A New Hybrid Model Using an Autoregressive Integrated Moving

A New Hybrid Model Using an Autoregressive Integrated Moving Average and a Generalized Regression Neural Network for the Incidence of Tuberculosis in Heng County, China

Wudi Wei,1† Junjun Jiang,1† Lian Gao,2 Bingyu Liang,1 Jiegang Huang,1 Ning Zang,1,3 Chuanyi Ning,1,3 Yanyan Liao,1,3 Jingzhen Lai,1 Jun Yu,1 Fengxiang Qin,1 Hui Chen,4 Jinming Su,1 Li Ye,1* and Hao Liang1,3* 1Guangxi Key Laboratory of AIDS Prevention and Treatment and Guangxi Universities Key Laboratory of Prevention, Guangxi Medical University, 22 Shuangyong Road, Nanning, Guangxi, China; 2Department of Infectious Diseases, Heng County Centers for Disease Control and Prevention, 16 Gongyuan Road, Heng County, China; 3Life Sciences Institute, Guangxi Medical University, 22 Shuangyong Road, Nanning, China; 4Geriatrics Digestion Department of Internal Medicine, The First Afﬁliated Hospital of Guangxi Medical University, 6 Shuangyong Road, Nanning, China

Abstract. It is a daunting task to eradicate tuberculosis completely in Heng County due to a large transient population, human immunodeficiency virus/tuberculosis coinfection, and latent infection. Thus, a high-precision forecasting model can be used for the prevention and control of tuberculosis. In this study, four models including a basic autoregressive integrated moving average (ARIMA) model, a traditional ARIMA–generalized regression neural network (GRNN) model, a basic GRNN model, and a new ARIMA–GRNN hybrid model were used to fit and predict the incidence of tuberculosis. Parameters including mean absolute error (MAE), mean absolute percentage error (MAPE), and mean square error (MSE) were used to evaluate and compare the performance of these models for fitting historical and prospective data. The new ARIMA–GRNN model had superior fit relative to both the traditional ARIMA–GRNN model and basic ARIMA model when applied to historical data and when used as a predictive model for forecasting incidence during the subsequent 6 months. Our results suggest that the new ARIMA–GRNN model may be more suitable for forecasting the tuberculosis incidence in Heng County than traditional models.

INTRODUCTION which is generally used to analyze the nonlinear data sets, could also be used.16–19 Combination models using the basic Guangxi Zhuang Autonomous Region has the second 1 assumptions of both ARIMA and GRNN have been used for largest burden of tuberculosis cases in China. Tuberculosis predicting disease incidence rates. The previous studies have fi has signi cant economic impact in Guangxi and may be an also shown that a combined model may provide the advan- ’ 2 obstacle to Guangxi s development. The tuberculosis epi- tages from both earlier models and increase forecasting demic of Heng County is particularly problematic in Guangxi. accuracy.20–23 However, Wu and others24 found that the fi The morbidity of tuberculosis was signi cant, and ranked predictions of the ARIMA–GRNN model were not as good as second in Nationally Notifiable Infectious Diseases of the 3,4 those of the ARIMA model for the incidence of hemorrhagic county. Heng County is a key area in the Guangxi Beibu Gulf fever with renal syndrome. Therefore, traditional ARIMA– Economic Zone, with a large transient population that pro- 5,6 GRNN models may not be useful for predicting the incidence motes the spread of tuberculosis. Currently, the prevalence of all infectious diseases, and the traditional hybrid models fi of human immunode ciency virus (HIV)-tuberculosis coin- may be improved under some circumstances. fections represent a serious new challenge to the public health To our knowledge there is no research with respect to system. In 2011, about 1.1 million (13%) of the 8.7 million forecasting the incidence of tuberculosis using these statis- people infected with tuberculosis worldwide were HIV- 7–9 tical models. In this study, we developed ARIMA models, positive. In addition, it is estimated that about one-third of – – 10 traditional ARIMA GRNN models, and new ARIMA GRNN people who are infected with tuberculosis are latent carriers. models to develop an optimal model for tuberculosis in- Furthermore, multidrug-resistant tuberculosis is prevalent, 11,12 cidence forecasting in Heng County. We expect that the especially in rural areas. To cope with the challenges to model will play an important role in prevention and control of the public health system posed by the spread of tuberculosis, tuberculosis. a series of preventative measures should be used in locations such as Heng County. Among these measures, a highly ac- curate epidemiologic model of the disease that can predict the MATERIALS AND METHODS trends in incidence that can help the public health authorities Materials source. We obtained all the necessary data in- control the epidemic would be useful. cluding the incidence of tuberculosis from January 2005 to At this time, the existing mathematical models, specifically December 2013 and population information from the Heng the autoregressive integrated moving average (ARIMA) model County Center for Disease Control and Prevention (CDC) and that is widely used to predict incidence of infectious – the Heng County Statistics Bureau. All tuberculosis cases diseases,13 15 are based on linear assumptions. Moreover, must be reported to the Heng County CDC through the net- the generalized regression neural network (GRNN) model, work system within 12 hours of diagnosis. Thus, the tuberculosis incidence data for Heng County are known to be reliable and complete because there is compulsory reporting. * Address correspondence to Hao Liang, Life Sciences Institute, The original data can be gained from the corresponding Guangxi Medical University, Guangxi, China, E-mail: lianghao@ author. gxmu.edu.cn or Li Ye, Guangxi Key Laboratory of AIDS Prevention and Treatment, School of Public Health, Guangxi Medical University, We divided the data into two parts so that we could compare Guangxi, China, E-mail: [email protected]. the modeling and predictive performances of the two com- † These authors contributed equally to this paper. bined models and the ARIMA model. The incidence rates from 799 800 WEI, JIANG AND OTHERS

FIGURE 1. Monthly incidence of tuberculosis from January 2005 to June 2013.

January 2005 to June 2013 were used to fit the models, regulation parameter, the smoothing factor. The equivalent whereas the rest were used to validate the four models. nonlinear regression formula shows how the GRNN model Any ethical considerations in this study were limited. No works: informed consent was required or was feasible because all the ! data used for the modeling came from publicly accessible n D2 + Y exp i secondary data sources, and no personal identifiers were i s2 ^ i¼1 2 entailed. YðXÞ¼ ! ARIMA model construction. n D2 The ARIMA (p,d,q) (P,D,Q)s + exp i s2 model is a frequently used time series model that contains i¼1 2 seven different parameters: 1) the order of autoregression (p), h i 2 ¼ ð ÞT ð Þ 2) the degree of difference (d), 3) the order of the moving av- Di X Xi X Xi erage (q), 4) the seasonal autoregression lags (P), 5) the degree ... of seasonal difference (D), 6) the seasonal moving average where X means the input vector (X1, X2, , Xi), which consists ^ lags (Q), and 7) the length of any cyclical pattern(s). We used of n predictor variables; Y denotes the output values predicted ^ ^ the augmented Dickey-Fuller (ADF) unit root test to first de- by the GRNN; YðXÞ is the expected value of the output Y given 26 termine the stationary value of the data. Second, log trans- an input vector X; and s is the smoothing factor. Thus, the formation and differencing were used to maintain the stationary smoothing factor is very significant contributor in the GRNN time series if the sequence is not stationary.23 Finally, we used model. the Box–Jenkins strategy to construct the ARIMA models in- In this study, the forecasted incidence of the ARIMA model cluding three steps: identification, estimation, and diagnostic and the actual incidence were used as the input and output checking.25 The R2, the Schwarz–Bayesian information crite- variables, respectively, to construct the traditional ARIMA– rion, and the Akaike information criterion were used to select GRNN model. The smoothing factor is the only parameter of the most appropriate model.24 the model that directly affects the precision of the predictions. Construction of the traditional ARIMA–GRNN model. Specht18 proposed a method to find the optimal smoothing The GRNN model is one of the various artificial neural network factor. We randomly selected two samples (each sample is models that were developed by Specht18; it contains four likely to be selected, and the probability is the same) as the layers, input layer, pattern layer, summation layer, and output testing data to develop the GRNN model. Subsequently, the layer. Compared with other networks, the GRNN model has model was examined by a series of variables to find the opti- strong ability for nonlinear mapping and good learning ability mal smoothing factor for which the minimum mean square skills. The forecasting performance of the network is good and error (MSE) of the model was the lowest. Finally, the stable even if the sample size is small. Moreover, the construction of the GRNN model is straightforward and only has a TABLE 2

