nonlinear manifold representations for functional data for functional data lying on an unknown nonlinear low dimensional space we study manifold learning and introduce the notions of manifold mean manifold modes of functional variation and of functional manifold components these constitute nonlinear representations of functional data that complement classical linear representations such as eigenfunctions and functional principal components our manifold learning procedures borrow ideas from existing nonlinear dimension reduction methods which we modify to address functional data settings in simulations and applications we study examples of functional data which lie on a manifold and validate the superior behavior of manifold mean and functional manifold components over traditional cross sectional mean and functional principal components we also include consistency proofs for our estimators under certain assumptions

regression analysis for a functional response functional responses are encountered when units are observed over time although the whole function itself is not observed a sufficiently large number of evaluations as is common with modern recording equipment are assumed to be available functional regression analysis relates the smooth functional response y t to known covariates x by a linear combination of parameter functions t which are to be estimated the model takes the standard form y t x t t t this approach provides an alternative to standard longitudinal data methods used in the biological sciences where less and noisier data necessitate parametric modeling the methodology is illustrated by an application in ergonomics

combining registration and fitting for functional models a registration method can be defined as a process of aligning features of a sample of curves by monotone transformations of their domain the aligned curves exhibit only amplitude variation and the domain transformations called warping functions capture the phase variation in the original curves in this article we precisely define a new type of registration process in which the warping functions optimize the fit of a principal components decomposition to the aligned curves the principal components are effectively the features that this process aligns we discuss the relationship of registration to closure of a function space under convex operations and define consistency for registration methods we define an explicit decomposition of functional variation into amplitude and phase partitions and develop an algorithm for combining registration with principal components analysis and apply it to simulated and real data

a fast algorithm for the unique decipherability of multivalued encodings abstract multivalued encodings constitute an interesting generalization of ordinary encodings in that they allow each source symbol to be encoded by more than one codeword in this paper the problem of testing the property of unique decipherability of multivalued encodings is considered we provide an efficient algorithm whose time complexity is o c where c is the number of codewords and is the sum of their lengths it is remarkable that the running time equals that of the fastest algorithms for testing the much simpler property of unique decipherability of ordinary encodings

incorporating parametric effects into functional principal components analysis summary the ideas of functional principal component analysis are extended to deal with data that are hybrids of functional and parametric effects the parametric effects may be more general than just the addition of a multiple of a given function a detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval and some remarks are made for the extension to a far more general case given data a procrustes fitting method can be used to estimate the parametric effects several possible ways of treating the estimated parameter values are discussed the methods are illustrated by reference to temperature data at 35 canadian weather stations

synchronizing sample curves nonparametrically more and more often the outcome of a study is not a random variable but a noisy function for each experimental unit resulting in a sample of curves typically the individual curves vary not only in amplitude or intensity but also with respect to the time axis different subjects experience certain events sooner or later analyzing such data involves finding out the time changes or curve registration among curves following our previous work where modified dynamic time warping is applied to align two curves we formulate a global minimization problem to align all curves in a sample and to compute the aligned average curve algorithms for solving the minimization problem are presented and tested with simulated and real data the test results are promising the method which involves kernel smoothing of regression functions estimates the time changes and the average of the aligned curves from noisy data large sample asymptotics is derived

on properties of functional principal components analysis summary functional data analysis is intrinsically infinite dimensional functional principal component analysis reduces dimension to a finite level and points to the most significant components of the data however although this technique is often discussed its properties are not as well understood as they might be we show how the properties of functional principal component analysis can be elucidated through stochastic expansions and related results our approach quantifies the errors that arise through statistical approximation in successive terms of orders n 1 2 n 1 n 3 2 where n denotes sample size the expansions show how spacings among eigenvalues impact on statistical performance the term of size n 1 2 illustrates first order properties and leads directly to limit theory which describes the dominant effect of spacings thus for example spacings are seen to have an immediate first order effect on properties of eigenfunction estimators but only a second order effect on eigenvalue estimators our results can be used to explore properties of existing methods and also to suggest new techniques in particular we suggest bootstrap methods for constructing simultaneous confidence regions for an infinite number of eigenvalues and also for individual eigenvalues and eigenvectors

nonparametric estimation of the mode of a distribution of random curves summary motivated by the need to develop meaningful empirical approximations to a typical data value we introduce methods for density and mode estimation when data are in the form of random curves our approach is based on finite dimensional approximations via generalized fourier expansions on an empirically chosen basis the mode estimation problem is reduced to a problem of kernel type multivariate estimation from vector data and is solved using a new recursive algorithm for finding the empirical mode the algorithm may be used as an aid to the identification of clusters in a set of data curves bootstrap methods are employed to select the bandwidth

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

stochastic modeling of large scale flow in heterogeneous unsaturated soils a simulation experiment is used to test the stochastic theory of unsaturated flow proposed by mantoglou and gelhar 1987a b c tension means and variances derived from this theory are compared to tension distributions obtained from a detailed three dimensional model developed by ababou 1988 the synthetically generated unsaturated hydraulic conductivity functions used in the detailed model vary significantly over relatively small distances results from a simulated infiltration event indicate that the stochastic theory s predictions reproduce the overall trend of the heterogeneous tension distribution deviations from the predicted mean generally lie within confidence intervals derived from the predicted variance the stochastic theory predicts less vertical moisture movement and somewhat more horizontal spreading than a comparable deterministic analysis this reflects the fact that the effective hydraulic conductivity function used in the stochastic approach is anisotropic the magnitude of this anisotropy increases with increasing tension

computing reproducing kernels with arbitrary boundary constraints smoothing with l splines involves the use of the penalty term x 2 2 int lx 2 t dt where l is an arbitrary linear differential operator of order m it also involves the choice of m co

model estimation in nonlinear regression under shape invariance given data from a sample of noisy curves we consider a nonlinear parametric regression model with unknown model function an iterative algorithm for estimating individual parameters as well as the model function is introduced under the assumption of a certain shape invariance the individual regression curves are obtained from a common shape function by linear transformations of the axes our algorithm is based on least squares methods for parameter estimation and on nonparametric kernel methods for curve estimation asymptotic distributions are derived for the individual parameter estimators as well as for the estimator of the shape function an application to human growth data illustrates the method

on multiset decipherable codes corresp in certain applications it is necessary to communicate a description of a sequence of events where the information of interest is which one of a set of possible events has occurred including multiplicity but where the order of occurrence is irrelevant typical examples are online compilations of inventories construction of histograms or updating of relative frequencies suitable codes for this purpose need not be uniquely decipherable ud in fact all that is required of such a code is that given a finite message over the code every possible parsing of the message into codewords must yield the same multiset of codewords a code with this property is referred to as a multiset decipherable msd code an msd code is said to be proper if it is not a ud code it is shown that for every n 3 there exist proper msd codes with n words for n 2 every msd code is necessarily a ud code and all evidence points to the same conclusion for n 3 it is further shown that no msd code contains a full prefix code or a full suffix code as a proper subcode and it is conjectured that despite the weaker decipherability condition every msd code satisfies the kraft inequality

a fast test for unique decipherability based on suffix trees corresp the classical algorithm for testing unique decipherability of codes is improved by using mccreight s algorithm for constructing suffix trees the complexity of the algorithm is o nm where n is the number of codewords and m is their total length efficiency is gained by avoiding repeatedly comparing subwords of the codewords

assessing the finite dimensionality of functional data summary if a problem in functional data analysis is low dimensional then the methodology for its solution can often be reduced to relatively conventional techniques in multivariate analysis hence there is intrinsic interest in assessing the finite dimensionality of functional data we show that this problem has several unique features from some viewpoints the problem is trivial in the sense that continuously distributed functional data which are exactly finite dimensional are immediately recognizable as such if the sample size is sufficiently large however in practice functional data are almost always observed with noise for example resulting from rounding or experimental error then the problem is almost insolubly difficult in such cases a part of the average noise variance is confounded with the true signal and is not identifiable however it is possible to define the unconfounded part of the noise variance this represents the best possible lower bound to all potential values of average noise variance and is estimable in low noise settings moreover bootstrap methods can be used to describe the reliability of estimates of unconfounded noise variance under the assumption that the signal is finite dimensional motivated by these ideas we suggest techniques for assessing the finiteness of dimensionality in particular we show how to construct a critical point 9q such that if the distribution of our functional data has fewer than q 1 degrees of freedom then we should be willing to assume that the average variance of the added noise is at least 9q if this level seems too high then we must conclude that the dimension is at least q 1 we show that simpler more conventional techniques based on hypothesis testing are generally not effective

nonparametric maximum likelihood estimation of the structural mean of a sample of curves a random sample of curves can be usually thought of as noisy realisations of a compound stochastic process x t z w t where z t produces random amplitude variation and w t produces random dynamic or phase variation in most applications it is more important to estimate the so called structural mean t e z t than the crosssectional mean e x t but this estimation problem is difficult because the process z t is not directly observable in this paper we propose a nonparametric maximum likelihood estimator of t this estimator is shown to be surd n consistent and asymptotically normal under the assumed model and robust to model misspecification simulations and a realdata example show that the proposed estimator is competitive with landmark registration often considered the benchmark and has the advantage of avoiding time consuming and often infeasible individual landmark identification

functional convex averaging and synchronization for time warped random curves data that can be best described as a sample of curves are now fairly common in science and engineering when the dynamics of development growth or response over time are at issue subjects or experimental units may experience events at different temporal paces for functional data where trajectories may be individually time transformed it is usually inadequate to use commonly used sample statistics such as the cross sectional mean or median or the cross sectional sample variance if one observes time warped curve data i e random curves or random trajectories that exhibit random transformations of the time scale then the usual l2 norm and metric typically are inadequate one may then consider subjecting each observed curve to a time transformation in an attempt to reverse the warping of the time scale before further statistical analysis dynamic time warping alignment curve registration and landmark based methods have been put forward with the goal of finding adequate empirical time transforma

assessing extrema of empirical principal component functions the difficulties of estimating and representing the distributions of functional data mean that principal component methods play a substantially greater role in functional data analysis than in more conventional finite dimensional settings local maxima and minima in principal component functions are of direct importance they indicate places in the domain of a random function where influence on the function value tends to be relatively strong but of opposite sign we explore statistical properties of the relationship between extrema of empirical principal component functions and their counterparts for the true principal component functions it is shown that empirical principal component funcions have relatively little trouble capturing conventional extrema but can experience difficulty distinguishing a shoulder in a curve from a small bump for example when the true principal component function has a shoulder the probability that the empirical principal component function has instead a bump is approximately equal to 1 2 we suggest and describe the performance of bootstrap methods for assessing the strength of extrema it is shown that the subsample bootstrap is more effective than the standard bootstrap in this regard a bootstrap likelihood is proposed for measuring extremum strength exploratory numerical methods are suggested

the finest homophonic partition and related code concepts let c be a finite set of n words having total length l where all words are taken over a r element alphabet the set c is called numerically decipherable if any two factorizations of the same word over the given alphabet into words in c have the same length an o nl sup 2 time and o n k l space algorithm is presented for computing the finest homophonic partition of c provided that this set is numerically decipherable the latter property can be decided by another algorithm requiring o nl time and o n k l space the two algorithms are based on a technique related to dominoes

nonparametric regression analysis of growth curves in recent years nonparametric curve estimates have been extensively explored in theoretical work there has however been a certain lack of convincing applications in particular involving comparisons with parametric techniques the present investigation deals with the analysis of human height growth where longitudinal measurements were collected for a sample of boys and a sample of girls evidence is presented that kernel estimates of acceleration and velocity of height and of height itself might offer advantages over a parametric fitting via functional models recently introduced for the specific problem considered both approaches are biased but the parametric one shows qualitative and quantitative distortion which both are not easily predictable data analytic problems involved with kernel estimation concern the choice of kernels the choice of the smoothing parameter and also whether the smoothing parameter should be chosen uniformly for all subjects or individually

on estimating functional average breakthrough curve using time warping technique and perturbation approach 1 simulated contaminant breakthrough curves btc are often used to predict mass arrival at compliance boundaries at waste storage sites in numerical simulations that involve uncertainties on input parameters such as randomly heterogeneous rock properties monte carlo simulations are commonly utilized and the mean breakthrough curve is often calculated from the arithmetic average of all realizations the arithmetic mean breakthrough curve in general overestimates the mass flow rate at early and late time but underestimates the peak mass flow rate the averaged breakthrough curve usually does not resemble any of individual breakthrough curves the reason is that btcs vary not only on amplitude but also on dynamics time and therefore it is not appropriate to take the arithmetic average directly in this study we consider each btc as a random curve and use time warping techniques to align all curves in a time warped space compute the sample mean of the curves in the time warped space and transform the means back to the original time space we show that all btcs are aligned based on the percentile of mass reaching the compliance boundary and the functional average is the percentile average of all btcs the confidence interval of the sample mean curve is estimated using the perturbation approach the functional average provides an additional metric that can be used to characterize the breakthrough behavior in addition to more traditional median and arithmetic average curves the method is illustrated using transport simulations at the material disposal area g los alamos national laboratory lanl in new mexico

convergence and consistency results for self modeling nonlinear regression on propose une methode d estimation simultanee et un algorithme iteratif simplifiant le calcul des estimateurs

cellular automaton fluids a model for flow in porous media numerical models of fluid flow through porous media can be developed from either microscopic or macroscopic properties the large scale viewpoint is perhaps the most prevalent darcy s law relates the chief macroscopic parameters of interest flow rate permeability viscosity and pressure gradient and may be invoked to solve for any of these parameters when the others are known in practical situations however this solution may not be possible attention is then typically focused on the estimation of permeability and numerous numerical methods based on knowledge of the microscopic pore space geometry have been proposed because the intrinsic inhomogeneity of porous media makes the application of proper boundary conditions difficult microscopic flow calculations have typically been achieved with idealized arrays of geometrically simple pores throats and cracks i propose here an attractive alternative which can freely and accurately model fluid flow in grossly irregular geometries this new method s

estimating multiple pathways of object growth using nonlongitudinal image data this article presents an infinite bayesian mixture of monotonic regression models for analyzing multiple growth pathways of star shaped objects growing over time by using nonlongitudinal data a motivating example is the analysis of nanocrystal growth processes a radius function representation used for the outlines of star shaped objects allows us to represent an object growth i e expansion of outlines by an increasing sequence of random radius functions we propose a monotonic regression model to fit the increasing sequence to nonlongitudinal data ensuring that the fitted radius functions always monotonically increase in the sequence to model the multiple pathways of object growth we propose the dirichlet infinite location mixture of multiple monotonic regression models and use a block gibbs sampler as a numerical solver we apply the proposed model to nonlongitudinal sets of microscopic image data for inferring multiple growth pathways of nanocrystals comparing the inference result with real nan

characteristics and controlling factors of dispersion in bounded heterogeneous porous media 1 in this study we present first order in terms of the log conductivity variance analytical solutions to displacement covariances xij and macrodispersion coefficients dij for transport of conservative solutes in two dimensional bounded heterogeneous porous media these solutions are presented as infinity series and are explicit functions of the statistics of the log hydraulic conductivity using numerical examples the convergence of these solutions in terms of the number of terms in truncated finite summations is first investigated and the accuracy of these solutions is examined by comparing them with results from monte carlo simulations and the semianalytical solutions of osnes 1998 through several examples the general features of time dependent xij and dij are discussed unlike in unbounded domains the longitudinal macrodispersivity d11 for bounded domains does not approach an asymptotic value at large dimensionless time but instead increases quickly near the downstream constant head boundary in addition d11 for bounded domains is always larger than that in unbounded domains and accordingly the transverse macrodispersivity d22 in bounded domains is smaller than that in unbounded domains these differences stem from the lateral no flow boundaries in our bounded domain models we also investigated the effect of domain sizes on xij and dij our study shows that both xij and dij depend not only on the dimensionless size of the domain but also on its aspect ratio the dependence of xij and dij on the particles initial location has also been investigated at detail our results indicate that while this dependence is very small for x11 and d11 the release location has a significant impact on both x22 and d22 our solutions to both displacement covariances and macrodispersivity are compared against those derived from tracer test data at the borden site the comparison shows that our solutions are quite close to observed data indicating that they may be applicable to predict solute transport at the field scale

conditional stochastic analysis of solute transport in heterogeneous variably saturated soils a method is developed for the conditional monte carlo simulation of steady stateflow and transient transport from point sources in variably saturated porous media it combines the geostatistical method a linearized approximation of the soil water tension perturbation solution and afinite element numerical model the method is used to investigate the usefulness of conditional simulation for predicting solute transport under a variety of sampling network designs applied to a number of hypothetical soils saturated hydraulic conductivity data yield the largest reduction of conditional uncertainty in relatively wet soils with mild heterogeneities in highly heterogeneous soils or under dry conditions soil water tension data by themselves taken at a sampling density of one to two correlation scales along the expected mean travel path can greatly reduce prediction uncertainty about solute concentration parameter uncertainty about statistical properties of the independent random variables becomes less important as the number of conditioning data increases however even with a very high number of sampling data uncertainty of predicted concentration levels remains significant

stochastic analysis of unsaturated flow in heterogeneous soils 1 statistically isotropic media steady unsaturated flow with vertical mean infiltration through unbounded heterogeneous porous media is analyzed using a perturbation approximation of the stochastic flow equation which is solved by spectral representation techniques the hydraulic conductivity k is related to the capillary pressure head by k ks exp where ks is the saturated conductivity and is a soil parameter a general formulation is presented for the case with ks and represented as statistically homogeneous spatial random fields in part 1 solutions are developed assuming is constant and representing ks variability by one dimensional and three dimensional isotropic random fields results are obtained for head variances and covariance functions effective hydraulic conductivities variances of the unsaturated hydraulic conductivity flux variances and variance of pressure gradient when the parameter is relatively large corresponding to coarse textured soils the head variance decreases and all of the results demonstrate a trend toward gravitationally dominated one dimensional vertical flow the effective conductivity is dependent on the correlation scale of ln ks and the mean hydraulic gradient

a comparison of three methods for selecting values of input variables in the analysis of output from a computer code two types of sampling plans are examined as alternatives to simple random sampling in monte carlo studies these plans are shown to be improvements over simple random sampling with respect to variance for a class of estimators which includes the sample mean and the empirical distribution function

a perturbation expansion for diffusion in a random velocity field we present a formal perturbation expansion for the large scale effective drift velocity and diffusion matrix of a medium with stationary random velocity v x and constant nonrandom diffusion matrix a on a small scale if mu denotes the mean of v x and we express v x as mu varepsilon u x then the expansion is in powers of varepsilon with mu u and a fixed we obtain explicit expressions up to second order in varepsilon which generalize the standard formulae to the case a ne 0

bayesian hierarchical curve registration functional data often exhibit a common shape but with variations in amplitude and phase across curves the analysis often proceeds by synchronization of the data through curve registration in this article we propose a bayesian hierarchical model for curve registration our hierarchical model provides a formal account of amplitude and phase variability while borrowing strength from the data across curves in the estimation of the model parameters we discuss extensions of the model by using penalized b splines in the representation of the shape and time transformation functions and by allowing temporal misalignment of the curves we discuss applications of our model to simulated data as well as to two data sets in particular we use our model in a nonstandard analysis aimed at investigating regulatory network in time course microarray data

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

alignment of curves by dynamic time warping when studying some process or development in different subjects or units be it biological chemical or physical we usually see a typical pattern common to all curves yet there is variation both in amplitude and dynamics between curves following some ideas of structural analysis introduced by kneip and gasser we study a method dynamic time warping with a proper cost function for estimating the shift or warping function from one curve to another to align the two functions for some models this method can identify the true shift functions if the data are noise free noisy data are smoothed by a nonparametric function estimate such as a kernel estimate it is shown that the proposed estimator is asymptotically normal and converges to the true shift function as the sample size per subject goes to infinity some simulation results are presented to illustrate the performance of this method

correlation model of transport in random fields this paper is an investigation of the correlation model for transport of nonreactive solutes in media with random porosity and permeability the method of perturbation is used to obtain a second order approximate nonlocal integrodifferential equation for mean concentration an approximate method of localization is used to convert to with the same order of approximation differential equations of transport exact averaged equations for one dimensional transport are examined and the question of the consistency and asymptotic behavior for approximate average equations is discussed a detailed investigation of transport in a stratified system has been carried out the second moment of concentration is examined the variance of the concentration is computed and cross correlation moments are obtained for fields of porosity and velocity including solute concentration 24 refs 4 figs

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

nonstationary flow and nonergodic transport in random porous media 1 transport processes taking place in natural formations are often characterized by the spatial nonstationarity of flow field and the nonergodic condition of solute spreading the former may originate from statistical inhomogeneity of porous media properties or from the influence of boundary conditions as well as from conditioning procedures on measured data while the latter is generally due to the finite size of the solute source in real world applications these aspects sometimes dominate the transport phenomena and known literature results based on statistical homogeneity and ergodicity assumptions cannot be properly applied in this paper a method is proposed to handle different and concurrent causes of the flow field nonstationarity and to describe nonergodic transport of inert solutes by spatial moments in a domain of finite size the goal is reached by expanding the steady state flow equation in taylor series limited to the first order and by the recursive application of finite element fe method the unknowns are the piezometric head mean values and its derivatives in respect to the fluctuating porous media hydraulic conductivity from the latter the velocity statistics are derived on the basis of darcy s law according to the first order approximation by neglecting the pore scale dispersion the spatial moments of a solute plume are a posteriori obtained by a consistent lagrangian analysis starting from the knowledge of the velocity field covariance matrices without any restriction regarding the statistical homogeneity of flow and or ergodicity conditions the proposed method is here applied to investigate the influence of boundary conditions on the solute transport developing in a limited domain the results obtained from different test cases give a velocity field behavior in agreement with literature findings and allow us to define the nonergodic transport in bounded domains the comparison of the results with some ad hoc developed monte carlo simulations ensures the applicability of the proposed approach at least to the small heterogeneity cases here considered

simulation of flow through a two dimensional random porous medium the cellular automata approximation of two dimensional hydrodynamics is used to model flow between randomly placed partially overlapping circles the flow resistance is first roughly proportional to the number of circles and then increases more strongly for higher numbers of such obstacles

design adaptive nonparametric regression abstract in this article we study the method of nonparametric regression based on a weighted local linear regression this method has advantages over other popular kernel methods moreover such a regression procedure has the ability of design adaptation it adapts to both random and fixed designs to both highly clustered and nearly uniform designs and even to both interior and boundary points it is shown that the local linear regression smoothers have high asymptotic efficiency i e can be 100 with a suitable choice of kernel and bandwidth among all possible linear smoothers including those produced by kernel orthogonal series and spline methods the finite sample property of the local linear regression smoother is illustrated via simulation studies nonparametric regression is frequently used to explore the association between covariates and responses there are many versions of kernel regression smoothers some estimators are not good for random designs such as in observational studies and

amplitude and phase variation of point processes we develop a canonical framework for the study of the problem of registration of multiple point processes subjected to warping known as the problem of separation of amplitude and phase variation the amplitude variation of a real random function y x x is an element of 0 1 corresponds to its random oscillations in the y axis typically encapsulated by its co variation around a mean level in contrast its phase variation refers to fluctuations in the x axis often caused by random time changes we formalise similar notions for a point process and nonparametrically separate them based on realisations of i i d copies pi i of the phase varying point process a key element in our approach is to demonstrate that when the classical phase variation assumptions of functional data analysis fda are applied to the point process case they become equivalent to conditions interpretable through the prism of the theory of optimal transportation of measure we demonstrate that these induce a natural wasserstein geometry tailored to the warping problem including a formal notion of bias expressing over registration within this framework we construct nonparametric estimators that tend to avoid over registration in finite samples we show that they consistently estimate the warp maps consistently estimate the structural mean and consistently register the warped point processes even in a sparse sampling regime we also establish convergence rates and derivev root n consistency and a central limit theorem in the cox process case under dense sampling showing rate optimality of our structural mean estimator in that case

eulerian lagrangian analysis of transport conditioned on hydraulic data 1 analytical numerical approach recently a unified eulerian lagrangian theory has been developed by one of us for nonreactive solute transport in space time nonstationary velocity fields we describe a combined analytical numerical method of solution based on this theory for the special case of steady state flow in a mildly fluctuating statistically homogeneous lognormal hydraulic conductivity field we take the unconditional mean velocity to be uniform but allow conditioning on measurements of log hydraulic conductivity or transmissivity and or hydraulic head this renders the velocity field nonstationary we solve the conditional transport problem analytically at early time and express it in pseudo fickian form at later time the deterministic pseudo fickian equations involve a conditional space time dependent dispersion tensor which we evaluate numerically along mean particle trajectories these equations lend themselves to accurate solution by standard galerkin finite elements on a relatively coarse grid the final step is an explicit numerical computation of lower bounds on conditional concentration prediction variance covariance and coefficient of variation travel time distribution cumulative mass release across a compliance surface the associated error and plume spatial moments our method also allows quantification of the uncertainty in the original source location of any solute particle located anywhere in the field at any time this paper describes the methodology and presents some unconditional results conditioning and more advanced computations are presented in the subsequent papers

a decision procedure for finite decipherability and synchronizability of multivalued encodings a multivalued encoding is a variable length coding system in which more than one codeword may correspond to each source symbol the properties of unique decipherability decipherability with finite delay and synchronizability for multivalued encodings are characterized in particular the property of unique decipherability is considered and a recent conjecture about decipherability with finite delay is proved to be true the notion of a synchronizable multivalued code is also introduced and a necessary and sufficient condition for a multivalued code to be synchronizable is provided some auxiliary results are also obtained

stochastic analysis of the velocity covariance and the displacement covariance tensors in partially saturated heterogeneous anisotropic porous formations considering steady state unsaturated flow in an unbounded domain of heterogeneous formation with three dimensional anisotropic structure first order approximations of the principal and off diagonal components of the velocity covariance tensor were derived for the general case in which the separation vector is inclined to the mean gradient vector j at arbitrary angles taking into account the texture of the soil materials constituting the heterogeneous formation the derived velocity covariance tensors were employed for modeling transport in the vadose zone for the general case in which j is inclined to the principal axes of the formation heterogeneity at arbitrary angles by the use of a lagrangian formulation results of this study suggest that under unsaturated flow for given statistics of the formation properties and given water saturation macrodispersion will diminish in formations of coarser texture when j does not coincide with the principal axes of the heterogeneous formation in this case and in formations of coarser texture the principal axes associated with the principal components of the dimensionless macrodispersion tensor d ij may be deflected in a direction opposite to that of j relative to the principal axes of the formation heterogeneity coarser texture may compensate for the enhanced lateral dispersion and the enhanced deflection of the principal axes of the principal components of d ij due to an increase in the statistical anisotropy of the heterogeneous formation for a formation of given statistics and texture and for a given the longitudinal component of d ij decreases while its transverse components increase when j is more inclined to the longitudinal axis of the formation heterogeneity the latter situation is pertinent to periods of redistribution between successive water applications in which increasing capillary pressure gradients combined with spatial heterogeneity in the formation properties may increase the transverse components of j

determining soil hydraulic properties by parameter estimation on the selection of a model for the hydraulic properties parameter estimation procedures involving solution of the inverse flow problem pertinent to a transient experiment constitute a powerful method of determining the soil hydraulic properties one of the problems associated with these procedures is the selection of an appropriate model to describe the soil hydraulic properties three different models were considered those of van genuchten vg and brooks and corey bc model and the exponential model of gardner for the hydraulic conductivity soil water pressure relationships coupled with a new expression for the soil water content soil water pressure relationship gr model for a given model the soil hydraulic properties were determined from simulated and measured outflow experiments supplemented with water content at a soil water pressure head of 15 000 cm h2o using the parameter estimation procedure of kool et al 1985a and data from two soils a hypothetical sandy loam soil assumed to be described by the vg model and a silt loam soil model validation tests were performed and the most appropriate model was selected from the candidate models by discrimination tests using the akaike information criterion aic in the case of the hypothetical sandy loam the vg model was found to be the most accurate and most consistent with the data the performance of both the bc and the gr models in terms of the aic was less good but very similar to each other in the case of the silt loam soil the vg model with additional parameter m which accounts for the correlation between pores and for the flow path tortuosity m 2 02 rather than a constant m 0 5 was found to be most accurate and most consistent with the data the performance of the bc and the gr models was similar when m was considered as an unknown parameter instead of a constant

statistical analysis of spatial variability in unsaturated flow parameters core scale estimates of soil parameters of the gardner russo and van gemachten models of the hydraulic conductivity and water retention functions were obtained for 417 undisturbed soil cores taken from a wall of a trench 20 m long 2 5 m deep using a procedure based on inverse problem methodology these estimates were used to evaluate the first two statistical moments of the underlying random space functions rsfs using the restricted maximum likelihood estimation procedure coupled with the weighted least squares procedure to estimate parameters of models of the covariance and the drift functions of the pertinent rsfs the fitted models were used to evaluate the mean and covariance functions of the hydraulic conductivity and water rentention functions for given water saturations covariance functions of log saturated conductivity log ks and the shape parameters of the gardner russo and van genuchten models exhibited statistical anisotropy characterized by aspect ratios that vary between 3 to 4 correlation scales of log ks were larger than those of the shape parameters consequently the product of the variance of log unsaturated conductivity and its correlation scale remained essentially invariant for a considerable range of water saturation the implications of these results regarding stochastic modeling of transport in heterogeneous porous formations and possible applications of the results of this study are discussed briefly

pairwise curve synchronization for functional data summary data collected by scientists are increasingly in the form of trajectories or curves often these can be viewed as realizations of a composite process driven by both amplitude and time variation we consider the situation in which functional variation is dominated by time variation and develop a curve synchronization method that uses every trajectory in the sample as a reference to obtain pairwise warping functions in the first step these initial pairwise warping functions are then used to create improved estimators of the underlying individual warping functions in the second step a truncated averaging process is used to obtain robust estimation of individual warping functions the method compares well with other available time synchronization approaches and is illustrated with berkeley growth data and gene expression data for multiple sclerosis

a multiresolution approach to time warping achieved by a bayesian prior posterior transfer fitting strategy summary warping is an approach to the reduction and analysis of phase variability in functional observations by applying a smooth bijection to the function argument we propose a natural representation of warping functions in terms of a new type of elementary functions named warping component functions or warplets which are combined into the warping function by composition the inverse warping function is trivial and explicit to obtain a sequential bayesian estimation strategy is introduced which fits a series of models and transfers the posterior of the previous fit into the prior of the next fit model selection is based on a warping analogue to wavelet thresholding combined with bayesian inference

resolution adaptive fixed rank kriging abstractthe spatial random effects model is flexible in modeling spatial covariance functions and is computationally efficient for spatial prediction via fixed rank kriging frk however the model depends on a class of basis functions which if not selected properly may result in unstable or undesirable results additionally the maximum likelihood ml estimates of the model parameters are commonly computed using an expectation maximization em algorithm which further limits its applicability when a large number of basis functions are required in this research we propose a class of basis functions extracted from thin plate splines the functions are ordered in terms of their degrees of smoothness with higher order functions corresponding to larger scale features and lower order ones corresponding to smaller scale details leading to a parsimonious representation of a nonstationary spatial covariance function with the number of basis functions playing the role of spatial resolution the proposed

warped functional regression a characteristic feature of functional data is the presence of phase variability in addition to amplitude variability existing functional regression methods do not handle time variability in an explicit and efficient way in this paper we introduce a functional regression method that incorporates time warping as an intrinsic part of the model the method achieves good predictive power in a parsimonious way and allows unified statistical inference about phase and amplitude components the asymptotic distribution of the estimators is derived and their finite sample properties are studied by simulation an application involving ground level ozone trajectories is presented

stochastic analysis of flow in a heterogeneous unsaturated saturated system 1 in this study we develop a stochastic model for transient unsaturated saturated flow in randomly heterogeneous media with the method of moment equations we first derive partial differential equations governing the statistical moments of the flow quantities by perturbation expansions and then implement these equations under general conditions with the method of finite differences the stochastic model developed is applicable to the entire domain of a bounded multidimensional unsaturated saturated system in the presence of random or deterministic recharge and sink source and in the presence of multiscale nonstationary medium features the unsaturated and saturated zones are coupled through the water table whose position is random in randomly heterogeneous porous media the presence of the water table renders the flow moments strongly nonstationary even in the absence of medium nonstationary features this finding is confirmed with monte carlo simulations the resulting first two moments of flow may be used to approximate confidence intervals for the flow quantities in addition the integrated stochastic flow model provides the prerequisite quantities for realistically studying contaminant transport in unsaturated saturated systems with a stochastic approach

power law for the permeability in a two dimensional disordered porous medium abstract we report numerical calculations of the permeability as a function of porosity for a two dimensional disordered porous medium this medium is modeled using the well known swiss cheese model the fluid is simulated using a cellular automata algorithm we find that for relatively high porosities the permeability decays exponentially with the density of obstacles as a consequence of this exponential behavior a power law dependence of the permeability as a function of the porosity is obtained for this model system we find that the power law exponent is given by the ratio between two characteristic scales one scale is given by the inverse of the area of one obstacle and is approximately equal to the density of obstacles necessary to reach the percolation threshold the other scale is equal to the average change in the density of obstacles necessary for the permeability to be reduced to about 1 e of its original value

integral shape averaging and structural average estimation a comparative study classical signal averaging is an important operation in signal processing applications but is meaningless if shape or time variability is present among the signals being averaged structural average estimation sae and integral shape averaging isa are recent methods that deal with these types of variability by providing a structural average for sae and a mean shape signal for isa that can act as a complement to the classical average in signal analysis in this paper these two techniques are regarded from the perspective of the time warping problem we approach isa using a new formalism that emphasizes the theoretical connections between the two techniques a simulation study is then carried out to compare the two methods in three kinds of time fluctuation two sae algorithms are employed namely dynamic time warping dtw and self modeling registration smr the discussion of the results obtained in this comparison offers specific guidelines for the application dependent use of each approach

calculation of the permeability of porous media using hydrodynamic cellular automata the permeability of two dimensional porous media is calculated numerically as a function of porosity using the hydrodynamic cellular automata lattice gas approach results are presented for systems with up to 22 million sites 8192 2688 for randomly distributed solid obstacles whose macroscopic dimensions are much longer than the mean free path of particles in the fluid the permeability varies with porositye as e 0 6 1 e fore 0 7 when the solid obstacles are much smaller than the mean free path of particles in the fluid i e when they form a dust of point objects then such a relationship no longer holds and the permeability is more than an order of magnitude smaller than for the former case the program used for the simulations is discussed and a listing is presented in the appendix which achieved a sustained speed of 185 million sites updated per second on a single processor of the cray ymp on a sun sparc workstation the same program ran about 100 times slower

solute spreading in nonstationary flows in bounded heterogeneous unsaturated saturated media 1 it is commonly assumed in stochastic solute advective transport models that either the velocity field is stationary statistically homogeneous or the mean flow is unidirectional in this study using a lagrangian approach we develop a general stochastic model for transport in variably saturated flow in randomly heterogeneous porous media the mean flow in the model is multidirectional and the velocity field can be nonstationary with location dependent statistics the nonstationarity of the velocity field may be caused by statistical nonhomogeneity of medium properties or complex boundary configurations the particle s mean position is determined using the mean lagrangian velocity particle spreading the displacement covariances is expressed in terms of the state transition matrix that satisfies a time varying dynamic equation whose coefficient matrix is the derivative of the mean lagrangian velocity field in the special cases of stationary velocity fields the transition matrix becomes the identical matrix and our model reduces to the well known model of dagan 1984 for nonstationary but unidirectional flow fields our model reduces to that of butera and tanda 1999 and sun and zhang 2000 the validity of the transport model is examined by comparisons with monte carlo simulations for the following three cases transport in a mean gravity dominated flow in an unsaturated flow with a water table boundary and in a saturated unsaturated flow an excellent agreement is found between our model results and those from monte carlo simulations

smoothed functional principal components analysis by choice of norm the principal components analysis of functional data is often enhanced by the use of smoothing it is shown that an attractive method of incorporating smoothing is to replace the usual l 2 orthonormality constraint on the principal components by orthonormality with respect to an inner product that takes account of the roughness of the functions the method is easily implemented in practice by making use of appropriate function transforms fourier transforms for periodic data and standard principal components analysis programs several alternative possible interpretations of the smoothed principal components as obtained by the method are presented some theoretical properties of the method are discussed the estimates are shown to be consistent under appropriate conditions and asymptotic expansion techniques are used to investigate their bias and variance properties these indicate that the form of smoothing proposed is advantageous under mild conditions indeed milder than those for existing methods of smoothed functional principal components analysis the choice of smoothing parameter by cross validation is discussed the methodology of the paper is illustrated by an application to a biomechanical data set obtained in the study of the behaviour of the human thumb forefinger system

clustering multivariate functional data with phase variation when functional data come as multiple curves per subject characterizing the source of variations is not a trivial problem the complexity of the problem goes deeper when there is phase variation in addition to amplitude variation we consider clustering problem with multivariate functional data that have phase variations among the functional variables we propose a conditional subject specific warping framework in order to extract relevant features for clustering using multivariate growth curves of various parts of the body as a motivating example we demonstrate the effectiveness of the proposed approach the found clusters have individuals who show different relative growth patterns among different parts of the body

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

estimating and depicting the structure of a distribution of random functions summary we suggest a nonparametric approach to making inference about the structure of distributions in a potentially infinite dimensional space for example a function space and displaying information about that structure it is suggested that the simplest way of presenting the structure is through modes and density ascent lines the latter being the projections into the sample space of the curves of steepest ascent up the surface of a functionaldata density modes are always points in the sample space and ascent lines are always one parameter structures even when the sample space is determined by an infinite number of parameters they are therefore relatively easily depicted our methodology is based on a functional form of an iterative data sharpening algorithm

stochastic analysis of steady state unsaturated flow in heterogeneous media comparison of the brooks corey and gardner russo models existing stochastic models of unsaturated flow and transport are usually developed using the simple gardner russo constitutive relationship though it is generally accepted that the more complex van genuchten and brooks corey relationships may perform better in describing experimental data in this paper we develop first order stochastic models for gravity dominated flow in second order stationary media with both the brooks corey and the gardner russo constitutive relationships these models also account for the spatial variability in effective water content while the spatial variability is generally neglected in most existing stochastic models analytical solutions are obtained for the case of one dimensional gravity dominated flow on the basis of the solutions we illustrate the differences between results from these two constitutive models through some one dimensional examples it is found that the impacts of the constitutive models on the statistical moments of suction head effective water content unsaturated hydraulic conductivity and velocity depend on the saturation ranges for example the mean head and the mean effective water content for the brooks corey model differ in a great manner with their counterparts for the gardner russo model near the dry and wet limits while the differences are small at the intermediate range of saturation this finding is confirmed with some two dimensional examples it is also found that the brook corey model has certain advantages over the gardner russo model in analyzing unsaturated flow in heterogeneous media for example the stochastic model developed based on the brooks corey function requires the coefficient of variation of head and soil parameter bc to be small 1 whereas that based on the gardner russo function assumes the one point cross covariance of head and olgr to be small 1 illustrative examples reveal that the latter condition may be violated because the one point covariance is found to increase rapidly to beyond unity as the soil becomes dry whereas the former may be readily satisfied