Variable sampling rate filtering for nonlinear time series

从不完善的测量中估计隐藏或潜在变量值的问题出现在许多科学分支中，包括工程、地球物理和金融建模。一个典型的例子是利用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。定期接收新的测量数据，用于更新位置和速度信息。从观测数据估计未观测变量的问题，与从观测到的大气数据估计当前和未来天气模式的问题，或从观测到的金融工具价格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，在这些分支中，人们必须处理描述涉及随机性的系统动力学的方程，以及从观察到的变量值推断未观察到的变量值的问题。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，并且系统的动力学是线性的，则称为卡尔曼滤波器的递归贝叶斯推理过程在学术文献和各个领域的实践中都得到了很好的建立。然而，对于许多实际相关的时间序列模型，系统动力学的线性假设是不成立的。即使是简单的滤波问题，例如从极坐标系中的噪声测量动态推断运动物体在笛卡尔坐标系中的位置，也具有严重的非线性。此外，在实践中，测量可能不会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物体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进行交易的金融资产的波动性进行建模，以非均匀率进行抽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估计非常差。线性化还要求系统动力学梯度的存在和评价，这是一个主要的约束条件。在过去的几年中，文献中出现了几种不依赖于非线性系统动力学线性化的非线性滤波启发式方法。本研究计划的目的是将这些新的滤波方法中的一些，这些方法最初是为均匀采样率而开发的，以适应可变采样率的情况。寻求支持首席研究员Date博士对印度巴特那印度理工学院进行为期一个月的访问，与Bhaumik博士进行合作研究。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）对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用于可变采样率下非线性滤波的通用滤波算法，并在模拟和实际实验数据上进行测试。研究合作将在访问结束后继续进行，并将包括双方学者的外部资助博士生。

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.