Estimation of the Quantization Period of Emergency Parameters with Allowance for Possible Losses
Abstract Generated abstract
This paper addresses the choice of a quantization period for centralized monitoring of emergency parameters when statistical information about the monitored process is limited. Modeling the parameter as a stationary random process with a specified Gaussian-type correlation function, the authors use excursion theory and a Poisson approximation for rare crossings of a critical level to relate the nonattainment time to the process variance. They then apply a Bayesian estimation approach with a simple loss function and an exponential prior for the variance, deriving a formula that updates the estimated quantization interval as new observations are obtained. The result is intended for implementation in monitoring systems with computing devices, allowing periodic refinement of the sampling period during operation.
Full Text
Estimation of the Quantization Period of Emergency Parameters Considering Possible Losses
Yu. M. Ageev, F. F. Idrisov
(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)
One of the tasks of centralized monitoring is the determination of the quantization period of a parameter. This problem can be solved using the Fokker—Planck—Kolmogorov equations [1, 2], or by means of the theory of excursions [3, 4]. The indicated methods presuppose the availability of sufficiently complete statistics on the parameter, which is not always possible. In addition, some technological processes are characterized by quantities such that, upon their reaching a critical level, a pre-emergency or emergency situation arises.
In the present work, the problem of determining the quantization period of parameters is solved with allowance for their specific features and for the insufficiency of the initial statistical data. When the initial statistics are insufficient, it is necessary to obtain the best estimate of the unknown statistical characteristics of the emergency parameter. However, the methods of maximum likelihood, maximum a posteriori probability, etc., do not take into account the losses due to inaccurate estimates. The Bayesian approach, however, makes it possible to construct estimates of statistical characteristics with allowance for losses from errors.
Let us assume that the emergency parameter is a stationary random process with correlation function
\[ K_x(\tau)=\sigma_x^2 \cdot e^{-\alpha^2 \tau^2}, \tag{1} \]
where
\[ \sigma_x^2 \]
is the variance of the process,
\[ \tau \]
is the correlation time.
We shall assume that the occurrence of successive excursions of the emergency parameter beyond the critical value \(a\) are independent rare events, i.e., the time between crossing instants is considerably greater than the correlation time. Then, regarding the process as centered and assuming that the number of excursions during time \(T\) obeys the Poisson law, we have, according to [4], the following expression for the probability \(P_0\) that during time \(T\) not a single excursion will occur:
\[ P_0=\exp\left\{-\frac{T}{2\pi}\sqrt{-\frac{\ddot K_x(\tau)}{K_x(\tau)}}\bigg|_{\tau=0}\cdot \exp\left(-\frac{a^2}{2\sigma_x^2}\right)\right\}, \tag{2} \]
where \(\ddot K_x(\tau)\) is the second derivative of the correlation function.
Having specified the probability \(P_0\) and taking (1) into account, we obtain the expression for the nonattainment time:
\[ T=-\frac{\sqrt{2}}{\alpha}\,\tau \ln P_0 \exp\left(\frac{a^2}{2\sigma_x^2}\right). \tag{3} \]
It is obvious that the error in finding \(T\) is determined mainly by the error in estimating \(\sigma_x^2\). Taking into account the nature of the consequences of errors in estimates of statistical characteristics for emergency parameters, we shall adopt a simple loss function
\[ \Pi(\hat{\sigma}_x^2,\sigma_x^2)=c-\delta(\hat{\sigma}_x^2-\sigma_x^2), \tag{4} \]
where \(c\) is some constant,
\(\delta(\cdot)\) is the Dirac delta function.
Taking into account the asymptotic independence of the posterior distribution from the prior distribution [5], let us specify the prior distribution of the variance according to the exponential law with parameter \(\sigma_{x0}^2\) [6]:
\[ W(\sigma_x^2)=\frac{1}{\sigma_{x0}^2}\exp\left(-\frac{\sigma_x^2}{\sigma_{x0}^2}\right). \tag{5} \]
To find the estimate corresponding to the simple loss function, the likelihood equation is written in the form
\[ \frac{\partial}{\partial \sigma_x^2}\ln W(\sigma_x^2)+ \sum_{i=1}^{n}\frac{\partial}{\partial \sigma_x^2}\ln W(\vec{x}_i,\sigma_{x0}^2)=0, \tag{6} \]
where \(\ln W(\sigma_x^2)\) is the likelihood function.
Solving equation (6) and substituting the result into (2), we obtain
\[ T \approx -\frac{\sqrt{2}}{\alpha}\,\tau \ln P_0 \exp\left\{ -\frac{a^2}{ n\sigma_{x0}^2 \left[ \sqrt{ 1+\frac{1}{n^2}\sum_{i=1}^{n}\left(\frac{x_i}{\sigma_{x0}}\right)^2 } -1 \right] } \right\}. \tag{7} \]
Expression (7) makes it possible to correct the estimate \(\hat{T}\) as information about the parameter is received. If the monitoring system contains a computing device, then formula (7) can be programmed, and the value \(\hat{T}\) will be periodically refined during the operation of the system.
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