Abstract
The starting point and focal point for this thesis was stochastic dynamical modelling
of neuronal imaging data with the declared objective of drawing inference,
within this model framework, in a large-scale (high-dimensional) data setting. Implicitly
this objective entails carrying out three separate but closely connected tasks;
i) probabilistic modelling, ii) statistical modeling and iii) implementation of an inferential
procedure. While i) - iii) are distinct tasks that range over several quite
different disciplines, they are joined by the premise that the initial objective can only
be achieved if the scale of the data is taken into consideration throughout i) - iii).
The strategy in this project was, relying on a space and time continuous stochastic
modelling approach, to obtain a stochastic functional differential equation on a
Hilbert space. By decomposing the drift operator of this SFDE such that each component
is essentially represented by a smooth function of time and space and expanding
these component functions in a tensor product basis we implicitly reduce
the number of model parameters. In addition, the component-wise tensor representation
induce a corresponding component-wise tensor structure in the resulting
statistical model. Especially, the statistical model is design matrix free and facilitates
an efficient array arithmetic. Using proximal gradient based algorithms, we combine
this computationally attractive statistical framework with non-differentiable regularization
to form computationally efficient inferential procedure with minimal memory
foot prints. As a result we are able to fit large scale image data in a mathematically
sophisticated dynamical model using a relatively modest amount of computational
resources in the process.
The contributions presented in this thesis are computational and methodological.
The computational contribution takes the form of solution algorithms aimed
at exploiting the array-tensor structure in various inferential settings. The methodological
contribution takes the form of a dynamical modelling and inferential framework
for spatio-temporal array data. This framework was developed with neuron
field models in mind but may in turn be applied to other settings conforming to the
spatio-temporal array data setup.
of neuronal imaging data with the declared objective of drawing inference,
within this model framework, in a large-scale (high-dimensional) data setting. Implicitly
this objective entails carrying out three separate but closely connected tasks;
i) probabilistic modelling, ii) statistical modeling and iii) implementation of an inferential
procedure. While i) - iii) are distinct tasks that range over several quite
different disciplines, they are joined by the premise that the initial objective can only
be achieved if the scale of the data is taken into consideration throughout i) - iii).
The strategy in this project was, relying on a space and time continuous stochastic
modelling approach, to obtain a stochastic functional differential equation on a
Hilbert space. By decomposing the drift operator of this SFDE such that each component
is essentially represented by a smooth function of time and space and expanding
these component functions in a tensor product basis we implicitly reduce
the number of model parameters. In addition, the component-wise tensor representation
induce a corresponding component-wise tensor structure in the resulting
statistical model. Especially, the statistical model is design matrix free and facilitates
an efficient array arithmetic. Using proximal gradient based algorithms, we combine
this computationally attractive statistical framework with non-differentiable regularization
to form computationally efficient inferential procedure with minimal memory
foot prints. As a result we are able to fit large scale image data in a mathematically
sophisticated dynamical model using a relatively modest amount of computational
resources in the process.
The contributions presented in this thesis are computational and methodological.
The computational contribution takes the form of solution algorithms aimed
at exploiting the array-tensor structure in various inferential settings. The methodological
contribution takes the form of a dynamical modelling and inferential framework
for spatio-temporal array data. This framework was developed with neuron
field models in mind but may in turn be applied to other settings conforming to the
spatio-temporal array data setup.
| Originalsprog | Engelsk |
|---|
| Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
|---|---|
| Antal sider | 167 |
| Status | Udgivet - 2017 |