Projective stochastic equations and nonlinear long memory

A projective moving average $\{X_t, t \in \mathbb{Z}\}$ is a Bernoulli shift written as a backward martingale transform of the innovation sequence. We introduce a new class of nonlinear stochastic equations for projective moving averages, termed projective equations, involving a (nonlinear) kernel $Q$ and a linear combination of projections of $X_t$ on "intermediate" lagged innovation subspaces with given coefficients $\alpha_i, \beta_{i,j}$. The class of such equations include usual moving-average processes and the Volterra series of the LARCH model. Solvability of projective equations is studied, including a nested Volterra series representation of the solution $X_t$. We show that under natural conditions on $Q, \alpha_i, \beta_{i,j}$, this solution exhibits covariance and distributional long memory, with fractional Brownian motion as the limit of the corresponding partial sums process

A Forward Branching Phase Space Generator for Hadron colliders

In this paper we develop a projective phase space generator appropriate for hadron collider geometry. The generator integrates over bremsstrahlung events which project back to a single, fixed Born event. The projection is dictated by the experimental jet algorithm allowing for the forward branching phase space generator to integrate out the jet masses and initial state radiation. When integrating over the virtual and bremsstrahlung amplitudes this results in a single K-factor, assigning an event probability to each Born event. This K-factor is calculable as a perturbative expansion in the strong coupling constant. One can build observables from the Born kinematics, giving identical results to tradi- tional observables as long as the observable does not depend on the infrared sensitive jet mass or initial state radiation.Comment: 16 pages, 10 figure

Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation

We present computer-assisted methods for analyzing stochastic models of gene regulatory networks. The main idea that underlies this equation-free analysis is the design and execution of appropriately-initialized short bursts of stochastic simulations; the results of these are processed to estimate coarse-grained quantities of interest, such as mesoscopic transport coefficients. In particular, using a simple model of a genetic toggle switch, we illustrate the computation of an effective free energy and of a state-dependent effective diffusion coefficient that characterize an unavailable effective Fokker-Planck equation. Additionally we illustrate the linking of equation-free techniques with continuation methods for performing a form of stochastic "bifurcation analysis"; estimation of mean switching times in the case of a bistable switch is also implemented in this equation-free context. The accuracy of our methods is tested by direct comparison with long-time stochastic simulations. This type of equation-free analysis appears to be a promising approach to computing features of the long-time, coarse-grained behavior of certain classes of complex stochastic models of gene regulatory networks, circumventing the need for long Monte Carlo simulations.Comment: 33 pages, submitted to The Journal of Chemical Physic

Improved Algorithms for Time Decay Streams

In the time-decay model for data streams, elements of an underlying data set arrive sequentially with the recently arrived elements being more important. A common approach for handling large data sets is to maintain a coreset, a succinct summary of the processed data that allows approximate recovery of a predetermined query. We provide a general framework that takes any offline-coreset and gives a time-decay coreset for polynomial time decay functions. We also consider the exponential time decay model for k-median clustering, where we provide a constant factor approximation algorithm that utilizes the online facility location algorithm. Our algorithm stores O(k log(h Delta)+h) points where h is the half-life of the decay function and Delta is the aspect ratio of the dataset. Our techniques extend to k-means clustering and M-estimators as well

An Iterative Projective Clustering Method

AbstractIn this article we offer an algorithm recurrently divides a dataset by search of partitions via one dimensional subspace discovered by means of optimizing of a projected pursuit function. Aiming to assess the model order a resampling technique is employed. For each number of clusters, bounded by a predefined limit, samples from the projected data are drawn and clustered through the EM algorithm. Further, the basis cumulative histogram of the projected data is approximated by means of the GMM histograms constructed using the samples’ partitions. The saturation order of this approximation process, at what time the components’ amount increases, is recognized as the “true” components’ number. Afterward the whole data is clustered and the densest cluster is omitted. The process is repeated while waiting for the true number of clusters equals one. Numerical experiments demonstrate the high ability of the proposed method