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11.4. Refinement of the results obtained by the method of linearization

Lecture



In some problems of practice, there is doubt about the applicability of the linearization method due to the fact that the range of changes of random arguments is not so small that within it the function can be linearized with sufficient accuracy.

In these cases, to verify the applicability of the linearization method and to refine the results obtained, a method can be applied based on preserving in the decomposition of a function not only linear members, but also some subsequent higher-order terms and an estimate of the errors associated with these members.

In order to clarify this method, we first consider the simplest case of a function of one random argument. Random value   11.4.  Refinement of the results obtained by the method of linearization there is a function of random argument   11.4.  Refinement of the results obtained by the method of linearization :

  11.4.  Refinement of the results obtained by the method of linearization , (11.4.1)

and function   11.4.  Refinement of the results obtained by the method of linearization relatively little different from the linear on the branch of practically possible values ​​of the argument   11.4.  Refinement of the results obtained by the method of linearization , but still differs so much that there is doubt about the applicability of the linearization method. To check this circumstance, we apply a more accurate method, namely: decompose the function   11.4.  Refinement of the results obtained by the method of linearization in taylor series in a neighborhood of a point   11.4.  Refinement of the results obtained by the method of linearization and keep in decomposition the first three members:

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.2)

The same formula will obviously approximate random variables.   11.4.  Refinement of the results obtained by the method of linearization and   11.4.  Refinement of the results obtained by the method of linearization :

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.3)

Using the expression (11.4.3), we find the expectation and variance of   11.4.  Refinement of the results obtained by the method of linearization . Applying the theorems on numerical characteristics, we have:

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.4)

According to the formula (11.4.4) you can find the adjusted value of the expectation and compare it with the value   11.4.  Refinement of the results obtained by the method of linearization which is obtained by the linearization method; the amendment taking into account the nonlinearity of the function is the second term of formula (11.4.4).

Determining the variance of the right and left side of the formula (11.4.3), we have:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization , (11.4.5)

Where   11.4.  Refinement of the results obtained by the method of linearization - correlation moment of magnitudes   11.4.  Refinement of the results obtained by the method of linearization .

We express the quantities entering the formula (11.4.5) through the central moments of the quantities   11.4.  Refinement of the results obtained by the method of linearization :

  11.4.  Refinement of the results obtained by the method of linearization ,

  11.4.  Refinement of the results obtained by the method of linearization .

Finally we have:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.6)

Formula (11.4.6) gives a refined value of the variance compared to the linearization method; its second and third terms represent the correction for the nonlinearity of the function. In the formula, except for the variance of the argument   11.4.  Refinement of the results obtained by the method of linearization , includes the third and fourth central points   11.4.  Refinement of the results obtained by the method of linearization ,   11.4.  Refinement of the results obtained by the method of linearization . If these moments are known, then the correction to the variance can be found directly by the formula (11.4.6). However, it is often not necessary to define it precisely; it is enough to know its order. In practice, there are often random variables distributed approximately according to the normal law. For a random variable subject to normal law,

  11.4.  Refinement of the results obtained by the method of linearization ,   11.4.  Refinement of the results obtained by the method of linearization , (11.4.7)

and the formula (11.4.6) takes the form:

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.8)

Formula (11.4.8) can be used for an approximate estimate of the error of the linearization method in the case when the argument is distributed according to a law close to normal.

A completely similar method can be applied to the function of several random arguments:

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.9)

Decomposing function

  11.4.  Refinement of the results obtained by the method of linearization

in taylor series in a neighborhood of a point   11.4.  Refinement of the results obtained by the method of linearization and keeping in the expansion terms not higher than second order, we have approximately:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization ,

or by entering centered values,

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization , (11.4.10)

where is the index   11.4.  Refinement of the results obtained by the method of linearization still denotes that in a partial derivative expression instead of arguments   11.4.  Refinement of the results obtained by the method of linearization their mathematical expectations are substituted   11.4.  Refinement of the results obtained by the method of linearization .

Applying to the formula (11.4.10) the operation of mathematical expectation, we have:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization , (11.4.11)

Where   11.4.  Refinement of the results obtained by the method of linearization - correlation moment of magnitudes   11.4.  Refinement of the results obtained by the method of linearization .

In the most important case for practice, when the arguments   11.4.  Refinement of the results obtained by the method of linearization uncorrelated, the formula (11.4.11) takes the form:

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.12)

The second term of formula (11.4.12) is the correction for the nonlinearity of the function.

Let us proceed to the determination of the variance of   11.4.  Refinement of the results obtained by the method of linearization . To get the expression of dispersion in the simplest form, suppose that the quantities   11.4.  Refinement of the results obtained by the method of linearization not only uncorrelated, but also independent. Determining the variance of the right and left side (11.4.10) and using the theorem on the dispersion of a product (see   11.4.  Refinement of the results obtained by the method of linearization 10.2), we get:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.13)

For quantities distributed according to a law close to normal, you can use the formula (11.4.7) and convert the expression (11.4.13) to the form:

  11.4.  Refinement of the results obtained by the method of linearization

  11.4.  Refinement of the results obtained by the method of linearization . (11.4.14)

The last two terms in expression (11.4.14) are the “correction for non-linearity of the function” and can serve to estimate the accuracy of the linearization method when calculating the variance.


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis