probabilistic measures of adequacy of a numerical search for a global maximum abstract measures of the probability of all unobserved species are applied to the problem of assessing the adequacy of a search for a global maximum using random starting points the measures as used here estimate the probability that an iterative algorithm using a randomly selected starting point will find a solution not observed in previous random starting points the probability of an unobserved global maximum is less than or equal to this probability we used these measures to evaluate the adequacy of our search procedure for the maximum likelihood estimates of the parameters of a mixture of two normals these measures indicated that for most problems generated there was little chance that there were unobserved domains of convergence occasional problems however had appreciable estimated probabilities in such problems examination of the data suggested regions where a more focused search for unobserved domains of convergence was warranted

classification of generalized multiresolution analyses abstract we discuss how generalized multiresolution analyses gmras both classical and those defined on abstract hilbert spaces can be classified by their multiplicity functions m and matrix valued filter functions h given a natural number valued function m and a system of functions encoded in a matrix h satisfying certain conditions a construction procedure is described that produces an abstract gmra with multiplicity function m and filter system h an equivalence relation on gmras is defined and described in terms of their associated pairs m h this classification system is applied to mras and other classical examples in l 2 r d as well as to previously studied abstract examples

the construction of exact d optimum experimental designs with application to blocking response surface designs summary the properties of optimum designs are investigated when there are both qualitative and quantitative factors algorithms for the construction of exact designs are described which are of general applicability the case when the qualitative factors represent blocking variables is given special treatment a modified algorithm provides the opportunity to divide the experimental trials into blocks of specified sizes examples are given of second order models in the quantitative factors divided into two or three blocks of a variety of sizes use of the adjustment algorithm to improve such designs generated from a list of candidate points is demonstrated

low pass filters and representations of the baumslag solitar group we analyze representations of the baumslag solitar group bs 1 n u t utu 1 t n that admit wavelets and show how such representations can be constructed from a given low pass filter we describe the direct integral decomposition for some examples and derive from it a general criterion for the existence of solutions for scaling equations as another application we construct a fourier transform for some hausdorff measures

higher order interpolatory vector bases on pyramidal elements in the numerical solution of three dimensional 3 d electromagnetic field problems the regions of interest can be discretized by elements having tetrahedral brick or prismatic shape however such different shape elements cannot be linked to form a conformal mesh to this purpose pyramidal elements are required in this paper we define interpolatory higher order curl and divergence conforming vector basis functions on pyramidal elements with extension to curved pyramids and discuss their completeness properties a general procedure to obtain vector bases of arbitrary polynomial order is given and bases up to second order are explicitly reported these new elements ensure the continuity of the proper vector components across adjacent elements of equal order but different shape results to confirm the faster convergence of higher order functions on pyramids are presented

generalized simulated annealing for function optimization a generalized simulated annealing method has been developed and applied to the optimization of functions possibly constrained having many local extrema the method is illustrated in some difftcult pedagogical examples and used to solve a problem analyzed by bates technometrics 25 pp 373 376 1983 for which we identify an improved optimum the sensitivity of the solution to changes in the constraints and in other specifications of the problem is analyzed and discussed

direct limits multiresolution analyses and wavelets a multiresolution analysis for a hilbert space realizes the hilbert space as the direct limit of an increasing sequence of closed subspaces in a previous paper we showed how conversely direct limits could be used to construct hilbert spaces which have multiresolution analyses with desired properties in this paper we use direct limits and in particular the universal property which characterizes them to construct wavelet bases in a variety of concrete hilbert spaces of functions our results apply to the classical situation involving dilation matrices on l2 rn the wavelets on fractals studied by dutkay and jorgensen and hilbert spaces of functions on solenoids

decomposition of wavelet representations and martin boundaries we study a decomposition problem for a class of unitary representations associated with wavelet analysis wavelet representations but our framework is wider and has applications to multi scale exp

designing studies for dose response summary dose response refers to the regression of a response on a stimulus we review a number of options for doseresponse designs and compare various designs which may be used in practice we start with two group designs next we introduce basic optimal approximate design theory for simple linear and quadratic regression illustrating different criteria of optimality and their effect on the allocation of the levels of the dose then we obtain the efficiencies of these optimal approximate designs and some simple designs which have intuitive appeal symmetry equal spacing of treatments reduced numbers of observations at the highest and lowest doses

integer valued minimax robust designs for estimation and extrapolation in heteroscedastic approximately linear models abstract we present our findings on a new approach to robust regression design this approach differs from previous investigations into this area in three respects the use of a finite design space the use of simulated annealing to carry out the numerical minimization problems and in our search for integer valued rather than continuous designs we present designs for the situation in which the response is thought to be approximately polynomial we also discuss the cases of approximate first and second order multiple regression in each case we allow for possible heteroscedasticity and also obtain minimax regression weights the results are extended to cover extrapolation of the regression response to regions outside of the design space a case study involving dose response experimentation is undertaken the optimal robust designs which protect against bias as well as variance can be roughly described as being obtained from the classical variance minimizing designs by replacing replicates with clust

the application of the annealing algorithm to the construction of exact optimal designs for linear regression models this article reports the application of the annealing algorithm to the construction of exact d i and g optimal designs for polynomial regression of degree 5 on the interval 1 l and for the second order model in two factors on the design space 1 l 1 1 details of the perturbation scheme and the annealing schedules used are given and the method of implementation is illustrated by means of a simple example the algorithm is assessed by comparing its performance in terms of computer time and effkiency with the modified fedorov procedure and it is shown to be particularly effective in finding g optimal designs the salient features of the exact designs constructed in this study are also summarized

constrained d and d_1 optimal designs for polynomial regression in the common polynomial regression model of degree m we consider the problem of determining the d and d 1 optimal designs subject to certain constraints for the d 1 efficiencies in the models of degree m j m j 1 m k m j 0 k 0 given we present a complete solution of these problems which on the one hand allow a fast computation of the constrained optimal designs and on the other hand give an answer to the question of the existence of a design satisfying all constraints our approach is based on a combination of general equivalence theory with the theory of canonical moments in the case of equal bounds for the d 1 efficiencies the constrained optimal designs can be found explicitly by an application of recent results for associated orthogonal polynomials

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

model robust two level designs using coordinate exchange algorithms and a maximin criterion we propose a candidate list free exchange algorithm that facilitates construction of exact model robust two level experiment designs in particular we investigate two model spaces previously considered in the literature the first assumes that all main effects and an unknown subset of two factor interactions are active but that the experimenter knows the number of active interactions the second assumes that an unknown subset of the main effects and all associated two factor interactions are active previous literature uses two criteria for design construction first maximize the number of estimable models then differentiate between designs equivalent in estimability by choosing the design with the highest average efficiency we adopt a similar strategy but 1 do not impose orthogonality or factor level balance constraints resulting in generally equal or larger numbers of estimable models and 2 use a flexible secondary criterion that maximizes the minimum efficiency we provide results fo

columnwise pairwise algorithms with applications to the construction of supersaturated designs motivated by the construction of supersaturated designs we develop a class of algorithms called columnwise pairwise exchange algorithms they differ from the k exchange algorithms in two respects 1 they exchange columns instead of rows of the design matrix and 2 they employ a pairwise adjustment in the search for a better column the proposed algorithms perform very well in the construction of supersaturated designs both for a single criterion and for multiple criteria they are also applicable to the construction of designs that are not supersaturated

generalized filters the low pass condition and connections to multiresolution analyses abstract we study generalized filters that are associated to multiplicity functions and homomorphisms of the dual of an abelian group these notions are based on the structure of generalized multiresolution analyses we investigate when the ruelle operator corresponding to such a filter is a pure isometry and then use that characterization to study the problem of when a collection of closed subspaces which satisfies all the conditions of a gmra except the trivial intersection condition must in fact have a trivial intersection in this context we obtain a generalization of a theorem of bownik and rzeszotnik

computer aided blocking of factorial and response surface designs we consider the problem of blocking response surface and factorial designs when block sizes are prespecified often rendering standard approaches inapplicable a generalization of harville s 1974

a mean squared error criterion for the design of experiments in estimating a response surface over a design region of interest mean squared error can arise both from sampling variance and bias introduced by model inadequacy the criterion adopted here for experimental design attempts to protect against bias resulting from a large class of deviations from the assumed model two algorithms are proposed for the construction of either approximate or exact designs and an example is given

some guidelines for constructing exact d optimal designs on convex design spaces some problems unique to the construction of n point d optimal designs on convex design spaces are considered multiple point augmentation and exchange algorithms are shown to be more costly and less efficient than the analogous single point procedures moreover some recommendations for improving single point methods are given finally empirical evidence is found that supports the galil and kiefer option for constructing initial designs and powell s optimization method for design augmentation

a general criterion for factorial designs under model uncertainty motivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design we show that the qb criterion which aims to improve the estimation in as many models as possible by incorporating experimenters prior knowledge along with an approximation to the as criterion is more general and has a better statistical interpretation than many standard criteria the generalization and application of the criterion to different types of designs are presented the relationships between qb and other criteria for different situations are explored it is shown that the e s2 criterion is a special case of qb and several aberration type criteria are limiting cases of our criterion so that qb provides a bridge between alphabetic optimality and aberration the two case studies illustrate the potential benefits of the qb criterion r programs for calculating qb are available online as supplemental materials

