In the work the task of prediction of parameters at construction of systems of forecasting and management is considered. Existing solutions in the use of fuzzy neural networks are studied and analyzed. The structure of a fully coupled fuzzy artificial neural network without a layer of fuzzy rules, corresponding to the "classical" multilayer perceptron, is proposed. Tested for learning time and RMS error of different structures of the proposed fuzzy artificial neural network in predicting the performance consumed by a coal company. The results allow you to select the number of neurons in the hidden layers depending on the desired accuracy of the prediction of the output parameter.
Prediction issues are relevant for any human activity: weather forecasts, exchange rate forecasts,
socio-economic forecasts, etc. In order to make these forecasts, there is usually a sufficient amount of
data collected over many years, as well as data obtained from current observations or from experts.
However, there are a number of specific tasks: forecasting the demand for resources of an enterprise,
forecasting natural phenomena, etc., when information about an object, its parameters and states is
incomplete, uncertain and/or poorly formalized. Such types of problems arise when building control
systems of objects, the parameters of which are difficult to measure, or the relationship between them
is not uniquely established. An example is the water treatment system of an industrial enterprise, when
the number of indicators (concentration of dissolved salts and gases in water) reaches several dozens,
they are all interrelated, and there is no unambiguous function to select the number of reagents for
treatment depending on the initial chemical composition of water. Another example is the prediction of
electricity demand for a coal mining enterprise, the energy consumption of which is a complex
nonstationary process, which is influenced by a significant number of mining-geological, technological,
industrial, climatic and other factors [
Among the currently known models and forecasting methods we can distinguish [
Various kinds of regressions and models generated from them, as well as time series can be
effectively used in cases where the dependence of the predicted indicator in time is continuous, has a
smooth character and does not contain jumps and discontinuities. In the case of forecasting on the basis
of non-periodic data series, in order to obtain an acceptable accuracy (at least in units of percent), a
significant number of row members or regression coefficients have to be taken into account. Moreover,
when processing non-periodic signals, both regression and time series give adequate results only within
the interpolation interval. Artificial neural networks are more flexible than the aforementioned models,
variables, as well as by the possibility of introducing nonlinearity into the activation functions [
To date, more than a dozen varieties of fuzzy artificial neural networks are known:
Takagi-SugenoKang (TSK), Wang-Mendel (WM), adaptive ANFIS, FALCON, GARIC, NEFCON and FUN FANN,
fuzzy multilayer perceptron, hybrid neural network, as well as FANN, which are various modifications
of "classical" ANS (Kohonen fuzzy self-organizing network, fuzzy baseline radial network and others.)
their terms, and can be determined by the formula [
number of input variables, - number of terms
Consequently, for a network with five input variables, each of which is represented by three terms, the number of such rules will be 125. As the number of rules increases, the time spent on training of FANN and the algorithm of its training will also increase accordingly.
Some publications propose to involve experts to tune membership functions of a fuzzy layer [
We propose the structure of a fuzzy artificial neural network corresponding to the "classical"
multilayer perceptron [
,
(1)
Figure 1: Structure of the proposed FANN
The proposed FANN contains five layers: 1. input data layer, 2. first hidden fuzzy layer (phasing layer - fuzzy reduction), 3. second hidden layer containing neurons with linear activation function, performing summation of data obtained in the second layer multiplied by synaptic weights vector w l, 4. layer performing summation of data obtained in the third layer multiplied by synaptic weights vector w 2, 5. output data layer.
The number of neurons in the first hidden (fuzzy) layer is equal to the sum of the number of terms
of all input variables [
The number of neurons in the second hidden layer is determined in the process of FANN adjustment. At the initial stage this index can be chosen equal to the number of neurons in the first hidden layer.
The fourth layer is resultant, so it consists of one neuron with linear activation function.
Based on the notation system accepted in literature, we can write the structure of FANN shown in Figure 1: as 2-5-4-1. To explain the working principle of the first hidden layer of the proposed FANN, consider the membership functions - the terms of the first hidden layer for one input variable. Suppose there are three such terms, as shown in Figure 2:.
Then on the interval xl-x2 only the effect of the term T1 will appear, on the interval x2-x3 - the
terms and T2, on the interval x3-x4 - the terms T2 and T3, and on the interval x4-x5 - the term T3. This
means that when you change the values of the input variable x in the interval xl-x5, the values of the
coefficients by which the elements of the vector wl are multiplied, which then arrive to the neurons of
the second hidden layer, will change. The selection of membership functions for the first hidden layer
of FANN can be done in two ways. The first way is to distribute the accessory function carriers
uniformly in the carrier domain for the whole range of possible values of input variables, as shown in
Figure 2:, and the second way is based on the processing of initial statistical information [
To verify the proposed structure of FANN, let's take the data of the half-hour graph of the power consumption of the coal mining enterprise for a day. Figure 3: shows a graph of the original data, and predicted values, obtained with the proposed FANN with the structure 2-5-3-1 and with the structure 2-14-5-1.
№
To estimate the training time and RMS errors, NINSs with different structures were constructed. Since the number of input variables is constant in all cases (two variables), the parameters of tested NINSs are represented by the number of neurons in the first hidden layer N HL1 and the second hidden layer N HL2. The results are shown in Table 1 Table 1 Differ by several orders of magnitude, the error values on the ordinate axis are plotted in the logarithmic scale
RMS error , % 6.15 5.37 4.4 5.86 5.86 5.86 8.67 5.92 9 10 11 12 13 14 15 16 17 18 6 8 8 8 10 10 10 14 14 14 5 3 5 7 3 5 8 3 4 5 as a function of the number of neurons in the second hidden layer: I. network with 4 IF; II. a network with 5 IF; III. a network with 5 IF; IV. a network with 8 IF; V. a network with 10 IF; VI. a network with 14 IF.
From the above dependencies of Figure 4:, Figure 5: The number of neurons in the first hidden layer has the greatest influence on the learning time and the error value.
The accuracy of prediction obtained with the proposed FANN can be improved by adjusting the accessory functions for hidden layers. Thus, for example, a slight modification of the parameters of membership functions for the first hidden layer, shown in Figure 6:, Figure 7: for FANN with the structure 2-5-5-1, allows to reduce the error of 5,86% to 0,11% (version 1) and 0,19% (version 2).
Exclusion of fuzzy rules layer from the structure of fuzzy artificial neural network, built on the structure of perceptron with two hidden layers, allows to simplify the procedure of its training and does not limit the researcher in the number of fuzzy neurons. Exclusion of the rules layer will also allow to refuse the subjective component introduced by experts. The presence of fuzzy in the first hidden layer provides an opportunity to train a neural network with a small amount of input data.