Estimate parameters of the ARIMA (0,1,2)(2,1,2)12 model Variable Coefﬁcient t statistic P value TABLE 1 The AIC and SBC values of the ﬁve ARIMA models MA(1) −0.3060 −2.3990 0.0200 Model AIC SBC R2 MA(2) 0.2760 2.1760 0.0340 SAR(12) −0.4250 −3.2490 0.0020 ARIMA(1,1,0)(2,1,1)12 0.0811 0.2161 0.4734 SAR(24) −0.6060 −5.2310 0.0000 ARIMA(0,1,0)(2,1,1)12 0.1353 0.2357 0.4274 SMA(12) 0.4430 3.2100 0.0020 ARIMA(0,1,2)(2,1,2) −0.0080 0.1927 0.5476 12 MA(1) = moving average, lag1; MA(2) = moving average, lag2; SAR(12) = seasonal moving AIC = Akaike information criterion; ARIMA = autoregressive integrated moving average; average, lag12; SAR(24) = seasonal moving average, lag24; SMA(12) = season moving SBC = Schwarz Bayesian information criterion. average, lag12. A NEW HYBRID MODEL FOR TUBERCULOSIS INCIDENCE IN HENG COUNTY 801

forecasting data, the actual incidence, and the predicted absolute error from GRNN model, respectively. The flow chart of the new hybrid model was presented in the Supplemental Appendix 2. Construction of the GRNN model. The GRNN model with N-dimensional input and one-dimensional output was used to predict the absolute errors. As mentioned earlier, the smoothing factor directly affects the prediction precision of model. Thus, the two parameters (the N and the smoothing factor) play an important role in constructing the GRNN model. The values of N can be obtained graphically or via a series of tests. To determine the best value of the two parameters, the initial data in the set were used to develop the model, and the last two samples were used for data to test the model. We then developed several GRNN models, with a series of smoothing factors and values of N, to find the most suitable model in which the MSE was the lowest. The flow chart of the basic FIGURE 2. The selection of the traditional autoregressive integrated – GRNN model is presented in the Supplemental Appendix 3. moving average generalized regression neural network (GRNN) Comparison with the four models for performance model. The smoothing factor between 0.001 and 0.010 with an interval of 0.001 were selected to find the minimum mean square error (MSE) for the in simulations. We selected mean square error (MSE), mean GRNN model. The GRNN model had lowest MSE when the smoothing absolute error (MAE), and mean absolute percentage error factor was 0.004. (MAPE) as model evaluation indexes to estimate the fitting and prediction effects of the four models. EViews 8.0 (IHS Global, predictions made by ARIMA model were used as the input Inc.,Southeastern,PA) was used to create the ARIMA model, values of the combined model, and then those values were and the traditional and new ARIMA–GRNN models were optimized. These values were used as the forecasting values performed with Matlab2012b (MathWorks, Natick, MA). of the ARIMA–GRNN model. The flow chart of the traditional hybrid model is presented in the Supplemental Appendix 1. RESULTS Construction of the new ARIMA–GRNN model. Unlike traditional ARIMA–GRNN model construction, in the new hy- Basic ARIMA model. We used the monthly incidence from brid models, the absolute error sequence from the ARIMA January 2005 to June 2013 to develop the model (Figure 1). model is first generated with the formula et = Yt At, where The ADF test showed that the sequence was not stationary et, Yt, and At denote the absolute error, the actual incidence, (ADF test: t = −1.7075, P = 0.0830). A log transformation, and the forecasting incidence, respectively. Subsequently, nonseasonal (d = 1), and seasonal difference (D = 1) were used this absolute error sequence was used to construct the GRNN to eliminate numerical instabilities. Finally, the time series was model. We used GRNN model that was developed to predict stationary (ADF test: t = −12.7095, P < 0.0001). the absolute error of the next 6 months’ tuberculosis incidence We developed a series of possible ARIMA models to de- rate. Finally, we obtained the final forecasting data through the termine the best version. After removing the models that did formula Ht = Yt1 et1, where Ht, Yt1, and et1 denote the final not pass the parameter or residual test, three models were

