Since all information signals and interference are random and can only be predicted with a certain degree of probability, the probability theory is used to describe such signals. In this case, statistical characteristics are used, which are obtained by conducting numerous experiments under the same conditions.
All random phenomena studied by probability theory can be divided into three groups:
- random events;
— random variables;
- random processes.
random event is any fact that, as a result of experience, may or may not occur.
A random event is the appearance of interference at the input of the receiver or the receipt of a message with an error.
Random events are denoted by Latin letters A, B, C.
The numerical characteristics of a random event are:
1. Frequency of occurrence of a random event:
where m is the number of experiments in which this event occurred;
N is the total number of experiments performed.
As follows from expression (40), the frequency of occurrence of a random event cannot exceed 1, since the number of experiments in which this event occurred cannot exceed the total number of experiments.
2. Probability of occurrence of a random event:
That is, the probability of occurrence of a random event is the frequency of its occurrence with an unlimited increase in the number of experiments performed. The probability of the occurrence of an event cannot exceed 1. A random event with a probability equal to one is reliable, i.e. it will definitely happen, therefore, events that have already occurred have such a probability.
Random value is a quantity that changes randomly from experience to experience.
The random variable is the noise amplitude at the receiver input or the number of errors in the received message. Random variables are denoted by Latin letters X, Y, Z, and their values are x, y, z.
Random variables are discrete and continuous.
Discrete is a random variable that can take on a finite set of values (for example, the amount of equipment, the number of telegrams, etc., since they can only take the integer 1, 2, 3, ...).
Continuous is a random variable that can take any value from a certain range (for example, the noise amplitude at the receiver input can take any value, just like an information analog signal can take any value).
Numerical, statistical characteristics describing random variables are:
1.Probability distribution function.
F(x)=P(X ? x) (42)
This function shows the probability that a random variable X will not exceed a specific value of x. If the random variable X is discrete, then F(x) is also a discrete function, if X is a continuous value, then F(x) ? continuous function.
2. Probability density.
P(x)=dF(x)/dx (43)
This characteristic shows the probability that the value of a random variable will fall into a small interval dx in the vicinity of the point x', i.e., in the shaded area (figure).
3. Expected value.
where xi are the values of a random variable;
Р(хi) is the probability of occurrence of these values;
n is the number of possible values of the random variable.
where p(x) is the probability density of a continuous random variable.
In its meaning, the mathematical expectation shows the average and most probable value of a random variable, i.e., this value is most often taken by a random variable. Expression (44) is used if the random variable is discrete, and expression (45) if it is continuous. The notation M[X] is special for the mathematical expectation of the random variable given in square brackets, but the notation mх or m is sometimes used.
4. Dispersion.
Dispersion quantitatively characterizes the degree of dispersion of the results of individual experiments relative to the average value. The notation for the variance of a random variable D[X] is generally accepted, but the notation ??x can also be used. Expression (46) is used to calculate the variance of a discrete random variable, and (47) is used to calculate the variance of a continuous random variable. If you take the square root of the variance, you get a value called the standard deviation (?x).
All characteristics of a random variable can be shown using Figure 22.
Figure 22 - Characteristics of a random variable
random process is such a function of time t, the value of which, for any fixed value of time, is a random variable. For example, Figure 23 shows a diagram of some random process observed as a result of three experiments. If we determine the value of the functions at a fixed time t1, then the obtained values will turn out to be random variables.
Figure 23 - Ensemble of implementations of a random process
Thus, the observation of any random variable (X) in time is a random process X(t). For example, information signals (telephone, telegraph, data transmission, television) and noise (narrowband and broadband) are considered as random processes.
A single observation of a random process is called implementation xk(t). The set of all possible realizations of one random process is called an ensemble of realizations. For example, Figure 23 shows an ensemble of implementations of a random process, consisting of three implementations.
To characterize random processes, the same characteristics are used as for random variables: probability distribution function, probability distribution density, mathematical expectation and variance. These characteristics are calculated in the same way as for random variables. Random processes are of various types. However, in telecommunications, most random signals and noises are stationary ergodic random processes.
Stationary is a random process whose characteristics F(x), P(x), M[X] and D[X] do not depend on time.
Ergodic is a process in which the time averaging of one of the implementations leads to the same results as the static averaging over all implementations. Physically, this means that all implementations of the ergodic process are similar to each other, so measurements and calculations of the characteristics of such a process can be carried out using one (any) of the implementations.
In addition to the four characteristics above, random processes are also described by the correlation function and the power spectral density.
The correlation function characterizes the degree of relationship between the values of a random process at different times t and t+?. Where? time shifting.
where tn is the observation time of the realization xk(t).
Power Spectral Density- shows the distribution of the power of a random process by frequency.
where? P is the power of the random process per frequency band? f.
So the observation random event in time is a random process, its occurrence is a random event, and its value is a random variable.
For example, the observation of a telegraph signal at the output of a communication line for some time is a random process, the appearance of its discrete element “1” or “0” at the reception is a random event, and the amplitude of this element is a random value.
