Multidimensional database technology. STATISTICA Multivariate Exploratory Techniques Multivariate data analysis technologies

The theory of random variables studies probabilistic phenomena "in statics", considering them as some fixed results of experiments. Methods of classical probability theory are insufficient for describing signals that reflect random phenomena evolving in time. Such problems are studied by a special branch of mathematics called the theory of random processes.

By definition, a random process is a special kind of function, characterized by the fact that at any moment of time the values ​​it takes are random variables.

Ensembles of implementations.

When dealing with deterministic signals, we display them with functional dependencies or oscillograms. If we are talking about random processes, then the situation is more complicated. Fixing the instantaneous values ​​of a random signal at a certain time interval, we obtain only a single realization of a random process. A random process is an infinite collection of such realizations that form a statistical ensemble. For example, an ensemble is a set of signals that can be simultaneously observed at the outputs of exactly the same noise voltage generators.

It is not at all necessary that the implementations of a random process be represented by functions with complex, irregular behavior in time. It is often necessary to consider random processes formed, for example, by all kinds of harmonic signals, in which one of the three parameters is a random variable that takes a certain value in each implementation. The random nature of such a signal lies in the impossibility of determining the value of this parameter in advance, prior to the experiment.

Random processes formed by realizations that depend on a finite number of parameters are usually called quasi-deterministic random processes.

Probability densities of random processes.

Let be a random process, given by an ensemble of realizations, be some arbitrary moment in time. Fixing the values ​​obtained in individual implementations, we carry out a one-dimensional cross section of a given random process and observe a random variable.Its probability density is called the one-dimensional probability density of the process at the moment of time

According to the definition, the quantity is the probability that the realizations of the random process at the moment of time will take values ​​lying in the interval

The information that can be extracted from the one-dimensional density is insufficient to judge the nature of the development of realizations of a random process in time. Much more information can be obtained by having two sections of a random process at mismatched instants of time A two-dimensional random variable arising in such a thought experiment is described by a two-dimensional probability density This characteristic of a random process makes it possible to calculate the probability of an event that the implementation of a random process at takes place in a small neighborhood of a point and for - in a small neighborhood of the point

A natural generalization is the -dimensional section of a random process leading to the -dimensional probability density

The multidimensional probability density of a random process must satisfy the usual conditions imposed on the probability density of a collection of random variables (see § 6.2). In addition, the value should not depend on the order in which its arguments are located (symmetry condition).

Sometimes, instead of the -dimensional probability density, it is convenient to use the -dimensional characteristic function, which is related to the corresponding density by the Fourier transform:

The description of the properties of random processes using high-dimensional multidimensional probability densities can be very detailed. However, serious mathematical difficulties are often encountered along this path.

Moment function of random processes.

Less detailed, but, as a rule, quite satisfactory in a practical sense, characteristics of random processes can be obtained by calculating the moments of those random variables that are observed in the cross sections of these processes. Since, in the general case, these moments depend on time arguments, they are called moment functions.

For statistical radio engineering, three lower-order moment functions are of greatest importance, called mathematical expectation, variance, and correlation function.

Expected value

is the average value of the process X (t) at the current time; averaging is carried out over the entire ensemble of realizations of the process.

Dispersion

makes it possible to judge the degree of scatter of instantaneous values ​​taken by individual realizations in a fixed section t, relative to the average value.

2D center moment

is called the correlation function of a random process This moment function characterizes the degree of statistical connection of those random variables that are observed when Comparing formulas (6.37), (6.38), we note that when the cross sections are combined, the correlation function is numerically equal to the variance:

Stationary stochastic processes.

So it is customary to call random processes, the statistical characteristics of which are the same in all sections.

They say that a random process is stationary in the narrow sense; if any of its -dimensional probability density is invariant with respect to the time shift

If we restrict the requirements so that the mathematical expectation and variance of the process do not depend on time, and the correlation function depends only on the difference -, then such a random process will be stationary in a broad sense. It is clear that stationarity in the narrow sense implies stationarity in the wide sense, but not vice versa.

