the integral 3 harmonic graphs abstract let g be a graph on n vertices v1 vn and let d vi be the degree of the vertex vi if d g d v1 d vn is an eigenvector of the 0 1 adjacency matrix a of g i e a g d g d g then g is said to be harmonic in this paper all connected integral 3 harmonic graphs are determined there are exactly 26 such graphs

integral graphs and k τ regular sets abstract a subset of the vertex set of a graph g w v g is a k regular set if it induces a k regular subgraph of g and every vertex not in the subset has neighbors in it in this paper we deal with the existence of k regular sets associated with all distinct eigenvalues we show some families that have this property and we give some results concerning the existence of such sets considering restrictions on the symbol of circulant graphs

on spectral integral variations of mixed graphs in this paper we characterize the mixed graphs with exactly one laplacian eigenvalue moving up by an integer and other laplacian eigenvalues remaining invariant when an edge is added the results extend those of fan linear and multilinear algebra 50 2002 133 for general graphs and so linear and multilinear algebra 46 1999 193 for simple graphs

on parametric density estimation summary the paper makes a critical assessment of aitchison s criterion of density estimation it seems not to satisfy certain intuitive requirements a new criterion based on decisions theoretic considerations is introduced it is shown through examples that the judgement about the choice of density estimates is seriously dependent on the criterion chosen some other density estimates are introduced which look similar to bayesian predictive density estimates without using a prior distribution

distance spectra and distance energy of integral circulant graphs the distance energy of a graph g is a recently developed energy type invariant defined as the sum of absolute values of the eigenvalues of the distance matrix of g there was a vast research for the pairs and families of non cospectral graphs having equal distance energy and most of these constructions were based on the join of graphs a graph is called circulant if it is cayley graph on the circulant group i e its adjacency matrix is circulant a graph is called integral if all eigenvalues of its adjacency matrix are integers integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer in this paper we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix d furthermore we calculate the distance spectra and distance energy of unitary cayley graphs in conclusion we present two families of pairs g1 g2 of integral circulant graphs with equal distance energy in the first family g1 is subgraph of g2 while in the second family the diameter of both graphs is three

regular graphs with four eigenvalues we study the connected regular graphs with four distinct eigenvalues properties and feasibility conditions of the eigenvalues are found several examples constructions and characterizations are given as well as some uniqueness and nonexistence results

on coherence in parametric density estimation summary in parametric density estimation the existence of a prior distribution on the parameter dictates the use of the corresponding predictive density function as estimate this paper argues that in a decision theory approach to parameter density estimation any loss function introduced should lead to this predictive result for situations where prior information is available the kullbackleibler directed divergence has this coherence property whereas the corresponding symmetric divergence does not the role of a recently suggested sequential improvement criterion in density estimation is also discussed

a parametric family of correlation structures for the analysis of longitudinal data in epidemiological settings we are often faced with numerous short time series and a parsimonious parametrization of the correlation structure is desired in order to optimize the efficiency of the estimation procedure we propose a damped exponential correlation structure for modeling multivariate gaussian outcomes the correlation between two observations separated by s units of time is modeled as gamma s theta where gamma is the correlation between elements separated by one s unit and theta is a damping parameter for theta 0 theta 1 and theta infinity the correlation structures of compound symmetry first order autoregressive and first order moving average processes are obtained although the ar 2 dependency structure and the combination of random effects and ar 1 errors are not special cases of the proposed parametric family these structures can be well approximated within the family for short time series maximum likelihood methods for parameter estimation and interpretations of intermediate models 0 less than theta less than 1 are discussed in the context of modeling pulmonary function in an adult population in the netherlands and t cell subsets in homosexual men infected with human immunodeficiency virus type i

further results on the perfect state transfer in integral circulant graphs for a given graph g denote by a its adjacency matrix and f t exp iat we say that there exist a perfect state transfer pst in g if f t a b 1 for some vertices a b and a positive real number t such a property is very important for the modeling of quantum spin networks with nearest neighbor couplings we consider the existence of the perfect state transfer in integral circulant graphs circulant graphs with integer eigenvalues some results on this topic have already been obtained by saxena et al 2007 5 basic et al 2009 6 and basic petkovic 2009 7 in this paper we show that there exists an integral circulant graph with n vertices having a perfect state transfer if and only if 4 n several classes of integral circulant graphs have been found that have a perfect state transfer for the values of n divisible by 4 moreover we prove the nonexistence of a pst for several other classes of integral circulant graphs whose order is divisible by 4 these classes cover the class of graphs where the divisor set contains exactly two elements the obtained results partially answer the main question of which integral circulant graphs have a perfect state transfer

goodness of prediction fit for multivariate linear models abstract using both frequentist and bayesian techniques predicting densities are derived for future observations from a multivariate linear model with matrix normal error terms all the candidates belong to a general location scale family of predicting densities an analytic comparison is undertaken using a kullback leibler loss by citing an optimal member of a subclass including most of these predicting densities as members the subclass is based on an invariant student t random matrix and the optimal member is the bayesian predictive density corresponding to a jeffreys noninformative prior information based numerical comparisons illustrate the nature of the dominance

convergence of a stochastic approximation version of the em algorithm the expectation maximization em algorithm is a powerful computational technique for locating maxima of functions it is widely used in statistics for maximum likelihood or maximum a posteriori estimation in incomplete data models in certain situations however this method is not applicable because the expectation step cannot be performed in closed form to deal with these problems a novel method is introduced the stochastic approximation em saem which replaces the expectation step of the em algorithm by one iteration of a stochastic approximation procedure the convergence of the saem algorithm is established under conditions that are applicable to many practical situations moreover it is proved that under mild additional conditions the attractive stationary points of the saem algorithm correspond to the local maxima of the function presented to support our findings

a monte carlo implementation of the em algorithm and the poor man s data augmentation algorithms abstract the first part of this article presents the monte carlo implementation of the e step of the em algorithm given the current guess to the maximizer of the posterior distribution latent data patterns are generated from the conditional predictive distribution the expected value of the augmented log posterior is then updated as a mixture of augmented log posteriors mixed over the generated latent data patterns multiple imputations in the m step of the algorithm this mixture is maximized to obtain the update to the maximizer of the observed posterior the gradient and hessian of the observed log posterior are also expressed as mixtures mixed over the multiple imputations the relation between the monte carlo em mcem algorithm and the data augmentation algorithm is noted two modifications to the mcem algorithm the poor man s data augmentation algorithms which allow for the calculation of the entire posterior are then presented these approximations serve as diagnostics for the validity o

the complement of the path is determined by its spectrum abstract it is proved that a graph whose 0 1 adjacency matrix has the spectrum of p n the complement of the path on n vertices must be p n

a random effects model for multiple characteristics with possibly missing data abstract the use of random effects models for the analysis of longitudinal data with missing responses has been discussed by several authors this article extends the random effects model for a single characteristic to the case of multiple characteristics allowing for arbitrary patterns of observed data two different structures for the covariance matrix of measurement error are considered uncorrelated error between responses and correlation of error terms at the same measurement times parameters for this model are estimated via the em algorithm the set of equations for this estimation procedure is derived these equations are appropriately modified to deal with missing data the methodology is illustrated with an example from clinical trials

laplacian integral graphs in s a b abstract let g n m be the family of graphs with n vertices and m edges when n and m are previously given it is well known that there is a subset of g n m constituted by graphs g such that the vertex connectivity the edge connectivity and the minimum degree are all equal in this paper s a b classes of connected a b linear graphs with n vertices and m edges are described where m is given as a function of a b n 2 some of them have extremal graphs for which the equalities above are extended to algebraic connectivity these graphs are laplacian integral although they are not threshold graphs however we do build threshold graphs in s a b

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

perfect state transfer in integral circulant graphs of non square free order abstract this paper provides further results on the perfect state transfer in integral circulant graphs icg graphs the non existence of pst is proved for several classes of icg graphs containing an isolated divisor d 0 i e the divisor which is relatively prime to all other divisors from d d d 0 the same result is obtained for classes of integral circulant graphs having the nsf property i e each n d is square free for every d d a direct corollary of these results is the characterization of icg graphs with two divisors which have pst a similar characterization is obtained for icg graphs where each two divisors are relatively prime finally it is shown that icg graphs with the number of vertices n 2 p 2 do not have pst

on efficient probability forecasting systems we study the asymptotic behaviour of probability forecasting systems and discuss their usefulness as inferential tools for statistical problems such as model verification and selection our theoretical setting is the prequential or predictive sequential framework proposed by dawid 1984 we study especially the notion of prequential efficiency of a forecasting system and present some new results we focus on plug in or estimative forecasting systems where the forecast distribution is generated by replacing the unknown parameter with an estimate