MINISTRY OF EDUCATION AND SCIENCE OF THE RUSSIAN FEDERATION
NOVOSIBIRSK STATE TECHNICAL
UNIVERSITY
FACULTY OF AUTOMATICS AND COMPUTING ENGINEERING
Department of Data Acquisition and Processing Systems
LAB #12
RANDOM SIGNALS AND THEIR CHARACTERISTICS
Group: AT-73 Teacher: Assoc. Shchetinin Yu.I.
Student: Vitenkova S.E.
Novosibirsk
Goal of the work: studying the main characteristics of stationary random signals (average value, autocorrelation function, power spectral density) and acquiring practical skills in their calculation and analysis in the Matlab environment.
1. Generation of 500 random signal samplesX with zero mathematical expectation and unit variance and calculating mean and variance estimates forX .
Let's use the following script file to generate 500 samples of a random signal X with zero mathematical expectation and unit variance and plotting X.
The resulting graph is shown in fig. 1.
Rice. 1. Graph of a random signal X.
Random processes can be characterized by mathematical expectation and dispersion. Mathematical expectation is the average value of a random variable, and the dispersion characterizes the dispersion of the signal relative to its average value.
These characteristics can be approximated by knowing N signal samples, using expressions (1) and (2).
(1)
(2)
Let's use custom functions dispersion() And wait() to determine estimates of mathematical expectation and variance by expressions (1) and (2).
function D = dispersion(y)
% dispersion
m = expectation(y);
D = sum((y - m).^2)/(length(y)-1);
function m = expectation(y)
% expected value
m = sum(y)/length(y);
Get the score values:
During generation, zero mathematical expectation and unit variance were set. We see that the obtained values of the estimates are close to the given ones. The reason for their incomplete match is that a finite sample of N counts, and the estimates converge to the true values at .
2. Plotting a Probability Density Plot and Signal HistogramX .
Using the following script-file, we will plot the probability density of a normal random variable (according to expression (3)) and plot the histogram of the signal X using the function hist() .
(3)
f = (exp(-(x-m).^2/(2*D)))/(sqrt(2*pi*D));
title("Probability Density Plot");
set(gca,"FontName", "Times New Roman","FontSize", 10);
title("Histogram of random signal X");
The resulting graphs are shown in fig. 2.
Rice. 2. Distribution density plot
probabilities and histograms.
We see that the histogram of the random signal X is similar in shape to the graph of the probability distribution density. They do not match completely, because to construct the histogram, a finite sample of N readings. The histogram converges to a probability density plot for .
3. Determining the ACF of the system output signal analytically and using the functionconv().
One of the characteristics of a random signal is its autocorrelation function (ACF), which is determined by expression (4).
ACF determines the degree of dependence of signal samples separated from each other by an interval m.
White noise is a random process whose ACF is zero for any , i.e. interval separated values m do not depend on each other. The white noise ACF at is determined by expression (5).
The connection between the ACF of the discrete output and input signals of the system is determined by the expression
Using expression (6), we determine the ACF of the output signal of the system with the equation when white noise is applied to the input of the system.
Let us determine the impulse response of a given system by applying a single delta pulse to its input.
Rice. 3. Graphs , , .
At the ACF of white noise is 
. The convolution of any signal with a single pulse gives the original signal, which means that 
.
Using the geometric meaning of the convolution operation, we find .
Rice. 4. Graph of the ACF of the output signal of the system
when white noise is applied to the input.
We see that, in comparison with the ACF of the input signal, nonzero components appeared in the output signal at , i.e. the output signal is a correlated process in contrast to the input white noise.
Let us determine the ACF of the output signal of the system when a random signal is applied to the input X defined in clause 1.
Estimation of signal ACF X can be determined by the expression
The ACF estimate determined by expression (7) can be calculated using the function xcorr() matlab. Using this function, we find the estimate of the ACF of the signal X and construct a graph of this estimate.
Xcorr(X, "biased");
stem(lags, Kxx);
set(gca,"FontName", "Times New Roman Cyr", "FontSize", 10)
title("Estimating the ACF of signal X");
Rice. Fig. 5. Graph of estimation of the ACF of a random signal X.
We see that the signal estimate X The ACF is close to the ACF of white noise (Fig. 3), which means that the relationship between different signal values X small. The presence of components at is explained by the finiteness of the sample.
Using the function conv() Matlab, we determine the ACF of the output signal by expression (6).
h1 = ;
h2 = ;
c = conv(h1,h2);
Kyy = conv(c, Kxx);
stem(-(N+3):(N+3), Kyy)
|
Rice. 6. ACF of the output signal when a signal is applied to the input X.
In the enlarged fragment of Fig. 6 it can be seen that the ACF values of the output signal with the input signal X are close to the ACF values of the output signal when white noise is applied to the input (Fig. 4).
Using the following sequence of commands, we will plot the ACF of the input and output signals for their comparison.
stem(lags, Kxx);
set(gca,"FontName", "Times New Roman Cyr", "FontSize", 10)
title("Estimating the ACF of signal X");
stem(-(N+3):(N+3), Kyy)
set(gca,"FontName", "Times New Roman Cyr", "FontSize", 10)
title("ACF of the output signal");
Rice. 7. Graphs of the ACF of the input and output signals of the filter.
