Russia, Saratov, SGTU

The task of simulation by a method of reconstruction of dynamic systems from time series is discussed. The shortages of the existing approaches are marked. The possibility and advantages of a solution of this task on neural networks is exhibited. The singularities of the approach are researched on an example of reconstruction of a system from wave forms of pulses. The results of experiments are given for the Hopfield network.

Россия, Саратов, СГТУ

Рассмотрена задача моделирования методом реконструкции динамических систем по временным рядам. Отмечены недостатки существующих подходов. Показана возможность и преимущества решения подобной задачи на нейронных сетях. Особенности подхода исследованы на примере реконструкции системы по пульсограммам. Приведены результаты экспериментов на сети Хопфилда.

Many problems of control, prediction, pattern recognition are connected to the task of construction of mathematical models. It is possible to mark off two approaches to its solution. The first approach consists of construction of model on the basis of known principles and laws of functioning of a system to be modeled. At the second approach model is constructed from experimental time-series data. In this case it is supposed, that the mathematical model represents a system of the ordinary differential equations approximately describing evolution of a status variable of system. The construction of model, or reconstruction of a dynamic system (DS), is restoring the equations of an analyzable system, which with a given accuracy are capable to reproduce the experimentally obtained time series.

We shall consider in more detail second approach, which is applied to research of complex systems accessible to study only from time-series data. Most striking examples are the medico-biological systems, where the achievements of the theory of dynamic systems can discover the practical application. As an example we shall consider the task of reconstruction of a dynamic system from pulse waves with the purpose of diagnosis of illness. So, having pulse signal and deciding the task of reconstruction, it is possible to estimate a state of health of the man and to reveal pathologies.

In this case interest to pulse forms is stipulated by three reasons:

The existing methods of recognition of wave forms of pulses are founded on definition of characteristic points, curvings, magnitudes of peaks, spectral analysis. However human brain is badly adjusted to realization of arithmetical operations. The brain creates on its neural networks images and operates with them. With reliability it is possible to assume, that feeling pulses by fingers, Tibetan doctors renovated in their brain a dynamic model of human organs and systems to be investigated.

It is necessary to mark, that the observable time series represents such stable conditions, which are reached at t ® ? . It implies that a phase space of a system has an attracting limited set of points, i.e. attractor. Therefore reconstruction implies the representation of an approaching phase space and shaping of limited sets appropriate to the initial time series. There are many publications on this problem (ex.. [1] and references in it).

Explained in [1] the approaches to a problem of reconstruction of DS are characterized by following main shortages:

1. The choice of polynomial nonlinearity is not always justified.
2. The resulting equations receive not rasping, that is the small modification of their parameters leads a phase trajectory to infinity.
3. Small distortion of experimental data can essentially distort results.

The neural networks are more flexible means of simulation [2]. The approximation of nonlinearities is carried out neural networks. Each neuron implements the simple nonlinear function of one argument j (y). The neuron has some inputs, on which the vector of signals X comes. The significance of inputs of neuron is defined by a vector of weights w. If there are n neurons, on which inputs the vector of signals X comes, we shall have n vectors w, which will derive a matrix of weight coefficients W.

Argument of the nonlinear function of a neuron is (X, W), i.e. j (y) = j [(X, W)]. Here j is the sigmoid function of activation of a neuron, y=1/ (1+e^-y). The result of transformation is transmitted to inputs of other neurons. The nonlinear transformation of the initial image X={x₁,x₂,…,x_n} calculates by an equation

Neural network represents superpositions of simple functions of one variable and their linear combinations and makes approximations of nonlinear functions [ 3] . Also in a case with polynomials, the matrix W is defined from a condition of minimization of an error of approximation. The principal difference consists that the structure of a matrix W (symmetry and signs of elements) is set originally arbitrary. On the following steps of approximation the values of elements are defined, for example, by method of least squares. The redundancy of the information in nets connections ensures the roughness of approximation and high accuracy.

Except the approximation of nonlinearities the neural networks are a general purpose means of representation of an operator of evolution. In this case equation (1) of transformation of an image is substituted by a system of the nonlinear differential equations of neural dynamics

Here V_i(t) - state i - th neuron at moment t; b_i - an amplification; a matrix W_ij and function j have the same sense, as earlier. Depending on choice of coefficients b_i, c_i the networks of different types will be received.

For dynamic systems of a such type the theorem of the Cohen - Grossberg guarantees existence of areas of attraction - attractors. These areas are shaped during synthesis of a matrix W on the basis of the initial or standard images. The evolution of a state of renovated neural network leads it to the area or point of equilibrium. This area is characterized by a local minimum of an energy, which coordinates correspond to a standard image.

From the equation (2) it is possible to receive following equation

where b_i - feedback coefficients, c_i - external influence on i- th neuron. The obtained equation (3) describes a Hopfield network. This network is selected for reconstruction of dynamic system according to the wave forms of pulses.

The Hopfield network will be used for storage of static images, which represent points of attractors in a phase space of a neural system.

The time realizations of renovated systems represent cluster sets of other type. In the elementary case it is a limit cycle. There are possible the more complex formations also. Therefore singularity of reconstruction on Hopfield networks consists of shaping the limit cycle, but not equilibrium point. This singularity imposes additional conditions on rules of synthesis of a matrix W [4].

In particular, the matrix was selected asymmetrical. Except it, the network architecture took into account that the original signal is almost-periodic. The amount of neurons in a network was calculated from expression n = (T/t )+1, where T - period of the initial realization, t - discretization of pick up of the information.

Two series of experiments were carried out. The first series of experiments was based on the theorem Takens [ 5] . As standard images there were selected the initial realization in form

and its modifications obtained simple renumeration a_i = a_i+1. Thus, there are represented n samples to the networks.

The second series of experiments was grounded on consecutive differentiation of the initial time sequence. As state variables the values a(t), a'(t) and a''(t) were used. Therefore network contained 3n units. The choice of three-dimensional space is stipulated by that the estimation of Hausdorff dimension is 2,75.

A possibility of reconstruction of dynamic system on neural networks in both series of experiments is proved experimentally. The solutions of the renovated equations of model repeat qualitatively main singularities of an original signal, however likeness in details are exhibited not always. Besides the limit cycles arose only at definite numerical values of the initial realizations. Let's mark, that the second series of experiments is less sensitive to values.

The further researches are necessary for perfection of a technique of definition of a matrix of connection and network architecture, which is guaranteed result the limit cycles.

References

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2. Булдакова Т.И., Суятинов С.И. Нейрокомпьютерные системы: Учеб. пособие. - Саратов: Изд-во Сарат. Гос. Техн. Ун-та. - 1999.-96 с.
3. Нейроинформатика /А.Н.Горбань, В.Л.Дунин-Барковский, А.Н.Кирдин и др. - Новосибирск: Наука. Сибирское предприятие РАН, 1998. - 296 с.
4. Петрякова Е.А. Нейронные сети Хопфилда с несимметрической матрицей коэффициентов связи между нейронами //Известия вузов. Приборостроение. - 1994. - Т. 37. - № 3-4. - С.24-32.
5. Takens F. Detecting strange attractors in turbulence // Dynamical Systems and Turbulence / Eds. D. Rang and L.S. Young. Lect. Notes in Math. - 1980.- V. 898. - P. 366-381.