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On-line version ISSN 1518-8787Print version ISSN 0034-8910
Rev. Saúde Pública vol.38 n.3 São Paulo Jun. 2004
Maria Lúcia F Penna
Escola Nacional de Saúde Pública. Departamento de Endemias Samuel Pessoa. Rio de Janeiro, RJ, Brasil
To evaluate recurrent neural networks as a predictive technique for time-series
in the health field.
METHODS: The study was carried out during a cholera epidemic which took place in 1993 and 1994 in the state of Ceará, northeastern Brazil, and was based on excess deaths having 'poorly defined intestinal infections' as the underlying cause (ICD-9). The monthly number of deaths with due to this cause between 1979 and 1995 in the state of Ceará was obtained from the Ministry of Health's Mortality Information System (SIM). A network comprising two neurons in the input layer, twelve in the hidden layer, one in the output layer, and one in the memory layer was trained by backpropagation using the fist 150 observations, with 0.01 learning rate and 0.9 momentum. Training was ended after 22,000 epochs. We compare the results with those of a negative binomial regression.
RESULTS: ANN forecasting was adequate. Excessive mortality (number of deaths above the upper limit of the confidence interval) was detected in December 1993 and October/November 1994. However, negative binomial regression detected excess mortality from March 1992 onwards.
CONCLUSIONS: The artificial neural network showed good predictive ability, especially in the initial period, and was able to detect alterations concomitant and a subsequent to the cholera epidemic. However, it was less precise that the binomial regression model, which was more sensitive to abnormal data concomitant with cholera circulation.
Keywords: Neural networks (computer). Time series. Forecasting. Cholera, epidemiology. Epidemiologic surveillance.
The prediction of events in epidemiological surveillance is aimed at planning the future needs of Public health and at detecting of disturbances in behavior within a time series, which may indicate an excessive number of cases or deaths. Techniques employed to this end include adjustment to periodic functions, the ARIMA4 models, and Poisson's regression.1,4,19,16
One of the goals of epidemiological surveillance is the detection of epidemics. This is accomplished by analyzing either disease notification data, healthcare service usage or mortality data. From the statistical standpoint, to detect an epidemic is to detect the presence of abnormal values within a time series. The detection of abnormalities based on notification levels from the previous five years18 is used in certain countries the US, for example in routine epidemiological surveillance. When the goal is to detect excess mortality, longer periods are generally used.
Excess mortality was introduced into epidemiological surveillance in the context of evaluating the impact of influenza epidemics.5 This was due to difficulties in classifying influenza-related deaths, which were, in their majority, attributed to complications of the infection, such as pneumonia. The concept of excess mortality was also used in evaluating the impact of heat waves, atmospheric pollution, and of apparently benign epidemics.
Epidemiological surveillance in Brazil shows greater operational variability than in developed countries, which results in decreased precision (greater variance) of the data and consequently less sensitivity of the notification system in detecting new diseases and the occurrence of epidemics. Thus, the use of excess mortality techniques may prove useful for increasing the sensitivity of the surveillance system, since mortality registration has higher coverage and longer available time series than the notification system.
However, the statistical methods traditionally used require profound knowledge of statistics for the selection and evaluation of models, which prevents them from being used in a decentralized manner.3,17 Artificial Neural Networks (ANN), on the other hand, have the advantage of being applicable to several time series sequentially, without prior diagnosis of their behavior, providing good results.12,21
In most time series of interest to Public Health, variability is largely attributable to trends and seasonality. Their behavior is non-linear, and their cycles are irregular. The advantage of using neural networks to approximate non-linear functions is that this technique has been successful in analyzing series in which the mathematical knowledge of the stochastic process behind the series is either unknown, or difficult to be rationalized.2
Neural networks were developed initially as a strategy for simulating human mental processes such as image and sound recognition and later as an efficient technological instrument for performing a large number of different tasks.
A neural network is composed of neurons, or nodes, ant their connections, which may be inputs or outputs with respect to each individual neuron. The node or neuron is the site of mathematical processing. Such processing is divided into two steps: the weighted addition of inputs (S WiXi) and the submission the result to an activating function, which may generate an input for other neurons. This function is usually a logistic function or a hyperbolic tangent. Such functions are sigmoidal in shape, with very little variation for extreme values of x, which simulates the saturation of a biological neuron when incoming stimuli are too intense.
ANNs may be described as a strategy for the mathematical modeling of problems, which are conceived as systems with inputs and outputs. Unlike in other modeling strategies, one does not need to know the mathematical relationship between inputs and outputs. Thus, unlike multiple regression, the ANN model does not require a function to be proposed in advance,11 since certain neural networks are capable of approximating any function whatsoever. The network would therefore be able to solve any problem whose relationships it can represent.
