estimation of functional derivatives situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis predictors are often appropriately modeled as square integrable smooth random functions imposing minimal assumptions on the nature of the functional relationship we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions in statistical applications and data analysis functional derivatives provide a quantitative measure of the often intricate relationship between changes in predictor trajectories and those in scalar responses this approach provides a natural extension of classical gradient fields in vector space and provides directions of steepest descent we suggest a kernel based method for the nonparametric estimation of functional derivatives that utilizes the decomposition of the random predictor functions into their eigenfunctions these eigenfunctions define a canonical set of directions into which the gradient field is expanded the proposed method is shown to lead to asymptotically consistent estimates of functional derivatives and is illustrated in an application to growth curves

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

correlation based functional clustering via subspace projection a correlation based functional clustering method is proposed for grouping curves with similar shapes a correlation between two random functions defined through the functional inner product is used as a similarity measure curves with similar shapes are embedded in the cluster subspace spanned by a mean shape function and eigenfunctions of the covariance kernel the cluster membership prediction for each curve attempts to maximize the functional correlation between the observed and predicted curves via shape standardization and subspace projection among all possible clusters the proposed method accounts for shape differentials through the functional multiplicative random effects shape function model for each cluster which regards random scales and intercept shifts as a nuisance a consistent estimate is proposed for the random scale effect whose sample variance estimate is also consistent the derived identifiability conditions for the clustering procedure unravel the predictability of cluster membersh

iterated snowdrift game among mobile agents with myopic expected reward based decision rule numerical and analytical research abstract the iterated spatial game among the mobile population is an interesting problem in the fields of biological social and economic sciences inspired by some recent works this paper concentrates on the iterated snowdrift game among movable and myopic agents two different decision making schemes namely the utility maximum rule and the fermi rule are applied and examined in the former case cooperation is found to be enhanced by moving velocity with an upper bound the analytical results of the model are deduced at two extreme cases when agents cannot move or move quickly in the latter case the influences of velocity and temptation to defect are much more complicated these results allow a deeper insight of the related model as well as the emergence of cooperation

dynamical behavior of susceptible infected recovered susceptible epidemic model on weighted networks abstract we study susceptible infected recovered susceptible epidemic model in weighted regular and random complex networks we institute a pairwise type mathematical model with a general transmission rate to evaluate the influence of the link weight distribution on the spreading process furthermore we develop a dimensionality reduction approach to derive the condition for the contagion outbreak finally we analyze the influence of the heterogeneity of weight distribution on the outbreak condition for the scenario with a linear transmission rate our theoretical analysis is in agreement with stochastic simulations showing that the heterogeneity of link weight distribution can have a significant effect on the epidemic dynamics

random matrix theory and the sixth painleve equation a feature of certain ensembles of random matrices is that the corresponding measure is invariant under conjugation by unitary matrices study of such ensembles realized by matrices with gaussian entries leads to statistical quantities related to the eigenspectrum such as the distribution of the largest eigenvalue which can be expressed as multidimensional integrals or equivalently as determinants these distributions are well known to be functions for painleve systems allowing for the former to be characterized as the solution of certain nonlinear equations we consider the random matrix ensembles for which the nonlinear equation is the form of pvi known results are reviewed as is their implication by way of series expansions for the distributions new results are given for the boundary conditions in the neighbourhood of the fixed singularities at t 0 1 of pvi displayed by a generalization of the generating function for the distributions the structure of these expansions is related to jimbo s general expansions for the function of pvi in the neighbourhood of its fixed singularities and this theory is itself put in its context of the linear isomonodromy problem relating to pvi

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

mixed moments of characteristic polynomials of random unitary matrices following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle

positive correlation between strategy persistence and teaching ability promotes cooperation in evolutionary prisoner s dilemma games abstract strategy persistence and heterogeneous teaching ability are two factors that play important roles in promoting cooperation in this work we simultaneously take the strategy persistence and heterogeneous teaching ability into account in evolutionary prisoner s dilemma games we introduce the correlation between strategy persistence and teaching ability by partitioning a population into two groups in the positive negative correlation scheme individuals in one group are assigned with large and 1 reduced teaching ability 1 and those in the other with 0 and 1 1 the simulation results show that compared to negative correlation positive correlation between strategy persistence and teaching ability can greatly enhance cooperation and result in a much higher cooperation level furthermore we show that the finding is robust with respect to reduced teaching ability initial cooperator fraction f 0 amplitude of noise k and underlying network topology

functional varying coefficient models for longitudinal data the proposed functional varying coefficient model provides a versatile and flexible analysis tool for relating longitudinal responses to longitudinal predictors specifically this approach provides a novel representation of varying coefficient functions through suitable auto and cross covariances of the underlying stochastic processes which is particularly advantageous for sparse and irregular designs as often encountered in longitudinal studies existing methodology for varying coefficient models is not adapted to such data the proposed approach extends the customary varying coefficient models to a more general setting in which not only current but also recent past values of the predictor time course may have an impact on the current value of the response time course the influence of past predictor values is modeled by a smooth history index function while the effects on the response are described by smooth varying coefficient functions the resulting estimators for varying coefficient and history

single and multiple index functional regression models with nonparametric link fully nonparametric methods for regression from functional data have poor accuracy from a statistical viewpoint reflecting the fact that their convergence rates are slower than nonparametric rates for the estimation of high dimensional functions this difficulty has led to an emphasis on the so called functional linear model which is much more flexible than common linear models in finite dimension but nevertheless imposes structural constraints on the relationship between predictors and responses recent advances have extended the linear approach by using it in conjunction with link functions and by considering multiple indices but the flexibility of this technique is still limited for example the link may be modeled parametrically or on a grid only or may be constrained by an assumption such as monotonicity multiple indices have been modeled by making finite dimensional assumptions in this paper we introduce a new technique for estimating the link function nonparametrically and we suggest an approach to multi index modeling using adaptively defined linear projections of functional data we show that our methods enable prediction with polynomial convergence rates the finite sample performance of our methods is studied in simulations and is illustrated by an application to a functional regression problem

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

most predictive design points for functional data predictors we suggest a way of reducing the very high dimension of a functional predictor x to a low number of dimensions chosen so as to give the best predictive performance specifically if x is observed on a fine grid of design points t 1 t r we propose a method for choosing a small subset of these say t i1 t ik to optimize the prediction of a response variable y the values t ij are referred to as the most predictive design points or covariates for a given value of k and are computed using information contained in a set of independent observations x i y i of x y the algorithm is based on local linear regression and calculations can be accelerated using linear regression to preselect the design points boosting can be employed to further improve the predictive performance we illustrate the usefulness of our ideas through simulations and examples drawn from chemometrics and we develop theoretical arguments showing that the methodology can be applied successfully in a range of settings

memory based snowdrift game on networks we present a memory based snowdrift game mbsg taking place on networks we found that when a lattice is taken to be the underlying structure the transition of spatial patterns at some critical values of the payoff parameter is observable for both four and eight neighbor lattices the transition points as well as the styles of spatial patterns can be explained by local stability analysis in sharp contrast to previously reported results cooperation is promoted by the spatial structure in the mbsg interestingly we found that the frequency of cooperation of the mbsg on a scale free network peaks at a specific value of the payoff parameter this phenomenon indicates that properly encouraging selfish behaviors can optimally enhance the cooperation the memory effects of individuals are discussed in detail and some nonmonotonous phenomena are observed on both lattices and scale free networks our work may shed some new light on the study of evolutionary games over networks

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

evolutionary prisoner s dilemma game on newman watts networks maintenance of cooperation was studied for a two strategy evolutionary prisoner s dilemma game where the players are located on a one dimensional chain and their payoff comes from games with the nearest and next nearest neighbor interactions the applied host geometry makes it possible to study the impacts of two conflicting topological features the evolutionary rule involves some noise affecting the strategy adoptions between the interacting players using monte carlo simulations and the extended versions of dynamical mean field theory we determined the phase diagram as a function of noise level and a payoff parameter the peculiar feature of the diagram is changed significantly when the connectivity structure is extended by extra links as suggested by newman and watts

moments of the derivative of characteristic polynomials with an application to the riemann zeta function we investigate the moments of the derivative on the unit circle of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the riemann function on the critical line we do the same for the analogue of hardy s z function the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle our formulae are expressed in terms of a determinant of a matrix whose entries involve the i bessel function and alternately by a combinatorial sum

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

cooperative behavior in evolutionary snowdrift games with the unconditional imitation rule on regular lattices we study an evolutionary snowdrift game with the unconditional imitation updating rule on regular lattices detailed numerical simulations establish the structure of plateaus and discontinuous jumps of the equilibrium cooperation frequency f c as a function of the cost to benefit ratio r by analyzing the stability of local configurations it is found that the transitions occur at values of r at which there are changes in the ranking of the payoffs to the different local configurations of agents using different strategies nonmonotonic behavior of f c r at the intermediate range of r is analyzed in terms of the formation of blocks of agents using the cooperative strategy that are stabilized by agents inside the block due to the updating rule for random initial condition with 50 50 agents of different strategies randomly dispersed cooperation persists in the whole range of r and the level of cooperation is higher than that in the well mixed case in a wide range of r these results are in sharp contrast to those based on the replicator updating rule the sensitivity to initial states with different fractions of cooperative agents is also discussed the results serve to illustrate that both the spatial structure and the updating rule are important in determining the level of cooperation in a competing population when extreme initial states are used where there are very few agents of a strategy in a background of the opposite strategy the result would depend on the stability of the clusters formed by the initially minority agents

effect of memory on the prisoner s dilemma game in a square lattice we have studied the effect of memory on the evolution of the prisoner s dilemma game in square lattice networks based on extensive simulations we found that the density of cooperators was enhanced by an increasing memory effect for most parameters however we also observed that the density of cooperators decreased with an increased memory effect in the case of a large memory and moderate temptation it is interesting to note that memory makes cooperators immune from temptation the strength of protection reaches its maximal value only for a moderate memory effect

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

functional linear regression with points of impact the paper considers functional linear regression where scalar responses y 1 ldots y n are modeled in dependence of i i d random functions x 1 ldots x n we study a generalization of the classical functional linear regression model it is assumed that there exists an unknown number of points of impact that is discrete observation times where the corresponding functional values possess significant influences on the response variable in addition to estimating a functional slope parameter the problem then is to determine the number and locations of points of impact as well as corresponding regression coefficients identifiability of the generalized model is considered in detail it is shown that points of impact are identifiable if the underlying process generating x 1 ldots x n possesses specific local variation examples are well known processes like the brownian motion fractional brownian motion or the ornstein uhlenbeck process the paper then proposes an easily implementable method for estimating the number and locations of points of impact it is shown that this number can be estimated consistently furthermore rates of convergence for location estimates regression coefficients and the slope parameter are derived finally some simulation results as well as a real data application are presented

phase diagrams for an evolutionary prisoner s dilemma game on two dimensional lattices the effects of payoffs and noise on the maintenance of cooperative behavior are studied in an evolutionary prisoner s dilemma game with players located on the sites of different two dimensional lattices this system exhibits a phase transition from a mixed state of cooperators and defectors to a homogeneous one where only the defectors remain alive using monte carlo simulations and the generalized mean field approximations we have determined the phase boundaries critical points separating the two phases on the plane of the temperature noise and temptation to choose defection in the zero temperature limit the cooperation can be sustained only for those connectivity structures where three site clique percolation occurs

ordering and selecting components in multivariate or functional data linear prediction the problem of component choice in regression based prediction has a long history the main cases where important choices must be made are functional data analysis and problems in which the explanatory variables are relatively high dimensional vectors indeed principal component analysis has become the basis for methods for functional linear regression in this context the number of components can also be interpreted as a smoothing parameter and so the viewpoint is a little different from that for standard linear regression however arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention we give a theoretical argument which is applicable in a wide variety of settings justifying the conventional approach although our result is of minimax type it is not asymptotic in nature it holds for each sample size motivated by the insight that is gained from this analysis we give theoretical and numerical justification for cross validation choice of the number of components that is used for prediction in particular we show that cross validation leads to asymptotic minimization of mean summed squared error in settings which include functional data analysis copyright c 2010 royal statistical society

evolutionary prisoner s dilemma game on a square lattice a simplified prisoner s game is studied on a square lattice when the players interacting with their neighbors can follow two strategies to cooperate c or to defect d unconditionally the players updated in random sequence have a chance to adopt one of the neighboring strategies with a probability depending on the payoff difference using monte carlo simulations and dynamical cluster techniques we study the density c of cooperators in the stationary state this system exhibits a continuous transition between the two absorbing states when varying the value of temptation to defect in the limits stackrel ensuremath rightarrow c 0 and 1 we have observed critical transitions belonging to the universality class of directed percolation

calculation of some determinants using the s shifted factorial several determinants with gamma functions as elements are evaluated these kinds of determinants are encountered for example in the computation of the probability density of the determinant of random matrices the s shifted factorial is defined as a generalization for non negative integers of the power function the rising factorial or pochammer s symbol and the falling factorial it is a special case of a polynomial sequence of the binomial type studied in combinatorics theory in terms of the gamma function an extension is defined for negative integers and even complex values properties mainly composition laws and binomial formulae are given they are used to evaluate families of generalized vandermonde determinants with s shifted factorials as elements instead of power functions

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

stringing high dimensional data for functional analysis we propose stringing a class of methods where one views high dimensional observations as functional data stringing takes advantage of the high dimension by representing such data as discretized and noisy observations that originate from a hidden smooth stochastic process assuming that the observations result from scrambling the original ordering of the observations of the process stringing reorders the components of the high dimensional vectors followed by transforming the high dimensional vector observations into functional data established techniques from functional data analysis can be applied for further statistical analysis once an underlying stochastic process and the corresponding random trajectory for each subject have been identified stringing of high dimensional data is implemented with distance based metric multidimensional scaling mapping high dimensional data to locations on a real interval such that predictors that are close in a suitable sample metric also are located close to each

a functional data analytic approach to signal discrimination motivated by specific problems involving radar range profiles we suggest techniques for real time discrimination in the context of signal analysis the key to our approach is to regard the signals as curves in the continuum and employ a functional data analytic fda method for dimension reduction based on the fda technique for principal coordinates analysis this has the advantage relative to competing methods such as canonical variates analysis of providing a signal approximation that is best possible in an l2 sense for a given dimension as a result it produces particularly good discrimination we explore the use of both nonparametric and gaussian based discriminators applied to the dimensionreduced data

peter hall s contributions to nonparametric function estimation and modeling peter hall made wide ranging and far reaching contributions to nonparametric modeling he was one of the leading figures in the developments of nonparametric techniques with over 300 published papers in the field alone this article gives a selective overview on the contributions of peter hall to nonparametric function estimation and modeling the focuses are on density estimation nonparametric regression bandwidth selection boundary corrections inference under shape constraints estimation of residual variances analysis of wavelet estimators multivariate regression and applications of nonparametric methods

functional singular component analysis summary aiming at quantifying the dependence of pairs of functional data x y we develop the concept of functional singular value decomposition for covariance and functional singular component analysis building on the concept of canonical expansion of compact operators in functional analysis we demonstrate the estimation of the resulting singular values functions and components for the practically relevant case of sparse and noise contaminated longitudinal data and provide asymptotic consistency results expanding bivariate functional data into singular functions emerges as a natural extension of the popular functional principal component analysis for single processes to the case of paired processes a natural application of the functional singular value decomposition is a measure of functional correlation owing to the involvement of an inverse operation most previously considered functional correlation measures are plagued by numerical instabilities and strong sensitivity to the choice of smoothing parameters these problems are exacerbated for the case of sparse longitudinal data on which we focus the functional correlation measure that is derived from the functional singular value decomposition behaves well with respect to numerical stability and statistical error as we demonstrate in a simulation study practical feasibility for applications to longitudinal data is illustrated with examples from a study on aging and on line auctions

additive modelling of functional gradients summary we consider the problem of estimating functional derivatives and gradients in the framework of a regression setting where one observes functional predictors and scalar responses derivatives are then defined as functional directional derivatives that indicate how changes in the predictor function in a specified functional direction are associated with corresponding changes in the scalar response for a model free approach navigating the curse of dimensionality requires the imposition of suitable structural constraints accordingly we develop functional derivative estimation within an additive regression framework here the additive components of functional derivatives correspond to derivatives of nonparametric one dimensional regression functions with the functional principal components of predictor processes as arguments this approach requires nothing more than estimating derivatives of one dimensional nonparametric regressions and thus is computationally very straightforward to implement while it also provides substantial flexibility fast computation and consistent estimation we illustrate the consistent estimation and interpretation of the resulting functional derivatives and functional gradient fields in a study of the dependence of lifetime fertility of flies on early life reproductive trajectories

boundary conditions associated with the painleve iii and v evaluations of some random matrix averages in a previous work a random matrix average for the laguerre unitary ensemble generalizing the generating function for the probability that an interval 0 s at the hard edge contains k eigenvalues was evaluated in terms of a painleve v transcendent in form however the boundary conditions for the corresponding differential equation were not specified for the full parameter space here this task is accomplished in general and the obtained functional form is compared against the most general small s behaviour of the painleve v equation in form known from the work of jimbo an analogous study is carried out for the hard edge scaling limit of the random matrix average which we have previously evaluated in terms of a painleve iii transcendent in form an application of the latter result is given to the rapid evaluation of a hankel determinant appearing in a recent work of conrey rubinstein and snaith relating to the derivative of the riemann zeta function

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

classification using censored functional data we consider classification of functional data when the training curves are not observed on the same interval different types of classifier are suggested one of which involves a new curve extension procedure our approach enables us to exploit the information contained in the endpoints of these intervals by incorporating it in an explicit but flexible way we study asymptotic properties of our classifiers and show that in a variety of settings they can even produce asymptotically perfect classification the performance of our techniques is illustrated in applications to real and simulated data supplementary materials for this article are available online

peter hall functional data analysis and random objects functional data analysis has become a major branch of nonparametric statistics and is a fast evolving field peter hall has made fundamental contributions to this area and its theoretical underpinnings he wrote more than 25 papers in functional data analysis between 1998 and 2016 and from 2005 on was a tenured faculty member with a 25 appointment in the department of statistics at the university of california davis this article describes aspects of his appointment and academic life in davis and also some of his major results in functional data analysis along with a brief history of this area it concludes with an outlook on new types of functional data and an emerging field of random objects that subsumes functional data analysis as it deals with more complex data structures

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