NSF/ENG/ECCS-BSF: Vector-State Estimation and Control for Linear Systems with Additive Heavy-Tailed Distributions

钟形曲线在技术上被称为高斯概率密度函数(Pdf)，一直是处理数据和自动化所需操作的工程和金融算法的核心元素。例如，在空中交通管制中，动态环境中飞机的距离和方位是由主动雷达测量的。这种测量是不准确的，其值有不确定度或误差。高斯pdf没有很好地描述这种不确定性，因为钟形曲线远离零的部分，称为pdf的尾部，比雷达数据显示的要小得多：众所周知，高斯钟形曲线有一个轻的、迅速(指数级)衰减的尾巴，而雷达数据据说有一个厚重的尾巴。人们已经很好地认识到，依赖高斯pdf可能是危险的(参考：Nessim Taleb的《黑天鹅》)。在工程、经济、生物、金融运动、地震、大气湍流等领域，许多动态系统用高斯pdf描述得很差，而用重尾pdf描述得更好。然而，目前的大多数数据处理算法都是基于高斯pdf假设的，这主要是因为它导致了易于处理的实时实现。新提出的理论是一种范式转变，它提出了基于重尾pdf的新算法，称为Cauchy pdf。结果是一个更准确和可靠的自动化系统，即使对于不是柯西的概率模型，当概率环境是这样的时候，它也表现出了与高斯相似的性能。由于极值数据可能存在，估值器具有丰富的结构，因此计算强度高于高斯估值器。从更高的技术角度出发，针对具有加性重尾柯西过程和测量噪声的线性动态系统，需要开发一类新的可实现的实时矢量状态估值器和随机控制器。针对这类具有加性柯西噪声的矢量状态线性动态系统，提出了一种递推方法，用于给定测量历史的状态的非归一化条件概率密度函数的特征函数的解析测量更新和传播。通过谱变换，在基于模型预测结构的随机控制器的开发中显式地使用了ucpdf的特征函数。由于ucpdf的特征函数的丰富解析形式，这些结果需要显著的分析和数值复杂性，该函数在每次测量更新时产生增长的项的总和。提出的研究的主要目标是通过使用由于算法的基本结构而进行的简化来确定可实现的实时矢量状态估计器和随机控制器。寻求将保留字符功能的基本结构的近似，并将在诸如图形处理单元的当前计算硬件上实现。这项工作是与理工学院的一位同事在两国科学基金会(BSF)的资助下进行的。这一国际合作将在NSF/ENG/ECCS-BSF和BSF赠款下继续进行。

The bell shaped curve, known technically as the Gaussian probability density function (pdf), has been a central element in engineering and financial algorithms that process data and automate a desired operation. For example, in air traffic control the distance and bearing to an aircraft in a dynamic environment is measured by an active radar. This measurement is not exact, having an uncertainty or error in its value. This uncertainty is not described well by the Gaussian pdf because the portion of the bell shaped curve far from zero, called the tail of the pdf, is far smaller than the radar data suggests: the Gaussian bell shaped curve is known to have a light, rapidly (exponentially) decaying tail, while radar data is said to have a heavy tail. It has been well recognized that reliance on the Gaussian pdf can be dangerous (reference: The Black Swan by Nessim Taleb). Many dynamic systems in engineering, economics, biology, financial movements, earthquakes, atmospheric turbulence, etc., are poorly described by Gaussian pdfs and better described by heavy tailed ones. However, majority of current data processing algorithms are based on the Gaussian pdf assumption mainly because it leads to tractable, real-time implementations. The newly proposed theory is a paradigm shift, which proposes new algorithms based on a heavy tailed pdf, known as the Cauchy pdf. The result is a more accurate and reliable automated system, which even for probabilistic models that are not Cauchy has demonstrated comparable performance to the Gaussian when the probabilistic environment is such. Since extreme data is likely, the estimator is rich in structure and hence is computationally more intense than its Gaussian counterparts.From a more technical viewpoint, a new class of implementable real-time vector-state estimators and stochastic controllers for linear dynamic systems with additive heavy-tailed Cauchy process and measurement noises are to be developed. The estimation methodology for this vector-state, linear dynamic system with additive Cauchy noises was addressed by developing a recursion for the analytic measurement update and propagation of the character function of the unnormalized conditional probability density function (ucpdf) of the state given the measurement history. Through a spectral transformation, the character function of the ucpdf is used explicitly in the development of stochastic controllers, based on a model predictive structure. These results entail significant analytical and numerical complexities due to the rich analytic form of the character function of the ucpdf, which produces a sum of terms that grows at each measurement update. The primary goal of the proposed study is to determine implementable real-time vector-state estimators and stochastic controllers by using simplifications that are due to the fundamental structure of the algorithms. Approximations that will conserve the basic structure of the character function are sought, and will be implemented on current computational hardware, such as graphic processing units. This work was performed with a colleague at the Technion under a Bi-national Science Foundation (BSF) Grant. This international collaboration will continue under the NSF/ENG/ECCS-BSF and BSF grants.