On localization of source by hidden Gaussian processes with small noise
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On localization of source by hidden Gaussian processes with small noise Yury A. Kutoyants1,2 Received: 11 June 2020 © The Institute of Statistical Mathematics, Tokyo 2020
Abstract We consider the problem of identification of the position of some source by observations of K detectors receiving signals from this source. The time of arriving of the signal to the k-th detector depends of the distance between this detector and the source. The signals are observed in the presence of small Gaussian noise. The properties of the MLE and Bayesian estimators are studied in the asymptotic of small noise. Keywords Partially observed linear system · Parameter estimation · Hidden process · Small noise · MLE · BE
1 Introduction ( )⊤ Consider the problem of estimation of the position 𝜗0 = x0 , y0 of the source 𝕊0 by the observations of the signals from this source received by K detectors 𝔻1 , … , 𝔻K (see Fig. 1). ( )⊤ If we denote 𝜗k = xk , yk ∈ R2 the position of 𝔻k and suppose that the source starts emission ( ) at the moment t = 0 , then the signal arrives at this detector at the ‖ moment 𝜏k 𝜗0 = 𝜈 −1 ‖ 𝜈 > 0 is the rate of propagation of the signals ‖𝜗k − 𝜗0 ‖ . Here and ‖⋅‖ is Euclidean distance in R2 . The set 𝛩 ⊂ R2 is supposed to be open, convex and bounded. ( ) The k-th detector receives the signal Yk = Yk (t), 0 ≤ t ≤ T from the source 𝕊 and additive Gaussian noise according to equation ( ( )) dXk (t) = ak (t)𝜓̄ t − 𝜏k 𝜗0 Yk (t)dt + 𝜀𝜎k (t)dWk (t), Xk (0) = 0. (1)
* Yury A. Kutoyants kutoyants@univ‑lemans.fr 1
Le Mans University, Av. O. Messiaen, 72000 Le Mans, France
2
Tomsk State University, Tomsk, Russia
13
Vol.:(0123456789)
Y. A. Kutoyants Fig. 1 Model of observations. 𝕊0 is position of the source and 𝔻k , k = 1, … , 5 are positions of the sensors
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S0 D3 D4
Here ak (⋅), 𝜎k (⋅) and 𝜓(⋅) ̄ are known functions and Wk (⋅), k = 1, … , K are independent Wiener processes. The parameter 𝜀 > 0 controls the level of noise. In this work we study the properties of estimators of 𝜗0 in the asymptotic of small noise, i.e., as 𝜀 → 0 . This is equivalent to the situation with large signal, which is rather reasonable in many real situations. This problem is quite close to the inverse problem, where we have K sources 𝔻1 , … , 𝔻K of signals with known positions and known moments of emission and one detector 𝕊0 . The detector receives K signals X K and has to estimate its own position. This is typical situation in the global positioning system (GPS/INS). The algorithms calculating the positions of different objects (cars, jets, ships et cet.) used in GPS/ISN are based essentially on the adaptive Kalman filtering theory, see, e.g., Almagbile et al. (2010), Gustaffson (2000), Hutchinson (1984), Luo (2013), Wang et al. (2006) and references therein. The same time it seems that the mathematical theory of statistical estimation of the position was not yet sufficiently well developed. This work is continuation of the study initiated in the papers Chernoyarov and Kutoyants (2020), Chernoyarov et al. (2020), Farinetto
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