As follows from the definition, the correlation function of a stationary random process is even:

In addition, the absolute values ​​of this function for any do not exceed its value for:

The method of proof is as follows: from the obvious inequality

follows that

whence inequality (6.41) follows directly.

It is often convenient to use the normalized correlation function

for which .

To illustrate the concept of a stationary stochastic process, consider two examples.

Example 6.5. A random process is formed by realizations of the form where are known in advance, while the phase angle is a random variable uniformly distributed over the interval -

Since the probability density of the phase angle, the mathematical expectation of the process

Similarly, you can find the variance:

Finally, the correlation function

So, this random process satisfies all the conditions that are necessary to ensure stationarity in a broad sense.

Example 6.6. A random process has realizations of the form and, moreover, given numbers. - a random variable with an arbitrary distribution law. Expected value

will be independent of time only for. Therefore, in the general case, the considered random process will be nonstationary.

Ergodic property.

A stationary random process is called ergodic if, when finding its moment functions, averaging over a statistical ensemble can be replaced by averaging over time. The averaging operation is performed on a single implementation whose duration T can theoretically be arbitrarily long,

Denoting the averaging over time by angle brackets, we write the mathematical expectation of an ergodic random process:

which is equal to the constant component of the chosen implementation.

Dispersion of a similar process

Since the quantity is the average power of the realization, and the quantity is the power of the constant component, the variance has a visual meaning of the power of the fluctuation component of the ergodic process.

The correlation function is found in a similar way:

A sufficient condition for the ergodicity of a random process stationary in a broad sense is the tendency to zero of the correlation function with an unlimited increase in the time shift:

It has been shown in mathematics that this requirement can be somewhat relaxed. It turns out that a random process is ergodic if the Slutsky condition is satisfied:

Thus, equality (6.47) is valid for a harmonic process with a random initial phase (see Example 6.5).

Measurement of characteristics of random processes.

If a random process is ergodic, then its realization of sufficient length is a “typical” representative of a statistical ensemble. Studying this implementation experimentally, you can get a lot of information characterizing this random process.

The device for measuring the one-dimensional probability density of a random process can be performed as follows. The one-dimensional probability density of an ergodic random process is a quantity proportional to the relative residence time of its realization at the level between Suppose that there is a device with two inputs, one of which is supplied with the studied realization x (t), and the other is a reference constant voltage, the level of which can be regulate. At the output of the device, rectangular video pulses of constant amplitude appear, the beginning and end of which are determined by the moments in time when the current values ​​of the random signal coincide either with the level or with the level of this device will be proportional to the probability density

Any sufficiently inertial pointer device can be used to measure the mathematical expectation of a random process [see. formula (6.43)].

A device that measures the variance of a random process, as follows from (6.44), must have a capacitor at the input that separates the DC component. Further steps in the measurement process - squaring and averaging over time - are performed with an inertial quadratic voltmeter.

The principle of operation of the meter of the correlation function (correlometer) follows from the formula (6.45). Here, the instantaneous values ​​of the random signal after filtering the constant component, dividing into channels, are fed to the multiplier, and in one of the channels the signal is delayed for a time. To obtain the value of the correlation function, the signal from the output of the multiplier is processed by an inertial unit, which performs averaging.

Regardless of the size

Here, the same designations are adopted as in formula (6.26). The elements of the correlation matrix of this random process are determined by the normalized correlation function:

In what follows, we will often use the two-dimensional Gaussian density

A stationary Gaussian process occupies an exclusive place among other random processes - any of its multidimensional probability density is determined by two characteristics: the mathematical expectation and the correlation function.

Analytical forecasting of multidimensional processes.

Generalized parameter method.

Purpose of work: study of practical techniques for predicting the state of a multiparameter object.

Brief theoretical information:

A change in the state of technical systems can be considered as a process characterized by changes in a set of parameters. The position of the state vector in space determines the degree of system performance. The state of the system is characterized by a vector in k-dimensional space, where the coordinates of the space are k parameters of the system,.

Prediction of the state is reduced to periodic preliminary control of parameters; determination at times t i T 1 of monitoring the state function

Q = Q [ ] and calculating the values ​​of the state function Q in the time range T 2> T 1.

In this case, the further the state vector is located from the hypersurface of permissible values ​​of the degree of operability Q *, the higher the operability of the system being diagnosed. The smaller the difference *, the lower the performance level.

The use of analytical forecasting methods assumes the regularity of changes in the components of the process over time.

The idea of ​​the generalized parameter method is that a process characterized by many components is described by a one-dimensional function, the numerical values ​​of which depend on the controlled components of the process. Such a function is considered as a generalized process parameter. In this case, it may turn out that the generalized parameter does not have a specific physical meaning, but is a mathematical expression constructed artificially from the controlled components of the predicted process.

When generalizing the parameters characterizing the degree of operability of technical systems, it is necessary to solve the following tasks:

Determination of the relative values ​​of the primary parameters;

Assessing the significance of the primary parameter for assessing the state of the object;

Building a mathematical expression for a generalized parameter.

Determination of the relative values ​​of the primary parameters is necessary due to the fact that the state of an object can be characterized by parameters having different dimensions. Therefore, all monitored primary parameters should be reduced to a single system of calculation, in which they can be comparable. Such a system is the system of dimensionless (normalized) relative calculus.

In reality, for each parameter, s = 1, 2, ..., k, one can select the admissible value, *, upon reaching which the object loses its performance, and the optimal value of opt (often it is equal to the nominal value of n).

Let the condition be met during the operation of the object. If , it is enough to enter in the local parameter and then the required condition will be met.

Let's write the dimensionless (normalized) parameter in the form:

where , and at , and at .

Thus, using expression (1), the parameter is normalized, and the dimensionless normalized value changes over time from 1 to 0. From here, by the value, one can judge the degree of performance of the object by this parameter. Theoretically it can be, but this means that in practice the object is inoperable.

You can specify various normalized expressions that are convenient for solving particular problems, for example:

etc., where - respectively current, zero, mat. waiting for the S -th parameter.

The use of normalizing expressions allows one to obtain a set of dimensionless quantities that characterize the state of an object. However, quantitatively, the same change in these values ​​is not equivalent in terms of the degree of influence on the change in the object's performance, therefore, it is necessary to differentiate the primary parameters. This process is carried out using weighting coefficients, the values ​​of which characterize the importance of the corresponding parameters for the physical essence of the problem. In this case, let the parameters of the object corresponding weighting factors satisfying one or another given criteria, and .

The degree of performance of an object for a set of monitored parameters can be estimated using a generalizing expression

Where is the generalized parameter of the object.

Expression (2) is a linear average. It follows from the definition of the generalized parameter that the greater the value of and, the greater the contribution of the S -th term (parameter) to.

A generalized parameter can be defined using an expression of the form

, (3)

which is the non-linear average. For such a model, the condition is also met: the greater and, the greater the contribution is made by the term in magnitude.

In practice, other forms of recording the nonlinear mean are also used, for example:

, (4)

, (5)

where chooses so that (5) giving the best approximation to the results obtained experimentally.

When considering expressions for a generalized parameter, it was assumed that it does not change sign, that is, always. If it is necessary to take into account the sign, expression (2) is transformed to the form

, (6)

Thus, the use of a generalized parameter makes it possible to reduce the problem of predicting the state of a multiparameter object to predicting a one-dimensional time function.

Example. Tests of the object for 250 hours, in which 6 parameters were controlled, gave the results shown in Table 1.

