TY - BOOK

T1 - Spatio-Temporal Modeling of Neuron Fields

AU - Lund, Adam

PY - 2017

Y1 - 2017

N2 - The starting point and focal point for this thesis was stochastic dynamical modellingof neuronal imaging data with the declared objective of drawing inference,within this model framework, in a large-scale (high-dimensional) data setting. Implicitlythis objective entails carrying out three separate but closely connected tasks;i) probabilistic modelling, ii) statistical modeling and iii) implementation of an inferentialprocedure. While i) - iii) are distinct tasks that range over several quitedifferent disciplines, they are joined by the premise that the initial objective can onlybe 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 stochasticmodelling approach, to obtain a stochastic functional differential equation on aHilbert space. By decomposing the drift operator of this SFDE such that each componentis essentially represented by a smooth function of time and space and expandingthese component functions in a tensor product basis we implicitly reducethe number of model parameters. In addition, the component-wise tensor representationinduce a corresponding component-wise tensor structure in the resultingstatistical model. Especially, the statistical model is design matrix free and facilitatesan efficient array arithmetic. Using proximal gradient based algorithms, we combinethis computationally attractive statistical framework with non-differentiable regularizationto form computationally efficient inferential procedure with minimal memoryfoot prints. As a result we are able to fit large scale image data in a mathematicallysophisticated dynamical model using a relatively modest amount of computationalresources in the process.The contributions presented in this thesis are computational and methodological.The computational contribution takes the form of solution algorithms aimedat exploiting the array-tensor structure in various inferential settings. The methodologicalcontribution takes the form of a dynamical modelling and inferential frameworkfor spatio-temporal array data. This framework was developed with neuronfield models in mind but may in turn be applied to other settings conforming to thespatio-temporal array data setup.

AB - The starting point and focal point for this thesis was stochastic dynamical modellingof neuronal imaging data with the declared objective of drawing inference,within this model framework, in a large-scale (high-dimensional) data setting. Implicitlythis objective entails carrying out three separate but closely connected tasks;i) probabilistic modelling, ii) statistical modeling and iii) implementation of an inferentialprocedure. While i) - iii) are distinct tasks that range over several quitedifferent disciplines, they are joined by the premise that the initial objective can onlybe 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 stochasticmodelling approach, to obtain a stochastic functional differential equation on aHilbert space. By decomposing the drift operator of this SFDE such that each componentis essentially represented by a smooth function of time and space and expandingthese component functions in a tensor product basis we implicitly reducethe number of model parameters. In addition, the component-wise tensor representationinduce a corresponding component-wise tensor structure in the resultingstatistical model. Especially, the statistical model is design matrix free and facilitatesan efficient array arithmetic. Using proximal gradient based algorithms, we combinethis computationally attractive statistical framework with non-differentiable regularizationto form computationally efficient inferential procedure with minimal memoryfoot prints. As a result we are able to fit large scale image data in a mathematicallysophisticated dynamical model using a relatively modest amount of computationalresources in the process.The contributions presented in this thesis are computational and methodological.The computational contribution takes the form of solution algorithms aimedat exploiting the array-tensor structure in various inferential settings. The methodologicalcontribution takes the form of a dynamical modelling and inferential frameworkfor spatio-temporal array data. This framework was developed with neuronfield models in mind but may in turn be applied to other settings conforming to thespatio-temporal array data setup.

UR - https://soeg.kb.dk/permalink/45KBDK_KGL/fbp0ps/alma99122927018905763

M3 - Ph.D. thesis

BT - Spatio-Temporal Modeling of Neuron Fields

PB - Department of Mathematical Sciences, Faculty of Science, University of Copenhagen

ER -