}h(i)k=1,2,\mathrm{...},m$ where h(i) is the output of the ith node in the hidden layer; l is the number of nodes in the hidden layer; m is the number of nodes in the output layer; and ω_ij is the weight between the hidden layer and the output layer.

The weights of the network, the scale factor, and the displacement factor were estimated by the steepest descent method in WNN. The correction process of prediction used by WNN follows.

• Step 1. Calculate error of prediction. $e={\displaystyle \sum _{k=1}^{m}yn(k)}-y(k)$ where yn(k) is the expected output, namely the true value. y(k) is the forecasting output.

• Step 2. Correct the weight of the network and the coefficients of wavelet basis function according to the prediction error. ${\omega }_{n,k}^{(i+1)}={\omega }_{n,k}^i+\Delta {\omega }_{n,k}^{(i+1)}$ $a_k^{(i+1)}=a_k^i+\Delta a_k^{(i+1)}$ $b_k^{(i+1)}=b_k^i+\Delta b_k^{(i+1)}$ where $\Delta {\omega }_{n,k}^{(i+1)}=-\eta \frac{\partial e}{\partial {\omega }_{n,k}^{(i)}}$ $\Delta a_k^{(i+1)}=-\eta \frac{\partial e}{\partial a_k^{(i)}}$ $\Delta b_k^{(i+1)}=-\eta \frac{\partial e}{\partial b_k^{(i)}}$ and η is the learning rate.

The algorithm of WNN flow chart is shown in Fig. 4.

GRNN model

GRNN is a memory-based network that provides estimates of continuous variables and converges to the underlying (linear or nonlinear) regression surface (Specht, 1991). One advantage of it is the simplicity. The adjustment of one parameter, namely, the spreading factor, is sufficient for determining the network.

The topology structure of GRNN consists of four layers: the input layer, the pattern layer, the summation layer, and the output layer. The topology structure is shown in Fig. 5.

The number of neurons in the input layer is equal to the dimension of the input vector of the learning samples. Every neuron in the input layer is the simple distribution unit and directly transmits the input variables to the pattern layer.

The neurons in the pattern layer and the neurons in the input layer have the same number and every one of the neurons in the pattern layer corresponds to a different sample. The transfer function of neurons in the pattern layer is as follows: $P_i=e^{-\frac{{(X-X_i)}^T(X-X_i)}{2{\sigma }^2}}i=1,2,\mathrm{...},n$ where P_i is the output of neurons in the pattern layer; X = [x₁, x₂, …, x_n] is the input vector; X_i is the learning samples of the i-th neurons; n is the number of input; i is the number of neurons; and σ is the smoothness factor.

There are two kinds of summation for the neurons in the summation layer. The first one is that the arithmetic sum is calculated for the output of neurons in the pattern layer. The weight between the pattern layer and every neuron is 1. The transfer function is shown in formula as follows.

The second one is the weighted sum performed for the output of neurons in the pattern layer. The weight between the i-th neuron in the pattern layer and the j-th summation neuron is equal to the j-th element in the i-th output samples of Y_i. The transfer function is given below: $S_{Nj}={\displaystyle \sum _{i=1}^n{Y}_{ij}{P}_i={\displaystyle \sum _{i=1}^n{Y}_i}}{e}^{-\frac{{(X-X_i)}^T(X-X_i)}{2{\sigma }^2}}j=1,2,\mathrm{...},k$ where k is the dimension of the output vector.

The number of neurons in the output layer is equal to the dimension of the input vector of the learning samples. The output of the j-th neurons is shown in formula as follows.

The relationship between predictions and seasonal fluctuation index

The seasonal fluctuation index of incidence is used to reveal the fluctuations of incidence with seasons. The seasonal fluctuation index of the same month in eleven years from 2004 to 2014 can be calculated as: $SFI1=\frac{\left|{\overline{x}}_{same}-{\overline{x}}_{all}\right|}{{\overline{x}}_{all}}$ where ${\overline{x}}_{same}$ is the average incidence of the same month and ${\overline{x}}_{all}$ is the average incidence of all of the months from 2004 to 2014.

The seasonal fluctuation index of the every month in 2014 is calculated as: $SFI2=\frac{\left|x_i-\overline{x}\right|}{\overline{x}},i=1,\mathrm{...},12$ where x is the incidence in each month and $\overline{x}$ is the average incidences of all of the months in 2014.

Obviously, the greater the number that the seasonal fluctuation index is, the more seasonal volatility of incidence is. That is to say, the index changes reflect the disease variation in the different months. In order to compare the relationship between the seasonal fluctuation index of incidence and the three prediction results, the relative error of prediction is defined as: $R{E}_i=\frac{\left|{\widehat{y}}_i-y_i\right|}{y_i},i=1,2,\dots ,n$ where ${\widehat{y}}_i$ is the predicted value and y_i are the observed values.

The seasonal fluctuation index of incidence and the relative error of the three prediction results are shown in Fig. 8.

Looking at Fig. 8, it can be seen that: 1) hepatitis A, B, and E have obvious seasonal characteristics. For Hepatitis B, in particular, the incidence which happens annually in January and February is relatively high with a rapid decline in March. April to September is relatively stable, but from October to December it began to rise significantly; 2) the greater the seasonal fluctuation index of the every month in 2014, the greater the relative error, especially in hepatitis C and E, which shows that the greater the disease fluctuations, the worse the prediction results; 3) the absolute error of the BPNN-GA is smaller than that of the other two methods when the incidence data is stable, such as from April to August for hepatitis A; the absolute error of the GRNN is smaller than that of the other two methods when the incidence data has great fluctuation, such as March, July, and December for hepatitis C and August, October, November, and December for in hepatitis E; and 4) the absolute error of the GRNN is larger than that of the other two methods when the incidence data is larger, and the absolute error of the WNN is larger than that of the other two methods when the incidence data is smaller. The size relationship of the average incidence is: B > C > E > A. When the incidence data is large, such as the data for hepatitis B, the size relationship of the absolute error of three methods is: GRNN > BPNN-GA > WNN.

Comparison of evaluation indexes

The mean square error (MSE), root mean square error (RMSE), mean average error (MAE), mean average percentage error (MAPE), and sum of squared error (SSE), have been calculated and compared with the three methods. The performance indexes are defined as shown in the following. $MSE\text{=}\frac{1}{n}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}$ $MAE\text{=}\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|{\widehat{y}}_i-y_i\right|}$ $RMSE\text{=}\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}}$ $MAPE\text{=}\frac{1}{n}{\displaystyle \sum _{i=1}^n\left|{\widehat{y}}_i-y_i\right|}*100$ $SSE\text{=}{\displaystyle \sum _{i=1}^n{({\widehat{y}}_i-y_i)}^2}$ where ${\widehat{y}}_i$ is the predicted value and y_i are the observed value.

The main evaluation indexes that were calculated by these three methods are listed in Table 2.

From the definitions of the other evaluation indexes, including MSE, MAE, RMSE, SSE, and MAPE, we know that the smaller the values of these indexes are, the more accurate the prediction is. The BPNN-GA method had the smallest values of these evaluation indexes when it was used to predict hepatitis A and the WNN method had the smallest values of these indexes when it was used to predict hepatitis B; the same for the GRNN method when it was used to predict hepatitis E. It can be seen that the BPNN-GA and WNN methods were not superior to the others when they were used to predict hepatitis C, but they were all superior to GRNN method. According to Fig. 8, we know that: 1) hepatitis A, B, and E have a strong seasonal volatility, but hepatitis C fluctuates up and down monthly and does not have seasonal volatility; and 2) the incidence data of hepatitis A and B are the smallest and the largest, respectively. Hepatitis E increased slowly from January to December (except for March). That is to say, these three prediction methods have their advantages when they are used to predict seasonal fluctuation data. The BPNN-GA and WNN methods are suitable for predicting small and large data, respectively, while GRNN is suitable for predicting data that increases steadily. The BPNN-GA and WNN methods were not superior to the others when they were used to predict the data that fluctuated up and down monthly and does not have seasonal volatility, and the GRNN method is not suitable for predicting these types of data.

Comparison of statistical significance tests

Statistical significance of the obtained results was investigated using T-test; a p-value of < 0.05 was considered significant. The results are listed in Table 3.

The correlation will be better when the correlation coefficient is close to 1, namely, the predicted value is closer to the observed value. From Table 3, it can be seen that the BPNN-GA method has the best correlation when it was used to predict hepatitis B and C. The same in regard to the GRNN and WNN method; they had the best correlation when they were used to predict hepatitis E and A, respectively.

P < 0.01 are for all models from Table 3 which reveals that the difference is statistically significant between the predictive value and the original data.