A Non-Asymptotic Confidence Region with a Fixed Size for a Scalar Function Value: Applications in C-OTDR Monitoring Systems

Andrey V Timofeev

doi:10.4236/ojs.2014.48054

Abstract

In this paper we will investigate some non-asymptotic properties of the modified least squares estimates for the non-linear function f(λ*) by observations that nonlinearly depend on the parameter λ*. Non-asymptotic confidence regions with fixed sizes for the modified least squares estimate are used. The obtained confidence region is valid for a finite number of data points when the distributions of the observations are unknown. Asymptotically the suggested estimates represent usual estimates of the least squares. The paper presents the results of practical applications of the proposed method in C-OTDR monitoring systems.

Highlights

( ) In some practical cases there appears a necessity to estimate of the value of the function f λ∗ by observations that depend on the parameter λ∗
( ) totic properties of the modified least squares estimates for the non-linear function f λ∗ by observations that nonlinearly depend on the parameter λ∗
X (n) is the observed value of the signal energy in the n-th sensor at time n; ξ (n +1) —a noise component at moment (n +1), which appear due to the influence of the dispersing medium, and ξ (n) ≤ C, Eξ (n) = 0, Eξ 2 (n) = S ; H ( Rn,T, n, f ) is a scalar function that describes the absorption of the elastic vibrations, this function depends on following parameters: water temperature T; value Rn∗ = Rncosα, where Rn is the distance between the sensor number n and the sensor number (n +1) ; n-sensor number; f-frequency of the source of the hydroacoustic emissions (SHdE)

Summary

Introduction

( ) In some practical cases there appears a necessity to estimate of the value of the function f λ∗ by observations that depend on the parameter λ∗. Timofeev signals in shallow water when sensors of the C-OTDR monitoring system are used for measurements In this ( ) case, the absorption coefficient (target parameter f λ∗ ) depends nonlinearly on the water temperature (unobservable parameter λ∗ ) and on the frequency of sonar emissions. In this particular case, it is very important to get the guaranteed accuracy estimate of the absorption coefficient using only a limited number of observation steps (non-asymptotic statement of the problem). In this paper a sequential design is suggested that will make it possible to solve the problem ( ) of non-linear estimation of the function f λ∗ value for a wide class of stochastic processes by means of confidence regions in the non-asymptotic setting. The solution was obtained under condition of partial a priori definiteness as regards to the stochastic distribution of the observations

Statement of the Problem

The Main Result

Practical Example

Conclusions