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.
Publication:
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.
Publications:
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.
Publications:
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 .