Noise reduction in nonlinear time series analysis
Michael Davies · UCL Discovery (University College London) · 1993
Over the last decade a variety of new techniques for the treatment of chaotic time series has been developed. Initially these concentrated on the characterisation of chaotic time series but attention soon focused on the possibility of predicting the short term behaviour of such time series and we begin by reviewing the work in this area. This in turn has lead to a growing interest in more sophisticated signal processing tools based on dynamical systems theory. In this thesis we concentrate on the problem of removing low amplitude noise from an underlying deterministic signal. Recently there has been a number of algorithms proposed to tackle this problem. Originally these assumed that the dynamics were known a priori. One problem with such algorithms is that they appear to be unstable in the presence of homoclinic tangencies. We review the original work on noise reduction and show that the problem can be viewed as a root-finding problem. This allows us to construct an upper bound on the condition number for the relevant Jacobian matrix, in the presence of homoclinic tangencies. Alternatively the problem can be viewed as a minimisation task. In this case simple algorithm such as a gradient descent algorithm or a Levenberg-Marquardt algorithm can be using to efficiently reduce noise. Furthermore these do not necessarily become unstable in the presence of tangencies. The minimisation approach also allows us to compare a variety of ad hoc methods that have recently been proposed to reduce noise. Many of these can be shown to be equivalent to the gradient descent algorithm. The problem can then be extended to include the unknown parameters used to estimate the mapping function when the dynamics are unknown. Here we incorporate these into the noise reduction process and we show that this appears to improve the stability of the noise reduction algorithm. Finally we conduct a number of numerical investigations using our noise reduction algorithm. These include applications to data from the Lorenz equations and some experimental laser data.