稀少取樣下之組合式訊號測不準表示法研究與其在訊號平均值估計之應用

Abstract

在許多訊號處理應用中，量測訊號時往往無法單獨得出某種訊號成分，所以對於組合式訊號(combined signals)之量測(measurement)與表示有其基本必要性。現今對於組合式訊號之量測以及其相對不確定性(uncertainty)之表示及分析方法上，僅對於特定條件下之輸出訊號可進行詮釋。在最新JCGM 101 (The Joint Committee for Guides in Metrology) 2008年公開文獻中對於上述組合式訊號之量測與表示，仍然承襲過往GUM (Guide to the Expression of Uncertainty in Measurement)之範疇，以衍生分布(propagation of distribution)之模型描述組合訊號，量測結果則以JCGM 101所建議之表示法為標準，此表示法之成員有：平均值(mean)、標準誤(standard uncertainty)、母體涵蓋率之相對應覆蓋區間(coverage interval)、以及此覆蓋區間之端點(endpoints)位置。對於其中屬於標準誤之部分，JCGM 以 law of uncertainty of propagation (LUP)之概念處理組合式訊號輸出之聯合標準誤(associated standard uncertainty)，但是對於輸出型態之不確定性可能影響平均值和覆蓋區間之估計卻未提出較佳之克服方法。故本研究之主要範圍在於界定稀少取樣資料下之組合式訊號以最小估計誤差前提下之測不準現象最佳表示模型。 JCGM 所遺留下的基本問題在於組合訊號之平均值使用算術平均數(sample mean)計算，覆蓋區間則只能針對近似對稱之分布進行計算。有鑒於此，本研究針對JCGM於組合式訊號之量測問題所留下之難題提出可行的解決方式，並且以Monte Carlo method進行驗證提出以下幾種量測表示之優化解決方法：(1)首先確認組合式訊號之輸出型態為一近似常態分布之窗型機率密度函數型態 (quasi-normal signals with asymptotic window-shape distribution, QSAW)；(2)提出適合所有分布型態之覆蓋區間之pdf表示式，以pdf解釋偏態母體中所定義之the probably shortest CI在asymptotically symmetry pdf下就是the shortest CI；(3)將覆蓋區間之意義延伸至統計覆蓋區間(statistical coverage interval)，並且以 truncated normal 機率密度函數為基礎之聯合機率密度函數(variably truncated normal joint probability density function)模擬統計覆蓋區間，並進而估計組合訊號之平均值；(4)在以quantile為基礎之前提下，提出非線性quantile estimation之方法，藉以改良對QSAW組合式訊號之平均值估計；(5)運用使用於強健式統計法的“the asymptotic minimax principle”來改進對QSAW訊號之平均值估計；(6)使用the quantile mapping invariance (QMI) principle來增進quantile-based平均值估計器之效能，並將其應用至由取樣訊號所估計之相關矩陣訊號求取eigenvalue上限之問題。 實驗證明本研究所提出之嶄新數學架構模型可以完美補強JCGM在組合訊號中之描述不足部分。

In many signal processing applications, to measure and represent combined signals is a necessary and essential work because it is generally difficult to obtain individual components of a combined signal. So far, there are only few attempts on analyzing the measurement and/or representing the uncertainty of some special combined signals. JCGM (the Joint Committee for Guides in Metrology) coordinated the publication of measurement standard since 1995 and followed the GUM’s (Guide to the Expression of Uncertainty in Measurement) suggestion to publish a standard, JCGM 101, to outline the representation of combined signals by an additive model which models a combined signal as the result of the propagation of different input source signals. The suggested format of JCGM 101 includes the following four items: mean, standard uncertainty, coverage interval (CI) and its two endpoints. The JCGM standard uses the law of uncertainty of propagation to evaluate the associated standard uncertainty of a combined signal. But it does not provide the way to explore the effects of the output uncertainty on mean and coverage interval estimations. This motivates us in this study to exploit the optimal representation of the uncertainty of combined signals based on the minimal estimation error criterion under small sample size condition. One basic problem of the JCGM standard is the use of sample mean to estimate the mean of a combined signal. It therefore neglects the uncertainty resulted from the rough mean estimation when the sample size is small. Another problem is that it evaluates the coverage interval based on the assumption of asymptotically symmetric distribution. This study proposes several approaches to attacking these problems and examines them by the Monte Carlo simulations. Items studied include: (1) We verify that the output of a combined signal distributes like a quasi-normal signal with asymptotic window-shape distribution (QSAW). (2) We derive a unified probability density function (pdf) for CI to eliminate the need of skewness recognition before the evaluation of CI. (3) We extend the CI representation to the statistical CI representation and form the variably truncated normal joint probability density function. A robust quantile-based mean estimator is accordingly proposed. (4) We try a nonlinear modification of the proposed quantile-based mean estimator and verify its robustness with specially focusing on the case when the pdf of the combined signal approximates a rectangular pdf. (5) We follow the robust statistical method using “the asymptotic minimax principle” to refine the sample mean. (6) We employ the quantile mapping invariance (QMI) principle to improve the efficiency of the quantile-based mean estimator and apply it to the task of finding the upper bound of eigenvalues from the correlation matrix calculated from sparse observed samples. We believe that the proposed unified representation of CI and its application to the quantile-based mean estimation are very promising and can contribute to extend the usage of the JCGM standard.