Inferring diffusion in single live cells at the single molecule level

The movement of molecules inside living cells is a fundamental feature of biological processes. The ability to both observe and analyse the details of molecular diffusion in vivo at the single molecule and single cell level can add significant insight into understanding molecular architectures of diffusing molecules and the nanoscale environment in which the molecules diffuse. The tool of choice for monitoring dynamic molecular localization in live cells is fluorescence microscopy, especially so combining total internal reflection fluorescence (TIRF) with the use of fluorescent protein (FP) reporters in offering exceptional imaging contrast for dynamic processes in the cell membrane under relatively physiological conditions compared to competing single molecule techniques. There exist several different complex modes of diffusion, and discriminating these from each other is challenging at the molecular level due to underlying stochastic behaviour. Analysis is traditionally performed using mean square displacements of tracked particles, however, this generally requires more data points than is typical for single FP tracks due to photophysical instability. Presented here is a novel approach allowing robust Bayesian ranking of diffusion processes (BARD) to discriminate multiple complex modes probabilistically. It is a computational approach which biologists can use to understand single molecule features in live cells.Comment: combined ms (1-37 pages, 8 figures) and SI (38-55, 3 figures

Rough path recursions and diffusion approximations

In this article, we consider diffusion approximations for a general class of stochastic recursions. Such recursions arise as models for population growth, genetics, financial securities, multiplicative time series, numerical schemes and MCMC algorithms. We make no particular probabilistic assumptions on the type of noise appearing in these recursions. Thus, our technique is well suited to recursions where the noise sequence is not a semi-martingale, even though the limiting noise may be. Our main theorem assumes a weak limit theorem on the noise process appearing in the random recursions and lifts it to diffusion approximation for the recursion itself. To achieve this, we approximate the recursion (pathwise) by the solution to a stochastic equation driven by piecewise smooth paths; this can be thought of as a pathwise version of backward error analysis for SDEs. We then identify the limit of this stochastic equation, and hence the original recursion, using tools from rough path theory. We provide several examples of diffusion approximations, both new and old, to illustrate this technique.Comment: Published at http://dx.doi.org/10.1214/15-AAP1096 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Semi-local scaling exponent estimation with box-penalty constraints and total-variation regularisation

We here establish and exploit the result that 2-D isotropic self-similar fields beget quasi-decorrelated wavelet coefficients and that the resulting localised log sample second moment statistic is asymptotically normal. This leads to the development of a semi-local scaling exponent estimation framework with optimally modified weights. Furthermore, recent interest in penalty methods for least squares problems and generalised Lasso for scaling exponent estimation inspires the simultaneous incorporation of both bounding box constraints and total variation smoothing into an iteratively reweighted least-squares estimator framework. Numerical results on fractional Brownian fields with global and piecewise constant, semi-local Hurst parameters illustrate the benefits of the new estimators

An analysis of spending behaviour under liquidity constraints with an application to financial hedging

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Variable bit rate video time-series and scene modeling using discrete-time statistically self-similar systems

This thesis investigates the application of discrete-time statistically self-similar (DTSS) systems to modeling of variable bit rate (VBR) video traffic data. The work is motivated by the fact that while VBR video has been characterized as self-similar by various researchers, models based on self-similarity considerations have not been previously studied. Given the relationship between self-similarity and long-range dependence the potential for using DTSS model in applications involving modeling of VBR MPEG video traffic data is presented. This thesis initially explores the characteristic properties of the model and then establishes relationships between the discrete-time self-similar model and fractional order transfer function systems. Using white noise as the input, the modeling approach is presented using least-square fitting technique of the output autocorrelations to the correlations of various VBR video trace sequences. This measure is used to compare the model performance with the performance of other existing models such as Markovian, long-range dependent and M/G/(infinity) . The study shows that using heavy-tailed inputs the output of these models can be used to match both the scene time-series correlations as well as scene density functions. Furthermore, the discrete-time self-similar model is applied to scene classification in VBR MPEG video to provide a demonstration of potential application of discrete-time self-similar models in modeling self-similar and long-range dependent data. Simulation results have shown that the proposed modeling technique is indeed a better approach than several earlier approaches and finds application is areas such as automatic scene classification, estimation of motion intensity and metadata generation for MPEG-7 applications

A fast Monte Carlo scheme for additive processes and option pricing

In this paper, we present a very fast Monte Carlo scheme for additive processes: the computational time is of the same order of magnitude of standard algorithms for Brownian motions. We analyze in detail numerical error sources and propose a technique that reduces the two major sources of error. We also compare our results with a benchmark method: the jump simulation with Gaussian approximation. We show an application to additive normal tempered stable processes, a class of additive processes that calibrates ``exactly" the implied volatility surface.Numerical results are relevant. This fast algorithm is also an accurate tool for pricing path-dependent discretely-monitoring options with errors of one bp or below

Aspects of positive definiteness and gaussian processes on planet earth

This thesis studies characterisations and properties of spatial and spatio-temporal Gaussian processes defined over the sphere (or in the spatio-temporal case the product of the sphere and the real line). Such processes are of importance in global weather and climate science, where the geometry is necessarily spherical, but, especially in the dynamic setting, they are less well-studied than their Euclidean counterparts. Beginning with Brownian motion, we first look at characterising Gaussian randomness on the sphere and sphere-cross-line, and how it compares with the Euclidean setting -- we show that the characterisation theorems of Gaussian processes on spaces of types spanning the real line, the sphere and sphere-cross-line can be phrased as consequences of a powerful general theorem of harmonic analysis. We go on to find the answer to a recent question posed about dimension-hopping operators for positive-definite (i.e. covariance) functions on the sphere-cross-line, and consider how we could go about constructing dimension-hopping operators with the semi-group property on the sphere. Later, we address the theory of the path properties of these processes, extending a finite-dimensional result the the infinite-dimensional case and showing that a remarkably elegant approach for processes on Euclidean space carries over to our setting. We finish by finding the analogue of the powerful Ciesielski isomorphism for continuous functions on the two-sphere.Open Acces

Virtual Super Resolution of Scale Invariant Textured Images Using Multifractal Stochastic Processes

International audienceWe present a new method of magnification for textured images featuring scale invariance properties. This work is originally motivated by an application to astronomical images. One goal is to propose a method to quantitatively predict statistical and visual properties of images taken by a forthcoming higher resolution telescope from older images at lower resolution. This is done by performing a virtual super resolution using a family of scale invariant stochastic processes, namely compound Poisson cascades, and fractional integration. The procedure preserves the visual aspect as well as the statistical properties of the initial image. An augmentation of information is performed by locally adding random small scale details below the initial pixel size. This extrapolation procedure yields a potentially infinite number of magnified versions of an image. It allows for large magnification factors (virtually infinite) and is physically conservative: zooming out to the initial resolution yields the initial image back. The (virtually) super resolved images can be used to predict the quality of future observations as well as to develop and test compression or denoising techniques