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16.1. The idea of ​​the method of canonical decompositions. Representation of a random function as a sum of elementary random functions

Lecture



AT 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions 15.7 we got acquainted with the general rules of linear transformations of random functions. These rules are reduced to the fact that with a linear transformation of a random function, its expectation undergoes the same linear transformation, and the correlation function undergoes this transformation twice: with one and the other argument.

The rule of converting the mathematical expectation is very simple and does not cause difficulties in the practical application. As for the double transformation of the correlation function, in some cases it leads to extremely complex and cumbersome operations, which complicates the practical application of the described general methods.

Indeed, consider, for example, the simplest integral operator:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.1)

According to the general rule, the correlation function is transformed by the same operator twice:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.2)

It is often the case that the correlation function obtained from experience 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions It has no analytical expression and is given in tabular form; then the integral (16.1.2) has to be calculated numerically, defining it as a function of both limits. This is a very cumbersome and time-consuming task. Even if we approximate the integrand by some analytical expression, then in this case the integral (16.1.2) is often not expressed through known functions. This is the case even with the simplest form of a conversion operator. If, as often happens, the work of a dynamic system is described by differential equations, the solution of which is not expressed explicitly, the problem of determining the correlation function at the output is even more complicated: it requires the integration of partial differential equations.

In this connection, in practice, the application of the described general methods of linear transformations of random functions, as a rule, turns out to be too complicated and does not justify itself. In solving practical problems, other methods are used more often, leading to simpler transformations. One of them - the so-called method of canonical decomposition, developed by V. S. Pugachev, is the content of this chapter.

The idea of ​​the method of canonical decompositions is that a random function, over which you need to make certain transformations, is first represented as a sum of the so-called elementary random functions.

An elementary random function is a function of the form:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , (16.1.3)

Where 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - ordinary random variable 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - ordinary (non-random) function.

The elementary random function is the simplest type of random function. Indeed, in expression (16.1.3) only the multiplier is random 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions facing function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ; time dependence itself is not accidental.

All possible implementations of an elementary random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions can be obtained from the function graph 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions simple measurement of the scale on the ordinate axis (Fig. 16.1.1).

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions

Fig. 16.1.1.

In this case, the abscissa axis 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions also represents one of the possible implementations of the random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions implemented when a random variable 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions takes the value 0 (if this value belongs to the number of possible values ​​of 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ).

As examples of elementary random functions, we give the functions 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions (fig. 16.1.2) and 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions (fig. 16.1.3).

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions

Fig. 16.1.2.

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions

Fig. 16.1.3.

An elementary random function is characterized by the fact that it distinguishes two features of a random function: the entire randomness is concentrated in the coefficient 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , and the dependence on time - in the usual function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

We define the characteristics of an elementary random function (16.1.3). We have:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ,

Where 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - expectation of a random variable 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

If a 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , expectation of random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions is also zero, and the identity is:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

We know that any random function can be centered, i.e., brought to such a form, when its expectation is zero. Therefore, in the future we will consider only centered elementary random functions for which 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ; 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ; 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Define the correlation function of an elementary random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . We have:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ,

Where 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - variance of magnitude 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Above elementary random functions, all sorts of linear transformations are quite simply performed.

For example, let's differentiate the random function (16.1.3). Random value 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions independent of 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , will be a derivative sign, and we get:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Similarly

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

In general, if an elementary random function (16.1.3) is transformed by a linear operator 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , then with a random factor 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions as independent of 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , beyond the operator sign, but a non-random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions converted by the same operator 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions :

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.4)

So, if an elementary random function arrives at the input of a linear system, then the task of its transformation is reduced to the simple task of converting a single non-random function. 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . Hence the idea: if a random form of a general form arrives at the input of a dynamic system, then it can be represented, precisely or approximately, as a sum of elementary random functions and only then subjected to transformation. Such an idea of ​​decomposing a random function into a sum of elementary random functions is the basis of the method of canonical decompositions.

Suppose there is a random function:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.5)

Suppose that we were able - precisely or approximately - to represent it as a sum

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , (16.1.6)

Where 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - random variables with mathematical expectations equal to zero; 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - non-random functions; 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - expectation function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

We agree to call the representation of a random function in the form (16.1.6) a decomposition of a random function. Random variables 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions will be called the expansion coefficients, and non-random functions 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions - coordinate functions.

Define the reaction of a linear system with an operator 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions on random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions given as a decomposition (16.1.6). It is known that a linear system has the so-called property of superposition, consisting in the fact that the response of the system to the sum of several actions is equal to the sum of the reactions of the system to each individual action. Indeed, the system operator 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions Being linear, it can, by definition, be applied to the sum term by term.

Denoting 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions system response to random exposure 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , we have:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.7)

Let's give the expression (16.1.7) a slightly different form. Given the general rule of linear transformation of mathematical expectation, we make sure that

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Denoting

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions ,

we have:

16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions . (16.1.8)

Expression (16.1.8) is nothing more than a decomposition of a random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions on elementary functions. The coefficients of this expansion are the same random variables. 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , and the expectation and coordinate functions are obtained from the expectation and coordinate functions of the original random function by the same linear transformation 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions to which random function is exposed 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Summarizing, we obtain the following rule for the transformation of a random function given by the decomposition.

If random function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions given by the decomposition into elementary functions undergoes a linear transformation 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , the coefficients of decomposition remain unchanged, and the expectation and coordinate functions undergo the same linear transformation 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions .

Thus, the meaning of the decomposition of a random function is reduced to reducing the linear transformation of a random function to the same linear transformation of several non-random functions — expectation and coordinate functions. This allows us to significantly simplify the solution of the problem of finding the characteristics of a random function. 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions compared to the general solution given in 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions 15.7. Indeed, each of the non-random functions 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions , 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions in this case, it is converted only once as opposed to the correlation function 16.1.  The idea of ​​the method of canonical decompositions.  Representation of a random function as a sum of elementary random functions which, according to the general rules, is converted twice.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis