Assessment of prediction ability for reduced probabilistic neural network in data classification problems
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Abstract
One of the most important problems in probabilistic neural network (PNN) operation is the minimization of its structure. In this paper, two heuristic approaches of PNN’s pattern layer reduction are applied. The first method is based on a kmeans clustering procedure. In the second approach, the candidates for the network’s pattern neurons are selected on the basis of a support vector machines algorithm. Modified models are compared in the classification problems with the traditional PNN, four wellknown computational intelligence algorithms (single decision tree, multilayer perceptron, support vector machines, kmeans algorithm) and PNN trained by the stateoftheart procedures. Seven medical benchmark databases are investigated and one authors’ own real ovarian cancer data set. Comparison is performed on the basis of the global performance indices which depend on the accuracy, sensitivity and specificity. These indices are computed using the standard tenfold crossvalidation procedure. On the basis of the reported results, we show that the algorithm based on kmeans clustering is a better PNN structure reduction procedure. Furthermore, this algorithm is much less timeconsuming.
Keywords
Probabilistic neural network kMeans clustering Support vector machines Medical data classification Global performance index Accuracy Sensitivity Specificity1 Introduction
Probabilistic neural network, along with multilayer perceptron, radial basis function neural network or selforganizing map is one of the most popular models used in data classification problems. PNN was proposed by Specht (1990) and quickly found many devotees. Its main advantage is that it can quickly learn from input data. Probabilistic neural networks have found their implementation in a variety of classification fields. It was presented in image classification and recognition (Chtioui et al. 1996, 1998; Ramakrishnan and Selvan 2007; Wen et al. 2008), earthquake magnitude prediction (Adeli and Panakkat 2009), multiple partial discharge sources classification (Venkatesh and Gopal 2011), interval information processing (Kowalski and Kulczycki 2014) or medical diagnosis and prediction (Shan et al. 2002; Folland et al. 2004; Huang and Liao 2004; Temurtas et al. 2009; Mantzaris et al. 2011).
From its architecture point of view, PNN is a feedforward model composed of four layers: an input layer where each element corresponds to a data feature, a radial basis pattern layer which consists of as many neurons as training vectors, a summation layer having single neuron for every class and an output layer that provides the prediction for unknown sample. In the original form, PNN has no weights to be updated; therefore, the training process for this network seems to be feasible. The attention only has to be paid to the appropriate selection and computation of the smoothing parameter for the radial basis neurons. However, the major drawback of PNN lies in the requirement of having one neuron in the pattern layer for each training example (Specht 1992). Thus, for large data set classification problems, the structure of this model is complex.
In this article, we concentrate on the architecture reduction in the PNN. For this purpose, two alternative approaches of the structure minimization are applied. The first method is based on the application of a kmeans clustering algorithm to input data in order to determine the optimal number of centroids as the representation of the pattern layer neurons. The second solution consists in the use of a support vector machine procedure which, out of the entire training set, provides the subset of optimal vectors (support vectors) which in turn form the layer of pattern nodes of PNN. Both techniques are tested on the medical data sets. The presented study is a generalization of the results published by the authors in Kusy and Kluska (2013). In that work, the attention is only paid to computing the prediction error on 20 % cases extracted randomly from each of the investigated data sets. Such a solution we consider insufficient. Therefore in this article, the diagnostic accuracy parameters for optimized PNNs are determined by means of a tenfold crossvalidation method. The obtained results are compared to the outcomes achieved by the reference classification algorithms: single decision tree, multilayer perceptron, support vector machines and kmeans clustering procedure and, additionally, to the performance of the stateoftheart PNN training solutions. Furthermore, we propose the global performance index which has the form of weighted sum of the accuracy, sensitivity and specificity for both binary and multiclass classification problems. The use of such a measure is of a particular importance, especially in medical data classification tasks, as the ones used in this study.
This paper is composed of the following sections. In Sect. 2, we conduct an overview on various PNN reduction methods presented up to this date. Section 3 discusses probabilistic neural network highlighting its basics, structure and a principle of operations. In Sect. 4, the reduction in PNN structure by means of kmeans clustering and support vector machines algorithms is outlined. In this section, the global performance index is also proposed. Section 5 briefly describes the input data used in this research. The performance of the standard and the modified PNN models is verified in Sect. 6. In this section, we also compare prediction abilities of reduced PNNs with the results obtained by the reference classifiers and the stateoftheart PNN models. Finally, in Sect. 7, the conclusions are presented.