Our aim is to evaluate the appropriateness of the use of recurrent neural networks as a predictive technique for time series in Public Health.
The study was carried out during a cholera epidemic which took place in 1993 and 1994 in the state of Ceará, northeastern Brazil, and was based on excess deaths with 'poorly defined intestinal infections' as the underlying cause (ICD-9).
Excess mortality was attributed to non-diagnosed lethal cases of cholera.8 Ceará was chosen for having presented, for two consecutive years, the highest yearly incidence rates of cholera ever registered in Brazil namely 346.19 and 302.74 per 100,000 population, in 1993 and 1994, respectively.
The monthly number of deaths in the state of Ceará between 1979 and 1995 with "poorly-defined intestinal infections" (ICD-9) as the underlying cause were obtained from the Ministry of Health's Mortality Information System.
As a comparison we chose negative binomial regression, a statistical technique widely employed for this type of data in cases where Poisson regression proves inadequate.15
A recurrent network was built with two neurons in the input layer (corresponding to year and month), twelve in the hidden layer, one in the output layer, and one in the memory layer. The recurrent connection connects the output layer to the memory layer, which, in its turn, is connected to the hidden layer. All activating functions were based on the logistic function. Training was carried out through backpropagation, with 0.01 learning rate and 0.9 momentum. The criterion for training termination was reaching 22,000 epochs. NeuroShellâ software was used.23
Data from January 1979 to June 1991 totaling 150 observations were used for network training and model adjustment. This choice was aimed at ensuring the absence of Vibrio cholerae circulation in the adjustment period, given that the first case notified in the state of Ceará took place in February 1992, and non-detected cases may have occurred in the immediately preceding months.
Data for the 54-month period between August 1991 and December 1995 were predicted based on the network. Due to the change from ICD-9 to ICD-10 in January 1996, we chose not to prolong the extrapolation beyond this date in order to ensure the homogeneity of the series. The confidence interval for the implicit ANN model was estimated based on the distribution of residues from the training period, assuming a normal distribution with mean =0. Bootstrapping with 2,000 samples was performed in order to evaluate the adequacy of the parameters directly estimated from the 150 residues.
Data were also adjusted to a negative binomial distribution, deaths being a function of order of occurrence, representing month-year, and of month of occurrence, transformed into a dummy variable, representing the seasonal component, after verifying an over-dispersion in a Poisson regression model. STATA19 software was used for this analysis.
Table 1 shows the results of negative binomial regression. June was removed from the model for colinearity reasons, i.e., according to the adjusted model, June is the baseline month for the definition of the remaining parameters, which express the seasonal component. The months between July and November were not statistically significant, but were kept in the model to ensure its validity.
The parameters related to the residues of the neural network are presented in Table 2. The confidence intervals around the mean include zero.
Figure 1 shows estimates from both models for the adjustment period. Agreement between the two models is high, with a 0.95 Pearson correlation coefficient. The mean difference between the two estimates was 0.544740 (95%CI -294971 to 4.039190). Figure 2 presents estimates from both models extrapolated for the July 1991 December 1995 period. In this case there is less agreement between both estimates, with a 0.92 Pearson correlation coefficient. Mean difference is -6.56195 (95%CI -13.9088 to 0.784901). The estimate given by the ANN is higher than that obtained through regression. Although ANN estimates for the December May period the season with highest mortality were closer to the observed values, the values estimated through regression were closer to the observed values in the remaining months.
Figure 3 presents the upper limit of the confidence intervals of both estimates, observed data, and cholera occurrence, from July 1991 to December 1995. The regression model detected excess mortality in March-April 1992 shortly after the detection of the first cases of cholera in the state in February and in October-November 1992. From February 1993 onward, with the exception of August and September 1994, all points were above the upper limit of the regression model. Considering excess mortality as the difference between the number of deaths observed and the upper limit of the confidence interval, the excess mortality as defined by regression was 68 deaths in 1992, 266 in 1993, 285 in 1994, and 205 in 1995.
The neural network detected an excess mortality of only five deaths in December 1993, the month preceding the highest peak in the cholera epidemic, and of 17 deaths in November-December 1994, in the following season.
The results obtained indicate that excess mortality did occur in the state of Ceará in the period, and that it may be possible to detect this excess mortality based on monthly mortality data. In the studied period, 217 deaths due to cholera were registered in the state: 19 in 1992, 89 in 1993, 104 in 1994 and five in 1995. The neural network underestimated excess mortality. There are doubts concerning the considerable magnitude of the binomial regression estimate for 1995. Such magnitude may be due to the extended extrapolation period. Prolonging the series to encompass the period immediately following would perhaps clear this point. Unfortunately, the change from ICD-9 to ICD-10 would cast doubts on the homogeneity of the series, introducing a likely bias and hampering result interpretation.
It should be noted that, with respect to the season with the greatest number of deaths ascribed to poorly defined intestinal infections, point estimates provided by the neural network were closer to the observed values than those provided by negative binomial regression in the extrapolation period (Figure 2), but not in the model-adjustment period (Figure 1), suggesting that, for longer intervals, extrapolation through binomial regression is less reliable than that provided by neural networks. This potential characteristic of the neural network, however, is not advantageous, since interval estimation has low levels of precision.
Estimates from both models showed good agreement, indicating the adequacy of using ANNs for health-related time series. However, from a more pragmatic standpoint the usual approach in econometrics the choice between different prediction strategies must tend towards that which correctly predicts the outcome. In epidemiological surveillance, the subject of the prediction is the detection of abnormal values20 rather than the greater proximity between predicted and observed numbers which is the case with price prediction, for instance. In the present case, negative binomial regression was more adequate for the detection of excess deaths during the circulation of Vibrio cholerae due to its narrower variance.
The difference between negative binomial regression and Poisson regression resides in its estimation of variance, which incorporates an over-dispersion parameter alpha. This method was used only because residues registered after adjustment of a Poisson regression showed greater dispersion than their corresponding distributions. When alpha equals zero, negative binomial distribution is reduced to Poisson distribution.
Different strategies have been used for error estimation in neural network predictions.2 The one used in the present study is based on the assumption that, theoretically, an exact model for the time series can be found, but that, due to measurement errors and to the influence of uncontrollable and unknown factors, there is a randomly produced residual error, and the neural network is a quasi-optimal model. This is doubtless an area to be further explored and developed.
The neural networks most used are the non-recurrent or feed-forward networks. In this model each neuron in a given layer interacts with al neurons in adjacent layers, but not with those in the same layer, processing occurring always in one direction, from input to output. By contrast, recurrent neural networks, such as the one employed in the present study, are capable of learning sequences and are thus the best choice for dealing with time-series data. Whereas networks with standard connections respond to a given input always through the same output, a recurrent network can respond to a same input through different outputs at different moments, depending on the input previously presented.
When incorporating a neuron into the memory layer, the network is incorporating an auto-regressive component,22 in addition to the trend and seasonality components represented by the two input neurons, year and month. The auto-regressive component allows the values predicted at a given moment to influence the subsequent prediction, ascribing greater influence to recent values. This did not generate divergence between models, despite the negative binomial regression not considering auto-correlation in time.
The main difficulty faced in using neural networks is that, due to the novelty of the method, researchers in general have little familiarity with the process, compared to other statistical methods.14 The criterion for choosing between different networks is pragmatic, i.e., the network chosen is that which fulfils the objectives expected. Result reproducibility is also not guaranteed, since initial weights are random in each training session which may lead to different areas in the error surface , and since there are different convergence criteria which may result in different local minimums. These characteristics contribute towards a certain degree of uncertainty in using this instrument, which can be overcome by carrying out several training sessions and observing result distribution.
Another problem mentioned in the literature9 is over-training, by which the network captures quantitative relationships generated by noise from the data, with negative effects on generalization (external validity). An alternative is to compare networks with different training times.6,7 The network presented in the present study was the first one adjusted, no important differences having been detected in subsequent networks. One may say that the criterion for convergence was learning 'time', since convergence was determined by the number of times the entire set of data was presented during training. The 'training set' comprised the data from the period preceding the introduction of cholera in the country. A separate calibration set was not determined, since the size of the training set has important implications on generalization. In this case, the calibration method cross-validation would not compensate for the reduction in the size of the training set.
In this example, the neural network was less sensitive than negative binomial regression. Its main advantage was the lower level of statistical knowledge required for its application. It was clearly the technique most easily applicable in the present example, since it did not imply the recognition of models according to the behavior of the series, nor the evaluation of the adjusted model. The present results indicate promising aspects in the use of neural networks in epidemiological surveillance. However, there is still need for deepening theoretical knowledge of the statistical behavior of network residues, so as to allow for greater estimate precision.
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Maria Lúcia F Penna
Escola Nacional de Saúde Pública
Departamento de Endemias Samuel Pessoa
Rua Leopoldo Bulhões, 1480 Térreo
21041-210 Rio de Janeiro, RJ, Brasil
Reviewed on 28/11/2003
Approved on 19/1/2004