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

17.8. Characterization of an ergodic stationary random function in one implementation

Lecture



Consider the stationary random function   17.8.  Characterization of an ergodic stationary random function in one implementation which has an ergodic property, and suppose that we have only one implementation of this random function, but on a sufficiently large time interval   17.8.  Characterization of an ergodic stationary random function in one implementation . For an ergodic stationary random function, one implementation of a sufficiently long duration is practically equivalent (in terms of the amount of information about the random function) to the set of realizations of the same total duration; characteristics of a random function can be approximated not as averages over a variety of observations, but as averages over time.   17.8.  Characterization of an ergodic stationary random function in one implementation . In particular, with a sufficiently large   17.8.  Characterization of an ergodic stationary random function in one implementation expected value   17.8.  Characterization of an ergodic stationary random function in one implementation can be approximately calculated by the formula

  17.8.  Characterization of an ergodic stationary random function in one implementation . (17.8.1)

Similarly, the correlation function can be approximately found.   17.8.  Characterization of an ergodic stationary random function in one implementation at any   17.8.  Characterization of an ergodic stationary random function in one implementation . Indeed, the correlation function, by definition, is nothing more than the expectation of a random function   17.8.  Characterization of an ergodic stationary random function in one implementation :

  17.8.  Characterization of an ergodic stationary random function in one implementation (17.8.2)

This expectation can also, obviously, be approximately calculated as a time average.

We fix some value   17.8.  Characterization of an ergodic stationary random function in one implementation and calculate in this way the correlation function   17.8.  Characterization of an ergodic stationary random function in one implementation . For this, it is convenient to pre-center this implementation.   17.8.  Characterization of an ergodic stationary random function in one implementation , i.e., subtract the expectation from it (17.8.1):

  17.8.  Characterization of an ergodic stationary random function in one implementation . (17.8.3)

Calculate for a given   17.8.  Characterization of an ergodic stationary random function in one implementation expectation of a random function   17.8.  Characterization of an ergodic stationary random function in one implementation as time average. In this case, obviously, we will have to take into account not the whole time interval from 0 to   17.8.  Characterization of an ergodic stationary random function in one implementation , and somewhat smaller, since the second factor   17.8.  Characterization of an ergodic stationary random function in one implementation we are not known for everyone   17.8.  Characterization of an ergodic stationary random function in one implementation , but only for those for whom   17.8.  Characterization of an ergodic stationary random function in one implementation .

Calculating the time average of the above method, we get:

  17.8.  Characterization of an ergodic stationary random function in one implementation . (17.8.4)

By calculating the integral (17.8.4) for a number of values   17.8.  Characterization of an ergodic stationary random function in one implementation , you can approximately reproduce point by point the entire course of the correlation function.

In practice, the integrals (17.8.1) and (17.8.4) are usually replaced by finite sums. We show how this is done. We divide the recording interval of a random function into   17.8.  Characterization of an ergodic stationary random function in one implementation equal parts long   17.8.  Characterization of an ergodic stationary random function in one implementation and denote the midpoints of the obtained sections   17.8.  Characterization of an ergodic stationary random function in one implementation (fig. 17.8.1).

  17.8.  Characterization of an ergodic stationary random function in one implementation

Fig. 17.8.1.

Provide integral (17.8.1) as the sum of integrals over elementary parts   17.8.  Characterization of an ergodic stationary random function in one implementation and on each of them we will carry out the function   17.8.  Characterization of an ergodic stationary random function in one implementation from the integral sign, the average value corresponding to the center of the interval   17.8.  Characterization of an ergodic stationary random function in one implementation . We get approximately:

  17.8.  Characterization of an ergodic stationary random function in one implementation ,

or

  17.8.  Characterization of an ergodic stationary random function in one implementation . (17.8.5)

Similarly, you can calculate the correlation function for the values   17.8.  Characterization of an ergodic stationary random function in one implementation equal   17.8.  Characterization of an ergodic stationary random function in one implementation . We give, for example, the value   17.8.  Characterization of an ergodic stationary random function in one implementation value

  17.8.  Characterization of an ergodic stationary random function in one implementation

calculate the integral (17.8.4), dividing the integration interval

  17.8.  Characterization of an ergodic stationary random function in one implementation

on   17.8.  Characterization of an ergodic stationary random function in one implementation equal length plots   17.8.  Characterization of an ergodic stationary random function in one implementation and putting on each of them a function   17.8.  Characterization of an ergodic stationary random function in one implementation over the integral sign of the mean value. We get:

  17.8.  Characterization of an ergodic stationary random function in one implementation ,

or finally

  17.8.  Characterization of an ergodic stationary random function in one implementation . (17.8.6)

The calculation of the correlation function by the formula (17.8.6) is performed for   17.8.  Characterization of an ergodic stationary random function in one implementation consistently down to such values   17.8.  Characterization of an ergodic stationary random function in one implementation at which the correlation function becomes almost zero or starts to make small irregular fluctuations around zero. The general course of the function   17.8.  Characterization of an ergodic stationary random function in one implementation reproduced by individual points (Fig. 17.8.2).

  17.8.  Characterization of an ergodic stationary random function in one implementation

Fig. 17.8.2.

In order to expectation   17.8.  Characterization of an ergodic stationary random function in one implementation and correlation function   17.8.  Characterization of an ergodic stationary random function in one implementation were determined with satisfactory accuracy, it is necessary that the number of points   17.8.  Characterization of an ergodic stationary random function in one implementation was large enough (about a hundred, and in some cases even a few hundred). The choice of the length of the elementary area   17.8.  Characterization of an ergodic stationary random function in one implementation determined by the nature of the change of the random function. If the random function changes relatively smoothly, the plot   17.8.  Characterization of an ergodic stationary random function in one implementation You can choose more than when it makes a sharp and frequent fluctuations. The higher-frequency composition have oscillations that form a random function, the more often it is necessary to position the reference points during processing. Approximately we can recommend to choose an elementary area   17.8.  Characterization of an ergodic stationary random function in one implementation so that for the full period of the highest-frequency harmonic in the composition of the random function there were about 5-10 reference points.

Often the choice of reference points does not depend on the processing at all, but is dictated by the pace of the recording equipment. In this case, it is necessary to conduct processing directly obtained from the experience of the material, not trying to insert intermediate values ​​between the observed values, since the echo can not improve the accuracy of the result, and unnecessarily complicate the processing.

Example. In the conditions of horizontal flight of the aircraft, a vertical overload acting on the aircraft was recorded. Overload was recorded at a time interval of 200 seconds with an interval of 2 seconds. The results are shown in table 17.8.1.

Table 17.8.1

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

Overload

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

Overload

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

Overload

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

Overload

  17.8.  Characterization of an ergodic stationary random function in one implementation

0

1.0

50

1.0

100

1.2

150

0.8

2

1,3

52

1.1

102

1.4

152

0.6

four

1.1

54

1.5

104

0.8

154

0.9

6

0.7

56

1.0

106

0.9

156

1.2

eight

0.7

58

0.8

108

1.0

158

1,3

ten

1.1

60

1.1

110

0.8

160

0.9

12

1,3

62

1.1

112

0.8

162

1,3

14

0.8

64

1.2

114

1.4

164

1.5

sixteen

0.8

66

1.0

116

1.6

166

1.2

18

0.4

68

0.8

118

1.7

168

1.4

20

0.3

70

0.8

120

1,3

170

1.4

22

0.3

72

1.2

122

1.6

172

0.8

24

0.6

74

0.7

124

0.8

174

0.8

26

0.3

76

0.7

126

1.2

176

1,3

28

0.5

78

1.1

128

0.6

178

1.0

thirty

0.5

80

1.2

130

1.0

180

0.7

32

0.7

82

1.0

132

0.3

182

1.1

34

0.8

84

0.6

134

0.8

184

0.9

36

0.6

86

0.9

136

0.7

186

0.9

38

1.0

88

0.8

138

0.9

188

1.1

40

0.5

90

0.8

140

1,3

190

1.2

42

1.0

92

0.9

142

1.5

192

1,3

44

0.9

94

0.9

144

1.1

194

1,3

46

1.4

96

0.6

146

0.7

196

1.6

48

1.4

98

0.4

148

1.0

198

1.5

Considering the process of changing overload stationary, to determine the approximate expectation of overload   17.8.  Characterization of an ergodic stationary random function in one implementation variance   17.8.  Characterization of an ergodic stationary random function in one implementation and normalized correlation function   17.8.  Characterization of an ergodic stationary random function in one implementation . Approximate   17.8.  Characterization of an ergodic stationary random function in one implementation any analytical function, find and build the spectral density of a random process.

Decision. By the formula (17.8.5) we have:

  17.8.  Characterization of an ergodic stationary random function in one implementation .

We center the random function (Table 17.8.2).

Table 17.8.2

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

(sec)

  17.8.  Characterization of an ergodic stationary random function in one implementation

0

0.02

50

0.02

100

0.22

150

-0.18

2

0.32

52

0.12

102

0.42

152

-0.38

four

0.12

54

0.52

104

-0.18

154

-0,08

6

-0,28

56

0.02

106

-0,08

156

0.22

eight

-0,28

58

-0.18

108

0.02

158

0.32

ten

0.12

60

0.12

110

-0.18

160

-0,08

12

0.32

62

0.12

112

-0.18

162

0.32

14

-0.18

64

0.22

114

0.42

164

0.52

sixteen

-0.18

66

0.02

116

0.62

166

0.22

18

-0,58

68

-0.18

118

0.72

168

0.42

20

-0,68

70

-0.18

120

0.32

170

0.42

22

-0,68

72

0.22

122

0.62

172

-0.18

24

-0.38

74

-0,28

124

-0.18

174

-0.18

26

-0,68

76

-0,28

126

0.22

176

0.32

28

-0.48

78

0.12

128

-0.38

178

0.02

thirty

-0.48

80

0.52

130

0.02

180

-0,28

32

-0,28

82

0.02

132

-0.38

182

0.12

34

-0.18

84

-0.38

134

-0.18

184

-0,08

36

-0.38

86

-0,08

136

-0,28

186

-0,08

38

0.02

88

-0.18

138

-0,08

188

0.12

40

-0.48

90

-0.18

140

0.32

190

0.22

42

0.02

92

-0,08

142

0.52

192

0.32

44

-0,08

94

-0,08

144

0.12

194

0.32

46

0.42

96

-0.38

146

-0,28

196

0.62

48

0.42

98

-0,58

148

0.02

198

0.52

Squaring all values   17.8.  Characterization of an ergodic stationary random function in one implementation and dividing the amount by   17.8.  Characterization of an ergodic stationary random function in one implementation we obtain approximately the variance of the random function   17.8.  Characterization of an ergodic stationary random function in one implementation :

  17.8.  Characterization of an ergodic stationary random function in one implementation

and standard deviation:

  17.8.  Characterization of an ergodic stationary random function in one implementation .

Multiplying values   17.8.  Characterization of an ergodic stationary random function in one implementation separated by an interval   17.8.  Characterization of an ergodic stationary random function in one implementation and dividing the sum of the works respectively   17.8.  Characterization of an ergodic stationary random function in one implementation ;   17.8.  Characterization of an ergodic stationary random function in one implementation ;   17.8.  Characterization of an ergodic stationary random function in one implementation ; ..., we obtain the values ​​of the correlation function   17.8.  Characterization of an ergodic stationary random function in one implementation . Normalizing the correlation function by dividing by   17.8.  Characterization of an ergodic stationary random function in one implementation get the table of function values   17.8.  Characterization of an ergodic stationary random function in one implementation (tab. 17.8.3).

Table 17.8.3

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

  17.8.  Characterization of an ergodic stationary random function in one implementation

0

1,000

1,000

2

0,505

0.598

four

0.276

0.358

6

0.277

0.214

eight

0,231

0.128

ten

-0,015

0.077

12

0.014

0.046

14

0.071

0.027

Function graph   17.8.  Characterization of an ergodic stationary random function in one implementation presented in fig. 17.8.3 in the form of points connected by a dotted line.

  17.8.  Characterization of an ergodic stationary random function in one implementation

Fig. 17.8.3.

The not quite smooth course of the correlation function can be explained by the insufficient amount of experimental data (insufficient experience duration), and therefore random irregularities in the course of the function do not have time to smooth out. Calculation   17.8.  Characterization of an ergodic stationary random function in one implementation continued to such values ​​at which the actual correlation disappears.

In order to smooth clearly the irregular oscillations of the experimentally found function   17.8.  Characterization of an ergodic stationary random function in one implementation , we replace it approximately with a view function:

  17.8.  Characterization of an ergodic stationary random function in one implementation ,

where is the parameter   17.8.  Characterization of an ergodic stationary random function in one implementation we select the method of least squares (see   17.8.  Characterization of an ergodic stationary random function in one implementation 14.5).

Using this method, we find   17.8.  Characterization of an ergodic stationary random function in one implementation . Calculating function values   17.8.  Characterization of an ergodic stationary random function in one implementation at   17.8.  Characterization of an ergodic stationary random function in one implementation Let's build a graph of a smoothing curve. In fig. 17.8.3 he held a solid line. The last column of the table 17.8.3 shows the values ​​of the function.   17.8.  Characterization of an ergodic stationary random function in one implementation .

Using the approximate expression of the correlation function (17.8.6), we obtain (see   17.8.  Characterization of an ergodic stationary random function in one implementation 17.4, example 1) the normalized spectral density of a random process in the form:

  17.8.  Characterization of an ergodic stationary random function in one implementation .

The graph of the normalized spectral density is presented in Fig. 17.8.4.

  17.8.  Characterization of an ergodic stationary random function in one implementation

Fig. 17.8.4.


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