Table 1

After normalizing the parameter values ​​using expression (1), the table takes the form (table2)

Table 2

Module Multidimensional Exploration Analysis Technologies STATISTICA(one of the product modules STATISTICA Advanced) provides a wide range of exploration technologies ranging from cluster analysis to advanced classification tree methods, combined with a huge set of interactive visualization tools for building models. The module includes:

In the module Cluster Analysis a full set of methods for cluster analysis of data has been implemented, including methods of k-means, hierarchical clustering and two-input join. The data can come both in its original form and in the form of a matrix of distances between objects. Observations, variables and / or observations, and variables can be clustered using various distance measures (Euclidean, Euclidean square, city blocks (Manhattan), Chebyshev, power, nonconformity percentage and Pearson's 1-correlation coefficient) and different rules for combining (linking) clusters (single, full connection, unweighted and weighted pairwise mean for groups, unweighted, weighted distance between centers, Ward's method and others).

Distance matrices can be saved for further analysis in other modules of the system STATISTICA... When performing k-means cluster analysis, the user has complete control over the initial location of the cluster centers. Extremely large analysis plans can be executed: for example, with hierarchical (tree-like) linking, you can work with a matrix of 90 thousand distances. In addition to the standard cluster analysis results, the module also provides a diverse set of descriptive statistics and advanced diagnostic methods (full merging scheme with threshold levels for hierarchical clustering, ANOVA table for k-means clustering). Information about the belonging of objects to clusters can be added to the data file and used in further analysis. Graphic capabilities of the module Cluster Analysis Includes customizable dendrograms, two-way join plots, plots of join patterns, k-means clustering means and more.

Module Factor analysis contains a wide range of statistics and methods of factor analysis (as well as hierarchical factor analysis) with advanced diagnostics and a wide variety of exploratory and exploratory charts. Here you can perform principal component and principal factor analysis (general and hierarchical oblique) on datasets containing up to 300 variables (larger models can be explored using the module (SEPATH)).

Principal component analysis and classification

STATISTICA also includes a program for principal component analysis and classification. The outputs of this program are eigenvalues ​​(normal, cumulative and relative), factor loads and factor score coefficients (which can be added to the input data file, viewed in the pictograph and recoded interactively), as well as some more specialized statistics and diagnostics. The user has the following methods of rotation of factors: varimax, biquartimax, quartimax and equimax (according to normalized or initial loads), as well as oblique rotations.

The factor space can be viewed slice-by-slice visually in 2D or 3D scatterplots with data points marked; among other graphical tools - "scree" plots, various types of scatterplots, histograms, line plots, etc. After the factorial solution is determined, the user can calculate (reproduce) the correlation matrix and evaluate the consistency of the factor model by analyzing the residual correlation matrix ( or residual variance / covariance matrix). As input, you can use both raw data and correlation matrices. Confirmatory factor analysis and other related analyzes can be performed using the module Structural Equation Modeling(SEPATH) from block STATISTICA General Linear and Nonlinear Models where a special Confirmatory Factor Analysis Wizard will guide the user through all the stages of model building.

This module implements a full set of canonical analysis methods (complementary to canonical analysis methods built into other modules). You can work with both raw data files and correlation matrices; all standard statistics of canonical correlation are calculated (eigenvectors and eigenvalues, redundancy coefficients, canonical weights, loads, variances, significance criteria for each of the roots, etc.), as well as some extended diagnostics. For each observation, the values ​​of the canonical variables can be calculated, which can then be viewed on embedded icons (and also added to the data file).

This module includes a wide range of procedures for designing and evaluating sample studies and questionnaires. As in all modules of the system STATISTICA, extremely large data sets can be analyzed here (a scale of 300 positions can be processed in one program call).

It is possible to compute reliability statistics for all positions on the scale, interactively select subsets, and compare between subsets of positions using the "split-half" or "split-part" method. In one call, you can assess the reliability of the total scale and subscales. With interactive deletion of positions, the reliability of the resulting scale is calculated instantly without re-accessing the data file. As the analysis results, the following are issued: correlation matrices and descriptive statistics for positions, Cronbach's alpha, standardized alpha, average position-position correlation, a complete table of analysis of variance for the scale, a full set of statistics common to all positions (including multiple correlation coefficients), split- attenuation-corrected half-reliability and correlation between the two halves.

