positive integral operators in unbounded domains abstract we study positive integral operators k in l 2 r with continuous kernel k x y we show that if k x x l 1 r the operator is compact and hilbert schmidt if in addition k x x 0 as x k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and k is trace class replacing the first assumption by the stronger k 1 2 x x l 1 r then k l 1 r 2 and the bilinear series converges also in l1 sharp norm bounds are obtained and mercer s theorem is derived as a special case

fractional brownian motion and fractional gaussian noise in subsurface hydrology a review presentation of fundamental properties and extensions recent studies have shown that fractional brownian motion fbm and fractional gaussian noise fgn are useful in characterizing subsurface heterogeneities in addition to geophysical time series although these studies have led to a fairly good understanding of some aspects of fbm fgn a comprehensive introduction to these stochastic fractal functions is still lacking in the subsurface hydrology literature in this paper efforts have been made to define fbm fgn and present a development of their mathematical properties in a direct yet rigorous manner use of the spectral representation theorem allows one to derive spectral representations for fbm fgn even though these functions do not have classical fourier transforms the discrete and truncated forms of these representations have served as a basis for synthetic generation of fbm fgn the discrete spectral representations are developed and various implications discussed in particular it is shown that a discrete form of the fbm spectral representation is equivalent to the well known weierstrass mandelbrot random fractal function although the full implications are beyond the scope of the present paper it is observed that discrete spectral representations of fbm constitute stationary processes even though fbm is nonstationary a new and general spectral density function is introduced for construction of complicated anisotropic 3 d fractals including those characterized by vertical fgn and horizontal fbm such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes finally some basic properties of fractional levy motion and concepts of universal multifractals which can be considered as generalizations of fbm fgn are reviewed briefly

generalized partially linear single index models abstract the typical generalized linear model for a regression of a response y on predictors x z has conditional mean function based on a linear combination of x z we generalize these models to have a nonparametric component replacing the linear combination t 0x t 0z by 0 t 0x t 0z where 0 is an unknown function we call these generalized partially linear single index models gplsim the models include the single index models which have 0 0 using local linear methods we propose estimates of the unknown parameters 0 0 and the unknown function 0 and obtain their asymptotic distributions examples illustrate the models and the proposed estimation methodology

positive definite matrices and differentiable reproducing kernel inequalities abstract let i r be a interval and k i 2 c be a reproducing kernel on i by the moore aronszajn theorem every finite matrix k x i x j is positive semidefinite we show that as a direct algebraic consequence if k x y is appropriately differentiable it satisfies a 2 parameter family of differential inequalities of which the classical diagonal dominance is the order 0 case an application of these inequalities to kernels of positive integral operators yields optimal sobolev norm bounds

the weierstrass mandelbrot process revisited we derive a functional central limit theorem for quasi gaussian processes in particular we prove that the limit of the mandelbrot weierstrass process is a complex fractional brownian motion

stochastic interpolation of sparsely sampled time series via multipoint fractional brownian bridges we propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and or temporal scales to this end we extend the notion of a fractional brownian bridge defined as fractional brownian motion with a given scaling hurst exponent h and with prescribed start and end points to a bridge process with an arbitrary number of intermediate and nonequidistant points determining the optimal value of the hurst exponent h opt appropriate to interpolate the sparse signal is a very important step of our method we demonstrate the validity of our method on a signal from fluid turbulence in a high reynolds number flow and discuss the implications of the non self similar character of the signal the method introduced here could be instrumental in several physical problems including astrophysics particle tracking and specific tailoring of surrogate data as well as in domains of natural and social sciences

multifractal anisotropic scaling of the hydraulic conductivity 1 we analyzed the scaling properties of the hydraulic conductivity k at three sites in northern america made borden and cape cod we found that k at all sites exhibits multifractality fractal and multiscaling in both the vertical and horizontal directions though the multiscaling was within a range smaller than that of the maximum distance between measurements in the vertical direction the k data for made borden and cape cod were multiscaling from 0 15 to 1 35 m 0 05 to 0 5 m and 0 15 to 0 9 m respectively they were multiscaling in the horizontal direction from 9 to 45 m 1 to 10 m and 1 to 17 m respectively the multiscaling was also anisotropic evidence of scaling was poorest for the horizontal direction of the made site and it spanned half an order of magnitude such results compel one to treat multifractality in the horizontal direction of made as supported by heuristic arguments rather than by pure statistical evaluation we fitted a multifractal model to the data and estimated its parameters we found the underlying statistics of all data to be non gaussian and the model capable of reproducing the probability distribution of k data especially the negative skewness of log k we also generated two dimensional isotropic multifractal fields illustrating the role of the parameters of the selected multifractal model

fractional levy motion through path integrals fractional levy motion flm is the natural generalization of fractional brownian motion in the context of self similar stochastic processes and stable probability distributions in this paper we give an explicit derivation of the propagator of flm by using path integral methods the propagators of brownian motion and fractional brownian motion are recovered as particular cases the fractional diffusion equation corresponding to flm is also obtained

one sided cross validation abstract a new method of selecting the smoothing parameters of nonparametric regression estimators is introduced the method termed one sided cross validation oscv has the objectivity of cross validation and statistical properties comparable to those of a plug in rule the new method may be viewed as an application of the prequential model selection method of dawid as such our results identify a situation in which the prequential method is a more efficient model selector than cross validation an example simulations and theoretical results demonstrate the utility of oscv when used with local linear and kernel estimators

definition of percolation thresholds on self affine surfaces we study the percolation transition on a two dimensional substrate with long range self affine correlations we find that the position of the percolation threshold on a correlated lattice is no longer unique and depends on the spanning rule employed numerical results are provided for spanning across the lattice in specified horizontal or vertical either or both directions

solute transport in heterogeneous porous media with long range correlations transport of finite plumes in three dimensional heterogeneous and statistically isotropic aquifers is investigated where the log hydraulic conductivity is characterized by a fractional gaussian noise fgn covariance structure of hurst coefficient h leading order analytical expressions for velocity autocovariance functions uij one particle displacement covariance xij and macrodispersivity tensor ij are derived under ergodic conditions and mean uniform steady state flow nonergodic transport is then discussed for a line source of finite length either normal or parallel to the mean flow by evaluating time dependent ensemble averages of the second spatial moments zij aij aij 0 xij rij and the effective dispersivity tensor ij defined as 0 5 d aij dt where aij 0 is the initial value of the second spatial moments of a plume aij and rij is the plume centroid covariance the main finding is that in a fgn log k field the spreading of a solute plume is never ergodic as h increases effective dispersivity results differ more from their ergodic counterparts since a larger h implies the medium is more correlated the most interesting results are as follows for a source parallel to flow 22 does not decrease below zero at large time but remains strictly positive in variance with exponential or gaussian covariance for a source normal to flow 11 reaches a large time asymptote whose value depends on h as follows it decreases with h for a small source it increases reaches a peak and then decreases as h goes from 1 2 to 1 for intermediate and large sources for h 1 11 is zero irrespective of the source size

multifractal scaling of the intrinsic permeability existing fractal studies dealing with subsurface heterogeneity treat the logarithm of the permeability k as the variable of concern we treat k as a multifractal and investigate its scaling and fractality using measured horizontal k data from two locations in the united states the first data set was from a shoreline sandstone near coalinga california and the second was from an eolian sandstone goggin 1988 by applying spectral analyses and computing the scaling of moments of various orders using the double trace moment method lavallee 1991 lavallee et al 1992 we found that k is multiscaling i e scaling and multifractal we also found that the so called universal multifractal um schertzer and lovejoy 1987 model essentially a log levy multifractal was able to reproduce the multiscaling behavior reasonably well the um model has three parameters a s and h representing the multifractality index the codimension of the mean field and the distance to stationary multifractal respectively we found a 5 1 7 s 5 0 23 h 5 0 22 and a 5 1 6 s 5 0 11 h 5 0 075 for the shoreline and eolian data sets respectively the fact that a values were less than 2 indicates that the underlying statistics are non gaussian we generated stationary and nonstationary multifractals and illustrated the role of the um parameters on simulated fields studies that treated log k as the variable of concern have pointed out the necessity for large data records especially when the underlying distribution is levy stable our investigation revealed that even larger data records are required when treating k as a multifractal because log k is less intermittent or irregular than k

the fractional order governing equation of levy motion a governing equation of stable random walks is developed in one dimension this fokker planck equation is similar to and contains as a subset the second order advection dispersion equation ade except that the order a of the highest derivative is fractional e g the 1 65th derivative fundamental solutions are levy s a stable densities that resemble the gaussian except that they spread proportional to time 1 a have heavier tails and incorporate any degree of skewness the measured variance of a plume undergoing levy motion would grow faster than fickian plume at a rate of time 2 a where 0 a 2 the equation is parsimonious since the parameters are not functions of time or distance the scaling behavior of plumes that undergo levy motion is accounted for by the fractional derivatives which are appropriate measures of fractal functions in real space the fractional derivatives are integrodifferential operators so the fractional ade describes a spatially nonlocal process that is ergodic and has analytic solutions for all time and space

the screening effect in kriging when predicting the value of a stationary random field at a location x in some region in which one has a large number of observations it may be difficult to compute the optimal predictor one simple way to reduce the computational burden is to base the predictor only on those observations nearest to x as long as the number of observations used in the predictor is sufficiently large one might generally expect the best predictor based on these observations to be nearly optimal relative to the best predictor using all observations indeed this phenomenon has been empirically observed in numerous circumstances and is known as the screening effect in the geostatistical literature for linear predictors when observations are on a regular grid this work proves that there generally is a screening effect as the grid becomes increasingly dense this result requires that at high frequencies the spectral density of the random field not decay faster than algebraically and not vary too quickly examples demonstrate that there may be no screening effect if these conditions on the spectral density are violated

specific discharge covariance in a heterogeneous semiconfined aquifer in this study covariance functions of specific discharge for both one dimensional and two dimensional flows in a semiconfined aquifer were derived with mean flow parallel to the formation bedding spectral theory was used to analyze a unidirectional mean flow field of infinite extent which is perturbed by one and two dimensional statistically homogeneous hydraulic conductivity field relatively simple closed form analytical expressions requiring only some numerical integration for two dimensional flows were obtained for the longitudinal and transverse components of specific discharge covariance the spatial statistical structure of the specific discharge field was investigated in terms of the leakage factor and spatial structure of hydraulic conductivity field

a linear algebraic approach to holomorphic reproducing kernels in cn abstract let c n be a domain and k be a holomorphic reproducing kernel on by the moore aronszajn characterization every finite matrix k z i z j is positive semidefinite we show that as a direct algebraic consequence k z u satisfies an infinite 2 n parameter family of differential inequalities of which the classic diagonal dominance inequality for reproducing kernels is the order 0 case in addition the mixed hemisymmetric partial derivative of k with respect to any pair of homologous variables yields again a holomorphic reproducing kernel on these results are interpreted in terms of the general theory of reproducing kernels

establishing a direct connection between detrended fluctuation analysis and fourier analysis to understand methodological features of the detrended fluctuation analysis dfa using a higher order polynomial fitting we establish the direct connection between dfa and fourier analysis based on an exact calculation of the single frequency response of the dfa the following facts are shown analytically 1 in the analysis of stochastic processes exhibiting a power law scaling of the power spectral density psd s f f a higher order detrending in the dfa has no adverse effect in the estimation of the dfa scaling exponent which satisfies the scaling relation 1 2 2 the upper limit of the scaling exponents detectable by the dfa depends on the order of polynomial fit used in the dfa and is bounded by m 1 where m is the order of the polynomial fit 3 the relation between the time scale in the dfa and the corresponding frequency in the psd are distorted depending on both the order of the dfa and the frequency dependence of the psd we can improve the scale distortion by introducing the corrected time scale in the dfa corresponding to the inverse of the frequency scale in the psd in addition our analytical approach makes it possible to characterize variants of the dfa using different types of detrending as an application properties of the detrending moving average algorithm are discussed

the limiting distribution of the maximum rank correlation estimator han s maximum rank correlation mrc estimator is shown to be 4 consistent and asymptotically normal the proof rests on a general method for determining the asymptotic distribution of a maximization estimator a simple u statistic decomposition and a uniform bound for degenerate u processes a consistent estimator of the asymptotic covariance matrix is provided along with a result giving the explicit form of this matrix for any model within the scope of the mrc estimator the latter result is applied to the binary choice model and it is found that the mrc estimator does not achieve the semiparametric efficiency bound

detecting heteroscedasticity in nonparametric regression summary diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression the large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances new diagnostic plots are also developed and illustrated

score test for the first order autoregressive model with heteroscedasticity summary a score test is proposed for simultaneous testing of independence and homoscedasticity in the first order autoregressive model with nonconstant variance the relationships between the score test statistic and the local influence of minor perturbations on a statistical model are examined

perturbative expansion for the maximum of fractional brownian motion brownian motion is the only random process which is gaussian stationary and markovian dropping the markovian property i e allowing for memory one obtains a class of processes called fractional brownian motion indexed by the hurst exponent h for h 1 2 brownian motion is recovered we develop a perturbative approach to treat the non locality in time in an expansion in varepsilon h 1 2 this allows us to derive analytic results beyond scaling exponents for various observables related to extreme value statistics the maximum m of the process and the time t text max at which this maximum is reached as well as their joint distribution we test our analytical predictions with extensive numerical simulations for different values of h they show excellent agreement even for h far from 1 2

a fractal based stochastic interpolation scheme in subsurface hydrology real porosity and hydraulic conductivity data do not vary smoothly over space so an interpolation scheme that preserves irregularity is desirable such a scheme based on the properties of fractional brownian motion fbm and fractional gaussian noise fgn is presented following the methodology of hewett 1986 the authors test for the presence of fgn in a set of 459 hydraulic conductivity k measurements the use of rescaled range analysis strongly indicated the presence of fgn when applied to the natural logs of the k data and the resulting hurst coefficient h was determined to be 0 82 this h value was then used along with the methodology for successive random additions to generate a fbm k interpolation realization in the vertical cross section between two wells the results appeared realistic and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media copyright c 1993 american geophysical union

ubiquity of and geostatistics for nonstationary increment random fields 1 nonstationary random fields such as fractional brownian motion and fractional levy motion have been studied extensively in the hydrology literature on the other hand random fields that have nonstationary increments have seen little study a mathematical argument is presented that demonstrates processes with stationary increments are the exception and processes with nonstationary increments are far more abundant the abundance of nonstationary increment processes has important implications e g in kriging where a translation invariant variogram implicitly assumes stationarity of the increments an approach to kriging for processes with nonstationary increments is presented and accompanied by some numerical results

diffusion in one dimensional multifractal porous media we examine the scaling properties of one dimensional random walks on media with multifractal diffusivities which is a simple model for transport in scaling porous media we find both theoretically and numerically that the anomalous scaling exponent of the walk is dw 2 k 1 where k 1 is the scaling exponent of the reciprocal spatially averaged dressed resistance to diffusion since k 1 0 the walk is subdiffusive the walkers are effectively trapped in a hierarchy of barriers the trapping is dominated by contributions from a specific order of singularity associated with a phase transition between anomalous and normal diffusion we discuss the implications for transport in porous media

data driven bandwidth selection in local polynomial fitting variable bandwidth and spatial adaptation when estimating a mean regression function and its derivatives locally weighted least squares regression has proven to be a very attractive technique the present paper focuses on the important issue of how to select the smoothing parameter or bandwidth in the case of estimating curves with a complicated structure a variable bandwidth is desirable furthermore the bandwidth should be indicated by the data themselves recent developments in nonparametric smoothing techniques inspired us to propose such a data driven bandwidth selection procedure which can be used to select both constant and variable bandwidths the idea is based on a residual squares criterion along with a good approximation of the bias and variance of the estimator the procedure can be applied to select bandwidths not only for estimating the regression curve but also for estimating its derivatives the resulting estimation procedure has the necessary flexibility for capturing complicated shapes of curves this is illustrated via a large variety of testing examples including examples with a large spatial variability the results are also compared with wavelet thresholding techniques and it seems that our results are at least comparable i e local polynomial regression using our data driven variable bandwidth has spatial adaptation properties that are similar to wavelets

fractional gaussian noise enhanced information capacity of a nonlinear neuron model with binary signal input this paper reveals the effect of fractional gaussian noise with hurst exponent h 1 2 1 on the information capacity of a general nonlinear neuron model with binary signal input the fgn and its corresponding fractional brownian motion exhibit long range strong dependent increments it extends standard brownian motion to many types of fractional processes found in nature such as the synaptic noise in the paper for the subthreshold binary signal sufficient conditions are given based on the forbidden interval theorem to guarantee the occurrence of stochastic resonance while for the suprathreshold binary signal the simulated results show that additive fgn with hurst exponent h 1 2 1 could increase the mutual information or bits count the investigation indicated that the synaptic noise with the characters of long range dependence and self similarity might be the driving factor for the efficient encoding and decoding of the nervous system

temporal scaling of hydraulic head fluctuations nonstationary spectral analyses and numerical simulations 1 temporal scaling of the hydraulic head time series h t was found in a previous analysis of hourly measured head data this issue is further investigated in this paper with nonstationary spectral analyses and numerical simulations the results show that temporal scaling may indeed exist in h t which fluctuates like a fractional brownian motion in most aquifers on the basis of a linear reservoir model with a white noise recharge input we show that the variance and covariance of h t are functions of time the head variance increases with time and approaches a constant limit as time progresses while the covariance decreases with the separation time interval for a fixed time and approaches the typical exponential covariance as time increases the spectra of the simulated h t using a one dimensional transient groundwater flow model with a white noise recharge in both homogeneous and heterogeneous aquifers are shown to be proportional to f where f is frequency and 1 84 or h 0 42 heterogeneity in the hydraulic conductivity may affect the fractal dimension of h t in highly permeable aquifers but not in the low permeable aquifer simulated in this study

universal scaling of hydraulic conductivities and dispersivities in geologic media an interpretation is offered for the observation that dispersivities increase with scale apparent longitudinal dispersivity data from a variety of hydrogeologic settings are assumed to represent a continuous hierarchy of log hydraulic conductivity fields with mutually uncorrelated increments each field having its own exponential autocovariance associated integral scale and variance that increases as a power of scale such a hierarchy is shown theoretically to form a self similar random field with homogeneous increments regardless of whether or not the underlying assumption is valid one can justify interpreting the apparent dispersivities in a manner consistent with a recent quasi linear theory of non fickian and fickian dispersion in homogeneous media which supports the notion of a self similar hierarchy a posteriori the hierarchy is revealed to possess a semivariogram s cs where c is a constant and a fractal dimension d e 0 75 where e is the topological dimension of interest this can be viewed as a universal scaling rule about which large deviations occur due to local influences including the existence of discrete natural scales at which log hydraulic conductivity is statistically homogeneous as such homogeneity is at best a local phenomenon occurring intermittently over narrow bands of the scale spectrum one must question the utility of associating medium properties with representative elementary volumes and relying on fickian models of dispersion over more than relatively narrow scale intervals porous and fractured media appear to follow the same idealized scaling rule for both flow and transport raising a question about the validity of many distinctions commonly drawn between such media finally the data suggest that conditioning transport models through calibration against hydraulic measurements has the effect of filtering out large scale modes from the hierarchy

aquifer operator scaling and the effect on solute mixing and dispersion 1 since aquifer parameters may have statistical dependence structures that are present across a huge range of scales the concepts of fractional brownian motion fbm have been used in both analytic and numerical stochastic settings most previous models have used isotropic scaling characterized by a single scalar hurst coefficient any real world anisotropy has been handled by an elliptical stretching random k field we define a d dimensional extension of fbm in which the fractional order integration may take on different orders in the d primary possibly nonorthogonal scaling directions and the degree of connectivity and long range dependence is freely assigned via a probability measure on the unit sphere this approach accounts for the different scaling found in the vertical versus horizontal directions in sedimentary aquifers and allows very general degrees of continuity of k in certain directions it also allows for the representation of fracture networks in a continuum setting the eigenvectors of the scaling matrix describe the primary fracture scaling directions and discrete weights of fractional integration represent fracture continuity that may be limited to a small number of directions in a numerical experiment the motion of solutes through 2 d operator fractional gaussian fields depends very much on transverse hurst coefficients transverse scaling in the range of fractional gaussian noise engenders greater plume mixing and a transition to a fickian regime higher orders of integration in the range of fractional brownian motion are associated with thicker layering of aquifer sediments and more preferential unmixed transport therefore direct representation of the unique directional scaling properties of an aquifer is important for realistic simulation of transport

optimal smoothing in single index models single index models generalize linear regression they have applications to a variety of fields such as discrete choice analysis in econometrics and dose response models in biometrics where high dimensional regression models are often employed single index models are similar to the first step of projection pursuit regression a dimension reduction method in both cases the orientation vector can be estimated root n consistently even if the unknown univariate function or nonparametric link function is assumed to come from a large smoothness class however as we show in the present paper the similarities end there in particular the amount of smoothing necessary for root n consistent orientation estimation is very different in the two cases we suggest a simple empirical rule for selecting the bandwidth appropriate to single index models this rule is studied in a small simulation study and an application in binary response models

a note on studentizing a test for heteroscedasticity abstract breusch and pagan 1979 have recently proposed a convenient test for heteroscedasticity in general linear models this note derives the asymptotic distribution of their test under sequences of contiguous alternatives to the null hypothesis of homoscedasticity the test is shown to possess asymptotically incorrect size nominal significance level except in the case of strictly gaussian disturbances a slight modification of the test is proposed which corrects this defect

smoothing parameter selection in nonparametric regression using an improved akaike information criterion summary many different methods have been proposed to construct nonparametric estimates of a smooth regression function including local polynomial convolution kernel and smoothing spline estimators each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set in this paper an improved version of a criterion based on the akaike information criterion aic termed aicc is derived and examined as a way to choose the smoothing parameter unlike plug in methods aicc can be used to choose smoothing parameters for any linear smoother including local quadratic and smoothing spline estimators the use of aicc avoids the large variability and tendency to undersmooth compared with the actual minimizer of average squared error seen when other classical approaches such as generalized cross validation or the aic are used to choose the smoothing parameter monte carlo simulations demonstrate that the aicc based smoothing parameter is competitive with a plug in method assuming that one exists when the plug in method works well but also performs well when the plug in approach fails or is unavailable

hydraulic conductivity velocity and the order of the fractional dispersion derivative in a highly heterogeneous system 1 a one dimensional fractional order advection dispersion equation accurately models the movement of the core of the tritium plume at the highly heterogeneous made site an a priori estimate of the parameters in that equation including the order of the fractional dispersion derivative was based on the assumption that the observed power law heavy tail of the hydraulic conductivity k field would create a similarly distributed velocity field monte carlo simulations were performed to test this hypothesis results from the monte carlo analysis show that heavy tailed k fields do give rise to heavy tailed velocity fields however the exponent of the power law the tail parameter describing these two distributions is not necessarily the same the tail parameter that characterizes a velocity distribution is not solely dependent on the tail parameter that characterizes the k distribution the k field must also have long range dependence so that water may flow through relatively continuous high k channels

stochastic interpolation of aquifer properties using fractional levy motion this paper describes a new technique for conditional simulation of aquifer properties in unobserved regions the technique utilizes a recently introduced model for heterogeneity based on the levy stable family of probability distributions to achieve a higher degree of realism than is possible with gaussian based techniques specifically permeability and porosity variations are modeled using fractional levy motion a new algorithm for producing fractional levy motion conditioned to match known data is introduced two dimensional simulations using actual permeability measurements as input demonstrate that this algorithm produces random fields with the specified properties simulations using a subset of a detailed permeability map of an outcrop as input illustrate how the new method can be used to estimate important flow and transport properties and to quantify uncertainties in these estimates

nonparametric spline regression with autoregressive moving average errors summary we estimate by spline nonparametric regression an unknown function observed with autocorrelated errors when the errors are modelled by an autoregressive moving average model unknown parameters are estimated by either maximum likelihood cross validation or generalized cross validation by expressing the problem in state space form we obtain 0 n algorithms to estimate the function and its derivatives and evaluate the marginal likelihood and cross validation functions the finite sample properties of the function estimates are evaluated by an extensive simulation study and examples are given

score tests for the single index model the single index model is a generalization of the linear regression model with e y x g x where g is an unknown function the model provides a flexible alternative to the linear regression model while providing more structure than a fully nonparametric approach although the fitting of single index models does not require distributional assumptions on the error term the properties of the estimates depend on such assumptions as does practical application of the model in this article score tests are derived for three potential misspecifications of the single index model heteroscedasticity in the errors autocorrelation in the errors and the omission of an important variable in the linear index these tests have a similar structure to corresponding tests for nonlinear regression models monte carlo simulations demonstrate that the first two tests hold their nominal size well and have good power properties in identifying model violations often outperforming other tests testing for the need for ad

semiparametric least squares sls and weighted sls estimation of single index models abstract for the class of single index models i construct a semiparametric estimator of coefficients up to a multiplicative constant that exhibits 1 n consistency and asymptotic normality this class of models includes censored and truncated tobit models binary choice models and duration models with unobserved individual heterogeneity and random censoring i also investigate a weighting scheme that achieves the semiparametric efficiency bound

multifractal analyses of hydraulic conductivity distributions the concept of a universal multifractal a generalization of a monofractal is a recently developed scaling model for natural phenomena characterized by irregularity presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity k which have a significant effect on contaminant transport in the subsurface structure function analyses of four k data sets show that some vertical variations of k display multifractal structures while others are consistent with monofractal behavior in order to make multifractal concepts more useful multifractal noise is introduced and defined as the increments of a multifractal it is concluded that the multifractal formalism of schertzer and lovejoy 1987 has provided a rather general approach for modeling in k variations in the vertical with the exception of horizontal variations of the borden data all results fell within the domain of universal multifractal behavior which includes the monofractal case parameters were well defined in an empirical sense and easy to calculate indicating a robust formalism results were consistent with the recent finding of liu and molz 1997 also non gaussian and scale variant behavior in hydraulic conductivity distributions submitted to water resources research 1997 hereinafter referred to as submitted paper that k variations display increasing heterogeneity at decreasing scales

anisotropy lacunarity and upscaled conductivity and its autocovariance in multiscale random fields with truncated power variograms it has been shown by di federico and neuman 1997 1998a b that observed multiscale behaviors of subsurface fluid flow and transport variables can be explained within the context of a unified stochastic framework which views hydraulic conductivity as a random fractal characterized by a power variogram such a random field is statistically nonhomogeneous but possesses homogeneous spatial increments di federico and neuman 1997 have demonstrated that the power variogram and associated spectra of a statistically isotropic fractal field can be constructed as a weighted integral from zero to infinity an infinite hierarchy of exponential or gaussian variograms and spectra of mutually uncorrelated fields modes that are homogeneous and isotropic we show in this paper that the same holds true when the field and its constituent modes are statistically anisotropic provided the ratios between principal integral spatial correlation scales are the same for all modes we then analyze the effect of filtering out truncating modes of low high and intermediate spatial frequency from this infinite hierarchy in the real and spectral domains a low frequency cutoff renders the truncated hierarchy homogeneous the integral scales of the lowest and highest frequency cutoff modes are related to length scales of the sampling window domain and data support sample volume respectively taking the former to be proportional to the latter renders expressions for the integral scale and variance of the truncated field dependent on window and support scale in a manner previously shown to be consistent with observations in the isotropic case it also allows in principle bridging across scales at a specific locale as well as among locales by adopting either site specific or generalized variogram parameters the introduction of intermediate cutoffs allows us to account in a straightforward manner for lacunarity due to gaps in the multiscale hierarchy created by the absence of modes associated with discrete ranges of scales for example where textural and structural features are associated with distinct ranges of scale such as fractures having discrete ranges of trace length and density which dissect the rock into matrix blocks having corresponding ranges of sizes we explore mathematically and graphically the effects that anisotropy and lacunarity have on the integral scale variance covariance and spectra of a truncated fractal field we then develop an expression for the equivalent hydraulic conductivity of a box shaped porous block embedded within a multiscale log hydraulic conductivity field under mean uniform flow the block is larger than the support scale of the field but is smaller than a surrounding sampling window consequently its equivalent hydraulic conductivity is a random variable whose variance and spatial autocorrelation function conditioned on a known mean value of support scale conductivity across the window are given explicitly by our multiscale theory

testing for homogeneity of variance in time series long memory wavelets and the nile river 1 we consider the problem of testing for homogeneity of variance in a time series with long memorystructure wedemonstratethat atest whose nullhypothesis isdesigned tobewhite noisecan in fact be applied on a scale by scale basis to the discrete wavelet transform of long memory processes in particular we show that evaluating a normalized cumulative sum of squares test statistic using critical levels for the null hypothesis of white noise yields approximately the same null hypothesis rejection rates when applied to the discrete wavelet transform of samples from a fractionally differenced process the point at which the test statistic using a nondecimated version of the discrete wavelet transform achieves its maximum value can be used to estimate the time of the unknown variance change we apply our proposed test statistic on five time series derived from the historical record of nile river yearly minimum water levels covering 622 1922 a d each series exhibiting various degrees of serial correlation including long memory in the longest subseries spanning 622 1284 a d the test confirms an inhomogeneity of variance at short time scales and identifies the change point around 720 a d which coincides closely with the construction of a new device around 715 a d for measuring the nile river the test also detects a change in variance for a record of only 36 years index terms 1829 hydrology groundwater hydrology 1869 hydrology stochastic processes 3299 mathematical geophysics general or miscellaneous

the path integral formulation of fractional brownian motion for the general hurst exponent in 1995 sebastian 1995 j phys a math gen 28 4305 gave a path integral computation of the propagator of subdiffusive fractional brownian motion fbm i e fbm with a hurst or self similarity exponent h 0 1 2 the extension of sebastian s calculation to superdiffusion h 1 2 1 becomes however quite involved due to the appearance of additional boundary conditions on fractional derivatives of the path in this communication we address the construction of the path integral representation in a different fashion which allows us to treat both subdiffusion and superdiffusion on an equal footing the derivation of the propagator of fbm for the general hurst exponent is then performed in a neat and unified way

the significance of heterogeneity of evolving scales to transport in porous formations flow takes place in a heterogeneous formation of spatially variable conductivity which is modeled as a stationary space random function to model the variability at the regional scale the formation is viewed as one of a two dimensional horizontal structure a constant head gradient is applied on the formation boundary such that the flow is uniform in the mean a plume of inert solute is injected at t 0 in a volume v0 under ergodic conditions the plume centroid moves with the constant mean flow velocity u and a longitudinal macrodispersion coefficient dl may be defined as half of the time rate of change of the plume second spatial moment with respect to the centroid for a log conductivity covariance cy of finite integral scale i at first order in the variance y2 and for a travel distance l ut i dl y2ui and transport is coined as fickian ergodicity of the moments is ensured if l i where l is the initial plume scale some field observations have suggested that heterogeneity may be of evolving scales and that the macrodispersion coefficient may grow with l without reaching a constant limit anomalous diffusion to model such a behavior previous studies have assumed that cy is stationary but of unbounded integral scale with cy ar 1 0 for large lag r under ergodic conditions it was found that asymptotically dl aul1 i e non fickian behavior and anomalous dispersion the present study claims that an ergodic behavior is not possible for a given finite plume of initial size l since the basic requirement that l i cannot be satisfied for cy of unbounded scale for instance the centroid does not move any more with u but is random figure 1 owing to the large scale heterogeneity in such a situation the actual effective dispersion coefficient dl is defined as half the rate of change of the mean second spatial moment with respect to the plume centroid in each realization this is the accessible entity in a given experiment we show that in contrast with dl the behavior of dl is controlled by l and it has the fickian limit dl aul1 figure 3 we also discuss the case in which y is of stationary increments and is characterized by its variogram y then u and dl can be defined only if y is truncated equivalently an infrared cutoff is carried out in the spectrum of y however for a bounded u it is shown that dl depends only on y furthermore for y ar dl aul2l 1 i e dispersion is fickian for 0 1 whereas for 1 2 transport is non fickian since 2 dl cannot grow faster than l ut this is in contrast with a recently proposed model neuman 1990 in which the dispersion coefficient is independent of the plume size and it grows approximately like l1 5

slicing regression a link free regression method slicing regression a link free regression m e t h o d author s naihua duan and k e r c h a u l i s o u r c e the annals of statistics v o l 19 n o 2 t u n 1991 p p 5 0 5 5 3 0 p u b l i s h e d b y institute of mathematical statistics s t a b l e u r l http www jstor org stable 2242072 accessed 16 05 2011 your use of the jstor archive indicates your acceptance of jstor s terms and conditions of use available at http www jstor org page info about policies terms jsp jstor s terms and conditions of use provides in part that unless you have obtained prior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content in the jstor archive only for your personal non commercial use please contact the publisher regarding any further use of this work publisher contact information may be obtained at http www jstor org action showpublisher publishercode ims each copy of any part of a jstor transmission must contain the same copyright notice that appears on the screen or printed page of such transmission jstor is a not for profit service that helps scholars researchers and students discover use and build upon a wide range of content in a trusted digital archive we use information technology and tools to increase productivity and facilitate new forms of scholarship for more information about jstor please contact support jstor org http www jstor org

evidence for non gaussian scaling behavior in heterogeneous sedimentary formations vertical and horizontal fluctuations in permeability and porosity in sedimentary formations are analyzed and are found to be consistent with scaling models based on levy stable probability distributions the approach avoids the assumption of gaussian behavior and is supported by evidence from horizontal and vertical well logs and from permeability measurements on a sandstone outcrop and segments of core from a heterogeneous formation the incremental values in these measurement sequences are accurately modeled as having levy stable distributions the width of the distribution of increments depends on the spatial scale in a manner consistent with scaling behavior the width of the distribution is smaller for horizontal increments than for vertical increments reflecting the reduced variability in the horizontal direction the scaling parameters are in the range associated with antipersistence and are roughly the same magnitude in the vertical and horizontal directions the relationships between different physical properties are briefly studied and it is suggested that they be quantified through off diagonal terms in a multivariate levy width matrix simulations designed to reproduce the observed statistical features are also described these results have some fundamental implications as levy stable distributions require a different set of statistical tools and theoretical methods compared to finite variance distributions

contrasting stochasticity with chaos in a permutation lempel ziv complexity shannon entropy plane abstract this paper aims at introducing the lempel ziv permutation complexity vs permutation entropy plane ec plane as a tool to analyze time series with diverse dynamical nature these two quantities use the bandt and pompe representation to quantify a continuous state time series the main strength of this approach lies in the fact that this plane combines two different perspectives to study a signal one being purely statistic the permutation entropy and the other being algorithmic the lempel ziv complexity the results allow us to conclude that the ec plane constitutes an appropriate framework for i characterizing non linear chaotic maps ii distinguishing deterministic from stochastic processes and iii to discriminate between fractional brownian motion fractional gaussian noise and k noise

choice of bandwidth for kernel regression when residuals are correlated summary standard techniques for selecting the bandwidth of a kernel estimator from the data in a nonparametric regression model perform badly when the errors are correlated in this paper we propose a modified version of an existing plug in bandwidth selector the method is generalized to stationary error variables by estimating a functional of the residual covariance function the proposed bandwidth selector shows good properties in asymptotic theory and in simulations without assuming a parametric model for the error process

ensemble solute transport in two dimensional operator scaling random fields 1 motivated by field measurements of aquifer hydraulic conductivity k recent techniques were developed to construct anisotropic fractal random fields in which the scaling or self similarity parameter varies with direction and is defined by a matrix ensemble numerical results are analyzed for solute transport through these two dimensional operator scaling fractional brownian motion ln k fields both the longitudinal and transverse hurst coefficients as well as the radius of isotropy are important to both plume growth rates and the timing and duration of breakthrough it is possible to create operator scaling fractional brownian motion fields that have more continuity or stratification in the direction of transport the effects on a conservative solute plume are continually faster than fickian growth rates highly non gaussian shapes and a heavier tail early in the breakthrough curve contrary to some analytic stochastic theories for monofractal k fields the plume growth rates never exceed a mercado s 1967 purely stratified aquifer growth rate of plume apparent dispersivity proportional to mean distance apparent superstratified growth must be the result of other demonstrable factors such as initial plume size

discrimination of fractional brownian movement and fractional gaussian noise structures in permeability and related property distributions with range analyses recent studies have shown that a useful scheme to characterize subsurface heterogeneity is to use fractional brownian motion fbm and fractional gaussian noise fgn commonly used techniques to detect these fractal distributions include rescaled range spectral and semivariogram analyses although rescaled range analysis is considered to be superior to spectral or semivariogram techniques it cannot be used to clearly discriminate fgn and fbm from a given data set in some cases on the basis of the fact that both adjusted range and rescaled range analyses for a true fgn data set should yield the same slope value in the relevant log log plots a simple scheme is developed to discriminate fgn and fbm using a combination of the two analyses the usefulness of the proposed scheme is evaluated with one dimensional computational experiments and by an application to a data set of observed air permeability logs lack of clear discrimination may have contributed to inconsistent past results and it is recommended that old data sets be reanalyzed

effective medium equations for fractional fick s law in porous media abstract this paper studies reaction diffusion phenomena in disordered porous media with non fickian diffusion effects the aim is to obtain an effective medium equation of the concentration dynamics having a fractional fick s law description for the particles flux since the methodology is based on a volume averaging approach a fractional spatial averaging theorem is developed to interchange averaging integration and fractional differentiation model structure simplifications are made on the basis of an order of magnitude analysis from physical insights the closure problem associated with the effective diffusivity definition is also developed showing that the macroscale diffusion parameter is affected by i the scaling from mesoscales to macroscales and ii by the disordered structure of the porous medium