On fig. 7 we see that the output signal is more correlated than the input signal, because there are more non-zero components and there is a dependence between the values of the output signal.
4. Building scatter diagrams of the output signalY system.
An analysis of various tasks in radio engineering shows that essentially any signal that carries information can be considered as random (stochastic). This is due, on the one hand, to random distortions of the signal during its propagation and the presence of various (external and internal) interference, and on the other hand, to the imperfection of the radio engineering devices and systems used. A number of processes that affect their technical level and quality belong to the category of random. The experimental analysis of such processes is also associated with the measurement of the characteristics of the corresponding random signals.
The study of the properties and characteristics of random signals is based, as you know, on the theory of probability and mathematical statistics. The need for this led to the development of methods and tools that make up the content statistical measurements. They are based on the general principles of measuring signal parameters, but have their own specifics and a number of fundamental features arising from the theory of random processes. We recall the initial definitions and information about the characteristics of random signals and clarify the main tasks of the technique of statistical measurements.
GENERAL INFORMATION
Random a signal is called, the instantaneous values of which change in time randomly. In this regard, it is described by a random function of time X(t), which can be considered as an infinite collection (ensemble) of functions x i (t), each of which represents one of the possible realizations of X(t). On fig. 8.1 shows a set of implementations as an example x i (t), Where x i (t j)-instantaneous signal value X(t), corresponding to the value i th implementation in j-th point in time.
A complete description of random signals can be produced using the system probabilistic characteristics. Any of them is determined either by averaging over the set of realizations x i (t), or by averaging over time for one realization X(t). In the general case, the results of such averagings are not the same, they may or may not depend on the time or realization number. The presence or absence of this dependence determines such fundamental signal properties as stationarity and ergodicity. Stationary A signal is called, the probabilistic characteristics of which do not depend on time. Accordingly, the probabilistic characteristics ergodic signals do not depend on the implementation number.
The classification of random signals according to the signs of stationarity and ergodicity allows us to distinguish the following types: stationary ergodic, stationary non-ergodic, non-stationary ergodic and non-stationary non-ergodic. Within the framework of the course, we will confine ourselves to considering methods and means for measuring the probabilistic characteristics of random signals of the first type, as the most simple and typical. For such signals, averaging any probabilistic characteristic over a set of realizations is equivalent to averaging over the time of one theoretically infinitely long realization. In other words, all probabilistic characteristics of a stationary ergodic signal can be obtained from one of its implementations. It is clear that it is much easier to carry out measurements with one realization of a signal than with a set of realizations.
For practical applications, the most important are the following probabilistic characteristics of stationary ergodic signals with a realization duration T(GOST 16465-70):
average value (mathematical expectation), characterizing, like
Rice. 8.1. A set of realizations of a random signal.
for deterministic signals (see § 3.1), the constant component of the signal
(8.1)
average power, characterizing the energy level of the signal,
(8.2)
dispersion, characterizing the average power of the variable component (fluctuation) of the signal,
or standard deviation his
distribution function, defined as the cumulative probability that the value X i (t j) below some given level X,
(8.5)
i.e. for stationary ergodic signals Fx characterized by the relative residence time of the implementation values below the level X ( - i-th stay interval; P- number of intervals);
one-dimensional probability density, also called the differential distribution law,
(8.6)
where is the distance between adjacent levels X i (t j) And , called differential corridor, A - i-th interval of stay of the implementation within this corridor;
correlation function, characterizing the stochastic relationship between the instantaneous values of a random signal separated by a given time interval ,
or normalized correlation function
(8.8)
mutual correlation function characterizing the stochastic relationship between the instantaneous values of two random signals X(t) And Y(t), separated by a time interval ,
and the corresponding normalized cross-correlation function
spectral density power, which determines the average signal power per unit of frequency band. The frequency distribution of the average power characterizes energy spectrum signal. It can be defined per implementation x i (t) according to general rules (see § 7.8). It turns out that for stationary random signals, the power spectral density function is related to the correlation function by a pair of Fourier transforms (Wiener-Khinchin theorem):
(8.11)
If we have two stationary signals X(t) And Y(t), they can be characterized mutual power spectral density, which is generally a complex quantity . Therefore, in practice, the functions of the real and imaginary components are determined :
(8.12)
When calculating according to formulas (8.11) and (8.12), you can use the values and . Then the power spectral density functions will be normalized.
As follows from formulas (8.1) - (8.12), all probabilistic characteristics, which are nonrandom numbers or functions, defined by one implementation X(t) infinite duration. In practice, the duration T, called the duration of the analysis, always limited. Therefore, in reality, any experimental characteristic differs from the corresponding probabilistic (theoretical) characteristic and can only be its evaluation. Estimates obtained by hardware are called statistical characteristics and are denoted by the sign “ ” (see § 1.3). In this sense, the measurement of the characteristics of random signals is always accompanied by statistical errors. The rest of the metrological characteristics analyzers statistical characteristics are similar to the characteristics of devices of other subgroups and are regulated by GOST 8.251-77. Statistical analyzers are included in the subgroup X (see § 2.1), where they form the form X6.
The mathematical model of the measurement information transmission process is a model of a random process with a probability density . Useful signals and interference signals acting on the information-measuring system are random processes that can be characterized by statistical averages and characteristics.
A random process is a more complex random phenomenon than a random variable, but its definition can be given through a random variable. A function (Fig. 4) is called a random process if its instantaneous values are random variables. Just as a random variable cannot be characterized by a separate value, so a random process cannot be defined by any single, albeit complex, function. A random process is a set of implementations (functions of time). Implementation x i (t)– a fragment of a random process X(t) fixed as a result i-th experiment of limited duration T, therefore, realization is understood as one of the possible outcomes of a random process. Random variable corresponding to i th implementation and j-th point in time, is an instantaneous (sample) value - a special case of a random process, and the probabilistic characteristics of a random process are based on the characteristics of random variables included in this process. A set of instantaneous values corresponding to the values of different implementations at the same time tj, is called j th process sequence X(t). When solving applied problems, they often turn to implementations than to sequences.
Experimentally, an ensemble of realizations of a random process can be obtained as a result of simultaneous registration of output parameters x i (t) at the outputs of objects of the same type, for example, measuring instruments, during a fixed time interval.
If the argument t changes continuously, dependency X(t) is continuous random process(for example, a change in the error of a measuring device over a long time of its operation), if the argument t is a discrete quantity - a random sequence or time series(an array of error measurement results at known time points). Process X(t) that takes a countably limited number of values is called discrete random process(for example, the sequence of operation states of the equipment of information-measuring systems or information-computer complexes).
By defining a random process by random variables, the probabilistic characteristics of the processes are found based on the probabilistic characteristics of these variables.
Fig.4. Graphical representation of a random process
The most complete description of a random process is the integral probability distribution function
and the differential probability distribution function
In the probability distribution functions of random processes, in contrast to the multidimensional distribution functions of the probability of random variables to the arguments x i variables are added tj, showing at what points in time the readings were taken.
For an approximate description of random processes, as well as for describing random variables, such numerical characteristics as mathematical expectation, variance, etc. are used. Moreover, these numerical characteristics are also functions of time.
The most commonly used probabilistic characteristics are.
1M mathematical expectation ,
the estimate of the mathematical expectation of a random function is its mean value.
2. D dispersion- non-random function
where is a centered random process; dispersion values for each tj equal to the variance of the random variable x i (t j).
The variance of a random function can be found through the differential function of the probability distribution of the random function
An estimate of the variance is its empirical value
Random processes with the same mathematical expectations and variances can differ significantly in shape (Fig. 4).
3. Autocorrelation function characterizes the statistical relationship between the instantaneous values of a random process at different points in time. The smaller the value of the autocorrelation function, the less the value of the measuring signal at time t 1 depends on the value at time t 2. . It is determined by one of the following relations
Where t 1 , t 2 – fixed moments of time at which the sections of the random function are determined.
Since at t 1 \u003d t 2, for the same cross sections the correlation function turns into the variance of a random function.
For each pair of time moments, the autocorrelation function is equal to the correlation moment, the statistical estimate of which is
In the formulas that determine the empirical estimates of the variance and the correlation function, the number of realizations n is reduced by one to obtain an unbiased estimate;
4. Cross-correlation function determines the statistical relationship of two signals X(t) And Y(t+τ)
The study of the properties of random processes using correlation functions is called the correlation theory of random processes.
5. Spectral density- a non-random function that establishes the distribution density of its dispersion in frequency ω, is equal to the Fourier transform of the corresponding correlation function
The correlation function can be expressed in terms of the spectral density by a relation of the inverse Fourier transform type.
The relations that make it possible to transform the spectral density into a correlation function and vice versa are called the Khinchin-Wiener theorem.
The use of fuzzy logic methods to determine the classification characteristics of random processes
1 2 A.M. Prokhorenkov, N.M. rocked
1 Polytechnic Faculty, Department of Automation and Computer Engineering
Faculty of Economics, Department of Information Systems
Annotation. The paper considers the issues of the need to classify random processes that take place in process control systems, analyzes the informative features and existing approaches to the classification of processes. An approach is proposed in which the classification features are the class of the process (stationary, non-stationary), the type of process (additive, multiplicative, additive-multiplicative) and the type of the deterministic component. An algorithm for classifying random processes by one implementation is proposed, based on the use of nonparametric criteria, the Hurst exponent, the Bayesian classification procedure, and fuzzy logic.
abstract. In the paper the necessity of random processes" classification in industrial control systems have been considered. Informative signs and existent methods for the classification have been analyzed. The new approach has been suggested. According to it the process type (stationary or non-stationary), process kind (additive, multiplicative or additive-multiplicative) and deterministic constituent "s kind are classification signs. A realization-based algorithm for the random processes" classification has been proposed. It implies application of non-parametric criteria, Hurst items, Bayesian classifying procedure and fuzzy logic.
1. Introduction
At present, one of the main directions for improving automatic control systems (ACS) is to increase the accuracy of control and stabilization of technological parameters within fairly narrow limits.
An important role in solving the problem of increasing control accuracy is assigned to the measuring subsystem, which is part of the ACS. The random nature of disturbing influences and controlled variables implies the use of a procedure for statistical processing of measurement results, which causes the presence of such error components as statistical error and error caused by the inadequacy of the processing algorithm to a real random process. The reason for the latter type of error is the error in the classification of the observed process. For example, by classifying a non-stationary process as stationary, one can increase the methodological error in estimating the mathematical expectation by increasing the smoothing interval. In turn, the complication of the measurement algorithm in order to reduce the methodological error, as a rule, leads to an increase in the instrumental error. Establishing a priori the class of the process largely determines the algorithm for processing measurement results and hardware.
In ACS, the need to classify random processes is also due to the requirements of a reasonable transition from the analysis of an ensemble of implementations to the analysis of a single implementation. In addition, knowledge of the process class is necessary to describe its dynamics, predict its future values, and select control algorithms.
2. Analysis of informative features and approaches to the classification of random processes
A common approach in classifying objects of any nature, including random processes, is to identify informative features. The analysis performed showed that the informative features used in the classification of processes differ in variety and are determined by the classification goal set by the authors.
All observable processes X(t), which characterize physical phenomena, in the most general form can be classified as deterministic and random.
A deterministic process is defined by a single implementation described by a given time function. Due to the inevitable influence of various external and internal factors in relation to the control system, a deterministic process is an abstraction. In this regard, in the practice of studying processes, a quasi-deterministic process is considered,
realizations of which are described by time functions of a given type ab...,an), where ab...,th are time-independent random parameters.
In contrast to a deterministic process, a random process is represented as a random function X(t, t), where t is time, t0, 0 is the space of elementary events. The function X(/, m) at any time can take on different values with a known or unknown distribution law.
Assigning a process to the class of random can be due either to its physical nature or to the conditions of its study, leading to insufficient a priori data. If the classification is based on the causes of the occurrence of randomness, then non-singular and singular processes can be distinguished. The first group includes processes for which it is impossible to trace the nature of causal relationships, since they are the result of a superposition of a large number of elementary processes. For non-singular processes, it is fundamentally impossible to predict instantaneous values. For processes of the second group, in the presence of a certain amount of data, the prediction of their instantaneous values becomes reliable. Singular processes can be either random or deterministic. In control systems for technological objects, all processes should be considered as random, and for processing the results of observations in real time, the reason for the randomness of the process does not play a role.
In the theory of random processes, the most general classification is the classification "by time" and "by state" (Wentzel, Ovcharov, 2000; Kovalenko et al., 1983; Levin, 1989). According to these features, four classes can be distinguished: 1) processes with discrete states and discrete time; 2) processes with discrete states and continuous time; 3) processes with continuous states and discrete time; 4) processes with continuous states and continuous time.
The processes occurring in automatic control systems are random processes with continuous states and continuous time. The use of digital measuring technology leads to the need to consider processes at discrete times and assign them to the first or third class.
An exhaustive characteristic of a random process is a multidimensional distribution law:
^n(xb X2, /2; ... ; x^ 4) = P(X(^)< XI,Х^)< хъ...,Х(4)< хп}.
In practice, as a rule, one-dimensional or two-dimensional distribution laws of a random process are considered, since they contain a sufficient amount of information about the properties of a random process, and the increase in the amount of information when using probabilistic characteristics of a higher order turns out to be insignificant. In addition, the determination of multidimensional probabilistic characteristics is associated with great difficulties in the hardware implementation of algorithms for their calculation.
Taking into account the change in probabilistic characteristics over time, random processes are divided into stationary (SSP) and non-stationary processes (NSP). The probabilistic characteristics of the SSP are the same in all sections. The stationarity condition in the narrow sense is the invariance of the n-dimensional probability density with respect to the time shift m. The stationarity conditions in the broad sense are limited by the requirements that the mathematical expectation M[X(0] and variance B[X(()]) be independent of time and the dependence of the correlation function only on time shift t, that is:
M[X(0\=cosh1, t[X(0\=cosh1, Xx(b, t2)=Rx(m), m=^2 - 1.
In practice, in most cases, the correlation function is a fairly complete characteristic of the BSC, so they are usually limited to identifying the stationarity of the process in a broad sense.
The structure of a random process can be established by the correlation function or by the known distribution density.
Depending on the type of distribution laws, normal, uniform, Rayleigh, Poisson and other random processes can be distinguished. Deviations from the classical form of the distribution indicates the non-stationarity of the process. Based on one implementation of limited length, it is difficult to judge with sufficient accuracy the law of distribution of a random process, and in most applied cases of analysis, the researcher does not have information about the form of the distribution function. Then the type of process is either postulated, or the distribution function is not taken into account in the analysis.
More complete information about the dynamic properties of the process can be obtained from the correlation function. A typical correlation function of the BSC is a symmetrical decreasing function. The presence of fluctuations in the correlation function indicates the periodicity of the random process. If the correlation function is aperiodically damped, then
random process is considered broadband. A multiband random process is characterized by a triangular correlation function. Stationary - in a broad sense - processes have correlation functions that, with an unlimited increase in m, tend to a constant value or are periodic functions of m.
Stationary processes whose correlation functions include an exponent with a negative argument are ergodic. The tendency of the correlation function to some constant value other than zero is usually a sign of the non-ergodic process.
Determination of the statistical characteristics of random processes is fundamentally possible in two ways: determination by one realization and by an ensemble of realizations. If the probabilistic characteristics of the process obtained by averaging over time are equal to the similar characteristics found by averaging over the ensemble, then the random process is ergodic. Processes that do not have the ergodicity property can only be processed over an ensemble of realizations.
A priori knowledge of the ergodicity of the process greatly simplifies the algorithmic support of information-measuring and information-control complexes. Under the conditions of real technological processes and control systems, it is impossible to check the global ergodicity of processes, and it is accepted as a hypothesis.
Non-stationary processes are characterized by a change in time of their statistical characteristics, so this can be taken into account when performing the classification. From the point of view of this approach, processes are usually singled out that have a time-varying mean value; time-varying mean square, time-varying mean and mean-square, time-varying frequency structure (Bendat, Peirsol, 1989). Such a classification reflects the change in time estimates of probabilistic characteristics.
The above analysis showed that there cannot be a unified classification of processes due to the independence of classification features and the diversity of classification purposes. There are several approaches to the classification of processes. A significant part of the authors seek to systematize information about random processes in order to show all their diversity (Ventzel, Ovcharov, 2000; Kovalenko et al., 1983; Levin, 1989; Shakhtarin, 2002). The most general approach to the classification of both stationary and non-stationary processes is associated with their continuous or discrete representation (Wentzel and Ovcharov, 2000; Kovalenko et al., 1983; Levin, 1989).
In applied cases, the specifics of tasks are taken into account, the solution of which must be preceded by the classification of the observed processes. For example, in (Tsvetkov, 1973; 1984; 1986) a classification of processes in metrology was carried out according to the signs of stationarity and ergodicity in order to identify the causes and analyze their influence on the methodological error in measuring the statistical characteristics of random processes. In radio engineering, classification according to the spectral properties of signals is widely used (Levin, 1989). To justify the transition from the analysis of an ensemble of realizations to the analysis of individual realizations, in (Bendat, Peirsol, 1989) it is proposed to perform a classification by types of non-stationarity and, at the same time, the time behavior of estimates of statistical characteristics is considered.
Thus, the currently existing approaches to the classification of random processes do not allow developing an algorithm for their analysis in order to identify the nature of the non-stationarity of the process, the type of deterministic components and their characteristics necessary to solve the problems of operational control and management of technological processes, according to one implementation. In this regard, solutions aimed at generalizing and improving existing approaches to the classification of random processes are relevant.
3. Classification of random processes according to one implementation
Random processes occurring in control systems can be represented as the result of the combined action of a deterministic useful signal and stationary interference. In the general case, the effect of interference on the useful signal can be expressed by the operator Depending on the type of operator V, the following signal models are distinguished (Kharkevich, 1965):
additive model X(0 = + e(0; (1)
multiplicative model X(/) = φ2(/) e(/); (2)
additive-multiplicative model
where φ1(0, φ20) are deterministic functions of time, e(1) is a stationary random process with zero mathematical expectation ne = 0 and constant variance D.
An example of an additive process is the output signal of a measuring instrument when the useful signal is added to the instrument's internal noise. A change in the stiffness of the pressure gauge sensor membrane, a change in the gain of the amplifier, a change in the reference voltage in a digital voltmeter, and others are the causes of the multiplicative error of measuring systems, which is described by a multiplicative model. In many cases, the non-stationary process of errors can be described in the form of an additive-multiplicative model.
In engineering practice, processes are usually considered stationary in a broad sense, while the behavior of the mathematical expectation, variance, and correlation function is estimated in time. Therefore, when classifying non-stationary processes, one should proceed from the analysis of the same characteristics.
Taking into account the assumptions made, the mathematical expectation tX, the BX variance and the correlation function RX of random processes represented by models (1-3) have the following form:
additive
multiplicative
additive-multiplicative
mX(0 = f:(0; Du(0 = D;
Rx(tl, /2) = Rs(th /2);
mX(() = 0; Dy(0 = ^(OD; Rx(tl, /2) = ^(M^^^WHA, /2); mX(P) = φ1(/); Rx(tl, /2) = Ф2(bsh/2shb, /2).
It follows from the above relations that the mathematical expectation for the additive and additive-multiplicative models depends on the deterministic component φ1(/). The dispersion and correlation function of the additive model are fully characterized by the properties of the stationary noise. And for the multiplicative and additive-multiplicative models, these probabilistic characteristics are also determined by the deterministic component φ2(/).
