a stochastic version of center manifold theory random dynamical systems arise naturally if the influence of white or real noise on the parameters of a nonlinear determinstic dynamical system is studied in this situation lyapunov exponents attached to the linearized flow replace the real parts of the eigenvalues and describe the stability behavior of the linear system if at least one of them vanishes then it is possible to prove the existence of a stochastic analogue of the deterministic center manifold the asymptotic behavior of the entire system can then be derived from the lower dimensional system restricted to this stochastic center manifold a dynamical characterization of the stochastic center manifold is given and approximation results are proved

unified lasso estimation by least squares approximation we propose a method of least squares approximation lsa for unified yet simple lasso estimation our general theoretical framework includes ordinary least squares generalized linear models quantile regression and many others as special cases specifically lsa can transfer many different types of lasso objective functions into their asymptotically equivalent least squares problems thereafter the standard asymptotic theory can be established and the lars algorithm can be applied in particular if the adaptive lasso penalty and a bayes information criterion type tuning parameter selector are used the resulting lsa estimator can be as efficient as the oracle extensive numerical studies confirm our theory

manifold regularization a geometric framework for learning from labeled and unlabeled examples we propose a family of learning algorithms based on a new form of regularization that allows us to exploit the geometry of the marginal distribution we focus on a semi supervised framework that incorporates labeled and unlabeled data in a general purpose learner some transductive graph learning algorithms and standard methods including support vector machines and regularized least squares can be obtained as special cases we use properties of reproducing kernel hilbert spaces to prove new representer theorems that provide theoretical basis for the algorithms as a result in contrast to purely graph based approaches we obtain a natural out of sample extension to novel examples and so are able to handle both transductive and truly semi supervised settings we present experimental evidence suggesting that our semi supervised algorithms are able to use unlabeled data effectively finally we have a brief discussion of unsupervised and fully supervised learning within our general framework

a simple resampling method by perturbing the minimand suppose that under a semiparametric setting an estimator of a vector of parameters of interest is obtained by optimising an objective function which has a u process structure the covariance matrix of the estimator is generally a function of the underlying density function which may be difficult to estimate well by conventional methods in this paper we present a simple resampling method by perturbing the objective function repeatedly inferences of the parameters can then be made based on a large collection of the resulting optimisers we illustrate our proposal by three examples with a heteroscedastic regression model

model evaluation based on the sampling distribution of estimated absolute prediction error the construction of a reliable practically useful prediction rule for future responses is heavily dependent on the adequacy of the fitted regression model in this article we consider the absolute prediction error the expected value of the absolute difference between the future and predicted responses as the model evaluation criterion this prediction error is easier to interpret than the average squared error and is equivalent to the misclassification error for a binary outcome we show that the prediction error can be consistently estimated via the resubstitution and crossvalidation methods even when the fitted model is not correctly specified furthermore we show that the resulting estimators are asymptotically normal when the prediction rule is nonsmooth the variance of the above normal distribution can be estimated well with a perturbation resampling method with two real examples and an extensive simulation study we demonstrate that the interval estimates obtained from the above normal approximation for the prediction errors provide much more information about model adequacy than their point estimate counterparts

normal modes for non linear vibratory systems a methodology is presented which extends to non linear systems the concept of normal modes of motion which is well developed for linear systems the method is constructive for weakly non linear systems and provides the physical nature of the normal modes along with the non linear differential equations which govern their dynamics it also provides the non linear co ordinate transformation which relates the original system co ordinates to the modal co ordinates using this transformation we demonstrate how an approximate non linear version of superposition can be employed to reconstruct the overall motion from the individual non linear modal dynamics the results presented herein for non linear systems reduce to modal analysis for the linearized system when non linearities are neglected even though the approach is entirely different from the traditional one the tools employed are from the theory of invariant manifolds for dynamical systems and were inspired by the center manifold reduction technique in this paper the basic ideas are outlined a few examples are presented and some natural extensions and applications of the method are briefly described in the conclusions

a centre manifold description of containment dispersion in channels with varying flow properties high order asymptotic approximations to the equation governing the longitudinal dispersion of a passive contaminant in poiseuille channel flow are derived and their validity discussed the derivation uses centre manifold theory which provides a systematic and near rigorous approach to calculating each successive approximation it also enables the derivation of the correct initial conditions for the taylor model of shear dispersion approximations that are valid when the channel is of a varying cross section are systematically derived via this approach and the effects of time dependent flow and variable diffusivity are also investigated the resultant modifications to the effective advection velocity and the effective dispersion coefficient are calculated and some general trends indicated

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

bifurcations in fluctuating systems the center manifold approach bifurcations in fluctuating dynamical systems are studied using the ideas of center manifold reduction the method provides not only a systematic procedure for the reduction of the system to a small number of variables but also a classification scheme for the different kinds of dynamical behavior possible near bifurcation points the joint probability density factorizes into a stationary gaussian densityp v u in the fast variablesv and a time dependent densityp u f in the slow variablesu describing the dynamics on the center manifoldv v0 u p u t obeys a reduced fokker planck equation that can be written in a normal form by means of local nonlinear transformations both additive and multiplicative white noise are considered as is colored noise the results extend and formalize haken s concept of adiabatic elimination of fast variables

one step sparse estimates in nonconcave penalized likelihood models fan and li propose a family of variable selection methods via penalized likelihood using concave penalty functions the nonconcave penalized likelihood estimators enjoy the oracle properties but maximizing the penalized likelihood function is computationally challenging because the objective function is nondifferentiable and nonconcave in this article we propose a new unified algorithm based on the local linear approximation lla for maximizing the penalized likelihood for a broad class of concave penalty functions convergence and other theoretical properties of the lla algorithm are established a distinguished feature of the lla algorithm is that at each lla step the lla estimator can naturally adopt a sparse representation thus we suggest using the one step lla estimator from the lla algorithm as the final estimates statistically we show that if the regularization parameter is appropriately chosen the one step lla estimates enjoy the oracle properties with good initial estimators computationally the one step lla estimation methods dramatically reduce the computational cost in maximizing the nonconcave penalized likelihood we conduct some monte carlo simulation to assess the finite sample performance of the one step sparse estimation methods the results are very encouraging

how biased is the apparent error rate of a prediction rule abstract a regression model is fitted to an observed set of data how accurate is the model for predicting future observations the apparent error rate tends to underestimate the true error rate because the data have been used twice both to fit the model and to check its accuracy we provide simple estimates for the downward bias of the apparent error rate the theory applies to general exponential family linear models and general measures of prediction error special attention is given to the case of logistic regression on binary data with error rates measured by the proportion of misclassified cases several connected ideas are compared mallows s cp cross validation generalized cross validation the bootstrap and akaike s information criterion

homogenization of the linearized ionic transport equations in rigid periodic porous media in this paper we undertake the rigorous homogenization of a system of partial differential equations describing the transport of a n component electrolyte in a dilute newtonian solvent through a rigid porous medium the motion is governed by a small static electric field and a small hydrodynamic force which allows us to use o brien s linearized equations as the starting model we establish convergence of the homogenization procedure and discuss the homogenized equations even if the symmetry of the effective tensor is known from the literature j r looker and s l carnie transp porous media 65 107 2006 its positive definiteness does not seem to be known based on the rigorous study of the underlying equations we prove that the effective tensor satisfies onsager properties namely is symmetric positive definite this result justifies the approach of many authors who use onsager theory as starting point

electroosmosis law via homogenization of electrolyte flow equations in porous media by the homogenization approach we justify a two scale model of ion transport in porous media for one dimensional horizontal steady flows driven by a pressure gradient and an external horizontal electrical field by up scaling the electroosmotic flow equations in horizontal nanoslits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the poisson equation for induced vertical electrical field and onsager s reciprocity relations between global fluxes hydrodynamic and electric and forces horizontal pressure gradient and external electrical field in addition the two scale approach provides macroscopic mobility coefficients in the onsager relations

combining several screening tests optimality of the risk score the development of biomarkers for cancer screening is an active area of research while several biomarkers exist none is sufficiently sensitive and specific on its own for population screening it is likely that successful screening programs will require combinations of multiple markers we consider how to combine multiple disease markers for optimal performance of a screening program we show that the risk score defined as the probability of disease given data on multiple markers is the optimal function in the sense that the receiver operating characteristic roc curve is maximized at every point arguments draw on the neyman pearson lemma this contrasts with the corresponding optimality result of classic decision theory which is set in a bayesian framework and is based on minimizing an expected loss function associated with decision errors ours is an optimality result defined from a strictly frequentist point of view and does not rely on the notion of associating costs with misclassifications the implication for data analysis is that binary regression methods can be used to yield appropriate relative weightings of different biomarkers at least in large samples we propose some modifications to standard binary regression methods for application to the disease screening problem a flexible biologically motivated simulation model for cancer biomarkers is presented and we evaluate our methods by application to it an application to real data concerning two ovarian cancer biomarkers is also presented our results are equally relevant to the more general medical diagnostic testing problem where results of multiple tests or predictors are combined to yield a composite diagnostic test moreover our methods justify the development of clinical prediction scores based on binary regression

homogenization of the fluid flow and heat transfer in transpiration cooled multi layer plates to predict the aerothermal behavior of transpiration cooled plates a multi scale approach based on the homogenization method of periodic material structures is developed this method allows one to calculate effective equivalent thermophysical properties either for each layer or for the multi layer of superalloy bondcoat and tbc a general formulation is developed here for the fluid flow through a porous media which is able to deduce as equivalent macroscopic behavior either a darcy law with a constant permeability or a more general fochheimer law with a permeability function of the mean velocity effective darcy permeabilities are calculated by solving special stokes flow problems on a unit cell finally effective conductivities and permeabilities are determined for different configurations of cylindrical and shaped transpiration cooling channels

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

semi supervised approaches to efficient evaluation of model prediction performance in many modern machine learning applications the outcome is expensive or time consuming to collect whereas the predictor information is easy to obtain semi supervised ss learning aims at utilizing large amounts of unlabelled data along with small amounts of labelled data to improve the efficiency of a classical supervised approach though numerous ss learning classification and prediction procedures have been proposed in recent years no methods currently exist to evaluate the prediction performance of a working regression model in the context of developing phenotyping algorithms derived from electronic medical records we present an efficient two step estimation procedure for evaluating a binary classifier based on various prediction performance measures in the ss setting in step i the labelled data are used to obtain a non parametrically calibrated estimate of the conditional risk function in step ii ss estimates of the prediction accuracy parameters are constructed based on the estimated conditional risk function and the unlabelled data we demonstrate that under mild regularity conditions the estimators proposed are consistent and asymptotically normal importantly the asymptotic variance of the ss estimators is always smaller than that of the supervised counterparts under correct model specification we also correct for potential overfitting bias in the ss estimators in finite samples with cross validation and we develop a perturbation resampling procedure to approximate their distributions our proposals are evaluated through extensive simulation studies and illustrated with two real electronic medical record studies aiming to develop phenotyping algorithms for rheumatoid arthritis and multiple sclerosis

electro osmotic flows through topographically complicated porous media role of electropermeability tensor in the present work we consider a framework for characterizing electro osmotic flows in topographically complicated porous media and derive an effective up scaled transport parameter to quantify this we term this parameter the electro permeability which characterizes the electro osmotic flow through composite porous media in analogy with darcy s law the electro permeability tensor thus introduced serves a simple means of relating the volume flow rate with the applied electric field without going into the intricacies of the microstructure of the porous domain first we consider cases where the solid fractions have a fractal dimension generated by the mandelbrot set purely for the sake of demonstration based on such considerations we employ the method of homogenization to obtain the effective electro permeability parameter from the numerical simulations executed over a representative volume element our derived electro permeability tensor components exhibit functional relationships with the solid or liquid fraction as well as the topography of the porous medium having established these functional relationships we evaluate the tensor components for a binary composite porous medium in which one constituent has markedly high ensuremath zeta potential than the other constituent for illustration with potential relevance in microfluidics we establish the sensitivity of the electro permeability tensor on the domain morphology solid fraction ratio of solid fractions of the two phases having the two different ensuremath zeta potential values and the ensuremath zeta potential contrast and compare it with equivalent darcy permeability for the same we thus provide a simple mathematical framework that may be immensely helpful for devising a computationally efficient way of characterizing electro osmosis through topographically complicated porous media

manifold regularization and semi supervised learning some theoretical analyses manifold regularization belkin et al 2006 is a geometrically motivated framework for machine learning within which several semi supervised algorithms have been constructed here we try to provide some theoretical understanding of this approach our main result is to expose the natural structure of a class of problems on which manifold regularization methods are helpful we show that for such problems no supervised learner can learn effectively on the other hand a manifold based learner that knows the manifold or learns it from unlabeled examples can learn with relatively few labeled examples our analysis follows a minimax style with an emphasis on finite sample results in terms of n the number of labeled examples these results allow us to properly interpret manifold regularization and related spectral and geometric algorithms in terms of their potential use in semi supervised learning

on the low dimensional modelling of stratonovich stochastic differential equations abstract we develop further ideas on how to construct low dimensional models of stochastic dynamical systems the aim is to derive a consistent and accurate model from the originally high dimensional system this is done with the support of centre manifold theory and techniques aspects of several previous approaches are combined and extended adiabatic elimination has previously been used but centre manifold techniques simplify the noise by removing memory effects and with less algebraic effort than normal forms analysis of associated fokker plank equations replace nonlinearly generated noise processes by their long term equivalent white noise the ideas are developed by examining a simple dynamical system which serves as a prototype of more interesting physical situations