A self-organising mixture network for density modelling

A completely unsupervised mixture distribution network, namely the self-organising mixture network, is proposed for learning arbitrary density functions. The algorithm minimises the Kullback-Leibler information by means of stochastic approximation methods. The density functions are modelled as mixtures of parametric distributions such as Gaussian and Cauchy. The first layer of the network is similar to the Kohonen's self-organising map (SOM), but with the parameters of the class conditional densities as the learning weights. The winning mechanism is based on maximum posterior probability, and the updating of weights can be limited to a small neighbourhood around the winner. The second layer accumulates the responses of these local nodes, weighted by the learning mixing parameters. The network possesses simple structure and computation, yet yields fast and robust convergence. Experimental results are also presente

Hospital Treatment Rates and Spill-Over Effects: Does Ownership Matter?

This paper studies the effect of hospital ownership on treatment rates allowing for spatial correlation among hospitals. Competition among hospitals and knowledge spillovers generate significant externalities which we try to capture using the spatial Durbin model. Using a panel of 2342 hospitals in the 48 continental states observed over the period 2005 to 2008, we find significant spatial correlation of medical service treatment rates among hospitals. The paper also shows mixed results on the effect of hospital ownership on treatment rates that depends upon the market structure where the hospital is located and which varies by treatment type

Stochastic population growth in spatially heterogeneous environments: The density-dependent case

This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout $n$ patches whose population abundances are modelled as the solutions of a system of nonlinear stochastic differential equations living on $[0,\infty)^n$. We prove that $r$, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter $r$ can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $r>0$, the population abundances converge polynomially fast to a unique invariant probability measure on $(0,\infty)^n$, while when $r<0$, the population abundances of the patches converge almost surely to $0$ exponentially fast. This generalizes and extends the results of Evans et al (2014 J. Math. Biol.) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. As an example we fully analyze the two-patch case, $n=2$, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al.Comment: 43 pages, 1 figure, edited according to the suggestion of the referees, to appear in Journal of Mathematical Biolog

Alternative statistical-mechanical descriptions of decaying two-dimensional turbulence in terms of "patches" and "points"

Numerical and analytical studies of decaying, two-dimensional (2D) Navier-Stokes (NS) turbulence at high Reynolds numbers are reported. The effort is to determine computable distinctions between two different formulations of maximum entropy predictions for the decayed, late-time state. Both formulations define an entropy through a somewhat ad hoc discretization of vorticity to the "particles" of which statistical mechanical methods are employed to define an entropy, before passing to a mean-field limit. In one case, the particles are delta-function parallel "line" vortices ("points" in two dimensions), and in the other, they are finite-area, mutually-exclusive convected "patches" of vorticity which in the limit of zero area become "points." We use time-dependent, spectral-method direct numerical simulation of the Navier-Stokes equations to see if initial conditions which should relax to different late-time states under the two formulations actually do so.Comment: 21 pages, 24 figures: submitted to "Physics of Fluids