There is a large selection of graphs (including built-in scatterplots, histograms, line and other plots) and a set of interactive what-if procedures to help you design scales. For example, when adding a number of questions to the scale, the user can calculate the expected reliability, or estimate the number of questions that need to be added to the scale in order to achieve the desired reliability. In addition, you can correct for attenuation between the current scale and another dimension (given the reliability of the current scale).

Module systems STATISTICA contains the most complete implementation of recently developed methods of efficient construction and testing (the method of classification trees is a definite ("iterative") way of predicting the class to which an object belongs, based on the values ​​of predictor variables for this object). Classification trees can be built on categorical or ordinal predictors or a mixture of both types of predictors by branching on individual variables or on their linear combinations.

The module also implements: a choice between a brute-force search of branching options (as in the THAID and CART packages) and discriminant branching; unbiased selection of branch variables (as in the QUEST package); explicitly specifying stopping rules (as in the FACT package) or pruning from the leaves of the tree to its root (as in the CART package); clipping by the fraction of classification errors or by the deviation function; generalized measures of goodness chi-squared, G-squared and Gini index. The prior probabilities of class membership and the cost of classification errors can be set equal, estimated from the data, or set manually.

The user can also set the multiplicity of cross-checking during tree construction and for evaluating the error, the SE-rule parameter, the minimum number of objects at the clipping vertex, the seed for the random number generator and the alpha parameter for the selection of variables. Built-in graphics help explore the input and output data.

This module contains a complete implementation of methods for simple and multidimensional analysis of correspondences, it is possible to analyze tables of very large sizes. The program accepts the following types of data files: files containing categorized variables, which are used to build a contingency matrix (cross-classification); data files containing frequency tables (or any other measures of correspondence, relationship, similarity, disorder, etc.) and code variables that define (enumerate) the cells of the input table; data files containing frequencies (or other measures of conformity). For example, a user can directly create and analyze a frequency table. In addition, in the case of multivariate correspondence analysis, it is possible to directly specify the Bert matrix as input.

In the process, the program calculates various tables, including a table of percent by row, by column and percent of the total, expected values, differences between expected and observed values, standardized deviations, and contributions to the chi-square statistic. All of these statistics can be plotted on 3D histograms and viewed using a dedicated dynamic layering technique.

In the module generalized eigenvalues ​​and eigenvectors are calculated, and a standard set of diagnostic values ​​is produced, including singular values, eigenvalues ​​and a fraction of inertia attributable to each measurement. The user can either select the number of measurements or set a threshold for the maximum cumulative percentage of inertia.

The program calculates standard coordinates for row points and column points. The user can choose between standardization by row profile, by column profile, by row and column profile, or canonical standardization. For each dimension and for each row-point and column-point, the program calculates the values ​​of inertia, quality and cosine ** 2. Additionally, the user can display (in the results window) the matrices of generalized singular vectors. Like any data from the working window, these matrices are available for processing using programs in the language STATISTICA Visual Basic, for example, to use any non-standard methods for calculating coordinates.

The user can calculate coordinates and related statistics (quality and cosine ** 2) for additional points (-columns or -lines) and compare the results with the original row points and column points. Additional points can be used in multivariate matching analysis. In addition to 3D histograms, which can be calculated for all tables, the user can display the eigenvalue graph, one-, two- and three-dimensional plots for row points and column points. Row points and column points can be displayed simultaneously on the same chart, along with any additional points (each point type uses a different color and unique marker so that different points will be easily distinguishable on the charts). All points have markers, and the user has the ability to set the size of the marker.

In the module a full set of (non-metric) multidimensional scaling methods has been implemented. Here you can analyze the matrices of similarity, differences and correlations between variables, and the dimension of the scaling space can reach 9. The initial configuration can be calculated by the program (using principal component analysis) or specified by the user. The magnitude of stress and the coefficient of alienation are minimized using a special iterative procedure.