2 Related work
In general, there exist two categories of studies related to the reduction in PNN construction. The first category includes the clustering techniques. For example, the work reported in Burrascano (1991) presents the learning vector quantization approach for finding representative patterns to be used as neurons in PNN. This method defines a number of examples that are reference vectors which approximate the probability density functions of the pattern classes. The reference in Chtioui et al. (1996) presents the reduction in the size of the training data for PNN by hierarchical clustering. The idea consists in applying the reciprocal neighbors technique, which allows the gathering of examples which are closest to each other. In Zaknich (1997), the quantization method for PNN structure is proposed. The input space is divided into a fixedsize hypergrid, and within each hypercube representative cluster centers are computed. In this way, the number of training vectors in each hypercube is reduced to one. The work presented in Chang et al. (2008) introduces an expectation–maximization method as the training algorithm for PNN. This amounts to the predefinition of the number of clusters as the input data set. A global kmeans algorithm is used as the solution. In the contribution (Chandra and Babu 2011), an improved architecture for PNN is proposed. The network is designed with an aggregation function based on the fmeans of training patterns. Such an architecture reduces the number of layers and therefore computational complexity.
In the second category of the studies which focus on the architecture optimization of PNN, the authors utilize nonclustering methods. For example, the model described in Traven (1991) is designed so that it can use far fewer nodes than the training patterns. It is achieved by estimating probability density functions as a mixture of Gaussian densities with varying covariance matrices. In the reference Streit and Luginbuhl (1994), a maximum likelihood algorithm for training the network is presented as the generalization of Fisher method for nonlinear discrimination. It is shown that the proposed PNN requires significantly fewer nodes and interconnection weights than the original model. In Mao et al. (2000), a supervised PNN structure determination algorithm is introduced. This algorithm consists of two parts and runs in an iterative way: smoothing parameter computation by means of genetic algorithm and pattern layer neuron selection. The important nodes for the layer are chosen by employing an orthogonal algorithm. The research presented in Berthold and Diamond (1998) introduces the automatic construction of PNN by the use of a dynamic decay adjustment algorithm. The model is dynamically built during training, which automatically optimizes the number of hidden neurons.
It is important to emphasize that there also exists a third category of articles which are related to the probabilistic neural network. This category encompasses the papers which explore the problem of a smoothing parameter selection as the variable of probability density functions determined for the hidden neurons of the model. Four approaches are usually regarded: single parameter for whole PNN, single parameter for each class, separate parameter for each variable and separate parameter for each variable and class. In the research, diverse procedures have been developed to solve these tasks (Chtioui et al. 1998; Specht 1992; Mao et al. 2000; Georgiou et al. 2008; Gorunescu et al. 2005; Specht and Romsdahl 1994; Zhong et al. 2007; Kusy and Zajdel 2015).
3 Probabilistic neural network
In this section, the fundamentals of PNN model are presented. Since the principle of operation of this network stems from Bayesian theory, we start with a short description of a Bayes’ theorem. Then, it is highlighted how PNN forwards the input signal to succeeding layers to compute its output. The architecture of the network is also shown.
3.1 Bayesian classifier
3.2 Network’s structure
4 Proposed algorithms
This section introduces two approaches applied for PNN structure simplification. Both solutions consist in reducing the number of pattern neurons of the network. The first method is based upon a kmeans procedure. The second idea, originally hinted in Kluska (2009), truncates the data size by utilizing the support vectors for PNN training. Moreover, in this section the global performance indices are proposed as the indicators for the reduction of the neurons in the network’s pattern layer.
4.1 Smoothing parameter selection
4.2 PNN structure reduction by the use of kmeans clustering
 (a)
for \(G=2\) we take \(\left( \alpha ,\beta ,\gamma \right) =\left( 1,0,0\right) \) and the global performance index equals \(J=\mathrm{Acc}\).
 (b)
for \(G>2\) we take \(\left( \alpha _{g},\beta _{g},\gamma _{g}\right) =\left( \frac{1}{G},0,0\right) \) for \(g=1,\ldots ,G\) and the global performance index is the average accuracy computed over partial accuracies separately.
A similar solution is provided in Zaknich (1997) but that concept is dependent on various quantization levels of the input space. Here the number of clusters is determined by (7). The other difference lies in the choice of smoothing parameter. In Zaknich (1997), single \(\sigma \) is used for the model. In this paper, PNN adopts single \(\sigma \) for each variable and class. It is necessary to note that there exist a large number of other more sophisticated clustering choices which could be applied for data reduction in Algorithm 1, e.g., cluster labeling method for SV clustering (Lee and Lee 2005), spectral biclustering (Liu et al. 2006), weighted graphbased clustering (Lee and Lee 2006), parameterless clustering (Tseng and Kao 2005) or the approach which allows for large overlaps among clusters of the same class (Fu and Wang 2003). However, the use of original, basic clustering method in the present study provides satisfactory prediction results for reduced PNN; therefore, we decide to utilize this approach.
4.3 PNN structure reduction by means of support vector machines
Finally, note that although the SVMs described above are binary classifiers, they are easily combined to handle multiclass classification problems. The most widely used approaches combine multiple binary classifiers trained separately using either G oneagainstall (say, “one” positive, “rest” negative) or oneagainstone schemes (Hsu and Lin 2002).
4.3.1 The meaning of C constraint
The coefficient C in (15) is the parameter which introduces additional capacity control for the classifier. The adjustment of C provides a greater or smaller number of support vectors which, in turn, influences the classification accuracy.
In this research, by setting different values to C constraint, we are capable of obtaining different sets of support vectors. Depending on the considered data set and the value of C, the size of PNN varies.
4.3.2 The use of kernel function
4.3.3 The proposed approach
For the constraint C and the spread constant sc, the final sets of values \(A_{C}\) and \(A_{{sc}}\) are assumed, respectively. The grid search method for both C and sc is performed, where \(A_{C}=\{10^{1}\), \(10^{0}\), \(10^{1}\), \(10^{2}\), \(10^{3}\), \(10^{4}\), \(10^{5}\}\) and \(A_{{sc}}=\{1.2\), 1.5, 2, 5, 10, 50, 80, 100, 200, \(500\}\).
In order to solve multiclass classification problems by SVM in Algorithm 2, we utilize “oneagainst one” method. In this approach, for a data set with G classes, \(G(G1)/2\) binary classifiers are constructed where each one is trained using vectors from two classes. The prediction is performed on the basis of voting strategy by assigning an unknown test case to the class with the highest vote (Wang and Fu 2005).
5 Input data used to test the models

Wisconsin breast cancer (WBC) set with 683 patterns and 9 features. The data are divided into two groups: 444 benign cases and 239 malignant cases.

Pima Indians diabetes (PID) set with 768 patterns and 8 features. Two classes of data are considered: samples tested negative (500 women) and samples tested positive (268 women).

Haberman’s survival (HS) set with 306 patterns and 3 features. There are two input classes: patients who survived 5 years or longer (225 records) and patients who died within 5 years (81 records).

Cardiotocography (CTG) set with 2126 patterns and 22 features. The classes are coded into three states: normal (1655 cases), suspect (295 cases) and pathological (176 cases).

Thyroid (T) set with 7200 patterns and 21 features. Three classes are regarded: subnormal functioning (166 samples), hyperfunction (368 samples) and not hypothyroid (6666 samples).

Dermatology (D) set with 358 patterns and 34 features. Six data classes are considered: psoriasis (111 cases), seborrheic dermatitis (60 cases), lichen planus (71 cases), chronic dermatitis (48 cases), pityriasis rosea (48 cases) and pityriasis rubra pilaris (20 cases).

Diagnostic Wisconsin breast cancer (DWBC) set with 569 patterns and 30 features. Two medical states are regarded: malignant (212 instances) and benign (357 instances).
6 Results and discussion
This section presents the comparison of the prediction ability measured for the standard PNN model and the networks for which the number of pattern neurons is reduced by means of Algorithms 1 and 2. The prediction ability of the examined classifiers is assessed using the global performance indices J(s), Q(C, sc) which involve the models’ accuracy, sensitivity and specificity. These indices are determined on the basis of a tenfold crossvalidation procedure. Furthermore, the obtained results are compared to the prediction ability of the reference classifiers: single decision tree (SDT), multilayer perceptron (MLP), SVM and kmeans algorithm, and to the outcomes for the stateoftheart PNNs training procedures. At the end of this section, we highlight some aspects related to time effectiveness of the proposed algorithms.
