Deterministic Learning From Sampling Data And Its Applications

In recent years,machine learning in dynamical or non-stationary environments has become a hot frontier of artificial intelligence.The ”dynamical” herein means that the environment of such problems tends to change dynamically over time.For example,the physical processes involved in different fields,such as engineering,biology,medicine,economics,astronomy,and so on,naturally constitute an open and dynamical environment.However,in an open dynamical environment,since the closed static assumptions are no longer satisfied,the dynamical environment machine learning problem poses a greater challenge and broader application prospects to the current development of artificial intelligence.Nowadays,one of the main directions of machine learning in dynamical environments is to study the modeling,recognition and classification of time series data obtained through sampling in engineering,medicine and other dynamical processes.In this paper,the problem of machine learning in dynamical environments is studied from the perspective of system and control.This paper mainly studies the following four aspects:1)For time series with periodic or recurrent properties generated from nonlinear dynamical systems,modeling and regression method is proposed based on deterministic learning.From the perspective of system and control,this difficult problem belongs to nonlinear system identification.By solving the problem of satisfaction and verification of the persistent excitation(PE)condition,the deterministic learning method achieves the accurate NN approximation for nonlinear system dynamics in a local region along the periodic or recurrent state trajectory.In this paper,this problem is further extended to the deterministic learning from time series(sampling data).Through the weight update law design from the Lyapunov procedure,a discrete LTV error system,which is in a form corresponding to the continuous deterministic learning,is yielded.Though the exponential stability of continuous LTV systems under the PE condition is established by Anderson,Narendra,and so on,there are few existing results concerning the exponential stability problem of the discrete-time LTV case.To this end,a detailed proof is given in this paper by using the discrete ISS-small gain theorem.First,under the PE condition,the Lyapunov functions for each sub-system is constructed by using the converse Lyapunov theorem.Second,it is proven that the Lyapunov functions of both subsystems satisfy the property of ISS-Lyapunov,which means both subsystems are ISS.Finally,according to the ISS property of these two interconnected subsystems,the discrete-time LTV system is proved to be globally uniformly asymptotically stable(i.e.exponentially stable for linear systems)by using the ISS-small gain theorem.Further,it is guaranteed that the exponential convergence of neural weights along the trajectory of time series to their optimal values,which leads to the locally accurate identification of the inherent dynamics of time series(sampling data).The proposed method provides an effective solution to the regression and modeling problems of time series such as sampling data.In particular,this result is of great significance for the following applications,such as recognition and classification for time series.2)For time series with periodic or recurrent properties generated from nonlinear dynamical systems,recognition and classification problem is investigated(These time series with periodic or recurrent properties can also be called dynamical patterns.It is also a problem of dynamical pattern recognition).Since the dynamical patterns represented by these time series exist widely in a practical dynamic process,essential different from traditional static patterns,the study of this problem is important and promising,and meanwhile,this problem is a challenging task.In this paper,this problem is further extended to the problem of rapid recognition of time series obtained by sampling from continuous systems.In the sampled-data framework,the corresponding similarity definition and rapid recognition method are presented.In these rapid recognition schemes,the differences in system dynamics of dynamical patterns are indirectly reflected in the errors of dynamical recognition systems.For accurate recognition,it is often necessary to make a assumption on dynamics differences for the dissimilar dynamical patterns(i.e.It is needed to assume that the dynamics differences are larger than a given value,and the sign of dynamics differences can not be changed in a continuous time-interval).However,these requirements are hard to verify in practical applications,since it is difficult to calculate the dynamics differences between dynamical patterns by using continuous-time signals.In other words,even in the case when misrecognition occurs,the situation is difficult to analyze due to the lack of condition verification.In the sampled-data framework,the dynamics differences can be verified based on historical sampling data.The accuracy of the recognition result can also be verified.Therefore,this recognition method based on dynamics differences is interpretable,and will play an important role in engineering practice.3)The structural stability is a fundamental property of nonlinear dynamical systems,which provides a natural dynamical classification relation of dynamical systems and their perturbed systems.The principle is that,if a structurally stable system has a similar topology to its perturbed system,the two systems can be regarded as the same class of systems.Based on the concept of structural stability,a new method for dynamical pattern recognition is explored in this paper.However,since the partial derivative information of dynamics is hard to obtain,this concept is difficult to apply in practice.In order to solve this problem,the partial derivative information of system dynamics along the trajectory direction can be approximated based on the knowledge reuse mechanism of deterministic learning.Thus,the similarity definition based on the structural stability can be given to describe the similarity between two dynamical patterns,and the corresponding recognition method of dynamical patterns is also presented.Compared with the dynamical pattern recognition method considering the differences only in dynamics,the method herein performs better in the scenarios where the differences in system dynamics of dynamical patterns are small.4)Finally,for important clinical problems about early detection of myocardial ischemia/myocardial infarction,the application of the aforementioned machine learning in dynamical environments is carried out.Our research group has carried out studies on the detection of myocardial ischemia based on the cardiodynamicsgram(CDG)yielded by deterministic learning for many years.CDG is a 3-D visualization of the locally accurate modeling results for the ST-T segment of Electrocardiogram(ECG).The CDG is related to the dispersion of the cardiac repolarization process and has higher sensitivity compared with the traditional electrocardiogram diagnosis.In this paper,the quantitative indexes of CDG are first proposed to describe the dispersion of the CDG.On one hand,the time dispersion index of CDG is proposed based on the physical time-frequency characteristics of electrocardiogram signals.On the other hand,the space dispersion index of CDG is proposed based on the chaos analysis of nonlinear systems.The clinical trials in the fuwai hospital and the shihezi people's hospital showed that the quantitative indexes of CDG could accurately detect myocardial ischemia in patients presenting with nondiagnostic ECGs.