functional data clustering via piecewise constant nonparametric density estimation in this paper we present a novel way of analyzing and summarizing a collection of curves based on piecewise constant density estimation the curves are partitioned into clusters and the dimensions of the curves points are discretized into intervals the cross product of these univariate partitions forms a data grid of cells which represents a nonparametric estimator of the joint density of the curves and point dimensions the best model is selected using a bayesian model selection approach and retrieved using combinatorial optimization algorithms the proposed method requires no parameter setting and makes no assumption regarding the curves beyond functional data it can be applied to distributional data the practical interest of the approach for functional data and distributional data exploratory analysis is presented on two real world datasets

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

sales distribution of consumer electronics abstract using the uniform most powerful unbiased test we observed the sales distribution of consumer electronics in japan on a daily basis and report that it follows both a lognormal distribution and a power law distribution and depends on the state of the market we show that these switches occur quite often the underlying sales dynamics found between both periods nicely matched a multiplicative process however even though the multiplicative term in the process displays a size dependent relationship when a steady lognormal distribution holds it shows a size independent relationship when the power law distribution holds this difference in the underlying dynamics is responsible for the difference in the two observed distributions

scale space view of curve estimation scale space theory from computer vision leads to an interesting and novel approach to nonparametric curve estimation the family of smooth curve estimates indexed by the smoothing parameter can be represented as a surface called the scale space surface the smoothing parameter here plays the same role as that played by the scale of resolution in a visual system in this paper we study in detail various features of that surface from a statistical viewpoint weak convergence of the empirical scale space surface to its theoretical counterpart and some related asymptotic results have been established under appropriate regularity conditions our theoretical analysis provides new insights into nonparametric smoothing procedures and yields useful techniques for statistical exploration of features in the data in particular we have used the scale space approach for the development of an effective exploratory data analytic tool called sizer

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

practical use of higher order asymptotics for multiparameter exponential families recently developed asymptotics based on saddlepoint methods provide important practical methods for multiparameter exponential families especially in generalized linear models the aim here is to clarify and explore these attention is restricted to tests and confidence intervals regarding a single parametric function which can be represented as a natural parameter of a full rank exponential family excellent approximations to exact conditional inferences are often available in terms of simple adjustments to the signed square root of the likelihood ratio statistic

gait analysis and the bootstrap this paper is about random coefficient trigonometric regression models and their use in gait analysis here gait analysis means free speed walking on a level surface our study focuses on bootstrap based prediction regions for the angular rotation curves of test children when the relevant training data are gathered from normal children of comparable ages considerations that led to our choice of model and use of the bootstrap are given prediction regions and empirical bootstrap distributions are displayed as is their application to several test cases also included is a study of the almost sure asymptotic behavior of the theoretical bootstrap probability of the prediction regions

defining probability density for a distribution of random functions the notion of probability density for a random function is not as straightforward as in finite dimensional cases while a probability density function generally does not exist for functional data we show that it is possible to develop the notion of density when functional data are considered in the space determined by the eigenfunctions of principal component analysis this leads to a transparent and meaningful surrogate for density defined in terms of the average value of the logarithms of the densities of the distributions of principal components for a given dimension this density approximation is estimable readily from data it accurately represents in a monotone way key features of small ball approximations to density our results on estimators of the densities of principal component scores are also of independent interest they reveal interesting shape differences that have not previously been considered the statistical implications of these results and properties are identified and discussed and practical ramifications are illustrated in numerical work

the best test of exponentiality against singly truncated normal alternatives abstract we show that the likelihood ratio test of exponentiality against singly truncated normal alternatives is the uniformly most powerful unbiased test and can be expressed in terms of the sampling coefficient of variation this test is closely related to greenwood s statistic for testing departures from the uniform distribution we provide a way to approximate the critical points of the test using saddlepoint methods that gives a high degree of accuracy

circuit definitions of nondeterministic complexity classes we consider restictions on boolean circuits and use them to obtain new uniform circuit characterizations of nondeterministic space and time classes we also obtain characterizations of counting classes based on nondeterministic time bounded computations on the arithmetic circuit model it is shown how the notion of semiunboundedness unifies the definitions of many natural complexity classes

clustering functional data into groups by using projections we show that in the functional data context by appropriately exploiting the functional nature of the data it is possible to cluster the observations asymptotically perfectly we demonstrate that this level of performance can sometimes be achieved by the k means algorithm as long as the data are projected on a carefully chosen finite dimensional space in general the notion of an ideal cluster is not clearly defined we derive our results in the setting where the data come from two populations whose distributions differ at least in terms of means and where an ideal cluster corresponds to one of these two populations we propose an iterative algorithm to choose the projection functions in a way that optimizes clustering performance where to avoid peculiar solutions we use a weighted least squares criterion we apply our iterative clustering procedure on simulated and real data where we show that it works well