FIGURE 3. The absolute error sequence from February 2008 to June 2013. 802 WEI, JIANG AND OTHERS

FIGURE 4. The selection of the basic generalized regression neural network (GRNN) model (the value of N was 37). The smoothing factor between 0.1 and 3.0 with an interval of 0.1 was selected to ﬁnd the minimum mean square error (MSE) for the basic GRNN model. The GRNN model has lowest MSE when the smoothing factor came to 0.3.

selected: ARIMA (1,1,0)(2,1,1)12, ARIMA (0,1,0)(2,1,1)12, and from the first sequence. The absolute error of November 2012 ARIMA (0,1,2)(2,1,2)12. The parameters of the three models are and December 2012 were used as validation samples and the shown in Table 1. We selected the ARIMA (0,1,2)(2,1,2)12 rest of these data were used to model the model. We chose the model as the best one according to the results of the testing of sequence number of the cycles as the N value (N = 37). After parameters. The residual test of this model showed a white many repeated tests, when the smoothing factor was 0.3, the noise sequence (P > 0.05). The parameter test results are model had the lowest MSE (Figure 4). shown on Table 2. Basic GRNN model. We also used the monthly incidence Traditional ARIMA–GRNN model. The forecasting in- rates from January 2005 to June 2013 to develop the GRNN cidence values from the ARIMA model were selected as inputs model. The incidence rates from May to June 2013 were used of the GRNN model, and the actual incidence values were to test the model, and the balance of the incidence data used for subsequent testing of the final model. To determine were used to refine the model. A smoothing factor of 0.1–3.0 the best smoothing factor, the incidence of January 2005 and with an interval of 0.1, and an N value of 1–98 with an interval May 2013 were randomly used as the testing samples for the of 1, were selected to find the minimum MSE for the GRNN GRNN model. When the value of the smoothing factor was model. When the combinations of smoothing factor and N 0.004, the MSE of the model was lowest (Figure 2). Thus, we value approached 1.6 and 26, respectively, the minimum developed the GRNN in which the smoothing factor was MSE of the GRNN model was the lowest. Then the de- 0.004. Then we used the constructed hybrid model to predict veloped GRNN model was used to forecast the incidence of the disease incidence from July to December 2013. next 6 months. New ARIMA–GRNN model. The absolute error sequence Comparison with the four models in simulation from February 2008 to June 2013 is shown in Figure 3. The performance. Four models were selected to forecast the absolute error sequence contained two periodic sequences: tuberculosis incidence in Heng County from July 2013 to the sequence from February 2008 to December 2009 with a December 2013 (Table 3). The modeling and forecasting cycle of 6 months, and sequence from January 2010 to June curves of the four models are shown in Supplemental Figure 1 2013 with a cycle of 37 months. We selected the second se- and Figure 5. The performance parameters of the four models quence to construct the GRNN model to prevent interference in fitting and validation are shown in Table 4.

TABLE 3 The forecasting TB incidence of four models Time Actual Basic ARIMA Traditional ARIMA–GRNN New AEIMA–GRNN Basic GRNN July 2013 2.6565 3.7544 4.2936 2.7970 3.9034 August 2013 3.3005 3.5109 3.9038 2.9354 3.3529 September 2013 2.4955 3.8208 4.0404 3.2987 2.5826 October 2013 3.0590 3.8648 4.0549 3.1201 4.6532 November 2013 2.5760 3.0948 2.8316 2.3182 5.6945 December 2013 2.4150 3.1007 2.8316 2.4710 5.3643 ARIMA = autoregressive integrated moving average; GRNN = generalized regression neural network. A NEW HYBRID MODEL FOR TUBERCULOSIS INCIDENCE IN HENG COUNTY 803

FIGURE 5. The forecasting curves of the four models and the actual tuberculosis incidence series. This ﬁgure appears in color at www.ajtmh.org.

