Forecasting model for the incidence of hepatitis A based on artificial neural network

Abstract

AIM: To study the application of artificial neural network (ANN) in forecasting the incidence of hepatitis A, which had an autoregression phenomenon.

METHODS: The data of the incidence of hepatitis A in Liaoning Province from 1981 to 2001 were obtained from Liaoning Disease Control and Prevention Center. We used the autoregressive integrated moving average (ARIMA) model of time series analysis to determine whether there was any autoregression phenomenon in the data. Then the data of the incidence were switched into [0,1] intervals as the network theoretical output. The data from 1981 to 1997 were used as the training and verifying sets and the data from 1998 to 2001 were made up into the test set. STATISTICA neural network (ST NN) was used to construct, train and simulate the artificial neural network.

RESULTS: Twenty-four networks were tested and seven were retained. The best network we found had excellent performance, its regression ratio was 0.73, and its correlation was 0.69. There were 2 input variables in the network, one was AR(1), and the other was time. The number of units in hidden layer was 3. In ARIMA time series analysis results, the best model was first order autoregression without difference and smoothness. The total sum square error of the ANN model was 9090.21, the sum square error of the training set and testing set was 8377.52 and 712.69, respectively, they were all less than that of ARIMA model. The corresponding value of ARIMA was 12291.79, 8944.95 and 3346.84, respectively. The correlation coefficient of nonlinear regression (R_NL) of ANN was 0.71, while the R_NL of ARIMA linear autoregression model was 0.66.

CONCLUSION: ANN is superior to conventional methods in forecasting the incidence of hepatitis A which has an autoregression phenomenon.

Key Words: $[Keywords]

INTRODUCTION

Hepatitis A is an important infectious disease in the world[1-4]. Guidelines of vaccination versus hepatitis A call for correct estimation of the incidence of hepatitis A[5-9]. The occurrence of infectious diseases has its rules, affected by the speed of pathogen variation, susceptible accumulation and environmental changes. Early recognition of the epidemic rules is significantly important for the prevention and control of hepatitis. Among the available models, time series analysis[10,11] and regression analysis[12-14] are poorly suited for discovering the epidemic rules. The reason for this lies in the complexity of their relationship.

However, artificial neural networks (ANN) can recognize the rules to make right prediction and provide assistance for decision-making because they have the characteristics of self-organizing and self-learning processes[15-18]. Artificial neural networks are computation systems that process information in parallel, using a large number of simple units, and that excel in tasks involving pattern recognition. These intrinsic properties of the neural networks have been translated into higher performance accuracy in outcome prediction compared with expert opinion or conventional statistical methods[19-22].

Recently, backpropagation (BP) artificial neural network has been widely applied to a variety of problems in the field of medicine[23-26]. It is also called the front forward network because it is the front forward error propagation network without feedback. There is no mutual connection among the neurons in the same layer.

In the present work, we applied several artificial neural networks to the forecast of the incidence of hepatitis A. We also represented the basic principles and how to train the neural networks.

MATERIALS AND METHODS

Materials

The data of the incidence of hepatitis A in Liaoning Province from 1981 to 2001 were obtained from Liaoning Disease Control and Prevention Center. The highest incidence of hepatitis A during this period was 160.07 per one hundred thousand, and then the incidence descended progressively from 1981-1986 and rose from 1986 to 1989 (Figure 1). The low incidence could be seen from 1996. To switch these data into [0,1] intervals, we applied the following transformation: the network theoretical output was equal to the incidence of hepatitis A/200 by using 1/10⁵ as the unit of measurement.

Figure 1 Incidence of hepatitis A in Liaoning Province, China.

Construction, training and simulation of the network

Automatic network designer The automatic network designer determined a suitable architecture, using a combination of heuristics and sophisticated optimization strategies. It conducted a large number of tests, which were used to determine the best architecture. It could automatically compare linear, probabilistic neural network (PNN), generalized regression neural network (GRNN), radial basis function and multilayer perceptron networks (Figure 2), and automatically choose the smoothing factor for PNN or GRNN, the smoothing factor and the number of units for radial basis function networks, and the number of units for multilayer perceptrons. We could control the length and rigor of search iteration to be performed, and bias the search towards smaller networks using a unit penalty factor.

Figure 2 Topological structure of ANN.

