Bifurcation and stability in a low-dimensional model for multiple-frequency mode-locked lasers

Recent theoretical investigations have demonstrated that the stability of mode-locked solutions of multiple frequency channels depends on the degree of inhomogeneity in gain saturation. In this article, these results are generalized to determine conditions on each of the system parameters necessary for both the stability and the existence of mode-locked pulse solutions for an arbitrary number of frequency channels. In particular, we find that the parameters governing saturable intensity discrimination and gain inhomogeneity in the laser cavity also determine the position of bifurcations of solution types. These bifurcations are completely characterized in terms of these parameters. In addition to influencing the stability of mode-locked solutions, we determine a balance between cubic gain and quintic loss, which is necessary for the existence of solutions as well. Furthermore, we determine the critical degree of inhomogeneous gain broadening required to support pulses in multiple-frequency channels. © 2010 The American Physical Society.

A principled approach to interactive hierarchical non-linear visualization of high-dimensional data

Hierarchical visualization systems are desirable because a single two-dimensional visualization plot may not be sufficient to capture all of the interesting aspects of complex high-dimensional data sets. We extend an existing locally linear hierarchical visualization system PhiVis [1] in several directions: bf(1) we allow for em non-linear projection manifolds (the basic building block is the Generative Topographic Mapping -- GTM), bf(2) we introduce a general formulation of hierarchical probabilistic models consisting of local probabilistic models organized in a hierarchical tree, bf(3) we describe folding patterns of low-dimensional projection manifold in high-dimensional data space by computing and visualizing the manifold's local directional curvatures. Quantities such as magnification factors [3] and directional curvatures are helpful for understanding the layout of the nonlinear projection manifold in the data space and for further refinement of the hierarchical visualization plot. Like PhiVis, our system is statistically principled and is built interactively in a top-down fashion using the EM algorithm. We demonstrate the visualization system principle of the approach on a complex 12-dimensional data set and mention possible applications in the pharmaceutical industry.

GTM: the generative topographic mapping

This thesis describes the Generative Topographic Mapping (GTM) --- a non-linear latent variable model, intended for modelling continuous, intrinsically low-dimensional probability distributions, embedded in high-dimensional spaces. It can be seen as a non-linear form of principal component analysis or factor analysis. It also provides a principled alternative to the self-organizing map --- a widely established neural network model for unsupervised learning --- resolving many of its associated theoretical problems. An important, potential application of the GTM is visualization of high-dimensional data. Since the GTM is non-linear, the relationship between data and its visual representation may be far from trivial, but a better understanding of this relationship can be gained by computing the so-called magnification factor. In essence, the magnification factor relates the distances between data points, as they appear when visualized, to the actual distances between those data points. There are two principal limitations of the basic GTM model. The computational effort required will grow exponentially with the intrinsic dimensionality of the density model. However, if the intended application is visualization, this will typically not be a problem. The other limitation is the inherent structure of the GTM, which makes it most suitable for modelling moderately curved probability distributions of approximately rectangular shape. When the target distribution is very different to that, theaim of maintaining an `interpretable' structure, suitable for visualizing data, may come in conflict with the aim of providing a good density model. The fact that the GTM is a probabilistic model means that results from probability theory and statistics can be used to address problems such as model complexity. Furthermore, this framework provides solid ground for extending the GTM to wider contexts than that of this thesis.

Exploiting sensitivity of nonorthogonal joint diagonalisation as a security mechanism in steganography

We have recently proposed the framework of independent blind source separation as an advantageous approach to steganography. Amongst the several characteristics noted was a sensitivity to message reconstruction due to small perturbations in the sources. This characteristic is not common in most other approaches to steganography. In this paper we discuss how this sensitivity relates the joint diagonalisation inside the independent component approach, and reliance on exact knowledge of secret information, and how it can be used as an additional and inherent security mechanism against malicious attack to discovery of the hidden messages. The paper therefore provides an enhanced mechanism that can be used for e-document forensic analysis and can be applied to different dimensionality digital data media. In this paper we use a low dimensional example of biomedical time series as might occur in the electronic patient health record, where protection of the private patient information is paramount.

Synchronization processes in nonlinear systems and their relation to cortical oscillatory dynamics

This thesis was focused on theoretical models of synchronization to cortical dynamics as measured by magnetoencephalography (MEG). Dynamical systems theory was used in both identifying relevant variables for brain coordination and also in devising methods for their quantification. We presented a method for studying interactions of linear and chaotic neuronal sources using MEG beamforming techniques. We showed that such sources can be accurately reconstructed in terms of their location, temporal dynamics and possible interactions. Synchronization in low-dimensional nonlinear systems was studied to explore specific correlates of functional integration and segregation. In the case of interacting dissimilar systems, relevant coordination phenomena involved generalized and phase synchronization, which were often intermittent. Spatially-extended systems were then studied. For locally-coupled dissimilar systems, as in the case of cortical columns, clustering behaviour occurred. Synchronized clusters emerged at different frequencies and their boundaries were marked through oscillation death. The macroscopic mean field revealed sharp spectral peaks at the frequencies of the clusters and broader spectral drops at their boundaries. These results question existing models of Event Related Synchronization and Desynchronization. We re-examined the concept of the steady-state evoked response following an AM stimulus. We showed that very little variability in the AM following response could be accounted by system noise. We presented a methodology for detecting local and global nonlinear interactions from MEG data in order to account for residual variability. We found crosshemispheric nonlinear interactions of ongoing cortical rhythms concurrent with the stimulus and interactions of these rhythms with the following AM responses. Finally, we hypothesized that holistic spatial stimuli would be accompanied by the emergence of clusters in primary visual cortex resulting in frequency-specific MEG oscillations. Indeed, we found different frequency distributions in induced gamma oscillations for different spatial stimuli, which was suggestive of temporal coding of these spatial stimuli. Further, we addressed the bursting character of these oscillations, which was suggestive of intermittent nonlinear dynamics. However, we did not observe the characteristic-3/2 power-law scaling in the distribution of interburst intervals. Further, this distribution was only seldom significantly different to the one obtained in surrogate data, where nonlinear structure was destroyed. In conclusion, the work presented in this thesis suggests that advances in dynamical systems theory in conjunction with developments in magnetoencephalography may facilitate a mapping between levels of description int he brain. this may potentially represent a major advancement in neuroscience.

Watermarking biomedical time series data

This thesis addresses the problem of information hiding in low dimensional digital data focussing on issues of privacy and security in Electronic Patient Health Records (EPHRs). The thesis proposes a new security protocol based on data hiding techniques for EPHRs. This thesis contends that embedding of sensitive patient information inside the EPHR is the most appropriate solution currently available to resolve the issues of security in EPHRs. Watermarking techniques are applied to one-dimensional time series data such as the electroencephalogram (EEG) to show that they add a level of confidence (in terms of privacy and security) in an individual’s diverse bio-profile (the digital fingerprint of an individual’s medical history), ensure belief that the data being analysed does indeed belong to the correct person, and also that it is not being accessed by unauthorised personnel. Embedding information inside single channel biomedical time series data is more difficult than the standard application for images due to the reduced redundancy. A data hiding approach which has an in built capability to protect against illegal data snooping is developed. The capability of this secure method is enhanced by embedding not just a single message but multiple messages into an example one-dimensional EEG signal. Embedding multiple messages of similar characteristics, for example identities of clinicians accessing the medical record helps in creating a log of access while embedding multiple messages of dissimilar characteristics into an EPHR enhances confidence in the use of the EPHR. The novel method of embedding multiple messages of both similar and dissimilar characteristics into a single channel EEG demonstrated in this thesis shows how this embedding of data boosts the implementation and use of the EPHR securely.

Uncertainty and topographic visualisations

This thesis is a study of low-dimensional visualisation methods for data visualisation under certainty of the input data. It focuses on the two main feed-forward neural network algorithms which are NeuroScale and Generative Topographic Mapping (GTM) by trying to make both algorithms able to accommodate the uncertainty. The two models are shown not to work well under high levels of noise within the data and need to be modified. The modification of both models, NeuroScale and GTM, are verified by using synthetic data to show their ability to accommodate the noise. The thesis is interested in the controversy surrounding the non-uniqueness of predictive gene lists (PGL) of predicting prognosis outcome of breast cancer patients as available in DNA microarray experiments. Many of these studies have ignored the uncertainty issue resulting in random correlations of sparse model selection in high dimensional spaces. The visualisation techniques are used to confirm that the patients involved in such medical studies are intrinsically unclassifiable on the basis of provided PGL evidence. This additional category of ‘unclassifiable’ should be accommodated within medical decision support systems if serious errors and unnecessary adjuvant therapy are to be avoided.

Facile route to conformal hydrotalcite coatings over complex architectures:a hierarchically ordered nanoporous base catalyst for FAME production

An alkali- and nitrate-free hydrotalcite coating has been grafted onto the surface of a hierarchically ordered macroporous-mesoporous SBA-15 template via stepwise growth of conformal alumina adlayers and their subsequent reaction with magnesium methoxide. The resulting low dimensional hydrotalcite crystallites exhibit excellent per site activity for the base catalysed transesterification of glyceryl triolein with methanol for FAME production.

Discovering properties of new DNA-binding activity of proteins

Protein-DNA interactions are an essential feature in the genetic activities of life, and the ability to predict and manipulate such interactions has applications in a wide range of fields. This Thesis presents the methods of modelling the properties of protein-DNA interactions. In particular, it investigates the methods of visualising and predicting the specificity of DNA-binding Cys2His2 zinc finger interaction. The Cys2His2 zinc finger proteins interact via their individual fingers to base pair subsites on the target DNA. Four key residue positions on the a- helix of the zinc fingers make non-covalent interactions with the DNA with sequence specificity. Mutating these key residues generates combinatorial possibilities that could potentially bind to any DNA segment of interest. Many attempts have been made to predict the binding interaction using structural and chemical information, but with only limited success. The most important contribution of the thesis is that the developed model allows for the binding properties of a given protein-DNA binding to be visualised in relation to other protein-DNA combinations without having to explicitly physically model the specific protein molecule and specific DNA sequence. To prove this, various databases were generated, including a synthetic database which includes all possible combinations of the DNA-binding Cys2His2 zinc finger interactions. NeuroScale, a topographic visualisation technique, is exploited to represent the geometric structures of the protein-DNA interactions by measuring dissimilarity between the data points. In order to verify the effect of visualisation on understanding the binding properties of the DNA-binding Cys2His2 zinc finger interaction, various prediction models are constructed by using both the high dimensional original data and the represented data in low dimensional feature space. Finally, novel data sets are studied through the selected visualisation models based on the experimental DNA-zinc finger protein database. The result of the NeuroScale projection shows that different dissimilarity representations give distinctive structural groupings, but clustering in biologically-interesting ways. This method can be used to forecast the physiochemical properties of the novel proteins which may be beneficial for therapeutic purposes involving genome targeting in general.

Quantifying the complexity of chaos in multibasin multidimensional dynamics of molecular systems

The simulated classical dynamics of a small molecule exhibiting self-organizing behavior via a fast transition between two states is analyzed by calculation of the statistical complexity of the system. It is shown that the complexity of molecular descriptors such as atom coordinates and dihedral angles have different values before and after the transition. This provides a new tool to identify metastable states during molecular self-organization. The highly concerted collective motion of the molecule is revealed. Low-dimensional subspaces dynamics is found sensitive to the processes in the whole, high-dimensional phase space of the system. © 2004 Wiley Periodicals, Inc.

Bosonisation as The Hubbard-Stratonovich transformation

The problem of strongly correlated electrons in one dimension attracted attention of condensed matter physicists since early 50’s. After the seminal paper of Tomonaga [1] who suggested the first soluble model in 1950, there were essential achievements reflected in papers by Luttinger [2] (1963) and Mattis and Lieb [3] (1963). A considerable contribution to the understanding of generic properties of the 1D electron liquid has been made by Dzyaloshinskii and Larkin [4] (1973) and Efetov and Larkin [5] (1976). Despite the fact that the main features of the 1D electron liquid were captured and described by the end of 70’s, the investigators felt dissatisfied with the rigour of the theoretical description. The most famous example is the paper by Haldane [6] (1981) where the author developed the fundamentals of a modern bosonisation technique, known as the operator approach. This paper became famous because the author has rigourously shown how to construct the Fermi creation/anihilation operators out of the Bose ones. The most recent example of such a dissatisfaction is the review by von Delft and Schoeller [7] (1998) who revised the approach to the bosonisation and came up with what they called constructive bosonisation.

Visualisation of bioinformatics datasets

Analysing the molecular polymorphism and interactions of DNA, RNA and proteins is of fundamental importance in biology. Predicting functions of polymorphic molecules is important in order to design more effective medicines. Analysing major histocompatibility complex (MHC) polymorphism is important for mate choice, epitope-based vaccine design and transplantation rejection etc. Most of the existing exploratory approaches cannot analyse these datasets because of the large number of molecules with a high number of descriptors per molecule. This thesis develops novel methods for data projection in order to explore high dimensional biological dataset by visualising them in a low-dimensional space. With increasing dimensionality, some existing data visualisation methods such as generative topographic mapping (GTM) become computationally intractable. We propose variants of these methods, where we use log-transformations at certain steps of expectation maximisation (EM) based parameter learning process, to make them tractable for high-dimensional datasets. We demonstrate these proposed variants both for synthetic and electrostatic potential dataset of MHC class-I. We also propose to extend a latent trait model (LTM), suitable for visualising high dimensional discrete data, to simultaneously estimate feature saliency as an integrated part of the parameter learning process of a visualisation model. This LTM variant not only gives better visualisation by modifying the project map based on feature relevance, but also helps users to assess the significance of each feature. Another problem which is not addressed much in the literature is the visualisation of mixed-type data. We propose to combine GTM and LTM in a principled way where appropriate noise models are used for each type of data in order to visualise mixed-type data in a single plot. We call this model a generalised GTM (GGTM). We also propose to extend GGTM model to estimate feature saliencies while training a visualisation model and this is called GGTM with feature saliency (GGTM-FS). We demonstrate effectiveness of these proposed models both for synthetic and real datasets. We evaluate visualisation quality using quality metrics such as distance distortion measure and rank based measures: trustworthiness, continuity, mean relative rank errors with respect to data space and latent space. In cases where the labels are known we also use quality metrics of KL divergence and nearest neighbour classifications error in order to determine the separation between classes. We demonstrate the efficacy of these proposed models both for synthetic and real biological datasets with a main focus on the MHC class-I dataset.

Duality in multi-channel Luttinger Liquid with local scatterer

We have devised a general scheme that reveals multiple duality relations valid for all multi-channel Luttinger Liquids. The relations are universal and should be used for establishing phase diagrams and searching for new non-trivial phases in low-dimensional strongly correlated systems. The technique developed provides universal correspondence between scaling dimensions of local perturbations in different phases. These multiple relations between scaling dimensions lead to a connection between different inter-phase boundaries on the phase diagram. The dualities, in particular, constrain phase diagram and allow predictions of emergence and observation of new phases without explicit model-dependent calculations. As an example, we demonstrate the impossibility of non-trivial phase existence for fermions coupled to phonons in one dimension. © 2013 EPLA.

Topographic visual analytics of multibeam dynamic SONAR data

This paper considers the problem of low-dimensional visualisation of very high dimensional information sources for the purpose of situation awareness in the maritime environment. In response to the requirement for human decision support aids to reduce information overload (and specifically, data amenable to inter-point relative similarity measures) appropriate to the below-water maritime domain, we are investigating a preliminary prototype topographic visualisation model. The focus of the current paper is on the mathematical problem of exploiting a relative dissimilarity representation of signals in a visual informatics mapping model, driven by real-world sonar systems. A realistic noise model is explored and incorporated into non-linear and topographic visualisation algorithms building on the approach of [9]. Concepts are illustrated using a real world dataset of 32 hydrophones monitoring a shallow-water environment in which targets are present and dynamic.

Unfolding kernel embeddings of graphs:enhancing class separation through manifold learning

In this paper, we investigate the use of manifold learning techniques to enhance the separation properties of standard graph kernels. The idea stems from the observation that when we perform multidimensional scaling on the distance matrices extracted from the kernels, the resulting data tends to be clustered along a curve that wraps around the embedding space, a behavior that suggests that long range distances are not estimated accurately, resulting in an increased curvature of the embedding space. Hence, we propose to use a number of manifold learning techniques to compute a low-dimensional embedding of the graphs in an attempt to unfold the embedding manifold, and increase the class separation. We perform an extensive experimental evaluation on a number of standard graph datasets using the shortest-path (Borgwardt and Kriegel, 2005), graphlet (Shervashidze et al., 2009), random walk (Kashima et al., 2003) and Weisfeiler-Lehman (Shervashidze et al., 2011) kernels. We observe the most significant improvement in the case of the graphlet kernel, which fits with the observation that neglecting the locational information of the substructures leads to a stronger curvature of the embedding manifold. On the other hand, the Weisfeiler-Lehman kernel partially mitigates the locality problem by using the node labels information, and thus does not clearly benefit from the manifold learning. Interestingly, our experiments also show that the unfolding of the space seems to reduce the performance gap between the examined kernels.