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），一直是工程和金融算法的核心元素，处理数据并自动化所需的操作。 例如，在空中交通管制中，在动态环境中与飞机的距离和方位由主动雷达测量。这种测量并不精确，其值存在不确定性或误差。这种不确定性不能很好地用高斯概率密度函数来描述，因为钟形曲线远离零的部分，称为概率密度函数的尾部，远小于雷达数据所显示的：高斯钟形曲线被认为有一个轻的，快速（指数）衰减的尾部，而雷达数据被认为有一个沉重的尾部。 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）。 工程学、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，高斯分布的概率密度函数描述得不好，而重尾分布的概率密度函数描述得更好。然而，大多数当前的数据处理算法是基于高斯pdf假设，主要是因为它导致易于处理，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（称为Cauchy PDF）的新算法。其结果是一个更准确和可靠的自动化系统，即使对于概率模型，不是柯西已经证明了可比的性能高斯当概率环境是这样的。由于极端的数据是可能的，估计器是丰富的结构，因此是计算上更激烈的比其Gaussian counterpart.From一个更技术的观点，一类新的可实现的实时向量状态估计器和随机控制器的线性动态系统添加剂重尾柯西过程和测量噪声。通过建立一个解析测量更新和状态的非归一化条件概率密度函数（ucpdf）特征函数传播的递推公式，给出了具有加性柯西噪声的向量状态线性动态系统的估计方法. 通过谱变换，明确使用的特征函数的ucpdf的随机控制器的发展，基于模型预测结构。这些结果需要显着的分析和数值计算的复杂性，由于丰富的解析形式的特征函数的ucpdf，它产生了一个总和的条款，增长在每个测量更新。拟议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。寻求将保存字符函数的基本结构的近似，并且将在当前的计算硬件（诸如图形处理单元）上实现。这项工作是在两国科学基金会（BSF）资助下与Technion的一位同事一起完成的。这种国际合作将在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.