Computational approach based on use of multilayer neural networks in classification of rail line states

The article considers the issue of classification of the states of rail lines using multilayer neural networks. It is shown that with significant external disturbing influences, the most promising direction is the retraining of classification models using neural networks. The representativeness of the sample is proposed to be realized by applying the scaling of informative features. It is proposed to minimize classification errors by gradient optimization methods. In the work during classification of rail lines states by neural networks, at the first stage, artificial classes of states are formed. At the second stage, the classification of states by the trained neural network is considered, classification errors are identified. On the basis of these errors the network is retrained until acceptable results are obtained. The article presents neuron learning algorithms: Hebb’s algorithm, error backpropagation method (for training a multilayer neural network), Konohen’s, Hopfield’s, Hamming’s algorithm and network. In the presented revision of the classification model, the correction is carried out according to the Widrow-Hoff algorithm.


Introduction
Classification of rail line states refers to a complex of complex, multi-level tasks under conditions of uncertainty.A specific feature of the problem of classification of rail lines (RL) states is associated with the impossibility of creation of a working alphabet of features in conditions of experimental studies, since the processes occurring in rail lines have seasonal changes during the day, seasons, many years, due to climatic influences, aging ballast material, cross-sleepers, rails.Therefore, it is necessary to use the results of simulation modeling of the information transmission rail path on multi-pole equivalent circuits of rail lines.Sufficient satisfactory results have been obtained using six-pole equivalent circuits for rail lines.At the same time, the problem of modeling under uncertainty has a multidimensionality of informative features and destabilizing effects, and this significantly complicates the general problem of classification [1].

Materials and methods
The practice of using neural networks or networks for recognizing the states of rail lines shows that, as a rule , the available set of input training images of the states Хо=f(х1, х2, х3…хn) cannot be directly used for learning even after ensuring the requirements of uncorrelated features.The nature of some informative features is such that the values of the variables corresponding to them change in a wide range, the boundaries of which differ by several orders of magnitude, and the peak of the distribution of feature values is non-linear, and often turns out to be shifted to one of the boundaries of the feature change interval.Therefore, due to objective reasons, the requirement of representativeness may be violated.
It is possible to make a representative sample by applying the scaling of informative features.The method of scaling informative features can be their logarithm by natural or decimal logarithm.

Results
The system that classifies the states of rail lines, synthesized by the principle of constructing neural networks, with the arrangement of neurons in several layers, is shown in Fig. 1.Such a classical structure for constructing multilayer neural networks makes it possible to recognize complex classes under the influence of significant destabilizing factors, and, accordingly, it is very convenient for recognizing the states of rail lines when changing the total transverse conductivity of the insulation of rail lines Yп=G0+jωC0 of the total longitudinal resistance of the rail lines Zр=R0+jωL0, where, G0, C0,R0,L0 are the primary parameters of the rail line, ω is the circular frequency of the polling current of the rail lines, the sensitive element of the sensor of information on the state of the rail lines.The neurons of the input layer of the network (figure 1,a) perceive the input signals (informative features) and, having converted, transmit them through the branching points to the neurons of the next layer.As a rule, the output signal of the i -th neuron is transmitted to the inputs of all neurons of the subsequent (i+1) layer.In practice, to achieve this goal, it is sufficient to have a two-or three-layer neural network.In fully connected neural networks (figure 1b), the output of each individual neuron is associated with all inputs of other neurons, including its own input.
A neuron is a processing of single information of a primary feature in a network.A functional model of a neuron, which is the basis of neural networks is shown on figure 2. Model can be divided into 3 main elements: synapses (weights, or coefficients), an adder that forms a decisive function and an activation function.In the mathematical description of the functioning of the k th neuron, the function dк(X) is described by the equations where (Х)=f(  At the second stage the states are classified by a neural network that are pretrained at nominal values of the circuit parameters, a classification error is detected, and the network is retrained (new synaptic weights of neurons are determined) until acceptable results are obtained by measuring the actual values of informative features.
Hebb 's algorithm (for the case of two classes of states).Accordingly, the reference outputs of the network 1.The initial values of the vector of coefficients are activated i.e. assigned.For example, all coefficients are equal to 1: (0) W . W n M so that the root mean square error of misclassification is minimal, i.e.

