Variable sampling rate filtering for nonlinear time series

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.

The proposed research is highly multi-disciplinary and will enable end users in a variety of fields including tracking, geophysical modelling (in weather pattern prediction), audio signal processing and financial modelling (in estimation of volatility from the observed prices) to improve the estimation of unobserved variables from non-uniformly sampled measurements. Some end users from financial services are already involved in this research at the problem definition stage, since one of the main motivations behind PI's interest in variable sampling rate filtering for nonlinear systems came from a conversation with PhD alumni working in high frequency financial trading. The outcomes of the proposed research can lead to faster and more accurate estimates of unobserved financial variables, which means correpondingly faster and more consistent pricing of complex financial instruments. Despite recent setbacks, financial services contributes around 20% of the GDP for the United Kingdom and remains one of the most important industry sectors for the country. A speedy dissemination of research results to end users in financial modelling will lead to a possible early adoption and will eventually have a beneficial impact on the country's economy.

To reach out to a wide spectrum of end users within financial mathematics, we will present the results in an industry open day, which is organized annually by the Department for engaging with industry practitioners. We also aim to publish the numerical results of an application of our algorithm in financial modelling in a practitioner-oriented journal. The software developed during the proposed collaborative research will be made freely available on the web. Further, PI aims to organize a workshop on the mathematics of filtering and its applications for both industrial and academic researchers in 2015 (in conjunction with the industry open day mentioned above), which will serve as a forum to disseminate the results of this research to end users in financial mathematics as well as in other disciplines (such as tracking and geophysical modelling) which employ filtering. PI had organized a similar workshop in 2011 and is confident of engaging end users in these diverse communities through his academic as well as industry contacts for this planned workshop.

Finally, the proposal also provides input to an externally sponsored doctoral student at Brunel who is working under PI's supervision. The student is working on calibration of financial time series models, with a clear focus on employment in the financial industry after completion of doctoral studies. This research will further enable the student to make useful contribution to the industry in the future.