product integration in the presence of a singularity this paper considers the approximate evaluation of int a b k x f x dx where k is a fixed lebesgue integrable function by quadrature rules of the form sigma i 0 m n w in f x in normally rules of this kind are used only for smooth functions f and any singularities are incorporated into k here however we allow f to have a singularity either at an endpoint or in the interior general convergence properties are established for a wide class of product integration rules more detailed results are established for the class of rules which are exact if f belongs to a specified family of piecewise polynomials

local full likelihood estimation for the proportional hazards model summary in this paper a new approach to local likelihood estimation for censored data is proposed this method employs the full likelihood and alternates between estimating the baseline hazard and estimating the covariate effect the proposed methodology incorporates multidimensional data via additive models some results regarding inference for the covariate effect are also presented in this paper we propose new methodology for estimating a general covariate effect when the data arise from a proportional hazards model the methodology uses the full likelihood rather than the partial likelihood and is iterative in nature we note that techniques relying on the partial likelihood do not agree with the modeling assumptions and hence may be less desirable to begin some notation is introduced the data are observed as independent identically distributed triples ti bi zi where t is the minimum of the true failure time and the true censoring time 6 is an indicator of failure and z is a covariate that may be vector valued the true possibly unobserved failure time is assumed to have distribution function f the survivor function is denoted x the hazard x t and the integrated hazard a t further it is assumed that the data arise from a proportional hazards model that is the hazard x t z is assumed to be the product of a baseline hazard xo t and a covariate effect v z

identifying representative trees from ensembles tree based methods have become popular for analyzing complex data structures where the primary goal is risk stratification of patients ensemble techniques improve the accuracy in prediction and address the instability in a single tree by growing an ensemble of trees and aggregating however in the process individual trees get lost in this paper we propose a methodology for identifying the most representative trees in an ensemble on the basis of several tree distance metrics although our focus is on binary outcomes the methods are applicable to censored data as well for any two trees the distance metrics are chosen to 1 measure similarity of the covariates used to split the trees 2 reflect similar clustering of patients in the terminal nodes of the trees and 3 measure similarity in predictions from the two trees whereas the latter focuses on prediction the first two metrics focus on the architectural similarity between two trees the most representative trees in the ensemble are chosen on the basis of the average distance between a tree and all other trees in the ensemble out of bag estimate of error rate is obtained using neighborhoods of representative trees simulations and data examples show gains in predictive accuracy when averaging over such neighborhoods we illustrate our methods using a dataset of kidney cancer treatment receipt binary outcome and a second dataset of breast cancer survival censored outcome

positive finite rank elementary operators and characterizing entanglement of states in this paper some indecomposable positive finite rank elementary operators of order n n are constructed this allows us to give a simple necessary and sufficient criterion for the separability of pure states in bipartite systems of any dimension in terms of positive elementary operators of order 2 2 and obtain some new mixed entangled states that cannot be detected by the positive partial transpose criterion and the realignment criterion

on the convergence of spline product quadratures for cauchy principal value integrals abstract in a recent paper this journal 1990 the authors proposed product integration formulas for the numerical evaluation of the cauchy principal value integral f1 1u x f x x dx based on cubic spline interpolation of f and obtained convergence results for functions f ck 1 1 k 1 2 or 3 in this report the same rules are considered and their convergence is investigated for a larger class of functions f an error bound and some uniform convergence results are established in the case of equally spaced quadrature nodes for functions f satisfying a holder condition of order on 1 1 0

optimal entanglement witnesses from generalized reduction and robertson maps we provide a generalization of the reduction and robertson positive maps in matrix algebras they give rise to a new class of optimal entanglement witnesses their structural physical approximation is analyzed as a byproduct we provide new examples of ppt positive partial transpose entangled states

optimal entanglement criterion for mixed quantum states we develop a strong and computationally simple entanglement criterion the criterion is based on an elementary positive map phi which operates on state spaces with even dimension n or 4 it is shown that phi detects many entangled states with a positive partial transposition ppt and that it leads to a class of optimal entanglement witnesses this implies that there are no other witnesses which can detect more entangled ppt states the map phi yields a systematic method for the explicit construction of high dimensional manifolds of bound entangled states

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

relative risk trees for censored survival data a method is developed for obtaining tree structured relative risk estimates for censored survival data the first step of a full likelihood estimation procedure is used in a recursive partitioning algorithm that adopts most aspects of the widely used classification and regression tree cart algorithm of breiman et al 1984 classification and regression trees belmont california wadsworth the performance of the technique is investigated through stimulation and compared to the tree structured survival methods proposed by davis and anderson 1989 statistics in medicine 8 947 961 and therneau grambsch and fleming 1990 biometrika 77 147 160

numerical integration in the presence of an interior singularity abstract the convergence of sequences of integration rules applied to an improperly integrable function with an interior singularity is investigated the rules studied include the gauss jacobi rules and the interpolatory integration rules based on the zeros of 1 x 2 m p n 2m x m 0 1 for certain values of and the results are then applied to the study of the convergence of hunter s method for cauchy principal value integrals

residual based tree structured survival analysis extensions of various non parametric regression techniques for example additive models trees mars have been devised for right censored survival data these approaches directly handle the difficulties posed by censoring however it is possible to bypass these difficulties by utilizing standard non parametric regression procedures applied with say martingale residuals as outcome analytic correspondences between the direct and residual based approaches have been established for additive models while more qualitative comparisons have been provided for mars here we develop such correspondences for tree structured regression in particular we provide an analytic relationship between logrank and martingale residual sum of squares split functions that explains the widely observed similarity of the resultant trees further investigation is provided by simulation and an illustrative example using time to aids with data deriving from a western australian hiv cohort study

survival trees by goodness of split abstract a tree based method for censored survival data is developed based on maximizing the difference in survival between groups of patients represented by nodes in a binary tree the method includes a pruning algorithm with optimal properties analogous to the classification and regression tree cart pruning algorithm uniform convergence of the estimates of the conditional cumulative hazard and survival functions is discussed and an example is given to show the utility of the algorithm for developing prognostic classifications for patients

a random forest approach for competing risks based on pseudo values random forest is a supervised learning method that combines many classification or regression trees for prediction here we describe an extension of the random forest method for building event risk prediction models in survival analysis with competing risks in case of right censored data the event status at the prediction horizon is unknown for some subjects we propose to replace the censored event status by a jackknife pseudo value and then to apply an implementation of random forests for uncensored data because the pseudo responses take on values on a continuous scale the node variance is chosen as split criterion for growing regression trees in a simulation study the pseudo split criterion is compared with the gini split criterion when the latter is applied to the uncensored event status to investigate the resulting pseudo random forest method for building risk prediction models we analyze it in a simulation study of predictive performance where we compare it to cox regression and random survival forest the method is further illustrated in two real data sets copyright 2013 john wiley sons ltd

tree stability diagnostics and some remedies for instability stability aspects of recursive partitioning procedures are investigated using resampling techniques diagnostic tools to assess single split stability and overall tree stability are introduced to correct for the procedure s preference for covariates with many unique realizations corrected p values are used in the factor selection component of the algorithm finally methods to stabilize tree based predictors are discussed

trees and tracking epidemiologists have used the term tracking to connote an individual s maintenance of relative rank of some longitudinally measured characteristic over a given time span to assess the extent to which an attribute tracks we have first to summarize individual growth curves and second to quantify the notion of maintenance of relative rank both in the face of random error a sequence of papers appearing in 1981 provided differing methodologies for appraising tracking here we take a different approach to tracking by using regression trees for longitudinal data the above two concerns are simultaneously addressed in that the procedure identifies subgroups defined in terms of covariates within which the collection of growth curves is homogeneous after reviewing the existing approaches to tracking we describe the tree structured methodology and present an illustrative example pertaining to lung function growth in children

a series of absolutely indecomposable positive maps in matrix algebras abstract we provide examples of positive maps in m n c n 4 which cannot be decomposed into a sum of a 2 positive map and a 2 copositive map

trees for correlated survival data by goodness of split with applications to tooth prognosis in this article the regression tree method is extended to correlated survival data and applied to the problem of developing objective prognostic classification rules in periodontal research the robust logrank statistic is used as the splitting statistic to measure the between node difference in survival while adjusting for correlation among failure times from the same patient the partition based survival function estimator is shown to converge to the true conditional survival function tooth loss data from 100 periodontal patients 2 509 teeth was analyzed using the proposed method the goal is to assign each tooth to one of the five prognosis categories good fair poor questionable or hopeless after the best sized tree was identified an amalgamation procedure was used to form five prognostic groups the prognostic rules established here may be used by periodontists general dentists and insurance companies in devising appropriate treatment plans for periodontal patients

constructing positive maps from block matrices positive maps are useful for detecting entanglement in quantum information theory any entangled state can be detected by some positive map in this paper the relation between positive block matrices and completely positive trace preserving maps is characterized consequently a new method for constructing decomposable maps from positive block matrices is derived in addition a method for constructing positive but not completely positive maps from hermitian block matrices is also obtained

on the evaluation of cauchy principal value integrals by rules based on quasi interpolating splines abstract in this paper we consider the numerical evaluation of one dimensional cauchy principal value integrals of the form 1 1 k x f x x d x 1 by rules obtained by subtracting out the singularity and then applying quadrature formulas based on quasi interpolating splines

numerical integration based on approximating splines abstract convergence results are proved for product integration rules based on approximating splines these results are both for bounded and unbounded integrands pointwise and uniform convergence results are proved for sequences of cauchy principal values of these approximating splines

a tree based model for homogeneous groupings of multinomials the motivation of this paper is to provide a tree based method for grouping multinomial data according to their classification probability vectors we produce an initial tree by binary recursive partitioning whereby multinomials are successively split into two subsets and the splits are determined by maximizing the likelihood function if the number of multinomials k is too large we propose to order the multinomials and then build the initial tree based on a dramatically smaller number k 1 of possible splits the tree is then pruned from the bottom up the pruning process involves a sequence of hypothesis tests of a single homogeneous group against the alternative that there are two distinct internally homogeneous groups as pruning criteria the bayesian information criterion and the wilcoxon rank sum test are proposed the tree based model is illustrated on genetic sequence data homogeneous groupings of genetic sequences present new opportunities to understand and align these sequences copyright 2005 john wiley sons ltd

a characterization of positive linear maps and criteria of entanglement for quantum states let h and k be finite or infinite dimensional complex hilbert spaces a characterization of positive completely bounded normal linear maps from into is given which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras this characterization is then applied to give a representation of quantum channels operations between infinite dimensional systems a necessary and sufficient criterion of separability is given which shows that a state on h k is separable if and only if i 0 for all positive finite rank elementary operators examples of ncp and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the ppt criterion and the realignment criterion

a class of rank test procedures for censored survival data summary a class of linear rank statistics is proposed for the k sample problem with rightcensored survival data the class contains as special cases the log rank test mantel 1966 cox 1972 and a test essentially equivalent to peto peto s 1972 generalization of the wilcoxon test martingale theory is used to establish asymptotic normality of test statistics under the null hypotheses considered and to derive expressions for asymptotic relative efficiencies under contiguous sequences of alternative hypotheses a class of distributions is presented which corresponds to the class of rank statistics in the sense that for each distribution there is a statistic with some optimal properties for detecting location alternatives from that distribution some monte carlo results are displayed which present small sample behaviour

piecewise exponential survival trees with time dependent covariates survival trees methods are nonparametric alternatives to the semiparametric cox regression in survival analysis in this paper a tree based method for censored survival data with time dependent covariates is proposed the proposed method assumes a very general model for the hazard function and is fully nonparametric the recursive partitioning algorithm uses the likelihood estimation procedure to grow trees under a piecewise exponential structure that handles time dependent covariates in a parallel way to time independent covariates in general the estimated hazard at a node gives the risk for a group of individuals during a specific time period both cross validation and bootstrap resampling techniques are implemented in the tree selection procedure the performance of the proposed survival trees method is shown to be good through simulation and application to real data

local likelihood estimation abstract a scatterplot smoother is applied to data of the form x 1 y 1 x 2 y 2 xn yn and uses local fitting to estimate the dependence of y on x a simple example is the running lines smoother which fits a least squares line to the y values falling in a window around each x value the value of the estimated function at x is given by the value of the least squares line at x a smoother generalizes the least squares line which assumes that the dependence of y on x is linear in this article we extend the idea of local fitting to likelihood based regression models one such application is to the class of generalized linear models nelder and wedderburn 1972 we enlarge this class by replacing the covariate form 0 x 1 with an unspecified smooth function s x this function is estimated from the data by a technique we call local likelihood estimation the method consists of maximum likelihood estimation for 0 and 1 applied in a window around each x value multiple covariates are incor

recursively imputed survival trees we propose recursively imputed survival tree rist regression for right censored data this new nonparametric regression procedure uses a novel recursive imputation approach combined with extremely randomized trees that allows significantly better use of censored data than previous tree based methods yielding improved model fit and reduced prediction error the proposed method can also be viewed as a type of monte carlo em algorithm which generates extra diversity in the tree based fitting process simulation studies and data analyses demonstrate the superior performance of rist compared with previous methods

spline product quadrature rules for cauchy singular integrals abstract in this paper product quadrature rules based on cubic spline interpolation are obtained for the numerical evaluation of cauchy singular integrals and an error bound is proposed some numerical examples that test the performance of the rules are given

robust procedures for converting among lindblad kraus and matrix representations of quantum dynamical semigroups given a quantum dynamical semigroup expressed as an exponential superoperator acting on a space of n dimensional density operators eigenvalue methods are presented by which canonical kraus and lindblad operator sum representations can be computed these methods provide a mathematical basis on which to develop novel algorithms for quantum process tomography the statistical estimation of superoperators and their generators from a wide variety of experimental data theoretical arguments and numerical simulations are presented which imply that these algorithms will be quite robust in the presence of random errors in the data

separable states and positive maps abstract using the natural duality between linear functionals on tensor products of c algebras with the trace class operators on a hilbert space h and linear maps of the c algebra into b h we study the relationship between separability entanglement and the peres condition of states and positivity properties of the linear maps

a class of atomic positive linear maps in matrix algebras we find a large class of atomic positive linear maps between n dimensional matrix algebras and study their properties in connection with complete positivity 2 positivity and schwarz inequality

multivariate survival trees a maximum likelihood approach based on frailty models summary a method of constructing trees for correlated failure times is put forward it adopts the backfitting idea of classification and regression trees cart breiman et al 1984 in classification and regression trees the tree method is developed based on the maximized likelihoods associated with the gamma frailty model and standard likelihood related techniques are incorporated the proposed method is assessed through simulations conducted under a variety of model configurations and illustrated using the chronic granulomatous disease cgd study data

tree structured logistic model for over dispersed binomial data with application to modeling developmental effects this article proposes tree structured logistic regression modeling for over dispersed binomial data recursive partitioning is performed using a combination of statistical tests and residual analysis the splitting criterion in cross validation is based on the deviance function a nested grid algorithm to estimate the bootstrap parameters is developed the regression tree procedure provides a new approach for exploring in detail the relationship between the binomial response and explanatory variables the proposed procedure is used to model the relationship between the incidence of malformation and dose and fetal weight using data from a developmental experiment conducted at the national center for toxicological research a conditional gaussian chain model is used to account for the effect of fetal weight by dose

facial structures for unital positive linear maps in the two dimensional matrix algebra abstract we completely determine the lattice structure for the faces of the convex set of all unital positive linear maps between the 2 2 matrix algebras

on positive linear maps between matrix algebras abstract there is a concrete example of a positive linear map from m 2 to m 4 which is not decomposable modification of this map gives an explicit counterexample to a conjecture of woronowicz on the strong kadison inequality

product integration of singular integrands based on cubic spline interpolation at equally spaced nodes in this paper the convergence of product integration rules based on cubic spline interpolation at equally spaced nodes with not a knot end condition is investigated for integrand functions with a interior or endpoint singularity in the integration interval

entanglement witnesses arising from choi type positive linear maps we construct optimal pptes witnesses to detect 3 3 ppt entangled edge states of type 6 8 constructed recently by kye and osaka 2012 j math phys 53 052201 to do this we consider positive linear maps which are variants of the choi type map involving complex numbers and examine several notions related to the optimality of those entanglement witnesses throughout the discussion we suggest a method to check the optimality of entanglement witnesses without the spanning property

martingale based residuals for survival models summary graphical methods based on the analysis of residuals are considered for the setting of the highly used cox 1972 regression model and for the andersen gill 1982 generalization of that model we start with a class of martingale based residuals as proposed by barlow prentice 1988 these residuals and or their transforms are useful for investigating the functional form of a covariate the proportional hazards assumption the leverage of each subject upon the estimates of 13 and the lack of model fit to a given subject 1 1 model consider a set of n independent subjects such that the counting process ni ni t t o for the ith subject in the set indicates the number of observed events experienced over time t the sample paths of the ni are step functions with jumps of size 1 and with ni 0 0 we assume that the intensity function for ni t is given by yi t da t zi t yi t ep z t dao t 1 where yi t is a 0 1 process indicating whether the ith subject is a risk at time t 13 is a vector of regression coefficients zi t is a p dimensional vector of covariate processes and ao is the baseline cumulative hazard function several familar survival models fit into this framework the andersen gill 1982 generalization of the cox 1972 model arises when ao t is completely unspecified the further restriction that yi t 1 until the first event or censoring and is 0 thereafter yields the cox model the parametric form ao t t yields a poisson model or an exponential if restricted to a single event per subject and ao t tp a weibull model our attention will focus primarily on the andersen gill and cox models however the

a class of exposed indecomposable positive maps exposed positive maps in matrix algebras define a dense subset of extremal maps we provide a class of indecomposable positive maps in the algebra of 2n 2n complex matrices with n 2 it is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states