flexible longitudinal linear mixed models for multiple censored responses data in biomedical studies and clinical trials repeated measures are often subject to some upper and or lower limits of detection hence the responses are either left or right censored a complication arises when more than one series of responses is repeatedly collected on each subject at irregular intervals over a period of time and the data exhibit tails heavier than the normal distribution the multivariate censored linear mixed effect mlmec model is a frequently used tool for a joint analysis of more than one series of longitudinal data in this context we develop a robust generalization of the mlmec based on the scale mixtures of normal distributions to take into account the autocorrelation existing among irregularly observed measures a damped exponential correlation structure is considered for this complex longitudinal structure we propose an exact estimation procedure to obtain the maximum likelihood estimates of the fixed effects and variance components using a stochastic approximation of the em algorithm this approach allows us to estimate the parameters of interest easily and quickly as well as to obtain the standard errors of the fixed effects the predictions of unobservable values of the responses and the log likelihood function as a byproduct the proposed method is applied to analyze a set of aids data and is examined via a simulation study

regular bipartite graphs with three distinct non negative eigenvalues abstract we derive some structural and spectral properties of regular bipartite graphs with three distinct non negative eigenvalues next we consider the relations between these graphs and two class partially balanced incomplete block designs and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs as a consequence we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues and we also determine all such graphs with degree 3 and all such graphs on at most 20 vertices

on the chromatic number of integral circulant graphs integral circulant graphs are a generalization of unitary cayley graphs recently studied by klotz and sander the integral circulant graph x n d has vertices 0 1 n 1 and two vertices a and b are adjacent iff gcd x y n d where d d d n 1 d 3 we construct a family of counterexamples using exhaustive search algorithm for graph coloring and disprove the conjecture in this case

on algebraic connectivity and spectral integral variations of graphs abstract let g be a simple connected graph and l g be the laplacian matrix of g let a g be the second smallest eigenvalue of l g an eigenvector of l g corresponding to the eigenvalue a g is called a fiedler vector of g let y be a fiedler vector of g a characteristic vertex is a vertex u of g such that y u 0 and such that there is a vertex w adjacent to u satisfying y w 0 a characteristic edge is an edge u v such that y u y v

the integral graphs with index 3 and exactly two main eigenvalues a graph is called integral if the spectrum of its adjacency matrix has only integral eigenvalues an eigenvalue of a graph is called main eigenvalue if it has an eigenvector such that the sum of whose entries is not equal to zero in this paper we show that there are exactly 25 connected integral graphs with exactly two main eigenvalues and index 3

walks and regular integral graphs we establish a useful correspondence between the closed walks in regular graphs and the walks in infinite regular trees which after counting the walks of a given length between vertices at a given distance in an infinite regular tree provides a lower bound on the number of closed walks in regular graphs this lower bound is then applied to reduce the number of the feasible spectra of the 4 regular bipartite integral graphs by more than a half next we give the details of the exhaustive computer search on all 4 regular bipartite graphs with up to 24 vertices which yields a total of 47 integral graphs

predictive fit for natural exponential families summary the paper examines predictive distributions concentrating on measuring their fit to the true distribution by average kullback leibler divergence the notion of an averaged bootstrap predictive distribution is introduced this predictive distribution is shown to be asymptotically superior to the estimative distribution in terms of average kullbackleibler divergence when the true distribution is in a natural exponential family smallsample results are presented for the poisson and binomial distributions which suggest that the bootstrap distribution performs well in these cases

indecomposable laplacian integral graphs abstract a graph that can be constructed from isolated vertices by the operations of union and complement is decomposable every decomposable graph is laplacian integral i e its laplacian spectrum consists entirely of integers an indecomposable graph is not decomposable the main purpose of this note is to demonstrate the existence of infinitely many indecomposable laplacian integral graphs

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

infinite families of q integral graphs abstract in this article we characterize all signless laplacian integral graphs here called q integral graphs in the following classes complete split graphs multiple complete split like graphs extended complete split like graphs multiple extended split like graphs all these graphs were defined by hansen et al p hansen h melot d stevanovic integral complete split graphs univ beograd publ elektrotehn fak ser mat 13 2002 89 95 also we characterize all q integral graphs of the form k n 1 k n 2 k n 3 where k n i i 1 2 3 is a complete graph on n i vertices these characterizations allow us to exhibit many infinite families of q integral graphs

two spectral characterizations of regular bipartite graphs with five eigenvalues graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure we show that regular bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles this enables to determine from the spectrum alone the feasible families of numbers of common neighbors for each vertex with other vertices in its part for particular spectra such as 6 29 06 29 6 where exponents denote eigenvalue multiplicities there is a unique such family which makes it possible to characterize all graphs with this spectrum using this lemma we also to show that for r 2 a graph has spectrum r rr r 1 02 r 1 rr r 1 r if and only if it is a graph of a 1 resolvable transversal design td r r i e if it corresponds to the complete set of mutually orthogonal latin squares of size r in a well defined manner

exact minimax estimation of the predictive density in sparse gaussian models we consider estimating the predictive density under kullback leibler loss in an l0 sparse gaussian sequence model explicit expressions of the first order minimax risk along with its exact constant asymptotically least favorable priors and optimal predictive density estimates are derived compared to the sparse recovery results involving point estimation of the normal mean new decision theoretic phenomena are seen suboptimal performance of the class of plug in density estimates reflects the predictive nature of the problem and optimal strategies need diversification of the future risk we find that minimax optimal strategies lie outside the gaussian family but can be constructed with threshold predictive density estimates novel minimax techniques involving simultaneous calibration of the sparsity adjustment and the risk diversification mechanisms are used to design optimal predictive density estimates

bayesian semiparametric mixture tobit models with left censoring skewness and covariate measurement errors common problems to many longitudinal hiv aids cancer vaccine and environmental exposure studies are the presence of a lower limit of quantification of an outcome with skewness and time varying covariates with measurement errors there has been relatively little work published simultaneously dealing with these features of longitudinal data in particular left censored data falling below a limit of detection lod may sometimes have a proportion larger than expected under a usually assumed log normal distribution in such cases alternative models which can account for a high proportion of censored data should be considered in this article we present an extension of the tobit model that incorporates a mixture of true undetectable observations and those values from a skew normal distribution for an outcome with possible left censoring and skewness and covariates with substantial measurement error to quantify the covariate process we offer a flexible nonparametric mixed effects model within the tobit framework a bayesian modeling approach is used to assess the simultaneous impact of left censoring skewness and measurement error in covariates on inference the proposed methods are illustrated using real data from an aids clinical study

combinatorial designs with two singular values ii partial geometric designs in this and an earlier paper e r van dam e spence combinatorial designs with two singular values i uniform multiplicative designs j combin theory a 107 2004 127 142 we study combinatorial designs whose incidence matrix has two distinct singular values these generalize v k designs and include uniform multiplicative designs and partial geometric designs here we study the latter which are precisely the designs with constant replication and block size we collect most known results give new characterization results and we enumerate partly by computer all small ones

which graphs are determined by their spectrum for almost all graphs the answer to the question in the title is still unknown here we survey the cases for which the answer is known not only the adjacency matrix but also other types of matrices such as the laplacian matrix are considered

on prediction intervals based on predictive likelihood or bootstrap methods we argue that prediction intervals based on predictive likelihood do not correct for curvature with respect to the parameter value when they implicitly approximate an unknown probability density partly as a result of this difficulty the order of coverage error associated with predictive intervals and predictive limits is equal to only the inverse of sample size in this respect those methods do not improve on the simpler naive or estimative approach moreover in cases of practical importance the latter can be preferable in terms of both the size and sign of coverage error we show that bootstrap calibration of both naive and predictive likelihood approaches increases coverage accuracy of prediction intervals by an order of magnitude and in the case of naive intervals preserves that method s numerical and analytical simplicity therefore we argue the bootstrap calibrated naive approach is a particularly competitive alternative to more conventional but more complex techniques based on predictive likelihood

graphs with integral spectrum abstract it is shown that only a fraction of 2 n of the graphs on n vertices have an integral spectrum although there are several explicit constructions of such graphs no upper bound for their number has been known graphs of this type play an important role in quantum networks supporting the so called perfect state transfer

constructably laplacian integral graphs a graph is laplacian integral if the spectrum of its laplacian matrix consists entirely of integers we consider the class of constructably laplacian integral graphs those graphs that be constructed from an empty graph by adding a sequence of edges in such a way that each time a new edge is added the resulting graph is laplacian integral we characterize the constructably laplacian integral graphs in terms of certain forbidden vertex induced subgraphs and consider the number of nonisomorphic laplacian integral graphs that can be constructed by adding a suitable edge to a constructably laplacian integral graph we also discuss the eigenvalues of constructably laplacian integral graphs and identify families of isospectral nonisomorphic graphs within the class

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

integration of stochastic models by minimizing α divergence when there are a number of stochastic models in the form of probability distributions one needs to integrate them mixtures of distributions are frequently used but exponential mixtures also provide a good means of integration this letter proposes a one parameter family of integration called integration which includes all of these well known integrations these are generalizations of various averages of numbers such as arithmetic geometric and harmonic averages there are psychophysical experiments that suggest that integrations are used in the brain the divergence between two distributions is defined which is a natural generalization of kullback leibler divergence and hellinger distance and it is proved that integration is optimal in the sense of minimizing divergence the theory is applied to generalize the mixture of experts and the product of experts to the mixture of experts the predictive distribution is also stated in the bayesian framework