a variable selection heuristic for k means clustering one of the most vexing problems in cluster analysis is the selection and or weighting of variables in order to include those that truly define cluster structure while eliminating those that might mask such structure this paper presents a variable selection heuristic for nonhierarchical k means cluster analysis based on the adjusted rand index for measuring cluster recovery the heuristic was subjected to monte carlo testing across more than 2200 datasets with known cluster structure the results indicate the heuristic is extremely effective at eliminating masking variables a cluster analysis of real world financial services data revealed that using the variable selection heuristic prior to the k means algorithm resulted in greater cluster stability

real number computability and domain theory we present the different constructive definitions of real number that can be found in the literature using domain theory we analyse the notion of computability that is substantiated by these definitions and we give a definition of computability for real numbers and for functions acting on them this definition of computability turns out to be equivalent to other definitions given in the literature using different methods domain theory is a useful tool to study higher order computability on real numbers an interesting connection between scott topology and the standard topologies on the real line and on the space of continuous functions on reals is stated an important result in this paper is the proof that every computable functional on real numbers is continuous w r t the compact open topology on the function space

cost trade offs in graph embeddings with applications an embedding of the graph g in the graph h is a one to one association of the vertices of g with the vertices of h there are two natural measures of the cost of a graph embedding namely the dilation cost of the embedding the maximum distance in h between the images of vertices that are adjacent in g and the expansion cost of the embedding the ratio of the size of h to the size of g the main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost the first result establishes the following there is an embedding of n node complete ternary trees in complete binary trees with dilation cost 2 and expansion cost n where log3 4 3 but any embedding of these ternary trees in binary trees that has expansion cost 2 must infinitely often have dilation cost const log log log n the second result provides a stronger but less easily stated example of the same type of tradeoff the third result concerns generic binary trees that is complete binary trees into which all n node binary trees are efficiently embeddable there is a generic binary tree into which all n node binary trees are embeddable with dilation cost o 1 and expansion cost o nc for some fixed constant c if one insists on embeddings whose dilation cost is exactly 1 then these embeddings must have expansion cost n log n 2 if one insists on embeddings whose expansion cost is 2 then these embeddings must infinitely often have dilation cost const log log log n an interesting application of the polynomial size generic binary tree in the first part of this three part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of computation tree such as alternating and nondeterministic turing machines and context free grammars

a new approach to cluster analysis the clustering function based method the purpose of the paper is to present a new statistical approach to hierarchical cluster analysis with n objects measured on p variables motivated by the model of multivariate analysis of variance and the method of maximum likelihood a clustering problem is formulated as a least squares optimization problem simultaneously solving for both an n vector of unknown group membership of objects and a linear clustering function this formulation is shown to be linked to linear regression analysis and fisher linear discriminant analysis and includes principal component regression for tackling multicollinearity or rank deficiency polynomial or b splines regression for handling non linearity and various variable selection methods to eliminate irrelevant variables from data analysis algorithmic issues are investigated by using sign eigenanalysis

model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

a model of computation for vlsi with related complexity results a new model of computation for vlsi based on the assumption that time for propagating information is at least linear in the distance is proposed while accommodating for basic laws of physics the model is designed to be general and technology independent thus from a complexity viewpoint it is especially suited for deriving lower bounds and trade offs new results for a number of problems including fan in transitive functions matrix multiplication and sorting are presented as regards upper bounds it must be noted that because of communication costs the model clearly favors regular and pipelined architectures e g systolic arrays

a computational model for metric spaces for every metric space x we define a continuous poset bx such that x is homeomorphic to the set of maximal elements of bx with the relative scott topology the poset bx is a dcpo iff x is complete and continuous iff x is separable the computational model bx is used to give domain theoretic proofs of banach s fixed point theorem and of two classical results of hutchinson on a complete metric space every hyperbolic iterated function system has a unique non empty compact attractor and every iterated function system with probabilities has a unique invariant measure with bounded support we also show that the probabilistic power domain of bx provides an continuous computational model for measure theory on a separable complete metric space x

dynamical systems measures and fractals via domain theory abstract we introduce domain theory in dynamical systems iterated function systems fractals and measure theory for a discrete dynamical system given by the action of a continuous map f x x on a metric space x we study the extended dynamical systems vx vf ux uf and lx lf where v u and l are respectively the vietoris hyperspace the upper hyperspace and the lower hyperspace functors we show that if x f is chaotic then so is ux uf when x is locally compact ux is a continuous bounded complete dcpo if x is second countable as well then ux will be continuous and can be given an effective structure we show how strange attractors attractors of iterated function systems fractals and julia sets are obtained effectively as fixed points of deterministic functions on ux or fixed points of non deterministic functions on cux where c is the convex plotkin power domain we also show that the set m x of finite borel measures on x can be embedded in pux where p is the probabilistic power domain this provides an effective framework for measure theory we then prove that the invariant measure of an hyperbolic iterated function system with probabilities can he obtained as the unique fixed point of an associated continuous function on pux

