The asymptotic behaviour of threshold-based classification rules in case of three prescribed classes

Object classification by its numerical characteristic is an important theoretical problem and has practical significance, for example, the definition of a person as “not healthy”, if the temperature of its body exceeds 37°C. To solve this problem we consider the threshold-based rule. According to this rule, an object is classified to belong to the first class if its characteristic does not exceed a threshold 37°C; otherwise, an object is classified to belong to the second class. The empirical Bayes classification (EBC) (Devroye and Giorfi, 1985; Ivan ’ko and Maiboroda, 2002) and minimization of the empirical risk (ERM) (Vapnik, 1989; Vapnik, 1996) are widely used methods to estimate the best threshold. The case when the learning sample is obtained from a mixture with varying concentrations is considered in (Ivan ’ko and Maiboroda, 2006).


INTRODUCTION
Object classification by its numerical characteristic is an important theoretical problem and has practical significance, for example, the definition of a person as "not healthy", if the temperature of its body exceeds 37°C.
To solve this problem we consider the threshold-based rule.According to this rule, an object is classified to belong to the first class if its characteristic does not exceed a threshold 37°C; otherwise, an object is classified to belong to the second class.The empirical Bayes classification (EBC) (Devroye and Giorfi, 1985; Ivan 'ko and Maiboroda, 2002) and minimization of the empirical risk (ERM) (Vapnik, 1989;Vapnik, 1996) are widely used methods to estimate the best threshold.The case when the learning sample is obtained from a mixture with varying concentrations is considered in (Ivan'ko and Maiboroda, 2006).However, it is often necessary to classify an object in case of more than one threshold, for example, the definition of a person as «not healthy», if the temperature of its body exceeds 37°C or lower then 36°C.Another example: the person is sick, if the level of its haemoglobin exceeds 84 units or lower than 72 units.In particular, this problem is discussed in (Kubaychuk, 2008;Kubaychuk, 2010).
In all previous examples we have only two prescribed classes.The case of two thresholds and three prescribed classes deserves special attention.An example is the classification of the disease stages.Thus, during the diagnosis of breast cancer a tumor marker CA 15 -3 is used.If the value is less than 22 IU/ml, then the person is healthy; if its level is in the range from 22 to 30 IU/ml -precancerous conditions can be diagnosed; if the index is above 30 IU/mlpatient has cancer.When solving some technical problems it is needed to consider the substance in its various aggregate forms: gaseous, liquid, solid.The transition from state to state occurs at a specific temperature.According to this, a boiling point and a melting point are used.

THE SETTING OF THE PROBLEM
The problem of the classification of an object O from the observation after its numerical characteristic () O   is studied.We assume that the object may belong to one of the three prescribed classes.An unknown number of a class containing that assigns a value to () ind O by using characteristic  .In general, classification rule is defined as a general measurable function, but we restrict the consideration in this paper to the so -called threshold-based classification rules of the six forms The distributions i H are unknown, but they have continuous densities } Gg  t t  .The probability of error of such a classification rules are given by Analogically, Furthermore, Further, similarly ).The threshold B t for a Bayes classification rule is called the Bayes threshold: 1,6 i  we have: arg min ( , ) (arg min ( ),arg min ( )) , and Let us consider the threshold rule

One can apply kernel estimators to estimate the densities of distributions
where K is a kernel (the density of some probability distribution), 0 N k  is a smoothing parameter (Sugakova,   1998; Ivan'ko, 2003).
Let us construct the threshold estimator using EBC method (Kubaychuk, 2008).The empirical Bayes estimator is constructed as follows.First, one determines the sets T of all solutions of the equations

L t t p p H t p p H t p p H t p p H t p p
( , ) N L t t is the estimator for 1 12 ( , ) Let the densities i h exist and be s times continuously differentiable in some neighborhood of the points , where

MAIN RESULTS
In what follows we assume that: ( A ) the threshold B t defined by ( 1) exists and it is the unique point of the global minimum for 1 () L t (  ˆˆ( , ) Proof.According to Theorem 1 of (Sugakova, 1998), the assumptions of the theorem imply that ˆ( ) ( )

NN N u x p p h x p p h x u x p p h x p p h x
Since Lt are continuous functions on  , 1 1 ( ) 0 given the event For the proof next theorem we need some auxiliary result on the asymptotic behavior of the processes N 2347-1921 Vol u m e 12 N u m b e r 05 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6262 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h J une 2 0 1 6 w w w .c i r w o r l d .c o m , 12

ih
with respect to the Lebesgue measure.The family of classifiers is denoted by 2 { : N 2347-1921 Vol u m e 12 N u m b e r 05 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6263 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h J une 2 0 1 6 w w w .c i r w o r l d .c o m


a sample from a mixture with varying concentrations, where : concentration in the mixture of objects of the i -th class at the moment when an observation j is made(Maiboroda, 2003), is the ( , )ki main minor of N Ã .
N 2347-1921 Vol u m e 12 N u m b e r 05 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6264 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h J une 2 0 1 6 w w w .c i r w o r l d .c o m as an estimator for B t , where

Remark 1 .Тheorem 1 .
Condition ( B ) is sufficient for lim , Let conditions ( A ) and ( B ) hold.Assume that the densities i h exist and are continuous, 0 N k  as N k N , k is the continuous function, and c o u n c i l f o r I n n o v a t i v e R e s e a r c h J une 2 0 1 6 w w w .c i r w o r l d .c o m and continuous functions, it follows that () i N ut changes sign in the neighborhood of B i t .This means that there are i

Remark 2 .Remark 3 .
for sufficiently large N .This completes the proof of the theorem, since i The estimator k H (obtained by construction) is unbiased iff I S S N 2347-1921 Vol u m e 12 N u m b e r 05 J o u r n a l o f A d v a n c e s i n M a t h e m a t i c s 6266 | P a g e c o u n c i l f o r I n n o v a t i v e R e s e a r c h J une 2 0 1 6 w w w .c i r w o r l d .c o m Often, ˆk H is not a probability distribution, but it is not important.To estimate k H you can use the corrected weighted empirical distribution function, if necessary.(Kubaychuk, 2003;Maiboroda and Kubaichuk, 2003; Maiboroda and Kubaichuk, 2004).