The user has the ability to observe iterations and monitor changes in these values. The final configuration can be viewed in the table of results, as well as in 2D and 3D scatter plots in scale space with marked object points. Outputs are: non-standardized stress (F), Kruskal's stress factor S, and exclusion factor. The level of agreement can be assessed using Shepard plots (with "d with a cap" and "d with an asterisk"). Like all analysis results in the system STATISTICA, the final configuration can be saved as a data file.

Module contains a complete implementation of methods of stepwise discriminant analysis using discriminant functions. STATISTICA also includes a module General Discriminant Analysis Models (GDA) to fit ANOVA / ANCOVA-like designs of categorical dependent variables; or to perform different types of analyzes (eg, better choice of predictions, profiling of posterior probabilities).

The program allows you to perform analysis with stepwise inclusion or exclusion of variables or to enter into the model user-defined blocks of variables. In addition to numerous graphs and statistics describing the dividing (discriminating) function, the program also contains a large set of tools and statistics for classifying old and new observations (to assess the quality of the model). The outputs are: Wilkes lambda statistics for each variable, private lambda, F statistics for inclusion (or exclusion), p significance levels, tolerance values ​​and the square of the multiple correlation coefficient. The program performs a complete canonical analysis and returns all eigenvalues ​​(in direct form and cumulative), their significance levels p, coefficients of the discriminant (canonical) function (in a direct and standardized form), coefficients of the structural matrix (factor loads), average values ​​of the discriminant function, and discriminant weights for each object (they can be automatically added to the data file).

Built-in graphical support includes: histograms of canonical weights for each group (and common across all groups), special scatter plots for pairs of canonical variables (which indicate which group each observation belongs to), a large set of categorized (multiple) plots, allowing you to explore distribution and relationships between dependent variables for different groups (including: multiple plots such as plots, histograms, scatter plots and normal probability plots) and much more.

In the module you can also compute the standard classification functions for each group. The results of the classification of cases can be displayed in terms of Mahalanobis distances, posterior probabilities, and the classification results themselves, and the values ​​of the discriminant function for individual cases (canonical values) can be viewed on overview pictographs and other multidimensional diagrams available directly from the results tables. All of this data can be automatically added to the current data file for further analysis. You can also display the final classification matrix, which shows the number and percentage of correctly classified cases. There are various options for setting a priori probabilities of belonging to classes, as well as selection conditions that allow you to include or exclude certain observations from the classification procedure (for example, to then check its quality on a new sample).

General Discriminant Analysis Models (GDA)

Module STATISTICA General Discriminant Analysis Models (GDA) is an application and an extension General Linear Models to classify tasks. Just like the module Discriminant Analysis The GDA allows routine sequential discriminant analyzes to be performed. The GDA presents the problem of discriminant analysis as a special case of the general linear model and thus provides extremely useful new user analytic technologies.

As with conventional discriminant analysis, GDA allows you to select the desired categories of dependent variables. In the analysis, groups of elements are written as indicator variables, and all GRM methods can be easily applied. A wide variety of GRM and GLM residual statistics are available in the GDA results dialog.

The GDA provides a variety of powerful tools for data mining and applied research. GDA calculates all standard discriminant analysis results, including the coefficients of the discriminant function, canonical analysis results (standardized and raw coefficients, step tests of canonical roots, etc.), classification statistics (including, Mahalanobis distance, posterior probabilities, classification of observations in valid analyzes, misclassification matrices, etc.). For more information on the unique features of the GDA

A multidimensional stationary random process is defined as a set of stationary and stationary interconnected random processes ... Such a process is usually denoted as a random column vector, depending on time:

.

Multidimensional stochastic processes are used to describe multidimensional (multichannel) systems. In this section, we consider the problem of digital modeling of normal multidimensional stationary random processes. The result of solving this problem, as in the one-dimensional case, is an algorithm that makes it possible to form multidimensional discrete realizations of a given process on a digital computer. -dimensional continuous normal stationary random process is usually specified either in the form of its correlation matrix

or in the form of a spectral matrix

where - autocorrelation (at) and cross-correlation (at) functions of random processes - Fourier transform of. Moreover, since , the elements and spectral matrix are complex conjugate,

.

Discrete multidimensional normal random processes are defined similarly to continuous ones using correlation and spectral matrices (35, 70]

where , and .

It is expedient to formulate the problem of digital modeling of a multidimensional normal random process as follows. A correlation or spectral matrix of a random process is given. It is required to find an algorithm for forming discrete realizations of a random process with specified correlation (spectral) properties on a digital computer.

To solve this problem, we will use, as before, the idea of ​​a shaping linear filter. In this case, we are talking about the synthesis of a multidimensional shaping filter.

A measured line filter is defined as a linear dynamic system with inputs and outputs. If - input action and is the response of the system, then the connection between the input and output of the -dimensional linear continuous filter is described using the transfer matrix in the form

where and - images of the input and output signals, respectively, in the sense of the Laplace transform; - the transfer matrix of the -dimensional filter, in which the elements are the transfer functions of the channels -th input - -th output.

The input-output connection in discrete -dimensional linear filters is described in a similar way:

,

where and - images in the sense of discrete Laplace transform of input and output signals; - transfer matrix of a discrete -dimensional filter.

The block diagram of a multidimensional filter using a two-dimensional filter as an example is shown in Fig. 2.9, according to which

(2.107)

We see that each of the output signals and is the sum of linear operators from the input signals and. Similar relationships hold in the general case. This is the identification of the transfer matrices.

Let the influence at the input of a -dimensional linear filter is a -dimensional white noise, i.e., a random process with a correlation matrix of the form

for continuous time and

for discrete time, where - delta function. -dimensional white noise is defined here as a set of mutually independent -correlated random processes.

It can be shown (see, for example,) that when exposed to white noise, the spectral matrix of the process at the output - a dimensional filter for continuous and discrete time, respectively, is related to the transfer matrix of the filter by the relations

(2.108)

where the symbol denotes the transposed matrix.

Therefore, to obtain an -dimensional random process with a given spectral matrix, it is necessary to pass the -dimensional white noise through the -dimensional shaping filter, the transfer matrix of which satisfies equations (2.108). To find the transfer matrix for a given spectral matrix, it is required to split the latter into two factors of the form (2.108). This procedure is called spectral matrix factorization. It can be implemented using well-known algorithms.

Multivariate filtering of white noise is quite simple: each component a random process at the output of a -dimensional filter with a transfer matrix is ​​obtained by summing over the components input process, filtered by one-dimensional filters with transfer functions [see. formula (2.107)]. One-dimensional filtering algorithms are discussed above.

With this modeling method, two ways are possible: 1) a given spectral matrix of a continuous -dimensional random process can be directly factorized to obtain the transfer matrix of a continuous shaping filter, and then, using the exact or approximate methods of discretization of continuous filters described above, multivariate filtration of continuous white noise; 2) for a given spectral matrix of a continuous -dimensional process, using the -transformation, one can find the spectral matrix of the corresponding discrete random process (see § 2.3), then, by factorization, find the transfer function of the discrete shaping filter, and then perform multidimensional filtering of the discrete white noise.

The greatest difficulties are encountered in the factorization of spectral matrices. At present, algorithms have been developed for factorizing only rational spectral matrices, that is, those matrices whose elements are fractional rational functions of the arguments or.

Let us describe, omitting the proofs, one of the algorithms for factorizing rational spectral matrices, taken from.

Let a rational spectral matrix be given

.

The matrix can be reduced to the form

by the following transformations.

1. The rank of the matrix is ​​determined, then one of the main order minors is located in the upper left corner of the matrix.

2. The matrix is ​​reduced to a diagonal form. To do this, the first row multiplied by - is added to the th row of the matrix,, then the first column multiplied by; the matrix is ​​obtained

, (2.109)

where the elements of the matrix

have the form

(2.110)

The same transformations are performed with the matrix as with the original matrix ... Continuing this process at the th step gives the diagonal matrix

such that .

3. Find the auxiliary matrix

whose elements are as follows:

(2.111)

where are determined from the recurrence relations

(2.112)

4. Find auxiliary polynomials

where - zeros of polynomials lying in the lower half-plane, counted as many times as their maximum multiplicity, and are the denominators of fractional-rational functions that are elements of the matrix:

.

5. By the method considered in § 2.9, item 2, the fractional-rational functions

are presented in the form

,

where the polynomials and have no zeros in the lower half-plane.

This concludes the factorization process. The final transfer matrix of the shaping filter is written in the form

(2.113)

Here we describe an algorithm for factorizing rational spectral matrices of continuous multidimensional processes. The factorization of spectral matrices of discrete processes is carried out in a similar way, only instead of the roots located in the lower half-plane, the roots located in the unit circle are taken.

Example 1. Let a two-dimensional continuous stationary centered random process with a correlation matrix be given

, (2.114)

where are some positive constants, and .

The correlation matrix corresponding to the spectral matrix (2.114) has the form

, (2.115)

where and - autocorrelation and cross-correlation moments of processes and, respectively; - coefficient of cross-correlation of processes and coinciding moments of time. The coefficients and are in this case the width (at the 0.5 level) of the energy spectra and the mutual energy spectrum of processes and.

It is required to factorize the spectral matrix (2.114) to obtain the transfer matrix of the shaping filter.

We will carry out the factorization procedure step by step in accordance with the above factorization algorithm.

1. In this case, the rank of the spectral matrix.

2. It takes one step to make the matrix diagonal. By formulas (2.109) and (2.110), we obtain

.

3. In accordance with expressions (2.111) and (2.112), the auxiliary matrix has the form

4. In the case under consideration, you need to find only one auxiliary polynomial. To do this, you need to find the roots of the denominator of the matrix element, that is, the roots of the polynomial. These roots are equal

Hence,

.

5. At the final stage, it is required to factorize the fractional-rational functions

In this case, the roots of the numerators and denominators of fractional rational functions and are easy to calculate. Using the roots lying in the upper half-plane (roots with positive imaginary parts), we also obtain for the variable:

.

In fig. 2.9 shows a block diagram of a two-dimensional shaping filter, at the output of which a two-dimensional random process with the required spectral characteristics is formed if white noise acts on the filter input. Replacing a continuous two-dimensional filter with a corresponding discrete filter, we obtain an algorithm for forming on a digital computer discrete realizations of a two-dimensional random normal process, i.e., discrete realizations of two stationary and stationary-coupled normal random processes with exponential auto- and cross-correlation functions of the form (2.115).

In another approach to the synthesis of the shaping filter, one must first find the spectral matrix of the corresponding discrete multidimensional random process. In the example under consideration, this matrix has the form

And matrices (2.116).

The example considered shows that the factorization of spectral matrices is relatively easy if one can analytically find the zeros of the corresponding polynomials. When factoring the spectral matrix of a continuous two-dimensional process, this was not difficult, since to determine the zeros it was required to solve only quadratic and biquadratic equations. In the factorization of the spectral matrix of a discrete two-dimensional process, there were quadratic equations and a return equation of the fourth degree, which also admits an analytical solution.

In other, more complicated cases, the zeros of the polynomial cannot always be found analytically. In these cases, they resort to numerical methods for solving equations of the first degree. In general, the factorization process can be implemented on a digital computer as a standard program. For this purpose, in addition to the one given here, other factorization algorithms can also be used.

It should be noted that all currently existing algorithms for factorizing spectral matrices are, generally speaking, very laborious.