International Journal of Computational Mathematics
Volume 2015 (2015), Article ID 415146, 4 pages
http://dx.doi.org/10.1155/2015/415146

Research Article

Building Expert Medical Prognostic Systems Using Voronoi Diagram

Maria A. Ivanchuk¹ and Igor V. Malyk²

¹Department of Biological Physics and Medical Informatics, Bukovinian State Medical University, Kobylianska Street 42, Chernivtsi 58000, Ukraine
²Department of the System Analysis and Insurance and Financial Mathematics, Yuriy Fedkovych Chernivtsi National University, Universitetska Street 12, Chernivtsi 58012, Ukraine

Received 27 June 2014; Revised 14 December 2014; Accepted 1 February 2015

Academic Editor: Neeraj Mittal

Copyright © 2015 Maria A. Ivanchuk and Igor V. Malyk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The method of building expert systems for medical prediction of severity in patients is purposed. The method is based on using Voronoi diagrams. Examples of using the method are described in the paper.

1. Introduction

Development of mathematical approaches for prediction in medicine was developed by Fisher, the father of the linear discriminant analysis [1]. Currently, there are many approaches to solving this problem such as cluster analysis [2], the construction of predictive tables [3], image recognition, and linear programming. Cluster analysis is commonly used for solving the tasks of medical prediction. The aim of a cluster analysis is to partition a given set of data or objects into clusters. This partition should have the homogeneity within the clusters and heterogeneity between clusters [4]. But cluster analysis has a significant disadvantage. It refers to the methods without teacher.

We purpose the method of building the prognostic system, which uses available information for teaching the expert system.

Let us have two sets of points and in Euclidean space , where is the number of points in the set. Set is the training sample, which includes the patients with severity; set is the training sample, which includes the patients without severity. There are parameters (factors which affect the severity) known for each patient. Our task is to separate the space into two half-spaces (patients with severity) and (patients without severity) that for each point determine its belonging to one of the half-spaces with predetermined significance level . We will check the expert system using the control sample .

We must use a smaller number of parameters to obtain the largest plausibility value for expert system.

2. Methods

We will use the Voronoi diagram [5] to solve the task of expert system’s building. Let us concatenate sets and and build Voronoi diagram for the set . Let us have a point , . Let . The same reasoning will be for .

Let be Voronoi polygon for point . The points for which Voronoi polygons have neighboring facets with polygon will be called point ’s nearest neighbors. The set of all point ’s nearest neighbors will be denoted by .

Point will be called the internal point of set if all its nearest neighbors belong to set

There are the following cases for Voronoi polygon .

(1)All point ’s nearest neighborhoods belong to set

 In this case point is the internal point of set .(2)All point ’s nearest neighborhoods belong to set

 In this case point is the outlier of set .(3) There are points belonging to set and the points belong to set among point ’s nearest neighborhoods. In this case point is the boundary point of set or point with one or several neighborhoods being the outliers of set . If there is a way from point to any internal point of set passing only through the points of set , point is the boundary point of set . In a different case point is the outlier of set .

Let us eject outliers from set (patients with severity) and build new Voronoi diagram. The diagram separates space into two half-spaces (patients with severity) and (patients without severity). We will assign patient from control set to the patients with severity if point is in the Voronoi polygon of any point of set ; in the different case we will assign patient to the set of patients without severity.

Let us sort parameters according to Kulbak’s information measure [3] and build Voronoi diagrams for different space dimension . To find the best of expert systems we will use Zagoruiko’s likelihood measure [6].

Let point . Euclidian distance between point and the nearest point is equal to . Euclidian distance between point and the nearest point is equal to . Than similarity signed measure (charge) of point and set is

Similarity measure may range from −1 to 1. Point is assigned as the part of set if . The high value of the measure indicates the high similarity between point and set .

Let be the set of control group patients with severity; is the set of control group patients without severity, .

We denote   is the set of patients of which were correctly assigned as the patients with severity; is the set of patients of which were incorrectly assigned as the patients without severity (underdiagnosis cases); is the set of patients of which were correctly assigned as the patients without severity; is the set of patients of which were incorrectly assigned as the patients with severity (overdiagnosis cases).

Then is the sum of similarity measures of the points of set and set :

is the sum of similarity measures of the points of set and set :

is the sum of similarity measures of the points of set and set :

and is the sum of similarity measures of the points of set and set :

The likelihood measure of the expert system is

if the aim of the expert system is the differential diagnostic of two similar diagnoses.

And

if the aim of the expert system is finding the patients with severity.

We will use the Akaike information criterion [7]

to find the optimal ratio of the likelihood of model and the quantity of using parameters. The best expert system is the system with the least value of . In other words, the best expert system is the system which uses the least number of parameters to have the greatest likelihood.

Let us formulate the following.

Algorithm for Modelling the Prognostic System

(1)Sort parameters using Kulbak’s information measure [3].(2)For each ,

(i)build Voronoi diagram for the set in the space ;(ii)eject the outliers of set ;(iii)build the new Voronoi diagram;(iv)if ther are no outliers in set go to step (v); else return to step (ii);(v)check the system on the control set ;(vi)calculate the likelihood measure of the expert system;(vii)calculate the Akaike information criterion .

(3)Find .(4)As the expert system use Voronoi diagram in the space . If point , assign it to set . If point , assign it to set .

Let us calculate the complexity of the algorithm. There are several ways to find Voronoi diagrams, one of which is known as Fortune’s algorithm [8]. Its complexity is , where . The complexity of finding the likelihood measure is , because we must find the similarity measure for each point . Step is repeated times and the complexity of the algorithm is . Since , the total complexity of the algorithm is .

3. Results

3.1. The Expert System of Predicting the Presence of Severity in Abdominal Surgery Patients

We built expert system using 8 parameters. Training sample consists of 28 patients with severity and 15 patients without severity. Control test consists of 8 patients with severity and 5 patients without severity. The level of significance was . In this case the aim of the expert system is finding the patients with severity; therefore we use formula (10) to find likelihood measure.

Voronoi diagram for is represented on Figure 1. The results of calculations are represented in Table 1.

Table 1: Building the expert system for abdominal surgery patients.

Outliers

AIC
Underdiagnosis cases
Overdiagnosis cases

2
1 (1.4%)
0.575
3.108
1  (16%)
2 (40%)

3
0 (0%)
0.389
4.889
1 (16%)
0 (0%)

4
0 (0%)
0.352
6.091
1 (16%)
0 (0%)

5
0 (0%)
0.627
5.933
2 (30%)
0 (0%)

6
0 (0%)
0.639
6.897
2 (30%)
0 (0%)

7
0 (0%)
0.684
7.761
2 (30%)
0 (0%)

8
0 (0%)
0.611
8.987
2 (30%)
1 (20%)

The best expert system was built using parameters.

(a)

(b)

Figure 1: Voronoi diagram for abdominal surgery patients before (a) and after (b) ejecting the outliers, .

3.2. Researching the Anthropological Parameters in Teenagers

We built expert system using 8 parameters. Training sample consists of 38 girls and 14 boys. Control test consists of 8 girls and 5 boys. The level of significance was . In this case the aim of the expert system is the differential diagnostic of two similar diagnoses; therefore we use formula (9) to find likelihood measure.

Voronoi diagram for is represented on Figure 2. The results of calculations are represented in Table 2.

Table 2: Building the expert system for researching the anthropological parameters in teenagers.

Outliers

AIC
Errors in the control sample

2
2 (4.2%)
0.0271
9.216
2 (15.4%)

3
0 (0%)
0.1147
7.331
2 (15.4%)

4
0 (0%)
0.1007
8.591
2 (15.4%)

5
0 (0%)
0.1212
9.221
2 (15.4%)

6
0 (0%)
0.2587
8.704
3 (23%)

7
0 (0%)
0.2121
10.101
3 (23%)

8
0 (0%)
0.1946
11.274
3 (23%)

The best expert system was built using parameters.

(a)

(b)

Figure 2: Voronoi diagram for anthropological parameters in teenagers before (a) and after (b) ejecting the outliers, .

4. Conclusions

The method of building expert medical prognostic systems is brought forward. The method is based on building Voronoi diagram in Euclidean spaces of different dimensions. The resulting expert systems are checked on the test samples. The expert system with the least value of the Akaike information criterion is accepted as the best system.

The described method is applied in practice to predict the presence of severity in abdominal surgery patients and gives 84% correct results for the patients with severity from the control sample. The expert system for researching the anthropological parameters in teenagers with 84.6% correct results was built, using the introduced method.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

R. A. Fisher, Contributions to Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1950.

B. S. Everitt, S. Landau, M. Leese, and D. Stahl, Cluster Analysis, John Wiley & Sons, 5th edition, 2011.

E. V. Gubler and A. A. Genkin, Primenenie Neparametricheskih Kriteriev Statistiki v Medico-Biologicheskih Issledovanijah (Using the Nonparametric Statistic Criteria in Medical and Biological Researches), Leningrad, Russia, 1973.

F. Höppner, F. Klawonn, R. Kruse, and T. Runkler, Fuzzy Cluster Analysis: Methods for Classification, Data Analysis and Image Recognition, John Wiley & Sons, New York, NY, USA, 1999.

F. Preparata and M. Shamos, Computational Geometry—An Introduction, Springer, 1985.

N. G. Zagoruiko, I. A. Borisova, V. V. Dyubanov, and O. A. Kutnenko, “A quantitative measure of compactness and similarity in a competitive space,” Journal of Applied and Industrial Mathematics, vol. 5, no. 1, pp. 144–154, 2011.

H. Akaike, “A new look at the statistical model identification,” IEEE Transactions on Automatic Control, vol. 19, no. 6, pp. 716–723, 1974.

M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf, Computational Geometry, Springer, 2000.