Estimating a Polya frequency function_2

We consider the non-parametric maximum likelihood estimation in the class of Polya frequency functions of order two, viz. the densities with a concave logarithm. This is a subclass of unimodal densities and fairly rich in general. The NPMLE is shown to be the solution to a convex programming problem in the Euclidean space and an algorithm is devised similar to the iterative convex minorant algorithm by Jongbleod (1999). The estimator achieves Hellinger consistency when the true density is a PFF_2 itself.Comment: Published at http://dx.doi.org/10.1214/074921707000000184 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Diagrammatic perturbation theory for Stochastic nonlinear oscillators

We consider the stochastically driven one dimensional nonlinear oscillator $\ddot{x}+2\Gamma\dot{x}+\omega^2_0 x+\lambda x^3 = f(t)$ where f(t) is a Gaussian noise which, for the bulk of the work, is delta correlated (white noise). We construct the linear response function in frequency space in a systematic Feynman diagram-based perturbation theory. As in other areas of physics, this expansion is characterized by the number of loops in the diagram. This allows us to show that the damping coefficient acquires a correction at $O(\lambda^2)$ which is the two loop order. More importantly, it leads to the numerically small but conceptually interesting finding that the response is a function of the frequency at which a stochastic system is probed. The method is easily generalizable to different kinds of nonlinearity and replacing the nonlinear term in the above equation by $\mu x^2$ , we can discuss the issue of noise driven escape from a potential well. If we add a periodic forcing to the cubic nonlinearity situation, then we find that the response function can have a contribution jointly proportional to the strength of the noise and the amplitude of the periodic drive. To treat the stochastic Kapitza problem in perturbation theory we find that it is necessary to have a coloured noise.Comment: Page 23 ,5 figure

Theoretical quantification of shape distortion in fuzzy hough transform

We present a generalization of classical Hough transform in fuzzy set theoretic framework (called fuzzy Hough transform or FHT) in order to handle the impreciseness/ill-definedness in shape description. In addition to identifying the shapes, the methodology can quantify the amount of distortion present in each shape by suitably characterizing the parametric space. We extended FHT to take care of gray level images (gray FHT) in order to handle the gray level variation along with shape distortion. The gray FHT gives rise to a scheme for image segmentation based on the a priori knowledge about the shapes