You get a bonus - 1 coin for daily activity. Now you have 1 coin

7.5. Alignment of statistical series

Lecture



In any statistical distribution, inevitably there are elements of randomness, due to the fact that the number of observations is limited, that it was those, and not other experiments that gave precisely those, and not other results. Only with a very large number of observations, these elements of randomness are smoothed out, and the random phenomenon reveals a fully inherent pattern. In practice, we almost never deal with such a large number of observations and are forced to reckon with the fact that any statistical distribution is characterized by more or less random features. Therefore, when processing statistical material, it is often necessary to decide how to select a theoretical distribution curve for a given statistical series, expressing only the essential features of statistical material, but not randomness, associated with an insufficient amount of experimental data. Such a task is called the task of leveling (smoothing) the statistical series.

The task of alignment is to find a theoretical smooth distribution curve that, from one point of view, best describes this statistical distribution (Fig. 7.5.1).

  7.5.  Alignment of statistical series

Fig. 7.5.1

The problem of the best alignment of statistical series, as well as the problem of the best analytical representation of empirical functions in general, is a task that is largely uncertain, and its solution depends on what is agreed to be considered “best”. For example, when smoothing empirical dependencies, very often they proceed from the so-called principle or the method of least squares (see   7.5.  Alignment of statistical series 14.5), considering that the best approximation to the empirical dependence in this class of functions is one in which the sum of the squares of the deviations turns to a minimum. In this case, the question of which class of functions the best approximation should be sought for is decided not from mathematical considerations, but from considerations related to the physics of the problem being solved, taking into account the nature of the empirical curve obtained and the degree of accuracy of the observations made. Often, the fundamental nature of the function expressing the dependence under investigation is known in advance from theoretical considerations, but from experience only some numerical parameters are required to be included in the function expression; These parameters are selected using the method of least squares.

The situation is similar with the task of leveling statistical series. As a rule, the fundamental form of a theoretical curve is chosen in advance from considerations related to the essence of the problem, and in some cases simply with the appearance of the statistical distribution. The analytical expression of the selected distribution curve depends on some parameters; the task of leveling the statistical series goes into the problem of rational choice of those values ​​of the parameters for which the correspondence between the statistical and theoretical distributions turns out to be the best.

Suppose, for example, that the quantity studied   7.5.  Alignment of statistical series there is a measurement error resulting from the summation of the effects of a set of independent elementary errors; then from theoretical considerations we can assume that the value   7.5.  Alignment of statistical series obeys the normal law:

  7.5.  Alignment of statistical series (7.5.1)

and the alignment problem goes into the problem of the rational choice of parameters   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series in expression (7.5.1).

There are cases when it is known in advance that   7.5.  Alignment of statistical series it is distributed statistically approximately evenly over a certain interval; then you can pose the problem of a rational choice of the parameters of the law of uniform density

  7.5.  Alignment of statistical series

which can best replace (equalize) the specified statistical distribution.

It should be borne in mind that any analytical function   7.5.  Alignment of statistical series , with the help of which the statistical distribution is aligned, should have the main properties of the density distribution:

  7.5.  Alignment of statistical series (7.5.2)

Suppose that, based on certain considerations, we have chosen the function   7.5.  Alignment of statistical series satisfying conditions (7.5.2), with the help of the bark we want to equalize this statistical distribution; The expression of this function includes several parameters.   7.5.  Alignment of statistical series ; It is required to select these parameters so that the function   7.5.  Alignment of statistical series best described this statistical material. One of the methods used to solve this problem is the so-called method of moments.

According to the method of moments, the parameters   7.5.  Alignment of statistical series are chosen in such a way that several of the most important numerical characteristics (moments) of the theoretical distribution are equal to the corresponding statistical characteristics. For example, if the theoretical curve   7.5.  Alignment of statistical series depends only on two parameters   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series , these parameters are chosen so that the expectation   7.5.  Alignment of statistical series and variance   7.5.  Alignment of statistical series theoretical distribution coincided with the corresponding statistical characteristics   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series . If the curve   7.5.  Alignment of statistical series depends on three parameters, you can choose them so that the first three points coincide, etc. When aligning statistical series, a specially developed system of Pearson curves, each of which depends in general on four parameters, may be useful. When aligning, these parameters are selected so as to preserve the first four points of the statistical distribution (expectation, variance, third and fourth moments). The original set of distribution curves constructed by a different principle was given by N.A. Borodachev. The principle on which the N.A. Borodachev, lies in the fact that the choice of the type of theoretical curve is not based on external formal features, but on an analysis of the physical essence of a random phenomenon or process leading to a particular distribution law.

It should be noted that when aligning the statistical series, it is not rational to use moments of order higher than four, since the accuracy of the calculation of the moments drops sharply with increasing order.

