Measuring departures from Hardy-Weinberg: a Markov chain Monte Carlo method for estimating the inbreeding coefficient

Many well-established statistical methods in genetics were developed in a climate of severe constraints on computational power. Recent advances in simulation methodology now bring modern, flexible statistical methods within the reach of scientists having access to a desktop workstation. We illustrate the potential advantages now available by considering the problem of assessing departures from Hardy-Weinberg (HW) equilibrium. Several hypothesis tests of HW have been established, as well as a variety of point estimation methods for the parameter which measures departures from HW under the inbreeding model. We propose a computational, Bayesian method for assessing departures from HW, which has a number of important advantages over existing approaches. The method incorporates the effects-of uncertainty about the nuisance parameters--the allele frequencies--as well as the boundary constraints on f (which are functions of the nuisance parameters). Results are naturally presented visually, exploiting the graphics capabilities of modern computer environments to allow straightforward interpretation. Perhaps most importantly, the method is founded on a flexible, likelihood-based modelling framework, which can incorporate the inbreeding model if appropriate, but also allows the assumptions of the model to he investigated and, if necessary, relaxed. Under appropriate conditions, information can be shared across loci and, possibly, across populations, leading to more precise estimation. The advantages of the method are illustrated by application both to simulated data and to data analysed by alternative methods in the recent literature.

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.

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.

Business cycle asymmetries: characterisation and testing based on Markov-switching autoregressions

Tests for business cycle asymmetries are developed for Markov-switching autoregressive models. The tests of deepness, steepness, and sharpness are Wald statistics, which have standard asymptotics. For the standard two-regime model of expansions and contractions, deepness is shown to imply sharpness (and vice versa), whereas the process is always nonsteep. Two and three-state models of U.S. GNP growth are used to illustrate the approach, along with models of U.S. investment and consumption growth. The robustness of the tests to model misspecification, and the effects of regime-dependent heteroscedasticity, are investigated.

Threshold autoregressive and Markov switching models: an application to commercial real estate

Although financial theory rests heavily upon the assumption that asset returns are normally distributed, value indices of commercial real estate display significant departures from normality. In this paper, we apply and compare the properties of two recently proposed regime switching models for value indices of commercial real estate in the US and the UK, both of which relax the assumption that observations are drawn from a single distribution with constant mean and variance. Statistical tests of the models' specification indicate that the Markov switching model is better able to capture the non-stationary features of the data than the threshold autoregressive model, although both represent superior descriptions of the data than the models that allow for only one state. Our results have several implications for theoretical models and empirical research in finance.

Noisy Monte Carlo: convergence of Markov chains with approximate transition kernels

Monte Carlo algorithms often aim to draw from a distribution π by simulating a Markov chain with transition kernel P such that π is invariant under P. However, there are many situations for which it is impractical or impossible to draw from the transition kernel P. For instance, this is the case with massive datasets, where is it prohibitively expensive to calculate the likelihood and is also the case for intractable likelihood models arising from, for example, Gibbs random fields, such as those found in spatial statistics and network analysis. A natural approach in these cases is to replace P by an approximation Pˆ. Using theory from the stability of Markov chains we explore a variety of situations where it is possible to quantify how ’close’ the chain given by the transition kernel Pˆ is to the chain given by P . We apply these results to several examples from spatial statistics and network analysis.

Hidden Markov analysis describes dive patterns in semiaquatic animals

Existing methods of dive analysis, developed for fully aquatic animals, tend to focus on frequency of behaviors rather than transitions between them. They, therefore, do not account for the variability of behavior of semiaquatic animals, and the switching between terrestrial and aquatic environments. This is the first study to use hidden Markov models (HMM) to divide dives of a semiaquatic animal into clusters and thus identify the environmental predictors of transition between behavioral modes. We used 18 existing data sets of the dives of 14 American mink (Neovison vison) fitted with time-depth recorders in lowland England. Using HMM, we identified 3 behavioral states (1, temporal cluster of dives; 2, more loosely aggregated diving within aquatic activity; and 3, terminal dive of a cluster or a single, isolated dive). Based on the higher than expected proportion of dives in State 1, we conclude that mink tend to dive in clusters. We found no relationship between temperature and the proportion of dives in each state or between temperature and the rate of transition between states, meaning that in our study area, mink are apparently not adopting different diving strategies at different temperatures. Transition analysis between states has shown that there is no correlation between ambient temperature and the likelihood of mink switching from one state to another, that is, changing foraging modes. The variables provided good discrimination and grouped into consistent states well, indicating promise for further application of HMM and other state transition analyses in studies of semiaquatic animals.