Tanya Schmah

Comparing classification methods for longitudinal fmri studies

We compare 10 methods of classifying fMRI volumes by applying them to data from a longitudinal study of stroke recovery: adaptive Fishers linear and quadratic discriminant; gaussian naive Bayes; support vector machines with linear, quadratic, and radial basis function (RBF) kernels; logistic regression; two novel methods based on pairs of restricted Boltzmann machines (RBM); and K-nearest neighbors. All methods were tested on three binary classification tasks, and their out-of-sample classification accuracies are compared. The relative performance of the methods varies considerably across subjects and classification tasks. The best overall performers were adaptive quadratic discriminant, support vector machines with RBF kernels, and generatively trained pairs of RBMs.

SURROGATE DATA PATHOLOGIES AND THE FALSE-POSITIVE REJECTION OF THE NULL HYPOTHESIS

It is shown that inappropriately constructed random phase surrogates can give false-positive rejections of the surrogate null hypothesis. Specifically, the procedure erroneously indicated the presence of deterministic, nonlinear structure in a time series that was constructed by linearly filtering normally distributed random numbers. It is shown that the erroneous identification was due to numerical errors in the estimation of the signals Fourier transform. In the example examined here, the introduction of data windowing into the algorithm eliminated the false-positive rejection of the null hypothesis. Additional guidelines for the use of surrogates are considered, and the results of a comparison test of random phase surrogates, Gaussian scaled surrogates and iterative surrogates are presented.

Pattern classification of fMRI data: Applications for analysis of spatially distributed cortical networks

The field of fMRI data analysis is rapidly growing in sophistication, particularly in the domain of multivariate pattern classification. However, the interaction between the properties of the analytical model and the parameters of the BOLD signal (e.g. signal magnitude, temporal variance and functional connectivity) is still an open problem. We addressed this problem by evaluating a set of pattern classification algorithms on simulated and experimental block-design fMRI data. The set of classifiers consisted of linear and quadratic discriminants, linear support vector machine, and linear and nonlinear Gaussian naive Bayes classifiers. For linear discriminant, we used two methods of regularization: principal component analysis, and ridge regularization. The classifiers were used (1) to classify the volumes according to the behavioral task that was performed by the subject, and (2) to construct spatial maps that indicated the relative contribution of each voxel to classification. Our evaluation metrics were: (1) accuracy of out-of-sample classification and (2) reproducibility of spatial maps. In simulated data sets, we performed an additional evaluation of spatial maps with ROC analysis. We varied the magnitude, temporal variance and connectivity of simulated fMRI signal and identified the optimal classifier for each simulated environment. Overall, the best performers were linear and quadratic discriminants (operating on principal components of the data matrix) and, in some rare situations, a nonlinear Gaussian naïve Bayes classifier. The results from the simulated data were supported by within-subject analysis of experimental fMRI data, collected in a study of aging. This is the first study that systematically characterizes interactions between analysis model and signal parameters (such as magnitude, variance and correlation) on the performance of pattern classifiers for fMRI.

Models of knowing and the investigation of dynamical systems

Abstract We present three distinct concepts of what constitutes a scientific understanding of a dynamical system. The development of each of these paradigms has resulted in a significant expansion in the kind of system that can be investigated. In particular, the recently-developed ‘algorithmic modelling paradigm’ has allowed us to enlarge the domain of discourse to include complex real-world processes that cannot necessarily be described by conventional differential equations.

Left-invariant metrics for diffeomorphic image registration with spatially-varying regularisation

We present a new framework for diffeomorphic image registration which supports natural interpretations of spatially-varying metrics. This framework is based on left-invariant diffeomorphic metrics (LIDM) and is closely related to the now standard large deformation diffeomorphic metric mapping (LDDMM). We discuss the relationship between LIDM and LDDMM and introduce a computationally convenient class of spatially-varying metrics appropriate for both frameworks. Finally, we demonstrate the effectiveness of our method on a 2D toy example and on the 40 3D brain images of the LPBA40 dataset.

Stability for Lagrangian relative equilibria of three-point-mass systems

In the present paper we apply geometric methods, and in particular the reduced energy–momentum (REM) method, to the analysis of stability of planar rotationally invariant relative equilibria of three-point-mass systems. We analyse two examples in detail: equilateral relative equilibria for the three-body problem, and isosceles triatomic molecules. We discuss some open problems to which the method is applicable, including roto-translational motion in the full three-body problem.

Torus actions on symplectic orbi-spaces

Which 2n-dimensional orbi-spaces have effective symplectic ktorus actions? As shown by Lerman and Tolman (1997) and Watson (1997), this question reduces to that of characterizing the finite subgroups of centralizers of tori in the real symplectic group Sp(2n, IR). We resolve this question, and generalize our method to a calculation of the centralizers of all tori in Sp(2n, R).

Normal Forms for Lie Symmetric Cotangent Bundle Systems with Free and Proper Actions

We consider free and proper cotangent-lifted symmetries of Hamiltonian systems. For the special case of G = SO(3), we construct symplectic slice coordinates around an arbitrary point. We thus obtain a parameterisation of the phase space suitable for the study of dynamics near relative equilibria, in particular for the Birkhoff-Poincare normal form method. For a general symmetry group G, we observe that for the calculation of the truncated normal forms, one does not need an explicit coordinate transformation but only its higher derivatives at the relative equilibrium. We outline an iterative scheme using these derivatives for the computation of truncated Birkhoff-Poincare normal forms.

Diffeomorphic image matching with left-invariant metrics

The geometric approach to diffeomorphic image registration known as large deformation by diffeomorphic metric mapping (LDDMM) is based on a left action of diffeomorphisms on images, and a right-invariant metric on a diffeomorphism group, usually defined using a reproducing kernel. We explore the use of left-invariant metrics on diffeomorphism groups, based on reproducing kernels defined in the body coordinates of a source image. This perspective, which we call Left-LDM, allows us to consider non-isotropic spatially-varying kernels, which can be interpreted as describing variable deformability of the source image. We also show a simple relationship between LDDMM and the new approach, implying that spatially-varying kernels are interpretable in the same way in LDDMM. We conclude with a discussion of a class of kernels that enforce a soft mirror-symmetry constraint, which we validate in numerical experiments on a model of a lesioned brain.