Correction of all coefficients
(2) ( ) It is convenient to minimize the functional (2) by the gradient descent method, by adjusting the coefficients of each layer.In order to do it: 1.The initial values of the coefficients are set (2)   ( ) ,.., M WW 2. A training vector of features is fed to the inputs of the neurons of the input layer of the network and the values are calculated 3. Using formula for the first layer where n is the layer; j -neuron. 5. Similarly, by backpropagating the error according the synaptic weights on other layers are corrected.
Known and widely used in practice are the Kohonen algorithm and network [3] for cluster data analysis, the Hopfield algorithm and network [5] for associative recognition in the Euclidean metric for a bipolar input vector , the Hamming algorithm and network [6] for associative recognition with respect to the Hamming metric for two bipolar vectors

Discussion
The results of the research shall be illustrated on the example of learning based on error correction [7,8].The neural network of the forward propagation of a neuron k , shown in Fig. 3   Correction is carried out according to the Widrow-Hoff algorithm [9], or the delta rule.On Fig. 3 () where is some positive constant that determines the learning rate used while moving from one iteration to another.

Conclusion
A machine study of network learning using a mathematical model and the software package «Matrix Laboratory» (MATLAB software package) showed that the obtained classification results depend on the specified criteria requirements.One of the informative features of the neural network model is the angular degree measure of the phase of voltages and currents at the input and output of the rail line, which can take an integer value ranging from -5 to -130°.In this case, the values of the variable -5 and -130° are essentially identical, but the difference between the values is large, so that with further transformation they will be on opposite normalization limits.Therefore, in this case, it is convenient to replace informative features, -the phase difference of the signals (angular degree measure) with others (radian measure).Then the range of change in the dimension of the phases of the signals will be minimal, which corresponds to the physical meaning of the original informative feature and will be reflected in the more correct operation of the neural network model.

ky
is the output signal of the neuron.In accordance with the structure of the neuron model, the decision function of the k-th neuron is calculated as of a neural network for classifying the states of rail lines with a retrainable model when external influences or internal parameters of the circuit change is associated with the choice of an adequate principle for learning the network, or determining the coefficients of a neuron To train a neuron, it is necessary to have a training sample with a predetermined membership of the sample to a certain class.Then the training of the neuron is carried out by the procedure of sequential presentation of informative features from the training set and the calculation of the classification error, i.e. deviation of the image classification result from the class boundary to which the image belongs on the presented set of training images of true class boundaries.Minimization of the classification error of determination of the true coefficients and the number of iterations is possible by gradient optimization methods (for example by the Hooke-Jeeves method, which provides multi-objective optimization).While classifying RL states by neural networks, at the first stage artificial, virtual classes ij a with certain numerical values are formed using simulation modeling, and, the neural network learns the correct, error-free classification of all classes of RL states with separate class centers using the indicators ij p of each class.
kj n is the current value of the coefficient of the  kj neuron k corresponding to the sign () j xn of the image vector () Xnat the iteration step n .According to the Widrow-Hoff rule applied to the coefficient  kj , the change ()   kj n is assigned by the expression: Training of a multilayer neural network by the method of error back propagation.The goal of training a multilayer neural network is to determine the coefficients of the matrix 3. The condition for completing the adjustment for the found new value of the coefficients is checked by a predetermined criterion  , namely, for each pair, ( , ), ,   ii sy , ∀ 1, 2,..., , = iN then training stops, otherwise go to step 2.
is considered.The input of the neuron k receives a signal vector (informative features) () Xn.The output signal of the neuron k is denoted as () d Х .This signal is the output signal of the neural network.The output signal in the designed neural network will be continuously compared with the reference value of class indicators ijk p in this case, applying the above notation, () XNXSh XK Figure 3. Variant of the neural network tuning model.As a result, the classification error signal at n -iteration is obtained: