應用非線性轉換於雙變量橢圓對稱分布資料之散布矩陣估計

Abstract

在真實世界中，信號會受到脈衝式雜訊和離群值的影響，導致其分布呈現重尾 (heavy-tailed) 分布，使得極端值和常態分布比起具有較高的出現機率。重尾分布可以使用橢圓對稱分布 (elliptically symmetric distribution) 來模擬。橢圓對稱分布為常態分布的推廣版本，能涵蓋尾部機率較大或較小的情形。

在統計訊號處理中，如果資料屬於重尾分布，此時估計散布矩陣 (scatter matrix) 比估計傳統的共變異數矩陣 (covariance matrix) 更為適合。散布矩陣可視為共變異數矩陣的推廣版本，兩者之間只有一個倍數的差異。在資料為重尾分布時，散布矩陣比起共變異數矩陣能提供更穩建且正確的資訊，尤其是在如柯西分布 (Cauchy distribution) 這類無法定義共變異數矩陣的分布下。

現有的文獻中已提出許多估計散布矩陣的方法，如 M 估計量 (M-estimator) 及泰勒估計量 (Tyler’s estimator)。然而，M 估計量通常無法以封閉型式 (closed-form)表示，因此在實作上較為困難。為了解決此問題，這篇論文基於前人的研究，整合出一個估計散布矩陣的統計架構。在此架構中，我們先將資料通過一個非線性函數，接著再基於轉換後的資料計算共變異數矩陣。基於估計的共變異數矩陣，我們提出從此估計值推算散布矩陣的方法。此架構相對於 M 估計量在計算複雜度上有顯著的下降。

在本論文中，我們選擇了兩種不同的非線性函數來設計估計量。第一種函數為空間符號函數 (spatial sign function) 。此類函數會將空間上的點投影到單位球面。第二種函數為單位量化函數 (one-bit quantization function)。此種函數會將輸入值量化成 1 或 -1。這兩種函數能有效的壓抑離群值。透過上述兩種函數，我們成功設計出兩種能夠估計重尾分布資料的散布矩陣估計量。這些估計量能以封閉形式表示。模擬實驗顯示我們所提出的估計量在運算時間上優於 M 估計量，並且使用單位量化函數的估計量在估計散布矩陣時，其均方誤差（mean-squared error）優於泰勒估計量。此外，在固定取樣位元率的情況下，我們的估計器在估計散布矩陣時, 其均方誤差優於基於柯西分布的最大概似估計（maximum likelihoodestimation）方法。

Real-world measurements may be affected by impulsive noise or outliers, resulting in heavy-tailed distributions where extreme values have larger densities than normal distributions. A heavy-tailed distribution is usually modeled by an elliptically symmetric (ES) distribution, a generalized version of the normal distribution with heavier or lighter tail characteristics.

In statistical signal processing for heavy-tailed data, we estimate the scatter matrix rather than the covariance matrix. The scatter matrix is a generalized version of the covariance matrix, differing only in scaling. For heavy-tailed data, the scatter matrix provides more robust and accurate information than the covariance matrix, especially when the covariance matrix is undefined, such as the Cauchy distribution.

In the literature, researchers have proposed several estimation methods for the scatter matrix, such as the M or Tyler's estimator. However, the M-type estimators cannot be expressed in closed-form equations, increasing the difficulty of implementation. To tackle this challenge, we summarize existing estimation methods into an estimation scheme in this thesis. In the scheme, we first adjust the data with a nonlinear function and then calculate the sample covariance matrix (SCM) based on the adjusted data. From the estimated SCM, we propose a method for recovering the scatter matrix. The scheme reduces the computation load compared to the M-type estimators.

In this thesis, we select two nonlinear functions for estimator design. One is the spatial sign function, which projects the point onto the unit sphere. Another one is the one-bit quantization function, which quantizes the input signal into 1 or -1. These two functions are efficient in suppressing the outliers. With the two functions, we successfully design two estimators that can estimate the scatter matrix for some heavy-tailed distributed data. Those estimators are expressed in a closed-form expression. In the numerical results, we find that the computation time of our proposed estimators is faster than the M-type estimators. The estimator with the one-bit quantization function has a lower mean-squared error than Tyler's estimators in the estimation of the scatter matrix. Moreover, we find that the estimator with the one-bit quantization function performs better than the maximum likelihood estimation (MLE) method of Cauchy distribution in the estimation of the scatter matrix under a fixed bit rate.