Intelligent problem solver In ST NN, an intelligent problem solver was used to select the most excellent network. We chose the advanced version to customize the design process, the problem needed to be solved was a time series problem. Because many time series had a natural period, steps parameter equals to 1 was indicated here by ARIMA analysis. The output variables were transformed values of incidence of hepatitis A, and the input variables were time and AR (1) which meant former transforming values. The software could search for a useful subset of the specified variables. The cases in the data set were divided into three subsets, one for training the network, one for cross verification and the last one for testing. In practice, the data from 1981 to 1992 were used to train the network and the data from 1993 to 1997 were used as a verifying set. The performance of the network could be tested by estimating the data from 1998 to 2001.

Four types of networks, such as linear network, multilayer perceptron (3 layers), probability network or GRNN and radial basis function, needed to be considered. In the mean time, network complexity was determined automatically. A medium length of design procedure was selected, which conducted a fast search for an optimal network. The network saved would be one with balance performance against types and complexity. The best network found (taking account of diversity) was retained.

Choosing the number of hidden layers and units was the most critical problem. There were a few heuristics that could guide here. One hidden layer was sufficient for most problems when multilayer perceptrons were used. If the error could not be gotten down to an acceptable level even with a large number of hidden units, it might be worth trying two hidden layers.

A large number of hidden units could generally model a more complex problem, but required more training and were more prone to poor generalization. The changes in training and verification errors should be observed as the experiment was being done. If the addition of more hidden units caused a decrease in both, then the network was probably too small. However, if the verification error was significantly larger than the training error and, in particular, if it deteriorated during iterative training, then the network was probably too large.

The number of weights and thresholds in the network should be less than or much less than the number of training cases. Ideally, there should be about two to five as many training cases as weights. If the number of training cases was small, the smaller networks should be used because of no enough data to model a complex function. If the cases were fewer than the product of the number of inputs and outputs, only a linear model should be used. Lack of sufficient data was one reason why it was sometimes good to remove input variables. Even if the variables had some genuine information, the reduction in network size consequent upon removing them might improve network performance.

Performance statistic index

One index to evaluate the performance of networks and ARIMA was sum square of error (SSE):Math 1. The other was nonlinear determinant coefficient: Math 2, where y_i is the real value of case i and ＾y_i is the estimated value of case i.

Math 1

Math 1 Math(A1).

Math 2

Math 2 Math(A1).

RESULTS

Network structure and prediction

Finally 24 networks were tested, and 7 were retained. The best network we found was BP network, which had excellent performance, its regression ratio was 0.73, and its correlation was 0.69.

The structure of the network is shown in Figure 3. There were 2 input variables, one was AR (1), the other was time. The number of units in hidden layer was 3. The weight and threshold distribution of the best network is shown in Table 1.

Table 1 Weight and threshold distribution of the best network.

	Unit 1	Unit 2	Unit 3	Y
Threshold	0.9607	1.0829	0.6224	0.4850
AR (1)	0.2390	0.7930	1.0584
Time	0.4902	0.4344	0.4156
Unit 1				0.5281
Unit 2				0.7572
Unit 3				1.0723

Figure 3 Topological structure of the best network.

ARIMA model coefficient and its testing results

In ARIMA time series analysis results, the best model was first order autoregression without difference and smoothness. The coefficient of AR (1) and constant were 0.83 and 81.22, respectively (Table 2).

Table 2 ARIMA model coefficient and its testing results.

Variables	Coefficient	SE of coefficient	t	P
AR (1)	0.83	0.12	6.88	5.27 × 10-6
Constant	81.22	30.94	2.62	1.91

Comparison between BP network and ARIMA

The goal of training the network was to compare the performance with conventional methods by using inverse transforming predicted value obtained from the network. Predicted value and error are shown in Table 3, and its corresponding line graph is shown in Figure 4.

Table 3 Comparison between BP network and ARIMA.

Time (yr)	Incidence (1/100000)	Predicted value from ANN	Error from ANN	Predicted value from ARIMA	Error from ARIMA
1981	160.07	-	-	-	-
1982	136.68	101.58	35.10	146.87	-10.19
1983	104.12	94.63	9.49	127.39	-23.27
1984	50.50	83.87	-33.37	100.29	-49.79
1985	32.99	62.34	-29.35	55.65	-22.66
1986	26.74	53.26	-26.52	41.07	-14.33
1987	39.94	49.11	-9.17	35.86	4.08
1988	36.75	53.72	-16.97	46.85	-10.10
1989	96.92	50.42	46.50	44.20	52.72
1990	91.33	73.84	17.49	94.29	-2.96
1991	79.52	70.90	8.62	89.64	-10.12
1992	75.31	64.60	10.71	79.81	-4.50
1993	85.57	61.39	24.18	76.30	9.27
1994	62.54	64.00	-1.46	84.84	-22.3
1995	28.23	52.11	-23.88	65.67	-37.44
1996	29.14	35.20	-6.06	37.11	-7.97
1997	42.90	34.36	8.54	37.86	5.04
1998	21.25	38.01	-16.76	49.32	-28.07
1999	34.36	27.27	7.09	54.66	-20.30
2000	19.31	30.87	-11.56	59.11	-39.80
2001	39.09	22.88	16.21	62.81	-23.72

Figure 4 Incidence of hepatitis A and its predicted value.

The total sum square error of the ANN model was 9090.21, the sum square error of the training set and testing set was 8377.52 and 712.69, respectively, while the corresponding value of ARIMA was 12291.79, 8944.95 and 3346.84, respectively. The nonlinear determinant coefficient of ANN and the ARIMA linear autoregression model was 0.71 and 0.66, respectively.

DISCUSSION

From the above results, we can see that there was a first order autoregression phenomenon in the linear autoregression model. The nonlinear determinant coefficient of the ARIMA linear autoregression model was 0.66. The regression graph shows that there was an obvious linear tendency, but some values were still not perfectly fitted. Differences in the amount were great between values. The sum square error of the model was 12291.79, the sum square error of the training set and testing set was 8944.95 and 3346.84, respectively. While in the model based on artificial neural networks, the nonlinear determinant coefficient was 0.71, higher than that of linear autoregression model. The sum square error of the ANN model was 9090.21, the sum square error of the training set and testing set was 8377.52 and 712.69, respectively. In time series analysis, there was statistical significance in the test of coefficients, showing the existence of first order autoregression. Values from extrapolation forecasting based on ANN were mainly in concordance with the actual values, while values from ARIMA model forecasting were a little higher than the actual values of the incidence. These findings showed that ANN could recognize some rules by preliminary learning, but the effect of some special values over considering sum square error made the forecasting of low incidence inaccurate. Combined application of the traditional methods and artificial neural networks can make use of each other’s merits and raise the right prediction level.

Forecasting the incidence of infectious diseases is very important for their prevention and control. Solving this problem calls for the ability to learn the unknown mapping or function only by the existing examples[27,28]. However, after the mapping or a similar one is learned, artificial neural networks can be used to estimate the mapping when only several parts are known. Now artificial neural networks have been widely accepted as a potentially useful way in modeling complex nonlinear and dynamic systems[29-31]. Neural networks could remove neither the need for knowledge nor prior information about the systems of interest[32]. They just reduce the model’s reliance on the prior information while totally removing the need for the model builders to correctly specify the precise functional forms of the relationship that the model seeks to represent[33]. In addition, they offer real prospects for a cheaper, more flexible, less assumption-dependent and adaptive methodology.

Generally, there are two major ways to train the networks, one seeking high accuracy, and the other considering both the accuracy and performance. Through the former method, mapping was input from N-dimensional space and output to M-dimensional space. Thus, the phenomenon of overlearning occurred and the results of extrapolation were not perfect[34]. In the present work, we divided the original data into three parts, one for training, one for verification, and one for extrapolation and prediction. Thus the problem of overlearning was solved effectively. Time series analysis showed that there was a first order autoregression phenomenon in the data without exponent smoothing. Autoregression phenomenon might often exist in the data of the incidence. Traditional method of solving this is time series analysis[35]. However, it is difficult to practice because it requires to meet several conditions, such as white noise[36]. There are no such limitations in artificial neural network, and the software could voluntarily choose appropriate order of autoregression to fit and predict. In addition, R_NL has shown that it is reasonable to employ artificial neural networks in modeling complex nonlinear data.

In conclusion, ANN is likely to be an effective tool in processing time series data. The construction and explanation of the model should be further explored.