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process possesses zero mathematical expectations and the variance estimate is carried out based on investigation of independent samples. The characteristic of transformation y = g(x) can be presented by the polynomial function of the μth order

Substituting (13.79) into (13.59) instead of xi and carrying out averaging, we obtain the mathematical expectation of variance estimate:

The estimate bias caused by the difference between the transformation characteristic and the squarelaw function can be presented in the following form:

For the given transformation characteristic the coefficients ak can be defined before. For this reason, the bias of variance estimate caused by the coefficients a0 and a2 can be taken into consideration. The problem is to take into consideration the sum in (13.81) because this sum depends on the shape of pdf of the investigated stochastic process:

In the case of Gaussian stochastic process, the moment of the kth order can be presented in the following form:

Difference between the transformation characteristic and the square-law function can lead to high errors under definition of the stochastic process variance. Because of this, while defining the stochastic process variance we need to pay serious attention to transformation performance. Verification of the transformation performance is carried out, as a rule, by sending the harmonic signal of known amplitude at the measurer input.

While measuring the stochastic process variance, we can avoid the squaring operation. For this purpose, we need to consider two sign functions (12.208)

instead of the initial stochastic process ξ(t) with zero mathematical expectation, where μ1(t) and μ2(t) are additional independent stationary stochastic processes with zero mathematical expectations and the same pdf given by (12.205). At the same time, the condition given by (12.206) is satisfied.

The functions η1(t) and η2(t) are the stationary stochastic processes with zero mathematical expectations. In doing so, if the fixed value ξ(t) = x the conditional stochastic processes η1(t|x) and η2(t|x) are statistically independent, that is,

The unconditional mathematical expectation of product can be presented in the following form:

Denote the realizations of stochastic processes η1(t) and η2(t) by the functions y1(t) and y2(t), respectively. Consequently, if we consider the following value as the variance estimate,

then the variance estimate is unbiased.

The realizations y1(t) and y2(t) take the values equal to ±1. For this reason, the multiplication and integration in (13.88) can be replaced by individual integration of new unit functions obtained as a result of coincidence and noncoincidence of polarities of the realizations y1(t) and y2(t):

u2(t)

x(t)

FIGURE 13.4  Measurer based on additional signals.

The flowchart of measuring device based on the implementation of additional signals is shown in Figure 13.4.

In the case of discrete process, the variance estimate can be presented in the following form:

where y1i = y1(ti) and y2i = y2(ti). In this case, the integrator can be replaced by the summator in Figure 13.4.

Determine the variance of the variance estimate Var applied to investigation of stochastic process at discrete instants for the purpose of simplifying further analysis that the samples y1i and y2i are independent. Otherwise, we should know the two-dimensional pdfs of additional uniformly distributed stochastic processes μ1(t) and μ2(t). The variance of the variance estimate can be presented in the following form:

Based on a definition of sign functions, we can obtain from (13.84) the following condition:

Define the cumulative moment  η1iη1jη2iη2j|xixj  at the condition that ξ(ti) = xi and ξ(tj) = xj. Taking into consideration the statistical independence of the samples η1 and η2, the statistical mutual

independence of the stochastic values η1(t|x) and η2(t|x), and (12.209), the conditional cumulative moment can be presented in the following form:

Averaging (13.97) by possible values of independent random variables xi and xj, we obtain

Substituting (13.96) and (13.98) into (13.93), we obtain the variance of the variance estimate (13.92)

The variance estimate according to (13.92) envisages that the condition (12.206) is satisfied. Consequently, the inequality σ2 A2 must be satisfied. Because of this, as in the case of estimation of the mathematical expectation with employment of additional signals, the variance of the variance estimate Var is completely defined by the half intervals of possible values of additional random sequences. Comparing the obtained variance (13.99) with the variance of the variance estimate in (13.76) by N independent samples, we can find that

Thus, we can conclude that the method of measurement of the stochastic process variance using the additional signals is characterized by the higher variance compared to the algorithm (13.74). This is based on the example of definition of the variance estimate of stochastic sample with the uniform pdf coinciding with (12.205), σ2 = A2/3. As a result, we have

Var{Var} = 4. (13.101)

Var{Var }

The methods that are used to measure the variance assume the absence of limitations of instantaneous values of the investigated stochastic process. The presence of these limitations leads to additional errors while measuring the variance. Determine the bias of variance estimate of the Gaussian stochastic process when there is a limiter of the type (12.151) presented in Figure 12.6 applied to zero mathematical expectation and the true value of the variance σ2. The variance estimate is defined by (13.41). The ideal integrator h(t) = T−1 plays a role in averaging or smoothing filter. Substituting the realization y(t) = g[x(t)] into (13.41) instead of x(t) and carrying out averaging of the obtained formula with respect to x(t), the bias of variance estimate of Gaussian stochastic process can be written in the following form:

– b{Var*} σ2

1.0

0.8

0.6

0.4

0.2

FIGURE 13.5  Bias of variance estimate of Gaussian stochastic process as a function of the normalized limitation level.

where

is the normalized limitation level. At γ  1, we obtain that the bias of variance estimate tends to approach zero, which is expected. The bias of variance estimate of Gaussian stochastic process as a function of the normalized limitation level is shown in Figure 13.5.

13.4  ESTIMATE OF TIME-VARYING STOCHASTIC PROCESS VARIANCE

To define the current value of nonstationary stochastic process variance there is a need to have a set of realizations xi(t) of this process. Then the variance estimate of stochastic process at the instant t0 can be presented in the following form:

N

Var*(t0 ) = N1 ∑[xi (t0 ) − E(t0 )]2 , (13.104)

i=1

where

N is the number of realizations xi(t) of stochastic process

E(t0) is the mathematical expectation of stochastic process at the instant t0

As we can see from (13.104), the variance estimate is unbiased and the variance of the variance estimate under observation of N independent realizations can be presented in the following form:

that is, the variance estimate according to (13.104) is the consistent estimate.

Denoting MVar the maximal absolute value of the second derivative of the current variance Var(t) with respect to time, we obtain the high bound value of the variance estimate bias by absolute value

The maximum value of the second derivative of the time-varying variance Var(t) can be evaluated, as a rule, based on the analysis of specific physical problems.

To estimate the difference in the time-varying variance a square of the investigated realization of the stochastic process can be presented in the following form of two sums:

where

Var(t) is the true value of the stochastic process variance at the instant t

ζ(t) are the fluctuations of square of the stochastic process realization with respect to its mathematical expectation at the same instant t

Then the variation in the variance estimate of the investigated nonstationary stochastic process can be presented in the following form:

where

is the correlation function of the random component of the investigated squared stochastic process with respect to its variance at the instant t.

To define the approximate value of the variance of slowly varying in time variance estimate we can assume that the centralized stochastic process ζ(t) within the limits of the finite time interval [t0 − 0.5T, t0 + 0.5T] is the stationary stochastic process with the correlation function defined as