6.1 Results for the proposed approaches
Results for WBC classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{4}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  Acc  Sen  Spe  J  Time (s)  sc  SVs  Acc  Sen  Spe  Q  Time (s) 
1  68  0.971  0.958  0.977  0.968  0.94  1.2  65  0.677  0.767  0.500  0.669  0.41 
2  137  0.993  1.000  0.988  0.994  2.62  1.5  69  0.681  0.773  0.520  0.676  0.92 
3  205  0.966  0.972  0.962  0.967  5.74  2  80  0.750  0.818  0.600  0.740  1.14 
4  274  0.975  0.968  0.977  0.973  8.27  5  218  0.954  0.971  0.889  0.946  8.27 
5  342  0.976  0.983  0.973  0.978  12.09  10  293  0.966  0.885  0.987  0.946  19.08 
6  409  0.976  0.965  0.981  0.974  16.91  50  397  0.982  0.992  0.968  0.982  17.31 
7  478  0.981  0.988  0.977  0.982  25.27  80  432  0.969  0.983  0.953  0.970  20.30 
8  546  0.985  0.989  0.983  0.986  30.99  100  439  0.982  0.970  0.992  0.980  24.99 
9  615  0.985  0.991  0.983  0.986  31.64  200  449  0.982  0.987  0.976  0.982  29.53 
All  683  0.987  0.987  0.986  0.987  46.71  500  449  0.984  0.987  0.981  0.984  35.31 
Results for PID classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{2}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  Acc  Sen  Spe  J  Time (s)  sc  SVs  Acc  Sen  Spe  Q  Time (s) 
1  77  0.909  0.852  0.940  0.898  2.07  1.2  374  0.486  0.259  0.709  0.463  2.49 
2  154  0.759  0.611  0.840  0.731  4.45  1.5  385  0.525  0.262  0.773  0.496  3.97 
3  230  0.800  0.675  0.867  0.776  9.50  2  384  0.521  0.312  0.733  0.501  7.38 
4  307  0.801  0.626  0.895  0.767  15.51  5  386  0.588  0.345  0.794  0.556  12.32 
5  384  0.794  0.619  0.888  0.760  19.10  10  407  0.636  0.389  0.826  0.600  13.17 
6  461  0.757  0.528  0.880  0.713  19.92  50  664  0.738  0.574  0.847  0.711  31.18 
7  538  0.797  0.622  0.891  0.763  39.06  80  725  0.774  0.608  0.871  0.744  38.75 
8  614  0.764  0.556  0.875  0.724  40.84  100  742  0.784  0.876  0.623  0.779  40.15 
9  691  0.769  0.573  0.876  0.732  39.79  200  768  0.778  0.608  0.870  0.745  45.54 
All  768  0.778  0.608  0.870  0.745  44.02  500  768  0.778  0.608  0.870  0.745  45.51 
Results for HS classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{0}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  Acc  Sen  Spe  J  Time (s)  sc  SVs  Acc  Sen  Spe  Q  Time (s) 
1  31  0.677  0.250  0.826  0.579  0.11  1.2  170  0.524  0.000  1.000  0.462  0.64 
2  61  0.754  0.313  0.911  0.653  0.15  1.5  169  0.550  0.914  0.216  0.592  0.55 
3  92  0.761  0.083  1.000  0.605  0.17  2  169  0.538  0.049  0.988  0.481  0.64 
4  122  0.778  0.375  0.922  0.686  0.57  5  171  0.549  0.062  0.989  0.491  0.67 
5  154  0.747  0.268  0.920  0.638  0.61  10  174  0.528  0.025  0.957  0.463  0.70 
6  184  0.761  0.102  1.000  0.611  0.47  50  200  0.600  0.062  0.966  0.512  1.72 
7  215  0.744  0.140  0.962  0.606  0.71  80  215  0.637  0.148  0.933  0.550  2.01 
8  245  0.735  0.077  0.972  0.585  1.57  100  224  0.687  0.308  0.902  0.616  2.42 
9  276  0.768  0.246  0.956  0.649  1.65  200  243  0.695  0.259  0.914  0.608  2.70 
All  306  0.761  0.247  0.946  0.644  2.12  500  266  0.741  0.333  0.919  0.654  2.79 
Results for CTG classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{3}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  sc  SVs  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  Q  Time (s) 
1  214  0.981  0.920  0.979  0.955  9.26  1.2  230  0.925  0.892  0.941  0.941  12.17 
2  425  0.980  0.912  0.978  0.952  31.61  1.5  247  0.933  0.897  0.944  0.945  12.78 
3  639  0.982  0.916  0.974  0.957  84.11  2  288  0.928  0.886  0.942  0.933  20.92 
4  850  0.981  0.920  0.975  0.957  119.93  5  600  0.971  0.941  0.972  0.969  117.65 
5  1064  0.984  0.936  0.977  0.965  342.56  10  1069  0.981  0.959  0.981  0.978  215.08 
6  1276  0.990  0.968  0.986  0.986  348.41  50  1985  0.991  0.977  0.989  0.989  605.79 
7  1489  0.981  0.948  0.980  0.973  330.78  80  2070  0.979  0.917  0.979  0.954  610.80 
8  1701  0.985  0.939  0.974  0.968  583.02  100  2084  0.990  0.970  0.986  0.986  777.62 
9  1914  0.991  0.976  0.989  0.988  814.52  200  2098  0.992  0.975  0.989  0.988  745.51 
All  2126  0.991  0.973  0.989  0.987  887.88  500  2110  0.992  0.978  0.991  0.989  757.76 
Results for T classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{1}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  sc  SVs  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  Q  Time (s) 
1  721  0.963  0.669  0.787  0.825  52.36  1.2  944  0.955  0.921  0.966  0.951  159.35 
2  1440  0.985  0.880  0.934  0.944  112.96  1.5  943  0.947  0.913  0.959  0.944  180.56 
3  2160  0.989  0.939  0.979  0.974  375.21  2  1009  0.949  0.900  0.957  0.940  202.41 
4  2879  0.981  0.881  0.941  0.943  655.14  5  1187  0.954  0.913  0.964  0.948  289.59 
5  3600  0.982  0.895  0.957  0.953  1380.51  10  1262  0.959  0.905  0.965  0.949  319.21 
6  4321  0.982  0.891  0.950  0.950  1898.36  50  1963  0.977  0.936  0.974  0.964  635.75 
7  5040  0.982  0.874  0.939  0.944  2098.01  80  2365  0.980  0.939  0.978  0.968  965.03 
8  5760  0.983  0.903  0.951  0.953  2293.61  100  2598  0.981  0.929  0.973  0.964  1128.61 
9  6479  0.981  0.894  0.944  0.947  4338.36  200  3449  0.987  0.951  0.977  0.974  1500.32 
All  7200  0.994  0.963  0.985  0.985  7543.13  500  5021  0.991  0.960  0.980  0.980  2833.95 
Results for D classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{3}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  sc  SVs  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  Q  Time (s) 
1  36  0.991  0.917  0.995  0.975  0.35  1.2  257  0.999  0.996  0.999  0.998  23.34 
2  72  0.999  0.999  0.999  0.999  0.67  1.5  282  0.998  0.991  0.998  0.997  24.24 
3  106  0.991  0.967  0.994  0.986  2.82  2  319  0.999  0.997  0.999  0.999  29.96 
4  142  0.995  0.985  0.997  0.994  3.23  5  356  0.999  0.997  1.000  0.999  33.31 
5  180  0.998  0.993  0.999  0.997  7.11  10  358  0.997  0.990  0.999  0.996  31.86 
6  216  0.995  0.978  0.997  0.992  10.34  50  358  0.997  0.990  0.999  0.996  31.90 
7  252  0.999  0.996  0.999  0.998  13.64  80  358  0.997  0.990  0.999  0.996  32.28 
8  286  0.999  0.996  0.999  0.998  18.08  100  358  0.997  0.990  0.999  0.996  32.10 
9  322  0.997  0.990  0.998  0.996  24.99  200  358  0.997  0.990  0.999  0.996  32.13 
All  358  0.997  0.990  0.998  0.996  34.88  500  358  0.997  0.990  0.999  0.996  31.79 
Results for DWBC classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{1}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  Acc  Sen  Spe  J  Time (s)  sc  SVs  Acc  Sen  Spe  Q  Time (s) 
1  57  0.999  0.997  0.998  0.998  0.62  1.2  206  0.976  0.980  0.971  0.976  56.48 
2  113  0.991  1.000  0.976  0.991  2.67  1.5  211  0.976  0.963  0.990  0.975  62.20 
3  171  0.998  0.997  0.997  0.997  7.48  2  226  0.982  1.000  0.965  0.984  74.32 
4  228  0.997  0.995  0.994  0.996  15.78  5  335  0.976  0.976  0.976  0.976  107.65 
5  285  0.986  0.983  0.991  0.986  22.41  10  442  0.989  0.981  0.996  0.988  132.06 
6  341  0.982  0.995  0.961  0.982  30.78  50  568  0.993  0.986  0.997  0.992  183.98 
7  398  0.992  0.996  0.986  0.992  50.52  80  569  0.993  0.986  0.997  0.992  189.27 
8  456  0.991  0.993  0.988  0.991  92.91  100  569  0.993  0.986  0.997  0.992  189.27 
9  512  0.988  0.997  0.974  0.988  104.80  200  569  0.993  0.986  0.997  0.992  189.27 
All  569  0.993  0.997  0.986  0.993  181.80  500  569  0.993  0.986  0.997  0.992  189.27 
Results for OC classification task: Algorithm 1—the number of cluster centers (\(c_{s}\)); Algorithm 2—the spread constant and the number of support vectors for \(C^{*}=10^{1}\)
Algorithm 1  Algorithm 2  

s  \(c_{s}\)  Acc  Sen  Spe  J  Time (s)  sc  SVs  Acc  Sen  Spe  Q  Time (s) 
1  20  0.650  0.143  0.923  0.553  0.16  1.2  155  0.826  0.867  0.793  0.832  5.16 
2  40  0.990  1.000  0.985  0.992  0.47  1.5  163  0.883  0.882  0.884  0.883  6.52 
3  59  0.966  0.950  0.974  0.963  1.61  2  171  0.859  0.882  0.845  0.863  6.07 
4  79  0.975  0.963  0.981  0.973  2.28  5  183  0.853  0.794  0.887  0.842  5.92 
5  100  0.990  1.000  0.985  0.992  3.23  10  189  0.867  0.853  0.876  0.865  6.09 
6  120  0.933  0.902  0.949  0.927  3.48  50  195  0.892  0.867  0.905  0.887  9.86 
7  140  0.914  0.854  0.946  0.902  4.76  80  196  0.857  0.838  0.867  0.853  10.88 
8  159  0.906  0.907  0.905  0.906  9.42  100  196  0.857  0.838  0.867  0.853  11.18 
9  179  0.916  0.902  0.924  0.913  9.68  200  196  0.857  0.838  0.867  0.853  11.18 
All  199  0.864  0.867  0.863  0.865  11.89  500  197  0.878  0.824  0.907  0.868  9.91 
 (a)
the primary case when the global performance indices J and Q are the accuracies or the average accuracy values,
 (b)
the exemplary case, in which we assume the values of the weights for the accuracy, sensitivity and specificity according to the designer knowledge.
 1.
In case of Algorithm 1, in seven out of eight data classification cases, by reducing the number of pattern neurons of PNN, we observe a higher value of the global performance index J than the one computed with the use of all pattern neurons of the model. The exception is in T data set classification task.
 2.
In five classification tasks, by reducing the number of pattern neurons of PNN by means of Algorithm 2, we obtain a higher value of the global performance index Q than the one determined for full structure network.
 3.
The most gainful reduction ratio R defined in (8) by optimal \(s=s^{*}\) can be directly read from Tables 1, 2, 3, 4, 5, 6, 7 and 8 . For example, in DWBC data set classification problem, it takes the value of \(R=\frac{57}{569}\approx 0.1\). Thus, instead of all the original data cases, we can use their substitutes, but about 10 times smaller in number and we get a higher value of global performance index.
It needs to be stressed that in majority of classification tasks, Algorithm 1 applied to the training process of PNN provides higher values of the global performance index J in comparison with Q obtained by PNN trained by means of Algorithm 2.
Results for the reference classifiers in the classification tasks of WBC and PID data sets
Classifier  WBC  PID  

Acc  Sen  Spe  J  Time (s)  Acc  Sen  Spe  J  Time (s)  
MLP  0.968  0.949  0.977  0.964  3.94  0.769  0.578  0.872  0.732  3.81 
SDT  0.950  0.928  0.962  0.946  0.22  0.748  0.608  0.824  0.721  0.39 
SVM  0.972  0.979  0.968  0.973  11.33  0.772  0.548  0.892  0.729  291.98 
kmeans  0.956  0.925  0.973  0.950  335.16  0.691  0.425  0.834  0.640  1.74 
Results for the reference classifiers in the classification tasks of HS and CTG data sets
Classifier  HS  CTG  

Acc  Sen  Spe  J  Time (s)  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  
MLP  0.728  0.161  0.933  0.599  2.78  0.985  0.949  0.980  0.974  77.31 
SDT  0.748  0.395  0.875  0.668  0.21  0.991  0.977  0.986  0.990  0.38 
SVM  0.742  0.111  0.968  0.598  233.36  0.987  0.951  0.982  0.976  157.32 
kMeans  0.686  0.383  0.796  0.617  8.96  0.936  0.842  0.926  0.919  763.34 
Results for the reference classifiers in the classification tasks of T and D data sets
Classifier  T  D  

\(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  \(\overline{\mathrm{Acc}}\)  \(\overline{\mathrm{Sen}}\)  \(\overline{\mathrm{Spe}}\)  J  Time (s)  
MLP  0.966  0.645  0.806  0.825  146.03  0.988  0.963  0.993  0.984  5.16 
SDT  0.990  0.949  0.977  0.977  0.50  0.980  0.914  0.988  0.967  0.38 
SVM  0.986  0.868  0.936  0.945  451.57  0.991  0.969  0.994  0.987  12.50 
kMeans  0.895  0.634  0.797  0.794  122,040.77  0.966  0.885  0.980  0.951  410.75 
The symbol “\(\star \)” means that the global performance index of Algorithm i, (\(i=1,2\)) is lower than the one provided by original PNN. We observe that in the case of Algorithm 1, this quotient is greater than 1 in seven out of eight data classification tasks taking the highest value of 293.23 (DWBC problem). However, as it can be seen from Table 9, Algorithm 2 is much more timeconsuming in comparison with Algorithm 1.
6.2 Comparison to reference classifiers
All the reference classifiers utilized in the classification problems are trained and tested in DTREG software (Sherrod 2015). Below, the short description of the model’s settings is highlighted.
SDT is simulated with the entropy to evaluate the quality of splits in the process of tree construction. The depth of the tree is set to 10. The pruning algorithm is applied to find the optimal tree size. We prune the tree with respect to minimum crossvalidation error.
MLP is trained with one or two hidden layers. Linear or logistic transfer functions are used for activation of the neurons in hidden and output layers. The search for the optimal number of hidden layer neurons is performed in order to minimize the sum squared error of model. The scaled conjugate gradient algorithm is MLP’s training algorithm.
SVM algorithm is also applied in this research as the reference classifier. Multiclass classification tasks are solved using the oneversusone approach. In each classification problem, radial basis kernel function is utilized with experimental grid search for model’s parameters C and sc.
Similarly to SVM, the kmeans clustering algorithm is the reference model applied in this work for comparison purposes. The kmeans predictions for the unknown patterns are determined by using the category of the nearest cluster. In the experiments, we search for the number of clusters for which the highest testing accuracy is obtained.
Results for the reference classifiers in the classification tasks of DWBC and OC data sets
Classifier  DWBC  OC  

Acc  Sen  Spe  J  Time (s)  Acc  Sen  Spe  J  Time (s)  
MLP  0.975  0.957  0.986  0.972  6.81  0.814  0.808  0.817  0.813  2.41 
SDT  0.936  0.896  0.961  0.929  0.54  0.758  0.750  0.763  0.757  0.25 
SVM  0.975  0.958  0.986  0.972  6.11  0.849  0.808  0.870  0.841  4.97 
kmeans  0.891  0.778  0.958  0.871  98.17  0.758  0.779  0.748  0.762  0.59 
We can observe that in six out of eight data set classification tasks, PNN with the structure reduced by means of Algorithm 1 or Algorithm 2 yields a higher value of the global performance index.