time dependent tree structured survival analysis with unbiased variable selection through permutation tests incorporating time dependent covariates into tree structured survival analysis tssa may result in more accurate prognostic models than if only baseline values are used available time dependent tssa methods exhaustively test every binary split on every covariate however this approach may result in selection bias towards covariates with more observed values we present a method that uses unbiased significance levels from newly proposed permutation tests to select the time dependent or baseline covariate with the strongest relationship with the survival outcome the specific splitting value is identified using only the selected covariate simulation results show that the proposed time dependent tssa method produces tree models of equal or greater accuracy as compared to baseline tssa models even with high censoring rates and large within subject variability in the time dependent covariate to illustrate the proposed method is applied to data from a cohort of bipolar youth to identify subgroups at risk for self injurious behavior

how to construct indecomposable entanglement witnesses we present a very simple method for constructing indecomposable entanglement witnesses out of a given pair an entanglement witness w and the corresponding state detected by w this method may be used to produce new classes of atomic witnesses which are able to detect the weakest quantum entanglement actually it works perfectly in the multipartite case too moreover this method provides a powerful tool for constructing new examples of bound entangled states

distribution free inference on contrasts of arbitrary summary measures of survival we present an approach for inference on contrasts of clinically meaningful functionals of a survivor distribution e g restricted mean quantiles that can avoid strong parametric or semiparametric assumptions on the underlying structure of the data in this multistage approach we first use an adaptive predictive model to estimate conditional survival distributions based on covariates we then estimate nonparametrically one or more functionals of survival from the covariate specific survival curves and evaluated contrasts of those functionals we find that the use of an adaptive nonparametric tree based predictive model leads to minimal loss in precision when semiparametric assumptions hold and provides marked improvement in accuracy when those assumptions are invalid therefore this work as a whole supports the use of survival summaries appropriate to a given medical application whether that be for example the median or 75th percentile in some settings or perhaps a restricted mean in others the approach is also compared with the mayo r score for primary biliary cirrhosis prognosis copyright 2012 john wiley sons ltd

a family of indecomposable positive linear maps based on entangled quantum states abstract we introduce a new family of indecomposable positive linear maps based on entangled quantum states central to our construction is the notion of an unextendible product basis the construction lets us create indecomposable positive linear maps in matrix algebras of arbitrary high dimension

convergence results for piecewise linear quadratures for cauchy principal value integrals etablissement des conditions de convergence simple et uniforme des quadratures lineaires par morceaux pour les integrales a valeur principale de cauchy

adaptive regression splines in the cox model we develop a method for constructing adaptive regression spline models for the exploration of survival data the method combines cox s 1972 journal of the royal statistical society series b 34 187 200 regression model with a weighted least squares version of the multivariate adaptive regressi on spline mars technique of friedman 1991 annals of statistics 19 1 141 to adaptively select the knots and covariates the new technique can automatically fit models with terms that represent nonlinear effects and interactions among covariates applications based on simulated data and data from a clinical trial for myeloma are presented results from the myeloma application identified several important prognostic variables including a possible nonmonotone relationship with survival in one laboratory variable results are compared to those from the adaptive hazard regression hare method of kooperberg stone and truong 1995 journal of the american statistical association 90 78 94

mixtures of proportional hazards regression models this paper presents a mixture model which combines features of the usual cox proportional hazards model with those of a class of models known as mixtures of experts the resulting model is more flexible than the usual cox model in the sense that the log hazard ratio is allowed to vary non linearly as a function of the covariates thus it provides a flexible approach to both modelling survival data and model checking the method is illustrated with simulated data as well as with multiple myeloma data

a partitioning deletion substitution addition algorithm for creating survival risk groups summary accurately assessing a patient s risk of a given event is essential in making informed treatment decisions one approach is to stratify patients into two or more distinct risk groups with respect to a specific outcome using both clinical and demographic variables outcomes may be categorical or continuous in nature important examples in cancer studies might include level of toxicity or time to recurrence recursive partitioning methods are ideal for building such risk groups two such methods are classification and regression trees cart and a more recent competitor known as the partitioning deletion substitution addition partdsa algorithm both of which also utilize loss functions e g squared error for a continuous outcome as the basis for building selecting and assessing predictors but differ in the manner by which regression trees are constructed recently we have shown that partdsa often outperforms cart in so called full data settings e g uncensored outcomes however when confronted with censored outcome data the loss functions used by both procedures must be modified there have been several attempts to adapt cart for right censored data this article describes two such extensions for partdsa that make use of observed data loss functions constructed using inverse probability of censoring weights such loss functions are consistent estimates of their uncensored counterparts provided that the corresponding censoring model is correctly specified the relative performance of these new methods is evaluated via simulation studies and illustrated through an analysis of clinical trial data on brain cancer patients the implementation of partdsa for uncensored and right censored outcomes is publicly available in the r package partdsa