Expressions (4) and (6) show that for processes represented by additive and additive-multiplicative models, the mathematical expectation can be estimated from one implementation using one or another operation equivalent to low-pass filtering.
If the noise variance e(Г) is constant, then it is also possible to determine the mean square of the multiplicative and additive-multiplicative processes (and thus obtain an estimate of the variance) from a single implementation (Bendat and Peirsol, 1989).
Thus, for the processes represented by models (1-3), there is no need to check the ergodic properties of a non-stationary random process.
The accuracy of estimating the statistical characteristics depends on the type and parameters of the deterministic processes φ1(t) and φ2(t) (Proktorkinkoy, 2002), so the classification of processes according to the type of nonstationarity should be supplemented by the classification according to the type of deterministic processes.
Classification should be considered as a necessary preliminary stage in the study of random processes in order to identify their properties before conducting the main statistical processing, therefore, in a sense, the classification should reflect the algorithm for analyzing the observed process. Taking into account the above, a classification of random processes was developed in the presence of one realization of the process under study (Fig. 1). The class of the process, the type of non-stationarity were chosen as classification features: non-stationarity in mathematical expectation (MO), non-stationarity in dispersion, non-stationarity in correlation function (CF), as well as the laws of change of mathematical expectation and dispersion. In the proposed classification, the most common transients in engineering practice are considered as deterministic components: linear, exponential, periodic, periodic damping.
Implementation of a random process
Stationary in MO
Non-stationary in MO
SP by dispersion
NSP for KF
ERP by dispersion
SP for KF NSP for KF
Linear
ERP by dispersion
SP for KF NSP for KF
SP by dispersion
NSP for KF
Exponential
Periodic
Periodically damped
Rice. 1. Classification of random processes represented by one implementation
4. Statement of the problem of classification of random processes
In the general case, classification is understood as the division of the considered set of objects or phenomena into homogeneous, in a certain sense, groups, or the assignment of each of a given set of objects to one of the previously known classes. In the second case, we have a classification problem in the presence of training samples ("classification with training"). In the classical form, the solution to this problem is to perform a display of the form:
those. assignment of the object, given by the vector of informative features R = (rb r2, ..., rn), to one of the predetermined classes (d?b a2, ..., ab).
The processes represented by models of the form (1-3) belong to the class of non-stationary random processes. To identify non-stationary properties, it is proposed to use non-parametric criteria (Kendall, Stewart, 1976), the Hurst exponent (Feder, 1991) and correlograms, the results of which will form a vector of informative features R.
A significant majority of nonparametric criteria respond to changes in the estimate of the mathematical expectation. Thus, nonparametric criteria without preliminary processing of the observed series make it possible to single out two classes of processes "stationary in terms of mathematical expectation" and "nonstationary in terms of mathematical expectation".
By the value of the Hurst exponent, one can judge both the stationarity of the process in terms of mathematical expectation, and the form of the deterministic component. This allows us to consider a priori three classes of processes: stationary with respect to mathematical expectation; non-stationary in terms of mathematical expectation, changing according to a monotonic law; non-stationary in mathematical expectation, changing according to a periodic law.
As noted in Section 2, the correlation function carries information about the dynamic properties of the process under study. The output of the correlogram beyond the 95% confidence interval allows, to a certain extent, to judge how the process under study differs from white noise.
The impossibility of applying the classification procedure for the simultaneous selection of classes of processes that are non-stationary in terms of mathematical expectation and dispersion leads to the need to apply the classification procedure twice.
The second problem is that informative features are given on different scales. The result of applying each non-parametric criterion separately is measured on a dichotomous scale, and the attribute can take two values: "the random process does not contain a deterministic component" - "the process contains a deterministic component", or "0" and "1". And the Hurst exponent is measured on a quantitative scale and takes values in the range from zero to one.
Randomness tests have different efficiencies for different types of deterministic components of non-stationary random processes, therefore, in conditions of limited a priori information about the properties of the process under study, the decision on the class of the process should be made based on the results of applying a set of criteria. In this regard, it is proposed to obtain a certain generalized classification feature. The classification based on nonparametric criteria is proposed to be based on the Bayesian procedure for binary features (Afifi, Eisen, 1982). The estimates obtained in this way are further considered as a generalized result of applying nonparametric criteria, and the posterior probability is considered as a classification feature. In this case, the measurement scale becomes the same as for the Hurst exponent.
The third problem is related to the dependence of the values of the selected classification features on the length of the implementation and the parameters of the process under study, which are unknown at the stage of process classification. Therefore, one should look for an answer to the question: "To what extent does the process under study belong to this or that class?". Due to this formulation of the question, it is proposed to use fuzzy logic methods to classify processes.
5. Bayesian classification procedure
It is required to classify the process X(/) based on the presence or absence of n events. The number of events (features) is equal to the number of considered nonparametric criteria. Let us define for each y-th event (y = 1, 2, ..., n) a random variable:
In our case, Ty = 1, if in the process X(/) under study, according to the criterion y, there is a tendency to change the mathematical expectation, Ty = 0 - otherwise.
R = (rb r2, ..., rn) ^ye (di, d2, ..., dm),
1 if there is a y event, 0 if there is no y event.
The probability of an object belonging to a class, provided that the value of the attribute Ty is equal to one, is denoted by , y=1,2, ... n. Since non-parametric criteria allow us to divide the set of processes under study into stationary and non-stationary processes, then in this case m = 2.
The distribution law Tu for a class has the form:
/ (Tu) = RT (1 - Ru) 1-TU.
The results Tu of applying nonparametric criteria are independent, so the joint distribution law / (r) for the class can be written as:
/g (G) \u003d P /g (Tu).
Assume that the prior probabilities are the same *1 = q2 = 0.5 and the misclassification costs are equal. The cost of erroneous classification in this case is associated with the losses that may occur when classifying a stationary process as a non-stationary process or when classifying a non-stationary process as a stationary process. The conditional probability Pr(e, | r) that the process under study belongs to the class for a given vector of observations (a posteriori probability) is determined by the formula (Afifi, Eisen, 1982):
ъ P RT (1 - Ru)
Pr(e/ | r) = ■
P Rku (! - Rku) 1-
The process X(0) belongs to the class for which the value of Рг(ё, | г) is maximum. The values of py are estimated from the training sample from the processes belonging to all the considered models (1-3) and containing various types of deterministic components. Let 51 and 52 - the number of non-stationary and stationary in MO processes, respectively, 5 = 51 + 52. Denote by y the number of processes of class / for which non-stationarity in MO is revealed by the y criterion.
For each newly arriving process X(/), characterized by the vector of feature values (m1, ..., mn), the posterior probability estimate is as follows:
Pr(e/ | r) = ■
6. Proposed fuzzy classification procedure
Each classification attribute Ku is given by a linguistic variable characterized by a triple of elements<Ку, Ту, Пу>, where Ku - variable name; Tu - term-set, each element of which is represented as a fuzzy set on the universal set Pu.
The universal set of values of the Hurst exponent - ПН = . H values around 0.4< Н < 0,6 определяют собой область белого шума в нечетком смысле. Значения Н в окрестности 0,3±0,1 говорят о наличии в рассматриваемом временном ряду периодической компоненты. Значения Н, близкие к единице, характеризуют наличие монотонной компоненты в исследуемом процессе.
Let's define a term-set as the names of possible components of non-stationary random processes: "periodic", "stationary", "monotone". We define the membership functions as the difference of two Gaussian functions defined by the relation:
¿u(x, cg1, c1, cg2, c2) = e a" - e °2.
This membership function reflects the fact that each type of process is characterized by a certain range of values of the Hurst exponent - the core of the fuzzy set is non-empty. Studies have shown that the probability of error attributing a process containing a periodic component to noise
higher than the probability of the error of attributing to the noise of a monotonous noisy process. The asymmetric double Gaussian function makes it possible to reflect this point. The membership functions of the linguistic variable "Hurst exponent" before setting up the fuzzy model are shown in Fig. 2a.
The universal set of values of the a posteriori probability estimate (7) PRg = . Evaluation values close to one indicate the presence of a deterministic component in the series under study, and those close to zero indicate the randomness of the series. The term-set of the variable "non-parametric criteria" is defined as ("stationary", "non-stationary"). We can formalize the terms using the double Gaussian membership function (Fig. 2b).
Let's call the third linguistic variable "correlogram". The universal set of values of this variable Pk = - the weight coefficient of the rule with the number /p.
As a solution, choose the class with the maximum degree of membership:
Mdi(**), Md2(**), ..., Mäm(**)),
where the symbol * denotes the vector of values of the classification features of the process under study.
Tuning is finding the parameters of the membership functions of the input variables and the weighting coefficients of the rules that minimize the deviation between the desired and actual behavior of the fuzzy classifier on the training sample.
Proximity criteria can be defined in various ways. In this work, we used the criterion proposed in (Shtovba, 2002). The training sample is formed from L data pairs connecting the inputs X = (xb x2, ..., xn) with the output y of the dependence under study: (Xq, yq), q = 1, 2, ..., L. Let us introduce the following notation: P - vector of parameters of membership functions of input terms; W - vector of weight coefficients of knowledge base rules; F(Xq, P, W) - the result of inference on a fuzzy base with parameters (P, W) with the value of the inputs Xq; ßd(yq) - the degree of belonging of the value of the output variable y in the q-th pair of the training sample to the solution d,; цdi(Xq, P, W) - the degree of membership of the output of the fuzzy model with parameters (P, W) to the solution d, determined by formula (8) with the values of the inputs from the q-th pair of training sample. As a result, the optimization problem takes the following form:
1 L m t \ T Z Sq Z ((yq) - Mdi (Xq, P, W))
Rice. 3. Membership function of the linguistic variable "Hurst exponent" after tuning
= [ 1 if yq = F(Xq, P, W)
where q)
