APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS IN FORECASTING THE QUARTERLY MALARIA CASES IN NIGERIA

APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS IN FORECASTING THE QUARTERLY MALARIA CASES IN NIGERIA

BY

 ABSTRACT

The objective of the study is to train artificial neural networks (AMORE SOFTWARE) with a suitable data obtained from the WHO record collected from the national surveillance system (NSS). The architecture of the neural networks in use has two inputs nodes Yt-1 and Yt-4 with one hidden layer and one output layer Yt. They were used to forecast quarterly malaria cases in Nigeria. The data used were from 1990 to 2003. A total of 32 data series was used in the training and ends up with, and 24 used in the validation of the neural networks with. The forecasting accuracy was found to be R2= 0.9665, which validates the networks.

Key words: Artificial Neural Network, Plasmodium, nodes, training.

  INTRODUCTION

1.1    Malaria

Malaria is a serious infectious disease spread by certain mosquitoes called anopheles seeking for blood to produce their eggs. It is most common in tropical climates. It is characterized by recurrent symptoms of chills, fever, and an enlarged spleen. 

Malaria is caused by anopheles bite through four different species of a parasite belonging to genus Plasmodium: Plasmodium falciparum (the most deadly), Plasmodium vivax, Plasmodium malaria, and Plasmodium ovale. The last two are fairly uncommon. Falciparum malaria is far more severe than other types of malaria because the parasite attacks all red blood cells, not just the young or old cells, as do other types. It causes the red blood cells to become very "sticky." A patient with this type of malaria can die within hours of the first symptoms. The fever is prolonged. So many red blood cells are destroyed that they block the blood vessels in vital organs (especially the brain and kidneys), and the spleen becomes enlarged. There may be brain damage, leading to coma and convulsions. The kidneys and liver may fail. It was soon observed that the use of insecticide- treated bed nets (henceforth termed ITNs) provided adequate protection against malaria infections, particularly in children. The World Health Organization has adopted the use of ITNs as one of the main strategies for malaria control in their Roll Back Malaria programme (RBM 1999). At present ITNs are being applied in many malaria-endemic regions worldwide and their use has replaced the use of indoor house spraying with insecticides in many countries. The World Health Assembly advocated the large-scale use of insecticides for malaria control in 1955, and programmes were carried out to spray as many houses as possible with a residual deposit of insecticide [mostly organ chlorine compounds such as dichlorodiphenyl-trichloroethane (DDT) and dieldrin].

1.2 Neural Networks

An artificial neural network is an information processing paradigm that was inspired by the way biological nervous systems, such as the brain, process information. The key element of this paradigm is the novel structure of the information processing system. It is composed of a large number of highly interconnected processing elements (neurons) working in unison to solve specific problems. Artificial neural networks, like people, learn by example. An artificial neural networks can be configured for a specific application, such as pattern recognition or data classification through a learning process. Learning in biological systems involves adjustments to the synaptic connection that exist between the neurons. This is true of artificial neural networks as well. In fact, this is the basis of Artificial Neural Network Technology.

1.3 Characteristics of Neural Networks

An Artificial Neural Networks (ANN) is a model composed of several highly interconnected computational units, called neurons or nodes. Each node performs a simple operation on an input to generate an output that is forwarded to the next nodes in the sequence. This parallel processing allows for great advantages in data analysis. The three essential features of a neural network: (1) the network topology, (2) the computational functions of its elements, and (3) the training of a network (Stern, 1991). 

1.3.1 Network Topology

The network topology refers to the number and organization of the computing units, the types of connections between neurons, and the direction of information flow in the network. The node is the basic organizational unit of a neural network, and nodes are arranged in a series of layers to create the artificial neural network. According to their location and function within the network, nodes are classified as input, output, or hidden layer nodes (Stern, 1991). Input layer nodes receive information from sources external to the neural network, and output layer nodes transmit information out of the neural network (Duliba, 1991). Hidden layer neurons act as the computational nodes in the neural network, communicating between input nodes and other hidden layer or output nodes. 

1.3.2 Neural Network Computational Structure

The connections between neurons are classified as either excitatory or inhibitory, according to whether the neuron weights are negative or positive, respectively. The direction of information movement also distinguishes ANNs as either feed forward or feedback networks (Duliba, 1991). Feed forward networks permit data transfer in only one direction through the network, while feedback networks allow the data to cycle through the network (Stern, 1991).  

1.3.3 Training the Network

Once the network layout and computational characteristics of the network are established, the network’s adaptive learning strategy must be determined. Neural networks must “learn” to generate values consistent with the patterns in a given data set to make accurate projections. The learning process is when network weights change in response to a training data set. ANNs learn in two ways: supervised and unsupervised learning, which differ according to whether or not known answers are used to train the network. 

1.4 Scope and limitation of the study

This study is conducted within Nigeria and restricted to the patients of primary health care particularly those attending clinic. The findings are therefore limited to those patients coming to clinic for related cases during the collection of the data. Malaria reporting from national surveillance systems varies in quality and reporting completeness and may have limited value in understanding the actual malaria burden, but may be useful for understanding trends in the relative burden of malaria in the public health sector.

1.5 Data collection 

The data used reflect aggregation of malaria cases at the national level and are presented by gender, age and sub national level as submitted to W.H.O by National Surveillance Systems in 2005. It is arranged by month and year in the WHO reported cases of malaria in Nigeria.

2 METHODOLOGY 

2.1 The neural networks models

A 2×1×1(two input nodes, one hidden nodes, and one output node) artificial neural networks is used to forecast the quarterly malaria cases  

Input Nodes: Because Malaria cases time series is affected by seasonality, two input nodes were chosen, one for Yt-1   and the other one for Yt-4, one bias node.

Hidden Nodes: There is only one hidden node since seasonality is the strong pattern in malaria cases time series and one bias node.

Output node: There is also one output node. Here Yt is being forecasted as a function of Yt-1 and Yt-4. A second output node for Yt+1 could easily be added to the artificial neural networks. In that case, the goal is to minimize one and two period–ahead errors. 

 With respect to neural networks, the performance criterion is the minimization of squared error. Therefore, the total system error is expressed as follows: 

Error (2.1)

where i indexes units of output; p indexes the input-output pairs to be learned; tip refers to the desired output, and yip  is the network’s calculated output. This function is minimized, and if the output functions are differentiable, the task of blame assignment is simplified. We begin with a set of arbitrarily chosen weights WO and improve them by implementing the formula

h(X, W) = output function of the network 

(2.2)

Where η= learning rate 

∆j= vector containing partial derivatives of h with respect to Yt

Xt= vector of inputs 

W = weights

t = time index (Garson, 1998) 

dividing the data into training and validation set 

It is important to train and validate the ANNs using two different data sets .The first data set is used to train the network and the second data set is used to validate the model .In this work, we use 32 data series of the data to train the ANN and 24 data series for the validation set.

Scale all input variables

Scaling input variables is a necessary step because of the type of nonlinear transformation and logic built into an artificial neural network. The formula is given by

(2.3)

Where: Maximum=Expected maximum value in the application

            Minimum=Expected minimum value in the application

initial weights at start of a training epoch

The initial values of the training weight can influence the required number of epochs and the final result. Thus, when good starting values are unknown, the randomization of the initial training weights is often recommended, typically randomly distributed between -1 and 1.

Input scaled variables

In this step each neurode in the input layer has its value and distributes it to every node in the hidden layer. Thus, each neurode of the hidden layer receives all input values except that the signals are weighted to yield the input to the hidden neurodes. In this work, there is only one hidden layer neurodes, however, as mentioned; each signal to a neurode will have a different weight associated with it. To understand it better let us start the forecasting of the first quarter 

                           I0=O0

                            I1=O1

                             I2=O2

 Weight and the sum of output at the receiving nodes

The weights were determined by many iterations of the training process. In a very real sense, the weights in an ANN contain its memory and intelligence. For this work 500 training epochs were performed to yield the weight. At node 4, the inputs are summed using equation 2.4.

I4 =W24O2+W14O1+W04O0 (2.4)

In general, at each neurodes, a weighted average of all inputs is calculated 

Transform hidden input to outputs

The relationship between the input and the output of the nodes is expressed by a transfer function. Because the sigmoid transfer function (also called logistic function) is used, the input value results in a neurode output with a value of 0 through 1. However, the logistic transfer function transforms the input to an output level in the range of 0 to 1. Consider the output of node 4, given input value I4 as expressed in equation 2.5.

(2.5)

During the transformation of hidden node inputs to their output, some hidden neurodes will be activated (i.e. non zero) and others will not (i.e., will equal zero). More importantly, as the network learns, the weights leading to the hidden layer neurodes are adjusted through back propagation so as to minimize the ANN error.

Weight and sum of hidden node outputs at the output nodes 

This next step is to sum inputs at the final nodes, the output node. Equation 2.6 is used to accomplish this task. We see that two nodes send signals to the output node, nodes 4 and 3 

I5 = W45O4+W35O3 (2.6)

Node 4 is a hidden node and node 3 is a bias node. The bias node 3 is used to change I5 and therefore position the sigmoid function at a different location on the x-axis so that output O5 has different values. It is a constant term where the added constant is determined by the weight from the bias node to the next node. Thus W35 is trained like the other weights. 

Transform input at the ANN output nodes

The input value I5 is transformed to the output value, O5 using equation 2.7 below.

(2.7)

Calculate output error

The output error is simply the difference between O5 and the desired value, see equation 2.8.

E5 = D5-O4 (2.8)

            Back propagate errors to adjust weights

During this step, the weights are adjusted through a training process by the back propagation so as to yield the correct output variable. Continue the epoch process to train all input in the training data set once.

Calculate the epoch RMS

Completing one epoch where all observations in the training data set are input to the ANN.

(2.9)

Judge out-of –sample validity

This out of sample test is an essential step in an actual application, its consist of running the remaining data (half data) to validate the ANN 

Use the model in forecasting

To predict future output, simply input the scaled values of the input variables and calculate the output O5 using the previously fitted weights. This is about as easy as using other forecasting methods

The coefficient of determination R2

          (2.10)

3.0   ANALYSES AND RESULTS

Table 1 (appendix) shows the scaled inputs and output variable of the training set of the neural networks.

3.1 Result of the Training 

The table 2 (appendix) shows the output given by the neural networks and the desired output after the training set.

3.2 Validation of the Neural Network

In Table 3 and Table 4, the first four observations are missing because yt-4 is used. The Tables show the validation of the artificial neural networks, the second part of the data was used in the forecast in other to validate the neural networks.

The autoregressive models of orders given by the neural networksare given below; 

 

The number of malaria cases from 1990 to 2004 is shown in Figure 1 of the appendix. It is observed that the seasonal aspect of the cases is moderate from 1990 to 1998, and starts increasing considerably from 1999 to 2004 where it reached it pick.

The figure 2 in the appendix is a two-dimensional graph showing the difference between the output value and the desired output value of the first training set of the neural network.

Figure 3 also in the appendix shows the final result of the training. It is observe that after the 34th period, there is much difference between the output and the desired output of the trained neural network. This is due to the fact that the seasonality of the data around that quarter of those very years was increasing considerably.

4.0 CONCLUSION 

This paper has shown how to train an artificial neural network composition and capabilities in the field of organizational research in addition to or as an alternative to multiple linear regressions. Because of their dynamic composition, neural networks are efficient at analyzing problems where the data are incomplete or fuzzy, and accurate predictions are sought more heavily than explanations. Neural network research has grown remarkably in the last fifteen years. Though most of this growth has been in the areas of mathematical and computer sciences.

