confidence intervals for an effect size measure based on the mann whitney statistic part 1 general issues and tail area based methods for two random variables x and y pr y x pr y x is advocated as a general measure of effect size to characterize the degree of separation of their distributions it is estimated by u mn a generalization of the mann whitney u statistic derived by dividing u by the product of the two sample sizes it is equivalent to the area under the receiver operating characteristic curve it is readily visualized in terms of two gaussian distributions with appropriately separated peaks the effect of discretization of a continuous variable is explored tail area based confidence interval methods are developed which can be applied to very small samples or extreme outcomes copyright 2005 john wiley sons ltd

scaling and river networks a landau theory for erosion we propose a coarse grained theory for the formation of a river network in the form of a langevin equation for the erosion of the landscape coupled to a conservation law for the surface water flow we claim that this is the universal form for large scale behavior we show by simulations of a discrete model that represents the same dynamics that the slope area law the basin size distribution law and horton s laws agree with real rivers we discuss the relationship to optimal channel networks and to self organized criticality

gibbs like ensembles and the inference of pure states abstract arguments of the jaynes maximum entropy sort have proved to be surprisingly successful in providing one with approximate descriptions of pure states in a variety of scenarios entirely bypassing any consideration of schrodinger s equation thus far however the concomitant algorithm was unable to incorporate input information concerning expectation values of complementary observables we discuss here how to overcome this difficulty by taking recourse to ideas borrowed from gibbs ones concerning statistical ensembles the ensuing approach is able to yield starting from a reduced set of expectation values approximate wave functions of a rather good quality as illustrated by simple examples

the maximum entropy inference of solutions to pdes we use the maximum entropy variational technique to infer solutions to second order partial differential equations we restrict ourselves to problems with dirichlet boundary conditions first we construct a basis of moment functions in terms of which we set up the constraints for a maximum entropy inversion for the inference of solutions to elliptic equations then we extend the scheme to the inference of time dependence in evolution equations this extension is done by solving a small system of first order initial value differential equations for the moments of the solution vector

universality of jaynes approach to the evolution of time dependent probability distributions abstract in order to ascertain which are the essential requirements that allow for the application of maxent techniques in connection with evolution problems we study time dependent probability distributions governed by a general linear evolution equation that admits as particular instances the lioville diffusion and fokker planck ones a general variational principle is introduced and the properties of an appropriately devised maxent ansatz are investigated the functional relation that gives the time derivative of the entropy in terms of the time dependent maxent distribution function is seen to be always of the same form as the one that holds in the case of exact solutions

covariate adjustment in randomized trials with binary outcomes targeted maximum likelihood estimation covariate adjustment using linear models for continuous outcomes in randomized trials has been shown to increase efficiency and power over the unadjusted method in estimating the marginal effect of treatment however for binary outcomes investigators generally rely on the unadjusted estimate as the literature indicates that covariate adjusted estimates based on the logistic regression models are less efficient the crucial step that has been missing when adjusting for covariates is that one must integrate average the adjusted estimate over those covariates in order to obtain the marginal effect we apply the method of targeted maximum likelihood estimation tmle to obtain estimators for the marginal effect using covariate adjustment for binary outcomes we show that the covariate adjustment in randomized trials using the logistic regression models can be mapped by averaging over the covariate s to obtain a fully robust and efficient estimator of the marginal effect which equals a targeted maximum likelihood estimator this tmle is obtained by simply adding a clever covariate to a fixed initial regression we present simulation studies that demonstrate that this tmle increases efficiency and power over the unadjusted method particularly for smaller sample sizes even when the regression model is mis specified

solving differential equations by a maximum entropy minimum norm method with applications to fokker planck equations the method of maximum entropy minimum norm is utilized to produce a general method of solving differential equations the technique is a generalization and extension of previous work performed by baker jarvis j math phys 30 302 1989 it is found that introducing an additional constraint on the norm of the solution vector produces a probability distribution that is integrable over the entire real axis a number of simplifications occur in this extended method the lagrange multipliers and solution vector can be solved for explicitly thus eliminating the necessity of solving systems of nonlinear equations for the lagrange multipliers as was required in the previous approach it is shown that the solution obtained is equivalent to a minimum norm approximation the maximum entropy solution of differential equations with fourier moments is shown to be identical to a fourier series solution additionally the new method is applied to solving the random walk and fokker planck equations

approximate solution of fredholm integral equations by the maximum entropy method an approximate means of solving fredholm integral equations by the maximum entropy method is developed the fredholm integral equation is converted to a generalized moment problem whose approximate solution by maximum entropy methods has been successfully implemented in a previous paper by mead and papanicolaou l r mead and n papanicolaou j math phys 25 2404 1984 several explicit examples are given of approximate maximum entropy solutions of fredholm integral equations of the first and second kinds and of the wiener hopf type both the weaknesses and strengths of the method are discussed

using regression models to analyze randomized trials asymptotically valid hypothesis tests despite incorrectly specified models regression models are often used to test for cause effect relationships from data collected in randomized trials or experiments this practice has deservedly come under heavy scrutiny because commonly used models such as linear and logistic regression will often not capture the actual relationships between variables and incorrectly specified models potentially lead to incorrect conclusions in this article we focus on hypothesis tests of whether the treatment given in a randomized trial has any effect on the mean of the primary outcome within strata of baseline variables such as age sex and health status our primary concern is ensuring that such hypothesis tests have correct type i error for large samples our main result is that for a surprisingly large class of commonly used regression models standard regression based hypothesis tests but using robust variance estimators are guaranteed to have correct type i error for large samples even when the models are incorrectly specified to the best of our knowledge this robustness of such model based hypothesis tests to incorrectly specified models was previously unknown for poisson regression models and for other commonly used models we consider our results have practical implications for understanding the reliability of commonly used model based tests for analyzing randomized trials

quantal euclidean mappings abstract given a set of non commuting operators o 1 o n and the assumed knowledge of the expectation values of a subset containing just m of them we discuss in quite general terms what can be predicted about the behaviour of the expectation values of the remaining operators when the concomitant wave functions are not available not even in principle because the pertinent hamiltonian is unknown an r m r n m mapping ensues information theoretical tools are employed and some simple examples examined

the role of information measures in the determination of the maximum entropy minimum norm solution of the generalized inverse problem we discuss here the role of different information measures vis a vis the maximum entropy minimum norm method of baker jarvis et al bj for dealing with under determined inverse linear transforms it is shown that the structure of the approximate solutions provided by the bj approach does not depend upon the functional form of the information measure

maximum entropy method for solving the collisional vlasov equation abstract by means of the maximum entropy method the reduced vlasov fokker planck equation has been solved explicitly the behaviour of the approximate maximum entropy distribution functions are shown graphically and compared with that obtained by using lie group method the results reported in this article provide further evidence of the usefulness of both maximum entropy and lie group methods for obtaining time dependent solution in compact form

river landscapes and optimal channel networks we study tree structures termed optimal channel networks ocns that minimize the total gravitational energy loss in the system an exact property of steady state landscape configurations that prove dynamically accessible and strikingly similar to natural forms here we show that every ocn is a so called natural river tree in the sense that there exists a height function such that the flow directions are always directed along steepest descent we also study the natural river trees in an arbitrary graph in terms of forbidden substructures which we call k path obstacles and ocns on a d dimensional lattice improving earlier results by determining the minimum energy up to a constant factor for every d 2 results extend our capabilities in environmental statistical mechanics

probabilistic index models summary we present a semiparametric statistical model for the probabilistic index which can be defined as p y y a where y and y a are independent random response variables associated with covariate patterns x and x a respectively a link function defines the relationship between the probabilistic index and a linear predictor asymptotic normality of the estimators and consistency of the covariance matrix estimator are established through semiparametric theory the model is illustrated with several examples and the estimation theory is validated in a simulation study

exploring network scaling through variations on optimal channel networks metabolic allometry a common pattern in nature is a close to 3 4 power scaling law between metabolic rate and body mass in organisms across and within species an analogous relationship between metabolic rate and water volume in river networks has also been observed optimal channel networks ocns at local optima accurately model many scaling properties of river systems including metabolic allometry ocns are embedded in 2d space this work extends the model to three dimensions in this paper we compare characteristics of 3d ocns with 2d ocns and with organic metabolic networks studying the scaling behaviors of area length volume and energy we find that the 3d ocn has predictable characteristics analogous to those of the 2d version as well as scaling properties similar to metabolic networks in biological organisms

estimation of the mean of a k sample u statistic with missing outcomes and auxiliaries we propose estimators of the mean of a k sample u statistic when data on the outcomes of interest are missing in some sampled units and auxiliary variables are available in the entire sample the proposed estimators exploit the information available in the auxiliaries without requiring assumptions about the joint distribution of the auxiliaries and outcomes the properties of the proposed estimators are derived as a consequence of general results on efficient semiparametric estimation of the mean of a k sample u statistic with missing at random outcomes observed auxiliary variables and known missingness probabilities we illustrate the methods with the estimation of the area under a receiver operating characteristic curve when the biomarker data are observed only in subsamples of the diseased and healthy subjects

maximum entropy approach with higher moments for solving fokker planck equation abstract the maximum entropy method with higher number of moments is used to solve the fokker planck equation an adopted newton method is used to iterate the maximum entropy set of equations the method is used to calculate the probability density function of the fokker planck equation the calculations are carried out for three examples 1 the bistable systems of double well potential that is used in many problems related to the fluctuation and relaxation processes in far from equilibrium systems 2 the malthus verhulst model which is used to study the evolution of the number of individuals of an ecological species and the evolution of the intensity of the laser light 3 the black scholes equation used in financial market option pricing although the maximum entropy approach has several advantages it is not convergent at large times and so cannot be used to calculate the steady state solution

probabilistic index an intuitive non parametric approach to measuring the size of treatment effects effect sizes es tell the magnitude of the difference between treatments and ideally should tell clinicians how likely their patients will benefit from the treatment currently used es are expressed in statistical rather than in clinically useful terms and may not give clinicians the appropriate information we restrict our discussion to studies with two groups one with n patients receiving a new treatment and the other with m patients receiving the usual or no treatment the standardized mean difference e g cohen s d is a well known index for continuous outcomes there is some intuitive value to d but measuring improvement in standard deviations sd is a statistical concept that may not help a clinician how much improvement is a half sd a more intuitive and simple to calculate es is the probability that the response of a patient given the new treatment x is better than the one for a randomly chosen patient given the old or no treatment y i e p x y larger values meaning better outcomes this probability has an immediate identity with the area under the curve auc measure in procedures for receiver operator characteristic roc curve comparing responses to two treatments it also can be easily calculated from the mann whitney u wilcoxon or kendall statistics we describe the characteristics of an ideal es we propose p x y as an alternative index summarize its correspondence with well known non parametric statistics compare it to the standardized mean difference index and illustrate with clinical data copyright 2005 john wiley sons ltd

feasible optimality implies hack s law we analyze the elongation the scaling properties of drainage area with mainstream length in optimal channel networks ocns obtained through different algorithms searching for the minimum of a functional computing the total energy dissipation of the drainage system the algorithms have different capabilities to overcome the imprinting of initial and boundary conditions and thus they have different chances of attaining the global optimum we find that suboptimal shapes i e dynamically accessible states characterized by locally stationary total potential energy show the robust type of elongation that is consistently observed in nature this suggestive and directly measurable property is not found in the so called ground state i e the global minimum whose features including elongation are known exactly the global minimum is shown to be too regular and symmetric to be dynamically accessible in nature owing to features and constraints of erosional processes thus hack s law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dynamic origin of fractal forms in nature

maximum entropy principle and classical evolution equations with source terms abstract we devise a maximum entropy technique to construct approximate time dependent solutions to evolution equations endowed with source terms and consequently not preserving normalization in some special cases the method yields exact solutions it is shown that the present implementation of the maximum entropy prescription always even in the case of approximate solutions preserves the exact functional relationship between the time derivative of the entropy and the time dependent solutions of the evolution equation other properties of the maximum entropy solutions and some illustrative examples are also discussed

increasing the power of the mann whitney test in randomized experiments through flexible covariate adjustment the mann whitney u test is frequently used to evaluate treatment effects in randomized experiments with skewed outcome distributions or small sample sizes it may lack power however because it ignores the auxiliary baseline covariate information that is routinely collected wald and score tests in so called probabilistic index models generalize the mann whitney u test to enable adjustment for covariates but these may lack robustness by demanding correct model specification and do not lend themselves to small sample inference using semiparametric efficiency theory we here propose an alternative extension of the mann whitney u test which increases its power by exploiting covariate information in an objective way and which lends itself to permutation inference simulation studies and an application to an hiv clinical trial show that the proposed permutation test attains the nominal type i error rate and can be drastically more powerful than the classical mann whitney u test copyright 2014 john wiley sons ltd

quantum inference as an under determined linear problem and the collective of accessible states abstract a detailed mathematical analysis of the collective of accessible micro states compatible with a given set of a priori fixed expectation values generalized ensemble is undertaken to such an effect the problem of quantum inference is framed as an under determined linear problem the analysis is based upon the splitting of the pertinent statistical distribution the density matrix one is trying to infer into two orthogonal components one of the components happens to coincide with the minimum norm solution of the linear problem posed by the fulfillment of the a priori conditions on the given expectation values the remaining component belongs to the null space of the expectation values transformation

assortativity and act degree distribution of some collaboration networks empirical investigation results on weighted and un weighted assortativity act degree distribution degree distribution and node strength distribution of nine real world collaboration networks have been presented the investigations propose that act degree distribution degree distribution and node strength distribution usually show so called shifted power law spl function forms which can continuously vary from an ideal power law to an ideal exponential decay two parameters a and z can be used for description of the distribution functions another conclusion is that assortativity coefficient and the parameter a or z is monotonously dependent on each other by the collaboration network evolution model introduced in a reference p zhang et al physica a 360 2006 599 we analytically derived the spl distributions which typically appeared in general situations where nodes are selected partially randomly with a probability p and partially by linear preferential principle with the probability 1 p the analytic discussion gives an explicit expression on the relationship between the random selection proportion p and the parameters a and z the numerical simulation results by the model show a monotonic dependence of assortativity on the random selection proportion p the empirically obtained assortativity versus a or z curve for the four collaboration networks with small maximal act size t max shows a good agreement with the model prediction according to the curves one can qualitatively judge the random selection proportion of the real world network in its evolution process r 2007 elsevier b v all rights reserved

estimating exposure effects by modelling the expectation of exposure conditional on confounders in order to estimate the causal effects of one or more exposures or treatments on an outcome of interest one has to account for the effect of confounding factors which both covary with the exposures or treatments and are independent predictors of the outcome in this paper we present regression methods which in contrast to standard methods adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders in the special case of a univariate dichotomous exposure or treatment this conditional expectation is identical to what rosenbaum and rubin have called the propensity score they have also proposed methods to estimate causal effects by modelling the propensity score our methods generalize those of rosenbaum and rubin in several ways first our approach straightforwardly allows for multivariate exposures or treatments each of which may be continuous ordinal or discrete second even in the case of a single dichotomous exposure our approach does not require subclassification or matching on the propensity score so that the potential for residual confounding i e bias due to incomplete matching is avoided third our approach allows a rather general formalization of the idea that it is better to use the estimated propensity score than the true propensity score even when the true score is known the additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model to illustrate our methods we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2 713 adult white males we compare the results with those obtained using standard methods

statistical properties of aggregation with injection we study a generalized aggregation process in which charged particles diffuse and coalesce randomly on a lattice for one dimensional and mean field models we show that there exists a statistically invariant steady state when randomly charged particles are continuously injected the steady state charge distribution obeys a power law with the exponent depending both on the type of the injection and on the spatial dimension the response of the system to a perturbation i e relaxation is characterized by either a power law decay t 1 or a compressed exponential decay exp t 1

invasion percolation and global optimization invasion bond percolation ibp is mapped exactly into prim close quote s algorithm for finding the shortest spanning tree of a weighted random graph exploring this mapping which is valid for arbitrary dimensions and lattices we introduce a new ibp model that belongs to the same universality class as ibp and generates the minimal energy tree spanning the ibp cluster copyright ital 1996 the american physical society

statistical dynamics of early river networks abstract based on local erosion rule and fluctuations in rainfall geology and parameters of a river channel a generalized langevin equation is proposed to describe the random prolongation of a river channel this equation is transformed into the fokker plank equation to follow the early evolution of a river network and the variation of probability distribution of channel lengths the general solution of the equation is in the product form of two terms one term is in power form and the other is in exponent form this distribution shows a complete history of a river network evolving from its infancy to adulthood the infancy is characterized by the gaussian distribution of the channel lengths while the adulthood is marked by a power law distribution of the channel lengths the variation of the distribution from the gaussian to the power law displays a gradual developing progress of the river network the distribution of basin areas is obtained by means of hack s law these provide us with new understandings towards river networks

trees networks and hydrology this paper reviews theoretical and observational material on form and function of natural networks appeared in somewhat disparate contexts from physics to biology whose study is related to hydrologic research moving from the exact result that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution we discuss the properties and the dynamic origin of the scale invariant structure of river patterns and its relation to optimal selection we argue that at least in the fluvial landscape nature works through imperfect searches for dynamically accessible optimal configurations and that purely random or deterministic constructs are clearly unsuitable to properly describe natural network forms we also show that optimal networks are spanning loopless configurations only under precise physical requirements that arise under the constraints imposed by continuity in the case of rivers every spanning tree proves a local minimum of total energy dissipation this is stated in a theorem form applicable to generic networks suggesting that other branching structures occurring in nature e g scale free and looping may possibly arise through optimality to different selective pressures we thus conclude that one recurrent self organized mechanism for the dynamic origin of fractal forms is the robust strive for imperfect optimality that we see embedded in many natural patterns chief and foremost hydrologic ones copyright 2006 by the american geophysical union

causal inference for mann whitney wilcoxon rank sum and other nonparametric statistics the nonparametric mann whitney wilcoxon mww rank sum test is widely used to test treatment effect by comparing the outcome distributions between two groups especially when there are outliers in the data however such statistics generally yield invalid conclusions when applied to nonrandomized studies particularly those in epidemiologic research although one may control for selection bias by using available approaches of covariates adjustment such as matching regression analysis propensity score matching and marginal structural models such analyses yield results that are not only subjective based on how the outliers are handled but also often difficult to interpret a popular alternative is a conditional permutation test based on randomization inference rosenbaum pr covariance adjustment in randomized experiments and observational studies statistical science 2002 17 3 286 327 because it requires strong and implausible assumptions that may not be met in most applications this approach has limited applications in practice in this paper we address this gap in the literature by extending mww and other nonparametric statistics to provide causal inference for nonrandomized study data by integrating the potential outcome paradigm with the functional response models frm frm is uniquely positioned to model dynamic relationships between subjects rather than attributes of a single subject as in most regression models such as the mww test within our context the proposed approach is illustrated with data from both real and simulated studies copyright 2013 john wiley sons ltd

optimal transportation network with concave cost functions loop analysis and algorithms transportation networks play a vital role in modern societies structural optimization of a transportation system under a given set of constraints is an issue of great practical importance for a general transportation system whose total cost c is determined by c sigma c i j ij i ij with c ij i ij being the cost of the flow i ij between node i and node j banavar and co workers phys rev lett 84 4745 2000 proved that the optimal network topology is a tree if c ij proportional to parallel to i ij parallel to gamma with 0 1 the same conclusion also holds in the more general case where all the flow costs are strictly concave functions of the flow i ij to further understand the qualitative difference between systems with concave and convex cost functions a loop analysis of transportation cost is performed in the present paper and an alternative mathematical proof of the optimality of tree formed networks is given the simple intuitive picture of this proof then leads to an efficient global algorithm for the searching of optimal structures for a given transportation system with concave cost functions

exact and asymptotically robust permutation tests given independent samples from p and q two sample permutation tests allow one to construct exact level tests when the null hypothesis is p q on the other hand when comparing or testing particular parameters of p and q such as their means or medians permutation tests need not be level or even approximately level in large samples under very weak assumptions for comparing estimators we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability in finite samples when the underlying distributions are identical a quite general theory is possible based on a coupling construction as well as a key contiguity argument for the binomial and hypergeometric distributions the ideas are broadly applicable and special attention is given to a nonparametric k sample behrens fisher problem whereby a permutation test is constructed which is exact level under the hypothesis of identical distributions but has asymptotic rejection probability under the more general null hypothesis of equality of means a monte carlo simulation study is performed 2010 msc subject classifications primary 62e20 secondary 62g10

improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data which involve models for both the propensity score and the regression of outcome on covariates the usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations we propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods even with some estimated propensity scores close to zero

properties of maximum entropy approximate solutions to fredholm integral equations the properties of the maximum entropy approximate solutions to fredholm integral equations of the second kind are discussed it is proved that the approximate solutions exist and converge to the exact one for a wider class of problems than previously determined the correspondence between the maximum entropy technique and the galerkin method is established and on its basis an upper a posterior bound for a solution is found

geometry of river networks i scaling fluctuations and deviations this paper is the first in a series of three papers investigating the detailed geometry of river networks branching networks are a universal structure employed in the distribution and collection of material large scale river networks mark an important class of two dimensional branching networks being not only of intrinsic interest but also a pervasive natural phenomenon in the description of river network structure scaling laws are uniformly observed reported values of scaling exponents vary suggesting that no unique set of scaling exponents exists to improve this current understanding of scaling in river networks and to provide a fuller description of branching network structure here we report a theoretical and empirical study of fluctuations about and deviations from scaling we examine data for continent scale river networks such as the mississippi and the amazon and draw inspiration from a simple model of directed random networks we center our investigations on the scaling of the length of a subbasin s dominant stream with its area a characterization of basin shape known as hack s law we generalize this relationship to a joint probability density and provide observations and explanations of deviations from scaling we show that fluctuations about scaling are substantial and grow with system size we find strong deviations from scaling at small scales which can be explained by the existence of a linear network structure at intermediate scales we find slow drifts in exponent values indicating that scaling is only approximately obeyed and that universality remains indeterminate at large scales we observe a breakdown in scaling due to decreasing sample space and correlations with overall basin shape the extent of approximate scaling is significantly restricted by these deviations and will not be improved by increases in network resolution

on hack s law hack s law is reviewed emphasizing its implications for the elongation of river basins as well as its connections with their fractal characteristics the relation between hack s law and the internal structure of river basins is investigated experimentally through digital elevation models it is found that hack s exponent elongation and some relevant fractal characters are closely related the self affine character of basin boundaries is shown to be connected to the power law decay of the probability of total contributing areas at any link and to hack s law an explanation for hack s law is derived from scaling arguments from the results we suggest that a statistical framework referring to the scaling invariance of the entire basin structure should be used in the interpretation of hack s law

local minimal energy landscapes in river networks the existence and stability of a universality class associated with local minimal energy landscapes is investigated using extensive numerical simulations we first study the dependence on a parameter ensuremath gamma of a partial differential equation which was proposed to describe the evolution of a rugged landscape toward a local minimum of the dissipated energy we then compare the results with those obtained by an evolution scheme based on a variational principle the optimal channel networks it is found that both models yield qualitatively similar river patterns and similar dependences on ensuremath gamma however the aggregation mechanism is strongly dependent on the value of ensuremath gamma a careful analysis suggests that scaling behaviors may depend weakly on both ensuremath gamma and on initial conditions but in all cases are within observational data predictions consequences of our results are finally discussed and the most plausible scenario is presented

maximum entropy minimum norm method for the determination of level densities abstract a new simple maximum entropy minimum norm inversion method is devised for obtaining the energy density of states from some limited information about the partition function an improvement upon the saddle point approximation can be obtained with the present method at low temperatures where the saddle point approach is less accurate

solution to boundary value problems using the method of maximum entropy the method of maximum entropy is used to solve a class of linear boundary value problems the method is based on using various moments of the differential equation as constraints when maximizing the entropy various examples are presented and compared to exact solutions for varying numbers of moments it is found that the maximum entropy approximation is in many cases better than a fourier series solution for a given number of expansion terms and moments the method is very general and will find applications in many areas of physics a comparison of the amount of work necessary for the maximum entropy solution versus finite difference techniques is presented and it is found that the maximum entropy technique shows promise as an alternative solution technique

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

analytical and numerical study of optimal channel networks we analyze the optimal channel network model for river networks using both analytical and numerical approaches this is a lattice model in which a functional describing the dissipated energy is introduced and minimized in order to find the optimal configurations the fractal character of river networks is reflected in the power law behavior of various quantities characterizing the morphology of the basin in the context of a finite size scaling ansatz the exponents describing the power law behavior are calculated exactly and show mean field behavior except for two limiting values of a parameter characterizing the dissipated energy for which the system belongs to different universality classes two modified versions of the model incorporating quenched disorder are considered the first simulates heterogeneities in the local properties of the soil and the second considers the effects of a nonuniform rainfall in the region of mean field behavior the model is shown to be robust for both kinds of perturbations in the two limiting cases the random rainfall is still irrelevant whereas the heterogeneity in the soil properties leads to different universality classes results of a numerical analysis of the model are reported that confirm and complement the theoretical analysis of the global minimum the statistics of the local minima are found to resemble more strongly observational data on real rivers

maximum entropy in the problem of moments the maximum entropy approach to the solution of underdetermined inverse problems is studied in detail in the context of the classical moment problem in important special cases such as the hausdorff moment problem we establish necessary and sufficient conditions for the existence of a maximum entropy solution and examine the convergence of the resulting sequence of approximations a number of explicit illustrations are presented in addition to some elementary examples we analyze the maximum entropy reconstruction of the density of states in harmonic solids and of dynamic correlation functions in quantum spin systems we also briefly indicate possible applications to the lee yang theory of ising models to the summation of divergent series and so on the general conclusion is that maximum entropy provides a valuable approximation scheme a serious competitor of traditional pade like procedures

on network form and function abstract moving from the observation that drainage network configurations minimizing total energy dissipation are stationary solutions of the general equation describing landscape evolution we review theoretical and observational evidence on river patterns and their scale invariant structure exact results complemented by numerical annealing of the basic equation in the presence of additive noise suggest that configurations at or very close to the global minimum of energy dissipation differ from dynamically accessible states which have rather different scaling properties and conform much better to natural forms thus we argue that at least in the fluvial landscape nature works through imperfect searches for dynamically accessible optimal configurations we also show that optimal networks are spanning loopless configurations only under precise physical requirements this is stated in a form applicable to generic networks suggesting that other branching structures occurring in nature e g scale free and looping may possibly arise through optimality to selective pressures indeed we show that this is the case

search for the optimality signature of river network development whether the evolution of natural river networks pursues a certain optimal state has been a most intriguing and fundamental question there have been many optimality hypotheses proposed but it has yet to be proved which of these best serves as a quantitative signature of river network development here this fundamental question is investigated for the five hypotheses of minimum total energy expenditure minimum total energy dissipation rate minimum total stream power minimum global energy expenditure rate and minimum topological energy using simple example landscapes i examined whether any of these hypotheses pursues both the treelike river network formation and the concave stream longitudinal profile the two characteristic patterns of natural landscapes it is found that none of these hypotheses captures both patterns under the steady state condition where the balance between tectonic uplift and sediment loss is satisfied these findings are further verified through simulations of landscapes that satisfy given optimality criteria using an optimization method

scaling optimality and landscape evolution a nonlinear model is studied which describes the evolution of a landscape under the effects of erosion and regeneration by geologic uplift by mean of a simple differential equation the equation already in wide use among geomorphologists and in that context obtained phenomenologically is here derived by reparametrization invariance arguments and exactly solved in dimension d 1 results of numerical simulations in d 2 show that the model is able to reproduce the critical scaling characterizing landscapes associated with natural river basins we show that configurations minimizing the rate of energy dissipation optimal channel networks are stationary solutions of the equation describing the landscape evolution numerical simulations show that a careful annealing of the equation in the presence of additive noise leads to configurations very close to the global minimum of the dissipated energy characterized by mean field exponents we further show that if one considers generalized river network configurations in which splitting of the flow i e braiding and loops are allowed the minimization of the dissipated energy results in spanning loopless configurations under the constraints imposed by the continuity equations this is stated in the form of a general theorem applicable to generic networks suggesting that other branching structures occurring in nature may possibly arise as optimal structures minimizing a cost function

approximate time dependent solutions of partial differential equations the maxent minimum norm approach a new approach to inverse problems with incomplete data the maxent minimum norm scheme was recently introduced by baker jarvis bj this method is based on the application of jaynes s maxent principle to a probability distribution defined on the functional space of all the possible solutions of the problem in the present work we consider the application of these ideas to non equilibrium time dependent systems in the spirit of jaynes s information theory approach to statistical mechanics we focus on the behaviour of a small number of physically relevant mean values by recourse to the bj procedure the equations of motion of those mean values are closed and an approximate description of the system s evolution is obtained it is also shown that the bj method is closely related to curado s case i e q 2 and standard mean values of tsallis nonextensive thermostatistics

minimum energy and fractal structures of drainage networks this paper explores the similarities of digital elevation maps dems of natural river basins and optimal channel network ocn configurations obtained minimizing the total rate of energy expenditure in the system as a whole and in its parts striking similarities are observed for natural and optimal networks in their fractal aggregation structure and in certain multifractal structures found to be characteristic of river basins our results suggest upon critical assessment of the reliability of the identification of the attractor of the underlying dynamics implied by our optimality concepts that fractal structures are indeed possibly a product of least energy dissipation power laws emerging in the description of the distribution of aggregated quantities from both dems and ocns suggest a link with the framework of self organized criticality in the dynamics of natural channel network formation also the geomorphological description of ocns reveals surprising analogies with well known empirical or experimental results a comparison of peano s basins with ocns suggests that nature seems to reject the type of strict self similarity exhibited by peano s construct in favor of different shapes implying statistical self similarity not only because of chance acting through random conditions but also because of necessity as reflected by least energy expenditure considerations

probabilistic structure of the distance between tributaries of given size in river networks we analyze the distribution of the distances between tributaries of a given size or of sizes larger than a given area draining along either an open boundary or the mainstream of a river network by proposing a description of the distance separating prescribed merging contributing areas we also address related variables like mean or bankfull flow rates and channel and riparian area widths which are derived under a set of reasonable hydrologic assumptions the importance of such distributions lies in their ecological hydrologic and geomorphic implications on the spreading of species along the ecological corridor defined by the river network and on the propagation of infections due to water borne diseases particularly in view of exact theoretical predictions explicitly using the alongstream distribution of confluences carrying a given flow use is made here of real river networks suitably extracted from digital elevation models optimal channel networks and exactly solved tree like constructs like the peano and the scheidegger networks the results obtained redefine theoretically in a coherent and general manner and verify observationally the distribution function of the above distances and thus provide the general probabilistic structure of tributaries in river networks specifically we find that the probability of exceedence of the alongstream distance d of tributaries of size larger than a has the explicit form p d exp cd a h 1 h where c is a constant that depends on the choice of boundary conditions and h 1 is the hurst exponent

a note on overadjustment in inverse probability weighted estimation standardized means commonly used in observational studies in epidemiology to adjust for potential confounders are equal to inverse probability weighted means with inverse weights equal to the empirical propensity scores more refined standardization corresponds with empirical propensity scores computed under more flexible models unnecessary standardization induces efficiency loss however according to the theory of inverse probability weighted estimation propensity scores estimated under more flexible models induce improvement in the precision of inverse probability weighted means this apparent contradiction is clarified by explicitly stating the assumptions under which the improvement in precision is attained

error bounds in maximum entropy approximations a useful technique in underdetermined inverse problems is that of maximum entropy a simple error bound for averages over a distribution approximated by the maximum entropy method in the case of the undetermined hausdorff moment problem was devised under the conditions specified the error bound for averages over such an approximate distribution can be very tight numerical examples to illustrate are presented

theoretical studies of self organized criticality these notes are intended to provide a pedagogical introduction to the abelian sandpile model of self organized criticality and its related models the abelian group the algebra of particle addition operators the burning test for recurrent states equivalence to the spanning trees problem are described the exact solution of the directed version of the model in any dimension is explained the model s equivalence to scheidegger s model of river basins takayasu s aggregation model and the voter model is discussed for the undirected case the solution for one dimensional lattices and the bethe lattice is briefly described known results about the two dimensional case are summarized generalization to the abelian distributed processors model is discussed time dependent properties and the universality of critical behavior in sandpiles are briefly discussed i conclude by listing some still unsolved problems

mann whitney test with adjustments to pretreatment variables for missing values and observational study the conventional wilcoxon mann whitney test can be invalid for comparing treatment effects in the presence of missing values or in observational studies this is because the missingness of the outcomes or the participation in the treatments may depend on certain pre treatment variables we propose an approach to adjust the mann whitney test by correcting the potential bias via consistently estimating the conditional distributions of the outcomes given the pre treatment variables we also propose semiparametric extensions of the adjusted mann whitney test which leads to dimension reduction for high dimensional covariate a novel bootstrap procedure is devised to approximate the null distribution of the test statistics for practical implementations results from simulation studies and an economic observational study data analysis are presented to demonstrate the performance of the proposed approach

geometry of river networks ii distributions of component size and number the structure of a river network may be seen as a discrete set of nested subnetworks built out of individual stream segments these network components are assigned an integral stream order via a hierarchical and discrete ordering method exponential relationships known as horton s laws between stream order and ensemble averaged quantities pertaining to network components are observed we extend these observations to incorporate fluctuations and all higher moments by developing functional relationships between distributions the relationships determined are drawn from a combination of theoretical analysis analysis of real river networks including the mississippi amazon and nile and numerical simulations on a model of directed random networks underlying distributions of stream segment lengths are identified as exponential combinations of these distributions form single humped distributions with exponential tails the sums of which are in turn shown to give power law distributions of stream lengths distributions of basin area and stream segment frequency are also addressed the calculations identify a single length scale as a measure of size fluctuations in network components this article is the second in a series of three addressing the geometry of river networks

universal scaling of optimal current distribution in transportation networks transportation networks are inevitably selected with reference to their global cost which depends on the strengths and the distribution of the embedded currents we prove that optimal current distributions for a uniformly injected d dimensional network exhibit robust scale invariance properties independently of the particular cost function considered as long as it is convex we find that in the limit of large currents the distribution decays as a power law with an exponent equal to 2d 1 d 1 the current distribution can be exactly calculated in d 2 for all values of the current numerical simulations further suggest that the scaling properties remain unchanged for both random injections and by randomizing the convex cost functions