effective domain representations of h x the space of compact subsets this paper gives effective domain representations of spaces h x of non empty compact subsets of effective complete metric spaces x the domain representation of h x is constructed from a domain representation of x using the plotkin power domain construction as an application of the representation an effective version of a fundamental theorem on ifs iterated function system is shown

ideal models of spaces ideal domains have an elementary order theoretic structure every element is either compact or maximal despite this we establish that 1 they can model any space currently known to possess a countably based model and 2 the metric spaces with ideal models are exactly the completely metrizable spaces

computability of probability measures and martin lof randomness over metric spaces in this paper we investigate algorithmic randomness on more general spaces than the cantor space namely computable metric spaces to do this we first develop a unified framework allowing computations with probability measures we show that any computable metric space with a computable probability measure is isomorphic to the cantor space in a computable and measure theoretic sense we show that any computable metric space admits a universal uniform randomness test without further assumption

graphs that are almost binary trees this paper studies embeddings of graphs in binary trees the cost of such an embedding is the maximum distance in the binary tree between images of adjacent graph vertices several techniques for bounding the costs of such embeddings from above are derived notable among these is an algorithm for embedding any outerplanar graph in a binary tree with a cost that is within a factor of 3 of optimal a number of techniques for bounding the costs of such embeddings from below are developed notable here are two techniques for inferring the presence of large separators in graphs finally a number of characterizations are established of those families of graphs that are almost binary trees in the sense that every graph in the family is embeddable in a binary tree within bounded cost

embedding metric spaces into cpo s abstract we show that metric spaces with polish topology mp can be represented very well by bounded complete algebraic cpo s with continuous weight and distance cwd an operator cwd mp is defined such that 1 and 2 hold 1 1 d can be embedded topologically into d such that a partial function on d is metrically continuous if it can be extended to a total continuous function on d 2 2 every m mp is isometric to some d therefore the continuous cpo functions are exactly adequate for describing the continuous functions on the corresponding metric spaces the approach could be refined in order to study computability on metric spaces with polish topology examples of application are the euclidean space the space of total functions on the natural numbers baire space and spaces of infinite trees

domain representations of topological spaces a domain representation of a topological space x is a function usually a quotient map from a subset of a domain onto x several dierent classes of domain representations are introduced and studied it is investigated when it is possible to build domain representations from existing ones it is for example discussed whether there exists a natural way to build a domain representation of a product of topological spaces from given domain representations of the factors it is shown that any t0 topological space has a domain representation these domain representations are very large however smaller domain representations are also constructed for large classes of spaces for example each second countable regular hausdor space has a domain representation with a countable base domain representations of functions and function spaces are also studied c 2000 elsevier science b v all rights reserved

on embedding rectangular grids in square grids the main results in this paper demonstrate that there exist pairs of integers e d for area expansion and edge dilation respectively such that any n vertex rectangular grid can be embedded into a square grid having at most en vertices in such a way that images in the square grid of vertices that are adjacent in the rectangular grid are at most distance d apart several techniques for squaring up rectangular grids are presented sample values for the parameter pair e d are e 1 2 d 15 e 1 45 d 9 e 1 8 d 3 note that these values of e and d hold for all rectangular grids independent of number of vertices the quest for these results was motivated by the question of whether or not one could automatically square up circuit layouts having aspect ratios very far from unity without compromising the efficiency of the layout in terms of area and length of the longest run of wire the results reported here yield an affirmative answer to this question at least in an idealized setting one corollary of the embeddings presented here is that the side 2n square king s move grid contains as a subgraph every n vertex rectangular grid another way to think of this result is that this embellished grid can be programmed or personalized by setting switches to represent any n vertex rectangular grid

domain representations of partial functions with applications to spatial objects and constructive volume geometry a partial spatial object is a partial map from space to data data types of partial spatial objects are modelled by topological algebras of partial maps and are the foundation for a high level approach to volume graphics called constructive volume geometry cvg where space and data are subspaces of n dimensional euclidean space we investigate the computability of partial spatial object data types in general and in volume graphics using the theory of effective domain representations for topological algebras the basic mathematical problem considered is to classify which partial functions between topological spaces can be represented by total continuous functions between given domain representations of the spaces we prove theorems about partial functions on regular hausdorff spaces and their domain representations and apply the results to partial spatial objects and cvg algebras

universality considerations in vlsi circuits the problem of embedding the interconnection pattern of a circuit into a two dimensional surface of minimal area is discussed since even for some natural patterns graphs containing m connections may require m2 area in order to achieve compact embeddings restricted classes of graphs have to be considered for example arbitrary trees of bounded degree can be embedded in linear area without edges crossing over planar graphs can be embedded efficiently only if crossovers are allowed in the embedding

discrimillant analysis with discrete and continuous variables a paradigmatic methodologic approach does not exist for the problem of discriminant analysis with both discrete and continuous explanatory variables procedures that have been employed in this situation include fisher s linear discriminant function the quadratic discriminant function the linear discriminant furlction with higher order terms discriminant functions with logistic regression estimates of the coefficients and the location model average optimal error rates for these procedures are reported for the case of one normal variable and two binary variables these approaches are also compared for the prediction of two year survival following recovery from myocardial infarction employing resubstitution jackknife and independent sample estimates of the error rates recommendations for applications are presented

total objects of domains abstract the main result of this paper is a characterization of the space induced by scott topology on the set of total objects of a bounded complete cpo this characterization is simple with bounded complete algebraic cpo s where a family of clopen sets plays an essential role in the characterization in the more general case of continuity one can only produce distinct families of open sets and closed sets to axiomatize the essential properties of a total space besides the main characterization we also discuss its relation to continuous spaces studied in our earlier paper and the relation between compact t spaces and total spaces under the lower topology

two categories of effective continuous cpos this paper presents two categories of effective continuous complete partial orders cpos we define a new criterion on the basis of a cpo so as to make the resulting category of consistently complete continuous cpos cartesian closed we also generalise to continuous cpos the definition of a complete set which was used as a definition of effective bifinite domains in hamrin and stoltenberg hansen cartesian closed categories of effective domains in h schwichtenberg r steinbruggen eds proof and system reliability kluwer academic publishers dordrecht 2002 pp 1 20 and investigate the closure results that can be obtained

computability on subsets of metric spaces the notions recursively enumerable and recursive are the basic notions of effectivity in classical recursion theory in computable analysis these notions are generalized to closed subsets of euclidean space using their metric distance functions we study a further generalization of these concepts to subsets of computable metric spaces it appears that different characterizations which coincide in case of euclidean space lead to different notions in the general case however under certain additional conditions such as completeness and effective local compactness the situation is similar to the euclidean case we present all results in the framework of type 2 theory of effectivity which allows to express effectivity properties in a very uniform way instead of comparing properties of single subsets we compare corresponding representations of the hyperspace of closed subsets such representations do not only induce a concept of computability for single subsets but they even yield a concept of computability for operations on hyperspaces such as union intersection etc we complete our investigation by studying the special situation of compact subsets

power domains and iterated function systems we introduce the notion of weakly hyperbolic iterated function system ifs on a compact metric space which generalises that of hyperbolic ifs based on a domain theoretic model which uses the plotkin power domain and the probabilistic power domain respectively we prove the existence and uniqueness of the attractor of a weakly hyperbolic ifs and the invariant measure of a weakly hyperbolic ifs with probabilities extending the classic results of hutchinson for hyperbolic ifss in this more general setting we also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure extending the corresponding algorithms for hyperbolic ifss we then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent ifs and obtain an algorithm to generate the invariant distribution on the digitised screen the generalised riemann integral is used to provide a formula for the expected value of almost everywhere continuous functions with respect to this distribution for hyperbolic recurrent ifss and lipschitz maps one can estimate the integral up to any threshold of accuracy

static space times naturally lead to quasi pseudometrics the standard 4 dimensional minkowski space time of special relativity is based on the 3 dimensional euclidean metric in 1967 h busemann showed that similar static space time models can be based on an arbitrary metric space in this paper we search for the broadest possible generalization of a metric under which a construction of a static space time leads to a physically reasonable space time model it turns out that this broadest possible generalization is related to the known notion of a quasi pseudometric

clustering objects on subsets of attributes with discussion read before the royal statistical society at a meeting organized by the research section on wednesday may 5th 2004 professor j t kent in the chair summary a new procedure is proposed for clustering attribute value data when used in conjunction with conventional distance based clustering algorithms this procedure encourages those algorithms to detect automatically subgroups of objects that preferentially cluster on subsets of the attribute variables rather than on all of them simultaneously the relevant attribute subsets for each individual cluster can be different and partially or completely overlap with those of other clusters enhancements for increasing sensitivity for detecting especially low cardinality groups clustering on a small subset of variables are discussed applications in different domains including gene expression arrays are presented