Hence the computing world has a lot to gain from neural networks; their ability to learn by example makes them very flexible and powerful. Furthermore there is no need to derive an algorithm in order to perform a specific task; there is no need to understand the internal mechanisms of that task. They are also very well suited for real time systems because of their fast response and computational times, which are due to their parallel architecture. Neural networks also contribute a lot in area like forecasting where it is found to be more suited using quarterly data.

A coefficient of determination ( ) shows that the training of the data with the first set was successful, whereas the ( ) shows the second  set of data is low in relation to the first set because of  the high cases of malaria after the years following the data set used in the first training set. With  of the forecasting using the entire data means that we can validate the training of the neural networks and use it in a forecasting of the future.

It’s observed that malaria cases are highly concentrated during the rain season and the season immediately after it. That the cases of malaria are increasing every year due to resistance of anopheles to chloroquine and chloroquine related drugs.

 4.1 Recommendation

Malaria is a very dangerous disease and using mosquito treated bed net is the best solution to avoid it especially now that the mosquitoes are developing resistance to the drugs. Efforts should be made to eliminate favorable habitats where anopheles can produce their eggs in the surrounding. Cleaning of our environment can also help in reducing the cases of malaria.

In using Artificial Neural Network in forecasting, the following cautions should be considered

Dealing with Noisy Data: The training set of data might as well be quite "noisy" or "imprecise". The training process usually relies on some version of least squares technique, which ideally should abstract from the noise in the data. However, this feature of ANN is dependent on how optimal is the configuration of the net in terms of number of layers, neurons and, ultimately, weights.

 Avoiding under fitting and over fitting: The best way to avoid over fitting is to use lots of training data. If there were at least 30 times as many training cases as there are weights in the network, it is unlikely to suffer from over fitting. But the number of weights cannot be reduced for fear of under fitting. Given a fixed amount of training data, there are at least five effective approaches to avoiding under fitting and over fitting and hence getting good generalization:

i. Model selection

ii. Jittering

iii. Weight decay

iv. Early stopping

v. Bayesian estimation 

 APPENDIX

Table 1: Scaled input and output 

 

 

Input 1

Input 2

Output

Period

Quarter

Yt-1

Yt-4

Yt

1

1

0.002

2

2

0.002

0.1312

3

3

0.1312

0.4479

4

4

0.4479

0.2165

5

1

0.2165

0.002

0.1114

6

2

0.1114

0.1312

0.1383

7

3

0.1383

0.4479

0.3065

8

4

0.3065

0.2165

0.0108

9

1

0.0108

0.1114

0.1369

10

2

0.1369

0.1383

0.1473

11

3

0.1473

0.3065

0.459

12

4

0.459

0.0108

0.1682

13

1

0.1682

0.1369

0.0605

14

2

0.0605

0.1473

0.1237

15

3

0.1237

0.459

0.3128

16

4

0.3128

0.1682

0.1503

17

1

0.1503

0.0605

0.1046

18

2

0.1046

0.1237

0.1133

19

3

0.1133

0.3128

0.5019

20

4

0.5019

0.1503

0.1423

21

1

0.1423

0.1046

0.0973

22

2

0.0973

0.1133

0.1145

23

3

0.1145

0.5019

0.4441

24

4

0.4441

0.1423

0.1605

25

1

0.1605

0.0973

0.1062

26

2

0.1062

0.1145

0.1578

27

3

0.1578

0.4441

0.443

28

4

0.443

0.1605

0.1267

29

1

0.1267

0.1062

0.1063

30

2

0.1063

0.1578

0.1276

31

3

0.1276

0.443

0.4564

32

4

0.4564

0.1267

0.1423

Series

Observation

Mean

std Error

minimum

Maximum

Yt

32

0.1958

0.1422

0.002

0.5019

 Table 2: Output and desired output 

Period

 Ot 

Dt   

 et

5

0.1113

0.1114

5E-05

6

0.1627

0.1383

-0.024

7

0.2131

0.229

0.0159

8

0.0936

0.0856

-0.008

9

0.1504

0.1369

-0.013

10

0.1676

0.1663

-0.001

11

0.4125

0.421

0.0085

12

0.1829

0.1682

-0.015

13

0.0617

0.0605

-0.001

14

0.1305

0.1237

-0.007

15

0.2193

0.235

0.0157

16

0.1797

0.1503

-0.029

17

0.1165

0.1046

-0.012

18

0.1445

0.1428

-0.002

19

0.2201

0.2315

0.0114

20

0.1545

0.1423

-0.012

21

0.0766

0.08

0.0034

22

0.1488

0.1466

-0.002

23

0.2198

0.2314

0.0116

24

0.1717

0.1605

-0.011

25

0.1383

0.1353

-0.003

26

0.159

0.1578

-0.001

27

0.2198

0.2193

-6E-04

28

0.156

0.1475

-0.008

29

0.186

0.1769

-0.009

30

0.172

0.1685

-0.003

31

0.2195

0.2215

0.002

32

0.1615

0.1541

-0.007

RMS

 

 

0.0104

 Table 3: Validation of the artificial neural networks I

 

 

Input 1

Input 2

Output

Period

Quarter

Yt-1

Yt-4

Yt

33

1

0.674829

0.14233

0.21432

34

2

0.217044

0.224492

0.54123

35

3

0.369425

0.379393

0.6523

36

4

0.58415

0.637486

0.20166

37

1

0.570744

0.674829

0.3725

38

2

0.335729

0.217044

0.6123

39

3

0.40288

0.369425

0.7859

40

4

0.668758

0.58415

0.20822

41

1

0.902543

0.570744

0.51264

42

2

0.348072

0.335729

0.54123

43

3

0.446373

0.40288

0.8967

44

4

0.780034

0.668758

0.5123

45

1

0.487277

0.902543

0.4236

46

2

0.286083

0.348072

0.78954

47

3

0.411707

0.446373

0.3125

48

4

0.780061

0.780034

0.21612

49

1

0.8745

0.7854

0.51268

50

2

0.287185

0.286083

0.89456

51

3

0.98546

0.98755

0.456

52

4

0.9875

0.7801

0.21267

53

1

0.98754

0.78006

0.54621

54

2

0.9856

0.987

0.6548

55

3

0.781285

0.62541

0.8569

56

4

0.986318

0.86952

0.714562

Series

Observation

Mean

std Error

 

Yt

24

0.5203

0.2306

 

 Table 4: Validation of the artificial neural networks II

Period

 Ot 

Dt   

 et

33

0.21432

0.379393

0.0963

34

0.54123

0.637486

0.0225

35

0.6523

0.674829

0.0154

36

0.20166

0.217044

-0.003

37

0.3725

0.369425

-0.028

38

0.6123

0.58415

-0.215

39

0.7859

0.570744

0.1275

40

0.20822

0.335729

-0.11

41

0.51264

0.40288

0.1275

42

0.54123

0.668758

0.0058

43

0.8967

0.902543

-0.164

44

0.5123

0.348072

0.0228

45

0.4236

0.446373

-0.01

46

0.78954

0.780034

0.1748

47

0.3125

0.487277

0.07

48

0.21612

0.286083

-0.101

49

0.51268

0.411707

-0.115

50

0.89456

0.780061

0.5193

51

0.456

0.975297

0.0745

52

0.21267

0.287185

-0.144

53

0.54621

0.401808

0.1265

54

0.6548

0.781285

0.126485

55

0.8569

0.986318

0.129418

56

0.714562

0.6235

-0.09106

RMS

 

 

0.15016

 Figure 1: Number of Malaria Cases from 1990 to 2004

Figure 2: The output value and the desired output

Figure 3: Final Result of the Training

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