Detection and estimation of abrupt changes in Gaussian random processes with unknown parameters By Sai Si Thu Min Oleg V. Chernoyarov National Research University “Moscow Power Engineering Institute”, Moscow 1

2 The random process with abrupt change: where, Ω − bandwidth − intensity of the process

3 By considering the process ‘ Fast ’, the satisfied condition is used as follow: where, The random process is observed with the additive Gaussian white noise n(t) with one-sided spectral density and we get the observable realization :

4 − output signal function of the filter

5 By using maximum likelihood method : ‘c’ is threshold defined by set optimality criterion

6 of these unknown parameters To estimate these unknown parameters: maximum likelihood estimations (B) Estimations of abrupt changes − the abrupt change in mathematical expectation of random process is realized with probability 1 within the interval [0,T]

7, The measurers

8 Fig.1 Block diagram of the (A)detector/(B)measurer of abrupt change in the mathematical expectation of low-frequency Gaussian random process T Matched filter Ramp Generator Peak detector Thres hold Extrem- ator Sampling device Sampling device Sampling device (A) Detection (B) Estimations max(0,x)

9 Theoretical and experimental values of detection characteristics l 0 = 0.05 l 0 = l 0 = 0.5 l 0 = λ 0 /T

10 2.Abrupt change in the intensity of high-frequency random process − root-mean-square deviations of the process where, − stationary centered Gaussian random process possessing the spectral density The high-frequency random process with abrupt change in it’s intensity : Ω− bandwidth, ϑ >Ω/2 − the band center of the process v(t) By considering the process ‘ Fast ’, the satisfied condition is used as follow: where,

11 (1) (3) (2) Three possible ways to detect: (C) Detection algorithm of abrupt change in dispersion of Gaussian process

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13 − the output signal of the filter with the transfer function The maximum likelihood functionals:

14 The maximum likelihood detection algorithm of abrupt change of high-frequency random process

15 (D) Estimations of abrupt changes For maximum estimations of unknown parameters are shown as follow:

Matched filter T Ramp Gene. Log. Amplifier Log. Amplifier Peak detector Thresh- old Extrem- ator Sampling device Sampling device max(0,x) 16 Fig.2 Block diagram of the (C) detector/(D) estimation of abrupt change in the intensity of high-frequency Gaussian random process (C) Detection (D) Estimations max(0,x)

17 Conclusion  To detect the change-point time and to estimate the unknown parameters of abruptly changing fast fluctuating Gaussian processes, the maximum likelihood method can be effectively used.  Application of this method allows us, while neglecting the sizes of order of the analyzed random process correlation time, to receive detection and measurement algorithms of abrupt changing low-frequency and high-frequency random processes, technically more simple essentially in comparison with the known analogues.  The given algorithms can be found with the help of a local Markov approximation method.  By means of the statistical computer modeling, the received theoretical results well coincide with corresponding experimental data in a wide range of parameter values of the observable process.  Additional researches show that synthesized detectors and measurers can be also used for the analysis of non-Gaussian abrupt changing random processes without appreciable loss in functioning.

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