Building Expert Medical Prognostic Systems Using Voronoi Diagram

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 A = {a i = (a 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  = {  = ( 1  ,  2  , . . .,    ),  = 1,   } and  = {  = ( 1  ,  2  , . . .,    ),  = 1,   } 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  = {  = ( 1  ,  2  , . . .,    ),  = 1,   }.We must use a smaller number of parameters to obtain the largest plausibility value for expert system.

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   ∈ ,  = 1,   +   .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  ∀ ∈   :  ∈ . (1) There are the following cases for Voronoi polygon   .(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   = 2, . . ., .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  / () > 0. 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] AIC = 2 − 2 ln () 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 AIC.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.
(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 ( log ), where  =  1 +  2 .The complexity of finding the likelihood measure  is (  log ), because we must find the similarity measure  / () for each point  ∈ .Step (2) is repeated  − 1 times and the complexity of the algorithm is (( − 1) log ).Since  ≪ , the total complexity of the algorithm is ( log ).

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  = 0,01.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  = 2 is represented on Figure 1.The results of calculations are represented in Table 1.

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  = 0,01.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  = 2 is represented on Figure 2. The results of calculations are represented in Table 2.

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 best expert system was built using  = 3 parameters.the control sample.The expert system for researching the anthropological parameters in teenagers with 84.6% correct results was built, using the introduced method.

Figure 2 :
Figure 2: Voronoi diagram for anthropological parameters in teenagers before (a) and after (b) ejecting the outliers,  = 2.
(3)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

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