Sofia Olhede – Research Interests
I use tools from applied harmonic analysis to describe the characteristics of dependent and structured processes observed in time or space. This is an important area because of the number of scientifically interesting application fields were such methods can be used, but recent developments in applications have also produced completely new challenges, due to large volumes of longer and more heterogeneous processes. Estimating properties of dependent data is hard because we always have to average across a number of samples to produce stable estimates. Estimating nonstationary processes, namely those who evolve in structure over the period of observation, is even harder because unless we are very careful with how we average, the methods become strongly biased, and not useable. To understand the mathematical properties of the methods a consistent stochastic model must be posited for the full set of observations. To be useful the model has to be good enough, so that if we use it to generate new data, we can fool ourselves into thinking the data was “real”. Subsequently the statistical properties of the methods must be understood under suitable simplifying sampling scenarios. Often these are under vanishing parameter or increasing sample size regimes, where the latter must be chosen carefully for the regime to be suitable for the application, and for the mathematics to “work”.
The big current challenge in modelling nonstationary processes is to propose classes of flexible models that incorporate properties of real observed processes that have to be understood jointly and derive the properties of estimation methods theoretically. I work in this field. My challenge is to break away from a very traditional view of what a nonstationary process is, namely like a “normal” stationary process but evolving very slowly in time. This strongly constrains what sets of processes can be treated simultaneously, or at all. I work on introducing new classes of models which do not fit into this framework, and develop inference procedures for them, using assumptions of realistic sampling. The new models I develop have mainly been motivated by applications in oceanography and neuroscience, and contain important characteristics that previously could not be estimated; such as complex multivariate structure and inhomogeneous data characteristics.
Inference for Noisy Diffusions
Stochastic Differential Equations are used to model processes in Finance, Oceanography and a variety of other areas. Modern data acquisition protocols enable the high frequency sampling of such processes. Unfortunately most sampling procedures are not noise-free and so any analysis method must cope with additional noise structure. This can be a very hard problem to tackle in the time domain, as information across several different scales must be combined to get a good appreciation of the noise level. In the Fourier domain, the problem simplifies considerably and leads to simple and effective inference procedures.
Olhede S. C., Sykulski A. and Pavliotis G. Multiscale Inference for High-Frequency Data, Department of Statistical Science Reseach Report 290. Version available at http://arxiv.org/abs/0803.0392 To appear in SIAM MMS.
Analysis of Nonstationary Multiple Time series
Temporal phenomena are rarely observed in isolation, and often we wish to infer the joint properties of multiple phenomena. This is a very difficult problem, and standard notions of defining nonstationary multiple processes are less than perfectly elegant, requiring very strong assumptions on joint structure or smoothness. I have been developing flexible analysis methods for pairs of nonstationary signals observed in physical oceanography.
Lilly J. and Olhede S. C. Higher Order Properties of Analytic Wavelet IEEE Transactions on Signal Processing , 57, Page(s):146 – 160.
Lilly J. and Olhede S. C. Bivariate Instantaneous Frequency and Bandwidth, To appear in IEEE Transactions on Signal Processing, http://arxiv.org/abs/0902.4111 .
Lilly J. and Olhede S. C. Analysis of Modulated Multivariate Oscillations. http://arxiv.org/abs/1104.2060 , 60, 600-612,IEEE Transactions on Signal Processing.
Analysis of phenomena in 2-D and 3-D
Spatial phenomena are distinct from temporal processes. For temporal processes, time has not just structure but also directionality: most phenomena should not operate identically under time-reversal. Spatial processes usually do not exhibit any natural ‘ordering’, and so are fundamentally different from temporal phenomena. Spatial phenomena are often local, and highly heterogeneous. I have been developing methods of characterising local heterogeneous spatial structure using triplets of wavelet functions.
Metikas, G. and Olhede, S. C. Multiple Multidimensional Morse Wavelets. IEEE Transactions on Signal Processing, 55, 921—936.
Olhede, S. C. Hyperanalytic Denoising. IEEE Transactions on Image Processing, 16, 1522—1537.
Olhede, S. C. Localisation of Geometric Anisotropy. IEEE Transactions on Signal Processing, 56, 2133—2138.
Olhede S and Metikas G. The Monogenic Wavelet Transform, Appeared in IEEE Transactions on Signal Processing, 57, Pages: 3426-3441 .