自我類化訊流的合成及分散式辨識

Abstract

在本論文的第一部分，我們提出了一個濾波器型式的自我類化訊流合成器。此合成器能生成可調控變異性及相關性的長程相依訊流，同時也只需很少的輸入參數。與既有的其它自我類化訊流合成器﹙如RMD方法和Paxson的IFFT方法﹚相比，我們提出的濾波器型式的自我類化訊流合成器具有能即時生成訊流以及生成之訊流恆不為負值之優點。我們接著研究了相關係數﹙只能反映線性的相依關係﹚和交互訊息﹙能測量一般的相依關係﹚兩者之間的蘊含關係。本研究的結果建議，對於弱相關的隨機變數，如一個自我類化過程中具有長的時間差的不同二個時刻的值，相關係數的平方值的一半似可作為交互訊息的一個合理的近似。

基於長尾分佈和自我類化訊流之間的存在的基本關係，我們進而研究了此類訊流的分散式辨識問題。我們發現若干有趣的結果。首先，我們確證了全同感測器系統在指數分佈族的參數辨識問題上的最佳性。在此研究方向上的一個相關的結果是，在指數分佈族辨識問題上，串接式兩感測器系統和平行式兩感測器系統具有相同的最佳性能。這多少是令人感到意外的，因為一般認為串接式兩感測器系統比平行式兩感測器系統有更好的性能。

其次，對於更一般類別分佈族的參數辨識問題，我們提出數個命題可用來檢證全同感測器系統的最佳性。如採取直捷的手法來檢證全同感測器系統，通常將導致在被一組非線性聯立方程所定義的解空間之中搜尋所有的局域最小值。然而這種方法在一些情況下會是不可行的，而我們提出的命題可作為一個較佳的替代方案。此外，我們的研究也可應用到其它的分散式檢測問題上，如在倖存分析及損壞時間分析的壽命問題上，或是如在利用在地理上分散設備，對不同連結上的封包到達時間間隔加以量測，來決定整個網路的自我類化參數的問題上。

最後，藉由對函數及方程式的數值計算結果，我們確認了，在加法性高斯雜訊下二元信號的檢測問題上全同感測器系統的最佳性。

In the first part of this dissertation, we propose a filter-based generator for the synthesization of self-similar traffics. It can produce long range dependent traffics with adjustable levels of bustiness and correlation, and is parsimonious in the number of model parameters. By comparing it with existing self-similar traffic synthesizers, e.g., the RMD and the Paxson IFFT algorithms, the proposed filter-based synthesizer has the advantages that the synthetic traffic can be generated on the fly, and always produces non-negative-valued traffic. The implications between the correlation coefficient (a quantity that only measures the linear dependence) and mutual information (a quantity that can represent the general dependence) is subsequently investigated. The obtained results suggest that for weakly correlated random variables such as two instances of a self-similar process with a long time lag, half the square of the correlation coefficients might be a reasonable approximation to the mutual information.

Continuing from the synthesization of processes with heavy tails, we turn to study the impact of such processes on decentralized detection. Several interesting results are found. Firstly, the optimality of identical sensor system for the exponential distribution family has been verified. A side result along this research line is that the optimal performance of the serial two-sensor system is the same as that of the parallel two-sensor system for exponential sources. This is somewhat surprising because it is generally considered that the serial two-sensor system has better performance than the parallel two-sensor system.

Secondly, for a more general class of distribution families, we propose several propositions on the optimality of the identical system. A straightforward approach to test the optimality of identical sensor system often results in searching all local minimums in the solution space that is defined through a set of nonlinear equations. However, this approach is not tractable in certain situations. Our propositions then provide an alternative for optimality test of identical sensor system. Besides, they can be applied to other decentralized detection problems like the detection of lifetime encountered in survival analysis and failure time analysis or the determination of the degree of self-similarity of the whole network system based on geographically dispersed measurements of the packet inter-arrival times on different links.

Finally, with the help of numerical study on functions and equations, we analytically confirm the optimality of identical sensor system over Gaussian sources.