Example. 1. In   7.5.  Alignment of statistical series 7.3 shows the statistical distribution of lateral interference errors   7.5.  Alignment of statistical series when shooting from an airplane at a ground target. It is required to level this distribution using normal law:

  7.5.  Alignment of statistical series .

Normal law depends on two parameters:   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series . We select these parameters so as to preserve the first two points — the expectation and variance — of the statistical distribution.

Let us calculate the approximate statistical average of the error of the pickup using the formula (7.47), and for the representative of each digit we take its middle:

  7.5.  Alignment of statistical series

To determine the variance, we first calculate the second initial moment using the formula (7.4.9), assuming   7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

Using the expression of dispersion through the second initial moment (formula (7.4.6)), we get:

  7.5.  Alignment of statistical series

Choose parameters   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series normal law so that the conditions are met:

  7.5.  Alignment of statistical series

that is, take:

  7.5.  Alignment of statistical series .

Write the expression of the normal law:

  7.5.  Alignment of statistical series

Using the table. 3 applications, calculate the values   7.5.  Alignment of statistical series on the borders of discharges

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

Let us construct a histogram on one graph (fig. 7.5.2) and a distribution curve leveling it.

The graph shows that the theoretical distribution curve   7.5.  Alignment of statistical series while preserving, in general, the essential features of the statistical distribution, it is free from random irregularities in the course of the histogram, which, apparently, can be attributed to random reasons; a more serious justification of the last judgment will be given in the next paragraph.

  7.5.  Alignment of statistical series

Fig. 7.5.2

Note. In this example, when determining   7.5.  Alignment of statistical series , we used the expression (7.4.6) of statistical variance through the second initial moment. This technique can be recommended only in the case when the expectation   7.5.  Alignment of statistical series investigated random variable   7.5.  Alignment of statistical series relatively small; otherwise, the formula (7.4.6) expresses the variance   7.5.  Alignment of statistical series as a difference of close numbers and gives a very low accuracy. In the case that this is the case, it is recommended to either calculate   7.5.  Alignment of statistical series directly by the formula (7.4.3), or move the origin to some point close to   7.5.  Alignment of statistical series and then apply the formula (7.4.6). Using formula (7.4.3) is equivalent to transferring the origin to a point   7.5.  Alignment of statistical series ; this can be inconvenient because the expression   7.5.  Alignment of statistical series can be fractional and subtraction   7.5.  Alignment of statistical series of each   7.5.  Alignment of statistical series while unnecessarily complicates the calculations; therefore it is recommended to transfer the origin to some round value   7.5.  Alignment of statistical series close to   7.5.  Alignment of statistical series .

Example 2. In order to investigate the law of the distribution of the error in measuring the distance using a radio-range meter, 400 distance measurements were made. The results of the experiments are presented in the form of a statistical series:

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

0.140

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

Align the statistical series using the law of uniform density.

Decision. The law of uniform density is expressed by the formula

  7.5.  Alignment of statistical series

and depends on two parameters   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series . These parameters should be chosen so as to preserve the first two points of the statistical distribution - the expectation   7.5.  Alignment of statistical series and variance   7.5.  Alignment of statistical series . From example   7.5.  Alignment of statistical series 5.8 we have the expression of the expectation and variance for the law of uniform density:

  7.5.  Alignment of statistical series

In order to simplify the calculations associated with the determination of statistical moments, we move the origin to the point   7.5.  Alignment of statistical series and take for the representative of his rank his middle. The distribution series is:

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

  7.5.  Alignment of statistical series

Where   7.5.  Alignment of statistical series - the average for the discharge value of the radio range meter   7.5.  Alignment of statistical series with a new origin.

Approximate value of statistical average error   7.5.  Alignment of statistical series equally:

  7.5.  Alignment of statistical series

Second statistical moment of magnitude   7.5.  Alignment of statistical series equals:

  7.5.  Alignment of statistical series ,

whence statistical variance:

  7.5.  Alignment of statistical series .

Turning to the previous reference point, we get a new statistical average:

  7.5.  Alignment of statistical series

in the same statistical variance:

  7.5.  Alignment of statistical series .

The parameters of the law of uniform density are determined by the equations:

  7.5.  Alignment of statistical series .

Solving these equations for   7.5.  Alignment of statistical series and   7.5.  Alignment of statistical series , we have:

  7.5.  Alignment of statistical series ,

from where

  7.5.  Alignment of statistical series .

In fig. 7.5.3. shows the histogram and the law of uniform density equalizing it   7.5.  Alignment of statistical series .

  7.5.  Alignment of statistical series

Fig. 7.5.3


Comments


To leave a comment
If you have any suggestion, idea, thanks or comment, feel free to write. We really value feedback and are glad to hear your opinion.
To reply

Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis