Diss / 10

Substituting (12.250) into (12.116) and taking into consideration (12.252), we obtain the normalized variance of the mathematical expectation estimate of stochastic process:

As we can see from (12.253), based on (12.255) we obtain (12.116). Computations carried out, λC2ν−1, for the first five magnitudes ν show that at λ ≤ 1.0 the formula (12.255) is approximately true with the relative error less than 0.02. Taking into consideration this statement, we observe a limitation caused by the first term in (12.253) for the indicated magnitudes λ, especially for the reason that the contribution of terms with higher order in the total result in (12.253) decreases proportionally

to the factors λ2C22ν−1 /(2ν − 1)! Thus, if the quantization step is not more than the root-mean-square deviation of the observed Gaussian stochastic process, (12.116) is approximately true for the definition of the variance of mathematical expectation estimate.

12.7  OPTIMAL ESTIMATE OF VARYING MATHEMATICAL EXPECTATION OF GAUSSIAN STOCHASTIC PROCESS

Consider the estimate of varying mathematical expectation E(t) of Gaussian stochastic process ξ(t) based on the observed realization x(t) within the limits of the interval [0, T]. At that time, we assume that the centralized stochastic process ξ0(t) = ξ(t) − E(t) is the stationary stochastic process and the time-varying mathematical expectation of stochastic process can be approximated by a linear summation in the following form:

i=1

where

αi indicates unknown factors φi(t) is the given function of time

If the number of terms in (12.256) is finite, that is, if N is finite, there will be a difference between E(t) and expansion in series. However, with increasing N, the error of approximation tends to approach zero:

In this case, we say that the series given by (12.256) approaches the average.

Based on the condition of approximation error square minimum, we conclude that the factors αi are defined from the system of linear equations:

In the case of representation E(t) in the series form (12.256), the functions φi(t) are selected in such a way to ensure fast series convergence. However, in some cases, the main factor in selection of the functions φi(t) is the simplicity of physical realization (generation) of these functions. Thus, the problem of definition of the mathematical expectation E(t) of stochastic process ξ(t) by a single realization x(t) within the limits of the interval [0, T] is reduced to estimation of the coefficients αi in the series given by (12.256). In doing so, the bias and dispersion of the mathematical expectation estimate E*(t) of the observed stochastic process caused by measurement errors of the coefficients αi are given in the following form:

i=1, j =1

where α*i is the estimate of the coefficients αi. Statistical characteristics (the bias and dispersion) of the mathematical expectation estimate of stochastic process averaged within the limits of the observation interval [0, T] take the following form:

The functional of the observed stochastic process pdf with accuracy, until the terms remain independent of the unknown mathematical expectation E(t), can be presented by analogy with (12.4) in the following form:

with respect to the estimates αi . Based on (12.268), we can obtain the estimates of coefficients

Taking into consideration (12.272) and (12.273), the determinant given by (12.271) can be presented in the form of sum of two determinants

where

The determinant Cm is obtained from the determinant Bm by changing the mth column on the ­column consisting of the terms ∑qN=1,q≠m αqciq at q ≠ m. Since the mth column of the determinant Cm consists­

of two elements that are the linear combination of elements of other columns, then Cm = 0.

Let us present the determinant Am = Bm in the form of sum of products of all elements of the mth column on their algebraic supplement Am, that is,

the estimates of the coefficients αi of series given by (12.256) are unbiased. The correlation function of estimates of the coefficients αm and αq is defined by the following form:

As we can see from (12.281), the correlation function of the estimates αm and αq takes the following form:

Based on (12.260), (12.262), and (12.290), we can note that the current and averaged variances of the varying mathematical expectation estimate E*(t) can be presented in the following form:

As we can see from (12.292), the higher the number of terms under expansion in series in (12.256) used for approximation of the mathematical expectation, the higher the variance of time-varying mathematical expectation estimate at the same conditions averaged within the limits of the observation interval. In doing so, there is a need to note that in general, the number N of series expansion terms essentially increases corresponding to the increase in the observation interval [0, T], within the limits of which the approximation is carried out.

As applied to the correlation function given by (12.13), the effective spectral density of the centralized stochastic process ξ0(t) takes the following form:

and the variance of the mathematical expectation estimate takes a form given by (12.50). The formulae for the variance of estimates of the coefficients αm given by (12.282) and the variance of the time-varying mathematical expectation E(t) given by (12.260) and (12.262) are simplified essentially if the functions φi(t) satisfy the integral Fredholm equation of the second kind:

In the considered case, the coefficients λi and the functions φi(t) are called the eigenvalues and eigenfunctions of the integral equation, respectively. Comparing (12.295) and (12.266), we can see

In theory of stochastic processes [15–17], it is proved that if the functions φi(t) were to satisfy the Equation 12.295, then these functions are the orthogonal normalized (orthonormalized) functions and the eigenvalues λi > 0. In this case, the following equation

is true for the eigenfunctions φi(t), and the correlation function R(t1, t2) can be presented by the following expansion in series

Substituting (12.295) into (12.265) and taking into consideration (12.297), we obtain

At that, the matrix of coefficients cij = λi is the diagonal matrix. The determinant A and the algebraic supplements Am of this matrix can be presented in the following form, respectively,

As we can see from (12.301), (12.302), (12.269), and (12.256), the estimates of the coefficients α*i and the time-varying mathematical expectation estimate E*(t) can be presented in the following form:

i=1

FIGURE 12.15  Optimal measurer of time-varying mathematical expectation estimate in accordance with (12.304); particular case of the flowchart in Figure 12.14.

The optimal measurer of the time-varying mathematical expectation of Gaussian stochastic process operating in accordance with (12.304) is shown in Figure 12.15. The flowchart presented in Figure 12.15 is a particular case of the measurer depicted in Figure 12.14. Based on the well-known correlation function of the observed stochastic process, in accordance with (12.295), the generator “Gene” forms the functions φi(t) that are employed to form the coefficients αi . The estimates of coefficients are multiplied by the functions φi(t) again and obtained products come in at the summator input. The expected estimate E*(t) of the time-varying mathematical expectation is formed at the summator output. Delay is required to compute the coefficients αi .

The variance of estimate of the coefficients αi and the current and averaged variances of estimates E*(t) of the time-varying mathematical expectation are transformed in accordance with (12.282), (12.260), and (12.262) in the following form:

As we can see from (12.307), with increase in the number of terms under approximation E(t) by the series given by (12.256), the averaged variance of the time-varying mathematical expectation estimate E* also increases. As N → ∞, taking into consideration (12.299), we obtain

Thus, at a sufficiently large number of the eigenfunctions in the sum given by (12.256), the averaged variance of the time-varying mathematical expectation estimate E(t) is equal to the variance of the initial stochastic process. In doing so, the estimate bias caused by the finite number of terms in series given by (12.256) tends to approach zero. However, in practice, there is a need to choose the number of terms in series given by (12.256) in such a way that a dispersion of the time-varying mathematical expectation estimate caused both by the bias and by the estimate variance would be minimal.

12.8  VARYING MATHEMATICAL EXPECTATION ESTIMATE UNDER STOCHASTIC PROCESS AVERAGING IN TIME

The problems with using optimal procedure realizations to estimate the time-varying mathematical expectations of stochastic processes are common in sufficiently complex mathematical computations. For this reason, to measure the time-varying mathematical expectations of stochastic processes in the course of practical applications, the averaging in time of the observed stochastic process is widely used. In principle, to define the current value of the mathematical expectation of nonstationary stochastic process there is a need to have a set of realizations xi(t) of the investigated stochastic process ξ(t). Then, the estimate of searched parameter of the stochastic process at the instant t0 is determined in the following form:

N

E (t0 ) = N1 ∑xi (t0 ), (12.309)

i=1

where N is the number of investigated stochastic process realizations. As we can see from (12.309), the mathematical expectation estimate is unbiased and the variance of the mathematical expectation estimate can be presented in the following form:

in the case of independent realizations xi(t), where σ2(t0) is the variance of investigated stochastic process at the instant t = t0. Thus, the estimate given by (12.309) is the consistent since N → ∞ the variance of estimate tends to approach zero. However, as a rule, a researcher does not use a sufficient number of realizations of stochastic process; thus, there is a need to carry out an estimation of the mathematical expectation based on an analysis of the limited number of realizations and, sometimes, based on a single realization.

Under definition of estimation of the time-varying mathematical expectation of stochastic process by a single realization, we meet with difficulties caused by the definition of optimal time of averaging (integration) or the time constant of smoothing filter at the earlier-given filter impulse response. In doing so, two conflicting requirements arise. On one hand, there is a need to decrease the variance of estimate caused by finite time interval of measuring; this time interval must be large. On the other hand, for better distinguishing the mathematical expectation variations in time, there is a need to choose the integration time as short as possible. Evidently, there is an optimal averaging time or the bandwidth of the smoothing filter under the given impulse response, which corresponds to minimal dispersion of the mathematical expectation estimate of stochastic process caused by the aforementioned factors.

The simplest way to define the mathematical expectation of stochastic process at the instant t = t0 is an averaging of ordinates of stochastic process realization within the limits of time interval about the given magnitude of argument t = t0. In doing so, the mathematical expectation estimate is defined as

Averaging (12.311) by realizations, we obtain the mathematical expectation of estimate at the instant t = t0: