**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 type of data
often comes in extremely large volumes and is a modern type of BIG data. *

*Studying
dependent processes is important 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,
mainly due to large volumes of longer and more heterogeneous processes being
collected. Estimating properties of dependent data is hard because we always
have to average across a number of samples to produce stable estimates. *

*Estimating
nonstationary time series for example, namely processes
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
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.*

**BIG
Data and Inference of Network data**

Networks and relational data are ubiquitous. Their analysis
poses a number of hard foundational statistical questions,
that have an immediate effect in applications. I have been studying basic
network models, such as the stochastic block model, as well as degree based
models, and have been applying these concepts in bioinformatics and ecology.

*Publications:*

Olhede, S. C.
and Wolfe, P. J., Degree-based network models. http://arxiv.org/abs/1211.6537 .

Olhede, S. C.
and Wolfe, P. J., Order Statistics of Observed Network Degrees, http://arxiv.org/abs/1210.4377.

Wolfe, P. J. and Olhede, S. C.,
Nonparametric graphon estimation, http://arxiv.org/abs/1309.5936

**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:*

Sykulski, A. M., Olhede S. C., Lilly, J. M. and Early, J. J.,
The Whittle Likelihood for Complex-Valued Time Series,** http://arxiv.org/abs/1306.5993 .**

Olhede, S. C. and Ombao, H. Modelling and
Estimation of Covariance of Replicated Modulated Cyclical Time Series, http://arxiv.org/abs/1206.1955 , *IEEE Transactions on Signal Processing*, 61,
1944-1957.

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:*

Olhede S and Metikas
G. The
Monogenic Wavelet Transform, Appeared in *IEEE Transactions on Signal
Processing*, 57, Pages: 3426-3441 .

Simons, F. J. and Olhede, S. C.
Maximum-likelihood estimation of lithospheric flexural rigidity,
initial-loading fraction, and load correlation, under isotropy. http://arxiv.org/abs/1205.0773,
*Geophysical Journal International*, 193, 1300–1342.

Olhede, S. C., Ramirez, D. and Schreier, P. J. Measuring Directionality in Random Fields Using
the Monogenic Signal, http://arxiv.org/abs/1304.2998 .