DISCUSSION one-dimensional output could not make full use of the original information and might generate bias. With the new ARIMA– Until now, there were no validated prediction models for GRNN combined model, the linear and nonlinear information tuberculosis incidence in Heng County. In this study, we de- were extracted adequately. The performance of the fitting part veloped a basic ARIMA model, a basic GRNN model, a tra- and forecasting part of the model obviously improved. – – ditional ARIMA GRNN model, and a new ARIMA GRNN To our knowledge, this is the first study to develop a fi model to t the data and used these results to then predict mathematical model for Heng County’s tuberculosis in- the tuberculosis incidence rate. The MSE, MAE, and MAPE of cidence forecasting. Moreover, unlike in other regions, the – new ARIMA GRNN model were the lowest in forecasting. In- tuberculosis incidence curve of this county showed some terestingly, although the values of the three parameters of the distinctive characteristics. For instance, seasonal patterns – traditional ARIMA GRNN model were lower than the basic were not obvious and there was an obvious peak incidence in ARIMA model in the fitting stage, the forecasting parameters 2008. The specialization may make the traditional ARIMA– were the highest, results that are similar to those of previous GRNN models unsuitable for prediction. Therefore, the new 24 research. Although the basic GRNN model had apparent ARIMA–GRNN model is a powerful tool that may allow the perfect fitting ability, the forecast effect from that model was Heng County public health authorities to reduce the trans- not very good. Thus, the basic GRNN model was overfitting. mission and control outbreaks of tuberculosis. Only when This is a congenital defect and up to now we do not have a health policy makers can understand incidence trends of tu- good solution to fix it. berculosis can they distribute the necessary resources prop- The effect of new ARIMA–GRNN model was better than erly and organize effective medical services to promote the basic ARIMA model. In this new hybrid model, the liner in- health of the population of the county. If the ARIMA–GRNN formation was first extracted from the data by the ARIMA results indicate that the incidence of tuberculosis will increase, model. Unlike the traditional hybrid model, we directly focused more health resources can be mobilized in advance. Thus, on a residual sequence that contained the nonlinear in- the local government will be able to take the initiative in the formation. An improved method was used to construct the struggle against tuberculosis outbreaks with the help of the GRNN model. It gave full consideration to the two parameters new model. Moreover, the novel ARIMA–GRNN model can (the value of N and the smoothing factor) and developed the also evaluate the effects of new intervention measures, such network with a 37-dimensional input and one-dimensional as the use of a tuberculosis vaccine, directly observed anti- output to extract the information from the residual sequence. microbial treatment, or the effects of short courses of antibi- Previous studies only focused on the smoothing factor, which otic combinations, and/or other interventions. The measures – is hard to find the most suitable model.20 23,25 In this study, the will be shown to have been effective when the forecasted traditional hybrid model with a one-dimensional input and a values are higher than the actual values that are observed after

TABLE 4 The ﬁtting and forecasting performance of the four models Fitting part Validation part

Prediction error MSE MAE MAPE MSE MAE MAPE Basic ARIMA 1.5692 1.0396 0.1701 0.8558 0.7740 0.2928 Traditional ARIMA–GRNN 0.5993 0.4007 0.0720 1.0536 0.9089 0.3359 New ARIMA–GRNN 0.0007 0.0007 0.0002 0.3811 0.2826 0.1048 Basic GRNN 0.0000 0.0000 0.0000 2.7710 2.0939 0.7703 ARIMA = autoregressive integrated moving average; GRNN = generalized regression neural network; MAE = mean absolute error; MAE = mean absolute error; MAPE = mean absolute percentage error; MSE = the mean square error. 804 WEI, JIANG AND OTHERS an intervention has been tested in the population. In general, Highly Prevalent Disease, Guangxi Medical University, Nanning, Guangxi, the new combined model should make a positive contribution China, E-mails: [email protected] and [email protected]. to tuberculosis control measures and disease prevention in Heng County. After being documented, these benefits may be REFERENCES extended to other regions. Several limitations of the new hybrid model are possible. First, 1. 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