6.3 Comparison to stateoftheart procedures
The accuracy results in the classification of WBC, PID, HS, T, D and DWBC data sets for the proposed approaches, PNN trained with total number of pattern neurons, the reference classifiers and the stateoftheart PNN learning algorithms
Data set  Proposed approaches  Full PNN  Reference classifiers  Stateoftheart methods  

Algorithm 1  Algorithm 2  MLP  SDT  SVM  kMeans  Source  Result  
WBC  0.993  0.984  0.987  0.968  0.950  0.972  0.956  Georgiou et al. (2008)  0.989 
Azar and ElSaid (2013)  0.976  
PID  0.909  0.784  0.778  0.769  0.748  0.772  0.691  Temurtas et al. (2009)  0.781 
Georgiou et al. (2006)  0.753  
HS  0.778  0.741  0.761  0.728  0.748  0.742  0.686  Chandra and Babu (2011)  0.743 
T  0.989  0.991  0.994  0.966  0.990  0.986  0.895  Yeh and Lin (2011)  0.983 
Saiti et al. (2009)  0.968  
D  0.999  0.999  0.997  0.988  0.980  0.991  0.966  Chang et al. (2008)  0.935 
DWBC  0.999  0.993  0.993  0.975  0.936  0.975  0.891  Chang et al. (2008)  0.954 
In this table, for comparison purposes, we also present the accuracy values provided by PNN with all hidden neurons in the pattern layer and the ones for the reference classifiers. The best results are marked with bold. As shown, in each data classification case, the PNN models trained by means of Algorithm 1 or Algorithm 2 outperform PNNs trained using stateoftheart methods (Georgiou et al. 2006; Chang et al. 2008; Georgiou et al. 2008; Saiti et al. 2009; Temurtas et al. 2009; Chandra and Babu 2011; Yeh and Lin 2011; Azar and ElSaid 2013). Our algorithms also perform better than the reference classifiers in all considered data set classification problems. Only in for T data classification task, PNN with all neurons in the pattern layer yields the highest value of the accuracy. However, our result is worse only by a margin of 0.3 %.
In our paper, in both Algorithms 1 and 2, the smoothing parameters are determined experimentally in a way that the global performance indices achieve maximal value. The authors of Xu et al. (1994) provided a theorem on the selection of this parameter for designing Parzen window estimator particularly for probabilistic neural network. It seems advisable utilizing this interesting result in future.
7 Conclusions
This article constituted the generalization of the previous authors’ results in Kusy and Kluska (2013). In the current study, we conducted more comprehensive analysis on the problem of PNN structure reduction. Firstly, the prediction ability of reduced PNN was assessed by means of a tenfold crossvalidation procedure. Such an approach is commonly used for algorithms testing purposes. Secondly, we proposed the global performance index, which included the accuracy, sensitivity and specificity in order to determine the prediction ability of the considered models. The global performance index presented in this way is quite flexible since it can take the form of model’s accuracy or the form of weighted accuracy, sensitivity and specificity values for each class separately. The values of particular weights can be established by a designer (domain expert) according to his/her best knowledge. We would like to stress that this is particularly important especially in medical data classification problems, as the ones used in this study. Furthermore, the PNN classifiers with the number of pattern neurons reduced by means of kmeans clustering and SVM procedure were compared to wellknown computational intelligence algorithms: single decision tree, multilayer perceptron, support vector machines and kmeans clustering procedure. In six classification tasks, we achieved a higher value of the global performance index for PNN with reduced architecture than for the considered reference classifiers. Finally, we also made the comparison of the accuracy values of the reduced PNN models and PNNs trained by stateoftheart procedures. In all data classification cases, the accuracies obtained by means of our algorithms took a higher value.
Notes
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
Human and animal participants
Research is not involved with human participants and/or animals.
Ethical standards
Seven data sets used in this work are taken from UCI machine learning repository. There is also a single database obtained from the Clinical Department of Obstetrics and Gynecology of Rzeszow State Hospital in Poland.
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