Quantifying chaos: a tale of two maps

In many applications, there is a desire to determine if the dynamics of interest are chaotic or not. Since positive Lyapunov exponents are a signature for chaos, they are often used to determine this. Reliable estimates of Lyapunov exponents should demonstrate evidence of convergence; but literature abounds in which this evidence lacks. This paper presents two maps through which it highlights the importance of providing evidence of convergence of Lyapunov exponent estimates. The results suggest cautious conclusions when confronted with real data. Moreover, the maps are interesting in their own right.

Revealing ensemble state transition patterns in multi-electrode neuronal recordings using hidden Markov models

In order to harness the computational capacity of dissociated cultured neuronal networks, it is necessary to understand neuronal dynamics and connectivity on a mesoscopic scale. To this end, this paper uncovers dynamic spatiotemporal patterns emerging from electrically stimulated neuronal cultures using hidden Markov models (HMMs) to characterize multi-channel spike trains as a progression of patterns of underlying states of neuronal activity. However, experimentation aimed at optimal choice of parameters for such models is essential and results are reported in detail. Results derived from ensemble neuronal data revealed highly repeatable patterns of state transitions in the order of milliseconds in response to probing stimuli.

An improved lesion detection approach based on similarity measurement between fuzzy intensity segmentation and spatial probability maps

The application of automatic segmentation methods in lesion detection is desirable. However, such methods are restricted by intensity similarities between lesioned and healthy brain tissue. Using multi-spectral magnetic resonance imaging (MRI) modalities may overcome this problem but it is not always practicable. In this article, a lesion detection approach requiring a single MRI modality is presented, which is an improved method based on a recent publication. This new method assumes that a low similarity should be found in the regions of lesions when the likeness between an intensity based fuzzy segmentation and a location based tissue probabilities is measured. The usage of a normalized similarity measurement enables the current method to fine-tune the threshold for lesion detection, thus maximizing the possibility of reaching high detection accuracy. Importantly, an extra cleaning step is included in the current approach which removes enlarged ventricles from detected lesions. The performance investigation using simulated lesions demonstrated that not only the majority of lesions were well detected but also normal tissues were identified effectively. Tests on images acquired in stroke patients further confirmed the strength of the method in lesion detection. When compared with the previous version, the current approach showed a higher sensitivity in detecting small lesions and had less false positives around the ventricle and the edge of the brain

Persistence of UK Real Estate returns: a Markov chain analysis

The persistence of investment performance is a topic of perennial interest to investors. Efficient Markets theory tells us that past performance can not be used to predict future performance yet investors appear to be influenced by the historical performance in making their investment allocation decisions. The problem has been of particular interest to investors in real estate; not least because reported returns from investment in real estate are serially correlated thus implying some persistence in investment performance. This paper applies the established approach of Markov Chain analysis to investigate the relationship between past and present performance of UK real estate over the period 1981 to 1996. The data are analysed by sector, region and size. Furthermore some variations in investment performance classification are reported and the results are shown to be robust.

Markov models of aging: theory and practice

We review and structure some of the mathematical and statistical models that have been developed over the past half century to grapple with theoretical and experimental questions about the stochastic development of aging over the life course. We suggest that the mathematical models are in large part addressing the problem of partitioning the randomness in aging: How does aging vary between individuals, and within an individual over the lifecourse? How much of the variation is inherently related to some qualities of the individual, and how much is entirely random? How much of the randomness is cumulative, and how much is merely short-term flutter? We propose that recent lines of statistical inquiry in survival analysis could usefully grapple with these questions, all the more so if they were more explicitly linked to the relevant mathematical and biological models of aging. To this end, we describe points of contact among the various lines of mathematical and statistical research. We suggest some directions for future work, including the exploration of information-theoretic measures for evaluating components of stochastic models as the basis for analyzing experiments and anchoring theoretical discussions of aging.

Bayesian parameter estimation for latent Markov random fields and social networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.