Real time clustering of time series using triangular potentials

Motivated by the problem of computing investment portfolio weightings we investigate various methods of clustering as alternatives to traditional mean-variance approaches. Such methods can have significant benefits from a practical point of view since they remove the need to invert a sample covariance matrix, which can suffer from estimation error and will almost certainly be non-stationary. The general idea is to find groups of assets which share similar return characteristics over time and treat each group as a single composite asset. We then apply inverse volatility weightings to these new composite assets. In the course of our investigation we devise a method of clustering based on triangular potentials and we present associated theoretical results as well as various examples based on synthetic data.Comment: AIFU1

Ground states of stealthy hyperuniform potentials: I. Entropically favored configurations

Systems of particles interacting with "stealthy" pair potentials have been shown to possess infinitely degenerate disordered hyperuniform classical ground states with novel physical properties. Previous attempts to sample the infinitely degenerate ground states used energy minimization techniques, introducing algorithmic dependence that is artificial in nature. Recently, an ensemble theory of stealthy hyperuniform ground states was formulated to predict the structure and thermodynamics that was shown to be in excellent agreement with corresponding computer simulation results in the canonical ensemble (in the zero-temperature limit). In this paper, we provide details and justifications of the simulation procedure, which involves performing molecular dynamics simulations at sufficiently low temperatures and minimizing the energy of the snapshots for both the high-density disordered regime, where the theory applies, as well as lower densities. We also use numerical simulations to extend our study to the lower-density regime. We report results for the pair correlation functions, structure factors, and Voronoi cell statistics. In the high-density regime, we verify the theoretical ansatz that stealthy disordered ground states behave like "pseudo" disordered equilibrium hard-sphere systems in Fourier space. These results show that as the density decreases from the high-density limit, the disordered ground states in the canonical ensemble are characterized by an increasing degree of short-range order and eventually the system undergoes a phase transition to crystalline ground states. We also provide numerical evidence suggesting that different forms of stealthy pair potentials produce the same ground-state ensemble in the zero-temperature limit. Our techniques may be applied to sample this limit of the canonical ensemble of other potentials with highly degenerate ground states

SSA of biomedical signals: A linear invariant systems approach

Singular spectrum analysis (SSA) is considered from a linear invariant systems perspective. In this terminology, the extracted components are considered as outputs of a linear invariant system which corresponds to finite impulse response (FIR) filters. The number of filters is determined by the embedding dimension.We propose to explicitly define the frequency response of each filter responsible for the selection of informative components. We also introduce a subspace distance measure for clustering subspace models. We illustrate the methodology by analyzing lectroencephalograms (EEG).FCT - PhD scholarship (SFRH/BD/28404/2006)FCT - PhD scholarship (SFRH/BD/48775/2008

How Many Dissimilarity/Kernel Self Organizing Map Variants Do We Need?

In numerous applicative contexts, data are too rich and too complex to be represented by numerical vectors. A general approach to extend machine learning and data mining techniques to such data is to really on a dissimilarity or on a kernel that measures how different or similar two objects are. This approach has been used to define several variants of the Self Organizing Map (SOM). This paper reviews those variants in using a common set of notations in order to outline differences and similarities between them. It discusses the advantages and drawbacks of the variants, as well as the actual relevance of the dissimilarity/kernel SOM for practical applications

The Eyring-Kramers law for Markovian jump processes with symmetries

We prove an Eyring-Kramers law for the small eigenvalues and mean first-passage times of a metastable Markovian jump process which is invariant under a group of symmetries. Our results show that the usual Eyring-Kramers law for asymmetric processes has to be corrected by a factor computable in terms of stabilisers of group orbits. Furthermore, the symmetry can produce additional Arrhenius exponents and modify the spectral gap. The results are based on representation theory of finite groups.Comment: 39 pages, 9 figure

A guide to time-resolved and parameter-free measures of spike train synchrony

Measures of spike train synchrony have proven a valuable tool in both experimental and computational neuroscience. Particularly useful are time-resolved methods such as the ISI- and the SPIKE-distance, which have already been applied in various bivariate and multivariate contexts. Recently, SPIKE-Synchronization was proposed as another time-resolved synchronization measure. It is based on Event-Synchronization and has a very intuitive interpretation. Here, we present a detailed analysis of the mathematical properties of these three synchronization measures. For example, we were able to obtain analytic expressions for the expectation values of the ISI-distance and SPIKE-Synchronization for Poisson spike trains. For the SPIKE-distance we present an empirical formula deduced from numerical evaluations. These expectation values are crucial for interpreting the synchronization of spike trains measured in experiments or numerical simulations, as they represent the point of reference for fully randomized spike trains.Comment: 8 pages, 4 figure

Efficient Semidefinite Branch-and-Cut for MAP-MRF Inference

We propose a Branch-and-Cut (B&C) method for solving general MAP-MRF inference problems. The core of our method is a very efficient bounding procedure, which combines scalable semidefinite programming (SDP) and a cutting-plane method for seeking violated constraints. In order to further speed up the computation, several strategies have been exploited, including model reduction, warm start and removal of inactive constraints. We analyze the performance of the proposed method under different settings, and demonstrate that our method either outperforms or performs on par with state-of-the-art approaches. Especially when the connectivities are dense or when the relative magnitudes of the unary costs are low, we achieve the best reported results. Experiments show that the proposed algorithm achieves better approximation than the state-of-the-art methods within a variety of time budgets on challenging non-submodular MAP-MRF inference problems.Comment: 21 page

Inherent Structures for Soft Long-Range Interactions in Two-Dimensional Many-Particle Systems

We generate inherent structures, local potential-energy minima, of the "$k$-space overlap potential" in two-dimensional many-particle systems using a cooling and quenching simulation technique. The ground states associated with the $k$-space overlap potential are stealthy ({\it i.e.,} completely suppress single scattering of radiation for a range of wavelengths) and hyperuniform ({\it i.e.,} infinite wavelength density fluctuations vanish). However, we show via quantitative metrics that the inherent structures exhibit a range of stealthiness and hyperuniformity depending on the fraction of degrees of freedom that are constrained. Inherent structures in two dimensions typically contain five-particle rings, wavy grain boundaries, and vacancy-interstitial defects. The structural and thermodynamic properties of inherent structures are relatively insensitive to the temperature from which they are sampled, signifying that the energy landscape is relatively flat and devoid of deep wells. Using the nudged-elastic-band algorithm, we construct paths from ground-state configurations to inherent structures and identify the transition points between them. In addition, we use point patterns generated from a random sequential addition (RSA) of hard disks, which are nearly stealthy, and examine the particle rearrangements necessary to make the configurations absolutely stealthy. We introduce a configurational proximity metric to show that only small local, but collective, particle rearrangements are needed to drive initial RSA configurations to stealthy disordered ground states. These results lead to a more complete understanding of the unusual behaviors exhibited by the family of "collective-coordinate" potentials to which the $k$-space overlap potential belongs.Comment: 36 pages, 16 figure