optimal bayes classifiers for functional data and density ratios bayes classifiers for functional data pose a challenge this is because probability density functions do not exist for functional data as a consequence the classical bayes classifier using density quotients needs to be modified we propose to use density ratios of projections on a sequence of eigenfunctions that are common to the groups to be classified the density ratios can then be factored into density ratios of individual functional principal components whence the classification problem is reduced to a sequence of nonparametric one dimensional density estimates this is an extension to functional data of some of the very earliest nonparametric bayes classifiers that were based on simple density ratios in the one dimensional case by means of the factorization of the density quotients the curse of dimensionality that would otherwise severely affect bayes classifiers for functional data can be avoided we demonstrate that in the case of gaussian functional data the proposed functional bayes classifier reduces to a functional version of the classical quadratic discriminant a study of the asymptotic behavior of the proposed classifiers in the large sample limit shows that under certain conditions the misclassification rate converges to zero a phenomenon that has been referred to as perfect classification the proposed classifiers also perform favorably in finite sample applications as we demonstrate in comparisons with other functional classifiers in simulations and various data applications including wine spectral data functional magnetic resonance imaging fmri data for attention deficit hyperactivity disorder adhd patients and yeast gene expression data

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

sizer for exploration of structures in curves abstract in the use of smoothing methods in data analysis an important question is which observed features are really there as opposed to being spurious sampling artifacts an approach is described based on scale space ideas originally developed in the computer vision literature assessment of significant zero crossings of derivatives results in the sizer map a graphical device for display of significance of features with respect to both location and scale here scale means level of resolution that is bandwidth

truncated lognormal distributions and scaling in the size of naturally defined population clusters using population data of high spatial resolution for a region in the south of europe we define cities by aggregating individuals to form connected clusters the resulting cluster population distributions show a smooth decreasing behavior covering six orders of magnitude we perform a detailed study of the distributions using state of the art statistical tools by means of scaling analysis we rule out the existence of a power law regime in the low population range the logarithmic coefficient of variation test allows us to establish that the power law tail for high population characteristic of zipf s law has a rather limited range of applicability instead lognormal fits describe the population distributions in a range covering from a few dozen individuals to more than 1 10 6 which corresponds to the population of the largest cluster

positive and negative proofs for circuits and branching programs we extend the operator in a natural way and derive new types of counting complexity classes while in the case of c classes where c could be some circuit based class like nc 1 only proofs for acceptance of some input are being counted one can also count proofs for rejection the zap c complexity classes we propose here implement this idea we show that in certain cases zap c lies between c and gap c which could help understanding the relationship between c and gap c in particular we consider zap nc 1 and polynomial size branching programs of bounded and unbounded width finally we argue about negative proofs in turing machines and how those relate to open questions

on the asymptotics of constrained m estimation limit theorems for an m estimate constrained to lie in a closed subset of r d are given under two different sets of regularity conditions a consistent sequence of global optimizers converges under chernoff regularity of the parameter set a n consistent sequence of local optimizers converges under clarke regularity of the parameter set in either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value limit theorems for the optimal value are also obtained agreeing with chernoff s result in the case of maximum likelihood with global optimizers

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

componentwise classification and clustering of functional data the infinite dimension of functional data can challenge conventional methods for classification and clustering a variety of techniques have been introduced to address this problem particularly in the case of prediction but the structural models that they involve can be too inaccurate or too abstract or too difficult to interpret for practitioners in this paper we develop approaches to adaptively choose components enabling classification and clustering to be reduced to finite dimensional problems we explore and discuss properties of these methodologies our techniques involve methods for estimating classifier error rate and cluster tightness and for choosing both the number of components and their locations to optimize these quantities a major attraction of this approach is that it allows identification of parts of the function domain that convey important information for classification and clustering it also permits us to determine regions that are relevant to one of these analyses but not the other copyright 2012 oxford university press

non cancellative boolean circuits a generalization of monotone boolean circuits cancellations are known to be helpful in efficient algebraic computation of polynomials over fields we define a notion of cancellation in boolean circuits and define boolean circuits that do not use cancellation to be non cancellative non cancellative boolean circuits are a natural generalization of monotone boolean circuits we show that in the absence of cancellation boolean circuits require super polynomial size to compute the determinant interpreted over gf 2 this non monotone boolean function is known to be in p in the spirit of monotone complexity classes we define complexity classes based on non cancellative boolean circuits we show that when the boolean circuit model is restricted by withholding cancellation p and popular classes within p are restricted as well but classes np and above remain unchanged

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

non commutative arithmetic circuits depth reduction and size lower bounds abstract we investigate the phenomenon of depth reduction in commutative and non commutative arithmetic circuits we prove that in the commutative setting uniform semi unbounded arithmetic circuits of logarithmic depth are as powerful as uniform arithmetic circuits of polynomial degree and unrestricted depth earlier proofs did not work in the uniform setting this also provides a unified proof of the circuit characterizations of the class logcfl and its counting variant logcfl we show that ac1 has no more power than arithmetic circuits of polynomial size and degree no log log n improving the trivial bound of no log n connections are drawn between tc1 and arithmetic circuits of polynomial size and degree then we consider non commutative computation we show that over the algebra max concat arithmetic circuits of polynomial size and polynomial degree can be reduced to o log2 n depth and even to o log n depth if unbounded fanin gates are allowed this establishes that optlogcfl is in ac1 this is the first depth reduction result for arithmetic circuits over a non commutative semiring and it complements the lower bounds of kosaraju and nisan showing that depth reduction cannot be done in the general non commutative setting we define new notions called short left paths and short right paths and we show that these notions provide a characterization of the classes of arithmetic circuits for which optimal depth reduction is possible this class also can be characterized using the auxpda model finally we characterize the languages generated by efficient circuits over the semiring 2 union concat in terms of simple one way machines and we investigate and extend earlier lower bounds on non commutative circuits

bandwidth choice for nonparametric regression this paper is concerned with the problem of choosing a bandwidth parameter for nonparametric regression we analyze a tapered fourier series estimate and discuss the relationship of this estimate to a kernel estimate we first consider a method based on an unbiased estimate of mean square error and show that the bandwidth thus chosen is asymptotically optimal other methods are examined as well and are shown to be asymptotically equivalent a small simulation shows however that for small or moderate sample size the methods perform quite differently

principal components analysis of sampled functions this paper describes a technique for principal components analysis of data consisting ofn functions each observed atp argument values this problem arises particularly in the analysis of longitudinal data in which some behavior of a number of subjects is measured at a number of points in time in such cases information about the behavior of one or more derivatives of the function being sampled can often be very useful as for example in the analysis of growth or learning curves it is shown that the use of derivative information is equivalent to a change of metric for the row space in classical principal components analysis the reproducing kernel for the hilbert space of functions plays a central role and defines the best interpolating functions which are generalized spline functions an example is offered of how sensitivity to derivative information can reveal interesting aspects of the data

universality of power law exponents by means of maximum likelihood estimation power law type distributions are extensively found when studying the behavior of many complex systems however due to limitations in data acquisition empirical datasets often only cover a narrow range of observation making it difficult to establish power law behavior unambiguously in this work we present a statistical procedure to merge different datasets with two different aims first we obtain a broader fitting range for the statistics of different experiments or observations of the same system second we establish whether two or more different systems may belong to the same universality class by means of maximum likelihood estimation this methodology provides rigorous statistical information to discern whether power law exponents characterizing different datasets can be considered equal among them or not this procedure is applied to the gutenberg richter law for earthquakes and for synthetic earthquakes acoustic emission events generated in the laboratory labquakes different earthquake catalogs have been merged finding a gutenberg richter law holding for more than eight orders of magnitude in seismic moment the value of the exponent of the energy distribution of labquakes depends on the material used in the compression experiments by means of the procedure proposed in this manuscript we find that the gutenberg richter law for earthquakes and charcoal labquakes can be characterized by the same power law exponent whereas vycor labquakes exhibit a significantly different exponent

mixture inner product spaces and their application to functional data analysis we introduce the concept of mixture inner product spaces associated with a given separable hilbert space which feature an infinite dimensional mixture of finite dimensional vector spaces and are dense in the underlying hilbert space any hilbert valued random element can be arbitrarily closely approximated by mixture inner product space valued random elements while this concept can be applied to data in any infinite dimensional hilbert space the case of functional data that are random elements in the l 2 space of square integrable functions is of special interest for functional data mixture inner product spaces provide a new perspective where each realization of the underlying stochastic process falls into one of the component spaces and is represented by a finite number of basis functions the number of which corresponds to the dimension of the component space in the mixture representation of functional data the number of included mixture components used to represent a given random element in l 2 is specifically adapted to each random trajectory and may be arbitrarily large key benefits of this novel approach are first that it provides a new perspective on the construction of a probability density in function space under mild regularity conditions and second that individual trajectories possess a trajectory specific dimension that corresponds to a latent random variable making it possible to use a larger number of components for less smooth and a smaller number for smoother trajectories this enables flexible and parsimonious modeling of heterogeneous trajectory shapes we establish estimation consistency of the functional mixture density and introduce an algorithm for fitting the functional mixture model based on a modified expectation maximization algorithm simulations confirm that in comparison to traditional functional principal component analysis the proposed method achieves similar or better data recovery while using fewer components on average its practical merits are also demonstrated in an analysis of egg laying trajectories for medflies

a taxonomy of problems with fast parallel algorithms the class nc consists of problems solvable very fast in time polynomial in log n in parallel with a feasible polynomial number of processors many natural problems in nc are known in this paper an attempt is made to identify important subclasses of nc and give interesting examples in each subclass the notion of nc 1 reducibility is introduced and used throughout problem r is nc 1 reducible to problem s if r can be solved with uniform log depth circuits using oracles for s problems complete with respect to this reducibility are given for many of the subclasses of nc a general technique the parallel greedy algorithm is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in nc 2 solvable by uniform boolean circuits of depth o log 2 n and polynomial size the class logcfl is given a new characterization in terms of circuit families the class det of problems reducible to integer determinants is defined and many examples given a new problem complete for deterministic polynomial time is given namely finding the lexicographically first maximal clique in a graph this paper is a revised version of s a cook 1983 in proceedings 1983 intl found comut sci conf lecture notes in computer science vol 158 pp 78 93 springer verlag berlin new york

achieving near perfect classification for functional data summary we show that in functional data classification problems perfect asymptotic classification is often possible making use of the intrinsic very high dimensional nature of functional data this performance is often achieved by linear methods which are optimal in important cases these results point to a marked contrast between classification for functional data and its counterpart in conventional multivariate analysis where the dimension is kept fixed as the sample size diverges in the latter setting linear methods can sometimes be quite inefficient and there are no prospects for asymptotically perfect classification except in pathological cases where for example a variance vanishes by way of contrast in finite samples of functional data good performance can be achieved by truncated versions of linear methods truncation can be implemented by partial least squares or projection onto a finite number of principal components using in both cases cross validation to determine the truncation point we establish consistency of the cross validation procedure

pp is closed under truth table reductions beigel reingold and spielman j comput system sci 50 191 202 1995 showed that pp is closed under intersection and a variety of special cases of polynomial time truth table closure we extend their techniques to show that pp is closed under general polynomial time truth table reductions we also show that pp is closed under constant round truth table reductions

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

skew circuits of small width abstract a celebrated result of barrington 1985 proved that polynomial size width 5 branching programs bp are equivalent in power to a restricted form of branching programs polynomial sized width 5 permutation branching programs pbp which in turn capture all of nc 1 on the other hand it is known that width 3 pbps require exponential size to compute the and function no such lower bound is known for width 4 pbps however it is widely conjectured that width 4 pbps will not capture all of nc 1 in this work we study the power of bounded width branching programs by comparing them with bounded width skew circuits it is well known that branching programs of bounded width have the same power as skew circuit of bounded width the naive approach converts a bp of width w to a skew circuit of width w 2 we improve this bound and show that bp of width w 5 can be converted to a skew circuit of width 7 this also implies that skew circuits of bounded width are equal in power to skew circuits of width 7 for the other way we prove that for any w 2 a skew circuit of width w can be converted into an equivalent branching program of width w we prove that width 2 skew circuits are not universal while width 3 skew circuits are universal and that any polynomial sized cnf or dnf is computable by width 3 skew circuits of polynomial size it is known that parity does not have small cnfs or dnfs it is easy to see that parity has width 4 skew circuits we prove that a width 3 skew circuit computing parity requires exponential size this gives an exponential separation between the power of width 3 skew circuits and width 4 skew circuits

invariant exponential models applied to reliability theory and survival analysis abstract for a two parameter exponential model with increasing failure rate ifr or decreasing failure rate dfr distributions necessary and sufficient conditions of the existence of a solution of the likelihood equations are given also all of the scale invariant two parameter statistical models closed by raising to a power and by exponential tilting are introduced the conditions of existence of a solution of the likelihood equations are studied for these invariant models and the models are applied to obtain some uniformly most powerful unbiased tests of exponentially against alternatives in these models

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

a tale of two tails do power law and lognormal models fit firm size distributions in the mid victorian era abstract the paper explores the frequency and size distributions of firm size in a novel dataset for the mid victorian era from a recent extraction of the england and wales population censuses of 1851 1861 1871 and 1881 the paper contrasts the hypothesis of the power laws against the lognormal model for the tails of the distributions using maximum likelihood estimation log likelihood ratio clipped sample coefficient of variation umpu wilks test kolmogorov smirnov statistic among other state of the art statistical methods our results show that the power law hypothesis is accepted for the size distribution for the years 1851 and 1861 while 1871 is marginally non significant but for 1881 the test is inconclusive the paper discusses the process that generates these distributions citing recent literature that shows how after adding an i i d noise to the gibrat s multiplicative model one can recreate a power law behaviour overall the paper provides describes and statistically tests for the very first time a unique historical dataset confirming that the tails of the distributions at least for 1851 and 1861 follow a pareto model and that the lognormal model is firmly rejected