Variable sampling rate filtering for nonlinear time series

从不完美的测量中估计隐藏或潜在变量的值的问题出现在许多科学分支中，包括工程，物理学和金融建模。一个典型的例子是使用来自雷达或光学传感器的不完美测量来推断移动物体的位置和速度。测量值通常是传感器和物体之间的距离以及相对于参考方向的角度。周期性地接收用于更新位置和速度信息的新测量。这个从观测数据中估计未观测变量的问题与从观测的大气数据中估计当前和未来天气模式的问题，或从观测的金融工具价格中预测市场隐含波动率的问题没有什么不同。事实上，类似的“状态估计”问题出现在物理科学的许多其他分支中，其中一个必须处理描述系统动态的方程，包括随机性和从观测变量的值推断未观测变量的值的问题。通过将模型预测与观测相结合来推断未观测变量的递归过程称为“过滤器”。如果测量发生在均匀的时间间隔和系统的动态是线性的，递归贝叶斯推理过程，称为卡尔曼滤波器，是很好地建立在学术文献和实践中的各个领域。然而，关于系统动力学的线性假设对于许多实际相关的时间序列模型是无效的。即使是简单的滤波问题，例如从极坐标中的噪声测量动态推断笛卡尔坐标中的移动物体的位置，也具有严重的非线性。此外，测量在实践中可能不以统一的时间间隔发生，例如，来自光学传感器和来自卫星的移动对象的位置的测量更新可能以不同的速率到达。在某些应用中，例如对以不规则时间间隔交易的金融资产的波动性进行建模，以非统一速率进行采样测量实际上可能是有利的。目前，变采样率下的非线性时间序列滤波是通过时间序列动力学的局部线性化过程来处理的。根据非线性的严重程度，线性化过程可能导致对未观测变量的非常差的估计。线性化还要求系统动力学梯度的存在和估计，这是一个主要的约束。在过去的几年里，在文献中出现了几个非线性滤波算法，它们不依赖于非线性系统的动力学线性化。本研究建议的目的是适应这些新的滤波方法，最初开发的均匀采样率，可变采样率的情况下。寻求支持的主要研究者（PI）博士日期一个月的访问印度理工学院巴特那，印度，与博士Bhaumik合作研究。PI和Bhaumik博士都是经验丰富的学者，他们在不同的应用领域（PI在金融建模中，Bhaumik博士在跟踪中）为不依赖显式线性化的非线性滤波方法的发展做出了贡献。两位研究人员将共同开发通用滤波算法，用于可变采样率下的非线性滤波，并在模拟和真实的实验数据上进行测试。研究合作将在访问结束后继续进行，并将包括两个学者的外部赞助的博士生。

The problem of estimating the values of hidden or latent variables from imperfect measurements arises in many branches of science, including engineering, geophysics and financial modelling. A typical example would be inferring the position and the velocity of a moving object using imperfect measurements from a radar or an optical sensor. The measurements are typically the distance between the sensor and the object and the angle with respect to a reference direction. New measurements are received periodically which are used to update the location and the velocity information. This problem of estimation of unobserved variables from the observed data is not unlike the problem of estimating current and future weather patterns from observed atmospheric data, or predicting the market implied volatility from the observed prices of financial instruments. In fact, similar 'state estimation' problems arise in many other branches of physical science, where one has to deal with equations describing the dynamics of a system involving randomness and a problem of inferring the values of unobserved variables from the values of the observed ones. A recursive procedure to infer unobserved variables by combining model predictions with observations is called a 'filter'. If the measurements occur at uniform time intervals and the dynamics of the system is linear, a recursive Bayesian inference procedure, called the Kalman filter, is well established in the academic literature and in practice in various fields. However, the linearity assumption about the system dynamics is invalid for many practically relevant time series models. Even simple filtering problems, such as that of dynamically inferring the position of a moving object in Cartesian coordinates from noisy measurements in polar coordinates, have severe nonlinearities. Further, the measurements may not occur at uniform time intervals in practice, e.g. measurement updates for position of a moving object from an optical sensor and from a satellite may arrive at a different rate. In some applications, such as modelling volatility of a financial asset which is traded at irregular time intervals, sampling measurements at a non-uniform rate may actually be advantageous. Currently, nonlinear time series filtering under variable sampling rate is handled by a procedure of local linearization of time series dynamics. Depending on the severity of nonlinearity, linearization procedures can lead to very poor estimates of the unobserved variables. Linearization also requires the existence and evaluation of gradient of the system dynamics, which is a major constraint.In the last few years, several nonlinear filtering heuristics have emerged in the literature which do not rely on linearization of dynamics of the nonlinear system. The aim of this research proposal is to adapt some of these new filtering methods, which were originally developed for uniform sampling rate, to the variable sampling rate case. Support is sought for a one month visit of the principal investigator(PI) Dr Date to Indian Institute of Technology Patna, India, for collaborative research with Dr Bhaumik. Both PI and Dr Bhaumik are experienced academics who have contributed to development of nonlinear filtering methods which do not rely on explicit linearization, in different application domains (PI in financial modelling, Dr Bhaumik in tracking). The two researchers will work together to develop generic filtering algorithms for nonlinear filtering under variable sampling rate and test them on simulated as well as real experimental data. The research collaboration will continue after the end of the visit and will include externally sponsored doctoral students of both the academics.