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

钟形曲线，技术上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，动态环境中到飞机的距离和方位是通过有源雷达测量的。该测量并不精确，其值存在不确定性或误差。高斯 pdf 没有很好地描述这种不确定性，因为钟形曲线中远离零的部分（称为 pdf 的尾部）远小于雷达数据显示的值：众所周知，高斯钟形曲线具有轻且快速（指数）衰减的尾部，而雷达数据据说具有重尾部。 众所周知，依赖高斯概率密度函数可能是危险的（参考：Nessim Taleb 的《黑天鹅》）。 工程、经济学、生物学、金融运动、地震、大气湍流等领域的许多动态系统用高斯 pdf 很难描述，而用重尾 pdf 可以更好地描述。然而，当前的大多数数据处理算法都基于高斯 pdf 假设，主要是因为它可以实现易于处理的实时实现。 新提出的理论是一种范式转变，它提出了基于重尾概率密度函数（称为柯西概率密度函数）的新算法。结果是一个更准确、更可靠的自动化系统，即使对于非柯西的概率模型，在概率环境如此的情况下，它也表现出了与高斯模型相当的性能。由于可能存在极端数据，该估计器结构丰富，因此比高斯估计器的计算强度更大。从更技术的角度来看，将开发一类新的可实现的实时矢量状态估计器和随机控制器，用于具有加性重尾柯西过程和测量噪声的线性动态系统。通过开发用于分析测量更新的递归和给定测量历史的状态的非归一化条件概率密度函数 (ucpdf) 的特征函数的传播，解决了这种具有加性柯西噪声的矢量状态线性动态系统的估计方法。 通过谱变换，ucpdf 的特征函数在基于模型预测结构的随机控制器的开发中被明确使用。由于 ucpdf 特征函数的丰富分析形式，这些结果带来了显着的分析和数值复杂性，它产生的项之和在每次测量更新时都会增长。本研究的主要目标是通过使用算法基本结构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。寻求保留字符函数基本结构的近似值，并将在当前的计算硬件（例如图形处理单元）上实现。这项工作是在两国科学基金会 (BSF) 资助下与以色列理工学院的一位同事合作完成的。这项国际合作将在 NSF/ENG/ECCS-BSF 和 BSF 资助下继续进行。

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.