1. Introduction

Most of works in respect of neuronal networks of the scientific literature describe the construction method of the net structure, however comparatively fewer numbers of works are dedicated to the error-correction throughout the recognition process. In that case when error-recognition process is discussed, e.g. error back propagation algorithm, the reasons of which are not defined and their role (contribution) in the recognition of error. In this work the method of revealing recognition error and correction, which is based on the feedback of neurons are discussed, owing to them information about error is spread, which is followed by the parameters of neurons: weighing coefficients and in case of need changes of the threshold value. Algorithm of changes is iterational and proceeds up to the time until the error recognition is corrected.

Additionally, the method of evaluation of features is described, owing to that each feature or their group under viewpoint of correct recognition is evaluated.

2. Description of the Recognition Process by the Formal Neuron

Recognition of unknown or known set of patterns, for example realization of learning set, vectors from feature space get at the entry of recognition system, in our case neuron or neuronal network. As a result the coordinates of vectors are multiplied by the appropriate weight coefficients of the neuron. Each member of the received product is summed up by the summing element, whereof neuron reaction is received upon the realization of recognition vectors, NET number, with the help of which we establish appropriate meaning of the activation function F(NET). The meaning of the activation function is compared to Z neuron threshold. According to the comparison result the output of neuron - OUT is formed, the meaning of which is equal to zero, if the recognition is wrong, or to one if the recognition is correct.

Let us define with predicate “presentation” the appearance of recognizable vector at the input of neuron, mathematical analogue of which is graphically expressed by the symbol ” ”, where symbol expresses predicate and represents the result received through mathematical operation. This allows us to express recognition process compactly:

(2.1)

Where is a recognizable realization, Ne(W)-set of neurons with the weighing coefficients summing element; the meaning of activation function F(NET) is defined according to NET, that is compared with the value of threshold Z of the neuron and according to the comparison the meaning of output signal OUT of neuron is defined. In particular we have:

(2.2)

(2.3)

Expression (2.2) describes the process of making decision whilst presenting unknown realization. Let us note that the conclusion of the experimenter about rightfulness and incorrectness of the recognition process, especially while realization of the learning process, might be different from the formal procedure, which we are going to discuss afterwards.

Let us denote pattern set for recognition by A, while the element of this set by where , where represents realization; the coordinates of vector are the real numbers received via measuring the features:

3. Description of Correct Recognition Process

Correct recognition process consists of two parts. The first, when the realization of recognizable pattern is presented to this kind of recognizing neuron, e.g. realization is presented to neuron.We consider, that in such case realization appeared at the input of “Its” type neuron. The second, one type realization e.g. is presented to for recognition, that is “The other” type neuron. Both cases are described in (2.1) and (2.2) terms.

(3.1)

(3.2)

Where Let us denote that in (3.1) term OUT=1, but in (3.2) OUT=0; Despite this, in both cases recognition is correct, thereby 3.1 and 3.2 differ from 2.1 and 2.2 terms.

4. Description of Incorrect Recognition Process

We might have two cases among incorrect recognition processes: the first, when any kind of realization is presented to “Its” neuron and the second one type of realization is presented for recognition to “the other’s” type neuron. In order to describe the first case let us assume, that is presented to neuron, term (3.1), with the difference that the output is equal to zero;

(4.1)

(4.2)

Where terms (4.1) and (4.2) give us description of the incorrect recognition process. The aim of the following thesis is to correct this mistake. In order to achieve this aim, hereafter let us discuss in detail the process of making an error and we will work out the algorithms for its correction.

5. Error Correction Whilst Recognizing “Its” Realization

In order to examine the error made by term (4.1) let us discuss the second (last) part of the this term:

(5.1)

There is one way to correct the error: we must change neuron weighing coefficients of so that it changed the sign of inequality

(5.2)

Let us discuss value-receiving process. The function F(.) is called an activation function, which is various by different nonlinears, we choose one out of them. Activation functions have one similar function: increasing monotonously, existence of nearly linear segment (besides activation function of perceptron). So we can assume, that F(NET) function parameters, or neuron weighing coefficients are chosen in the way that, in case of any kind of change of weighing coefficients we stay on the linear segment of the activation function, so the following correlation is correct:

(5.3)

In case of term 5.3 is true k value represents scale coefficient, which is changed during the process of learning weighing coefficient. So we receive, that k=1 and accordingly instead of term 5.3, we will have:

accordingly instead of term 5.1 we will have: 5.4

(5.4)

In the process of recognition the error correction will take place if (5.1), i.e. instead of (5.4) term we will have inequality expressed by (5.2) term:

(5.5)

It is possible to get (5.5) inequality from inequality (4.4) through increasing of weighing coefficients. Therefor we should grant weighing coefficients with initial meanings. Let us assume that in realization set we have this type learning realization set for which the following condition is fulfilled: Let us count weighing coefficients initial meanings through learning set realization (vectors) coordinates; through so-called awarding procedures:

(5.6)

By conducting overlay procedure for any kind of neuron e.g. we take weighing coefficients, the values of which are placed at range. It is possible to increase weighing coefficient step-by-step using recurrent iteration procedure and heuristically i.e. all of a sudden by adding any constant/unvarying number. We should carry out increasing i.e. awarding procedure for the weighing coefficients values of which is close or equals to it means these characteristics are more important for the given pattern than any other. Let us discuss recurrent procedure:

(5.7)

If we use (5.7) iteration it will be necessary to choose that must be done by considering initial weighing coefficients. We check the recognition result of iteration at every step through the 5.5 term. We increase the number of steps until it becomes inequality 5.5 true/correct. Which corresponds to present incorrectly recognized realization for recognition for the second time. When weighing coefficients received from the previous step move unchangeably to the next step.

Those values of weighing coefficients, when inequality (5.5) becomes true, let’s designate it by In that case we will have:

(5.8)

The truthfulness of 5.8 term means that the error in recognition is corrected. The change of “i “and “m” indexes show that errors can be corrected for realizations of any learning set and for any pattern of A set.

The algorithm of error-correction in (5.8) term does not change correct recognition results received through changes of neuron weighing coefficients of different patterns.

Whilst changing of weighing coefficient we just use “Awarding” procedure expression (5.7); we choose equal quantities as neuron thresholds initial values:

(5.9)

Then by using (5.8) and (5.6) procedures we have an opportunity to recognize all the learning-set-realizations of pattern sets without errors, so that previously received recognition correct results stay unchanged.

6. Error Correction Whilst Recognizing Realizations of “Other’s” Pattern

The process of incorrectly recognition is described by (4.1) and (4.2) terms.

Let us discuss 4.1 expression, where instead of we write its term according to 5.1.

(6.1)

Where

In order to correct the error it is necessary to change inequality signs in (6.1) term, that can be done by increasing weighing coefficients, i.e. with the method described in the previous paragraph.

Let’s discus the right part of the (4.2) term:

(6.2)

Threshold must be increased for correcting the error, or weighing coefficient decreased. Because of the fact that according to the condition, we use just “awarding” procedure, that’s why we choose the procedure of increasing of the neuron threshold in the image similar to (5.7), where instead of weighing coefficient we will have the meaning of the threshold:

(6.3)

Required value of the neuron threshold when (6.2) sign of inequality is changed in an opposite way, we will denote it with

(6.4)

As the neuron pattern is one, which has one threshold, that’s why if we assume the increase of the threshold of the type neuron up to quantity, then we might receive errors in the recognition of “its” realizations, such opportunity especially will take place there, where the term is fulfilled: Precisely in order to find out those characteristics, upon which “awarding” procedure might be carried out, it is necessary to carry out the experiment of recognition for those pattern realizations for which the above mentioned inequality takes place. In case of an error we should repeat the procedure that was mentioned in the fifth paragraph.

7. Evaluation of the Features According to the Correct Recognition Criteria

For the evaluation of features basically clustering methods are used, for example “Theory of Rank Links” [1], [2], with their help the meanings of clusters are established for each pattern; also the degree of their compactness or non- compactness and etc. are established. In our case we consider recognition process by the formal neuron, but for the evaluation of features we use just the correct recognition criteria.

The correct recognition criterion implies determination of the features, changes of which provide correct recognition, or if the recognition is incorrect, corrects an error. recognition process of realization by neuron is described in 3.1 term. In the recognition process basically receiving of value, which may be used instead of value (5.3 term) in order to receive final result OUT of recognition. Hereinafter we will see that changes of neuron weighing coefficients F(.) take place on the linear segment of the function (5.3 term).

According to the assumption in 5.3 term the result of recognition process is correct, if inequality is fulfilled 5.5:

(7.1)

The value in 7.1 term consists of the sum of multiplications. Let us examine each term of the sum, which consists of n element and the appropriate weighing coefficient of the learning set realization. In order to fulfil the inequality of 7.1 term the meaning/value of the multiplication must be as high as possible. As realization coordinates are important for the signs received through measuring, which could not be changed, therefore we can change just weighing coefficients, i.e. elements of vector and choose initial meanings of weighing coefficients through learning set realization. Let us consider that If we denote elements of X set by but coordinates of then for the initial meanings/values of the weighing coefficients will be:

(7.2)

By fulfilling of 7.2 term we will get vectors for each pattern and neuron initial weighing coefficients. set, coordinates of which represent objective values, as they are received through the processing of the results of features.

It is obvious that as high of meaning/value in 7.1 term is as high its contribution in receiving value. By this way the reliability of correct recognition will increase. In order to evaluate weighing coefficients we have to ascertain lower limit of their meanings/values, according to which we can evaluate meanings/values of the given features, i.e. their contribution in the formation of NET value. We should choose lower limit of value according to maximum meaning of the weighing coefficients of features.

(7.3)

Where is a lower limit of the meanings/values of neuron weighing coefficients.

Definition 7.1 for feature is important for the given pattern, if the value received by measuring of which is more or equals to a lower limit of the weighing coefficient:

Let us assume that the term of 7.1 definition is fulfilled by the number of feature Let us calculate new weighing coefficients according to 7.2 term.

(7.4)

Where constitutes a number of those realizations, where inequality of 7.1 term was fulfilled.

It is clear that that’s why by using in 7.1 term, the inequality will strengthen, which means that the reliability of correct recognition will increase. In that case when the recognition is wrong, which means that it can be corrected. In case the error is not corrected, we should use algorithm described in chapter 5. Where we will award features defined by 7.1 term. We will isolate subsets from the feature sets by (7.3) and (7.4) terms, which is important and necessary for the correct recognition of “its” realisation of the given pattern.

Let us discuss the method and algorithm for the features definition, which gives us opportunity to correct the result of the incorrect recognition, which we might have of one type, for e.g. when presenting the realization for the other type Nej neuron. According to terms 7.2 we will have:

(7.5)

İn order to correct an error it is important to change the sign of 7.5 inequality reversely, so the right part of the inequality must be decreased/reduced. Reduction can be carried out in two ways: 1. Through the decreasing of weighing coefficients, which might cause errors. While recognizing of “Its” realizations; hence we cannot use this method because according to the term we use just awarding procedure.

2. Let us discuss the method of the reduction of the sum of the weighing coefficients i.e. value, which was described in (7.3) and (7.4) terms. Let us suppose that we have ascertained sub-level of weighing coefficients- Uj and through it let us set a definition (7.1) important features list for pattern and for neuron weighing coefficients. Let us calculate weighing coefficients according to (7.4) terms.

(7.6)

Where important features of neurons are number of those realizations, for which 7.3 term is fulfilled; those numbers of features, for which definition 7.1, i.e. inequality is true.

İn order to calculate value of the definition 7.5 instead of let us use its correlation from 7.5 terms and we will receive:

(7.7)

Because summing in 7.7 terms is done by index, but presumably as a rule that’s why the quantity of the components of sum 7.7 is reduced by term, which might change the sign of the inequality in 7.5 terms reversely:

(7.8)

İn that case if the sign of the inequality in 7.5, despite the attempt will not change, then we should use the procedure described in chapter 6, i.e increasing the threshold, for the compensation of which it is necessary to increase weighing coefficient. In order not to make new errors while increasing coefficients while recognising realizations, when defining features for awarding we should use the procedure described below.

Let us assume that and for neuron patterns and We have the vectors of weighing coefficients calculated by the 7.2 and 7.3 terms. Let us make binary vectors by means of them, the elements of which fulfill the term:

It is obvious that vectors will be the same dimensional as vectors. Let’s assume that for recognition of pattern realization we get an error pattern on neuron. Let us choose corresponding for as means of awarding, for which the term is satisfied.

(7.9)

İf when recognizing the realization we get an error to neuron then in order to correct an error we should choose appropriate for which the term is fulfilled:

(7.10)

İn both cases the inequality 7.8 will be srengthened, so that the recognition results received through 7.1 inequality will be unchanged.

8. Conclusions

The method of error-correction in the learning process of neuron and its net is described in the work. Theoretical and Applied Perspectives of learning method are elaborated. Two possibilities of making an error are mentioned and algorithmically described. First, the error which will be made through the recognition of the realizations of “Its” pattern. Second, the error which we have through the recognition of the realizations of “Other’s” pattern. Both cases require the formation of different theoretical and applied means. It is proved that error can be corrected through the means of changing of neuron weighing coefficients and thresholds values. Thereby just awarding procedure is used, which provides an error–correction, in that case, if necessary measures will be received in order to avoid neuron over-filling effect.

It has been proved theoretically that correction of one error does not cause other errors.