geometrical validity of curvilinear pyramidal finite elements a method to efficiently determine the geometrical validity of curvilinear finite elements of any order was recently proposed in 1 the method is based on the adaptive expansion of the jacobian determinant in a polynomial basis built using bezier functions that has both properties of boundedness and positivity while this technique can be applied to all usual finite elements triangles quadrangles tetrahedra hexahedra and prisms it cannot readily be applied to pyramids due to non polynomial nature of pyramidal finite element spaces in this short paper we extend the results from 1 to pyramidal elements by making use of the high order nodal pyramidal finite element proposed by bergot et al 10 which exhibits optimal convergence properties inh1 norm the paper is organized as follows we begin by briefly recalling the pyramidal finite element space in section 2 before constructing the function space of the jacobian determinant in section 3 section 4 then introduces a generalized bezier function basis which can be used to obtain adaptive bounds on the pyramidal jacobian determinant numerical results showing the sharpness of the estimates are given in section 5

optimal design in nonlinear multiresponse estimation poisson model for filter feeding summary d optimality criteria have been applied to construct locally optimal designs for a multiresponse nonlinear model simulated annealing was used to perform the needed numerical optimization calculations as this method can locate the global optimum of a function and can efficiently handle constraints in the independent variables the calculated optimal designs greatly reduce variances of model parameter estimates compared to variances from previously used empirical designs the effect of several design variables including the number of design points and the number of responses on the efficiency of the design was investigated and designs for various subsets of parameters were also calculated new directions for the design of future experiments were suggested by this analysis

the coordinate exchange algorithm for constructing exact optimal experimental designs we describe the cyclic coordinate exchange algorithm for constructing d optimal and linear optimal experimental designs the algorithm uses a variant of the gauss southwell cyclic coordinate descent algorithm within the k exchange algorithm to achieve substantive reductions in required computing among its advantages are the following candidate sets which grow exponentially in the number of factors need not be explicitly constructed or enumerated convex design spaces or mixed convex by discrete design spaces are handled directly without the need for sophisticated nonlinear programming routines or candidate set adjustment for design problems having 10 or more factors the reductions in execution time are typically two or more orders of magnitude when compared to standard candidateset based procedures such as k exchange yet the designs produced exhibit no loss of efficiency

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

least squares polynomial chaos expansion a review of sampling strategies abstract as non intrusive polynomial chaos expansion pce techniques have gained growing popularity among researchers we here provide a comprehensive review of major sampling strategies for the least squares based pce traditional sampling methods such as monte carlo latin hypercube quasi monte carlo optimal design of experiments ode gaussian quadratures as well as more recent techniques such as coherence optimal and randomized quadratures are discussed we also propose a hybrid sampling method dubbed alphabetic coherence optimal that employs the so called alphabetic optimality criteria used in the context of ode in conjunction with coherence optimal samples a comparison between the empirical performance of the selected sampling methods applied to three numerical examples including high order pce s high dimensional problems and low oversampling ratios is presented to provide a road map for practitioners seeking the most suitable sampling technique for a problem at hand we observed that the alphabetic coherence optimal technique outperforms other sampling methods specially when high order ode are employed and or the oversampling ratio is low

approximate inferences in multiresponse regression analysis summary we present an approximate inference procedure for the multiresponse regression model this can be used to determine approximate confidence regions or intervals and to formulate diagnostic tools such as standardized residuals and influence measures the overall approximation is based on approximations at two stages at the first stage the multiresponse regression model is replaced by a generalized least squares model using the estimated variance covariance matrix of the error terms at the second stage the nonlinear model functions are replaced by linear approximations

an adjustment algorithm for the construction of exact d optimum experimental designs existing algorithms for the construction of exact d optimum designs customarily search a list of candidate points this article describes a computationally simple method of improving such designs tables are given of designs for second order models with as many as five factors found by search over candidate points these are compared with designs already in the literature and used as the starting point for the adjustment algorithm

the invertibility of the isoparametric mapping for pyramidal and prismatic finite elements we consider the isoparametric transformation which maps a given reference element onto a global element given by its vertices for multi linear finite elements on pyramids and prisms we present easily computable conditions on the position of the vertices which ensure that the isoparametric transformation is bijective

wavelets for iterated function systems abstract we construct a wavelet and a generalised fourier basis with respect to some fractal measure given by a one dimensional iterated function system in this paper we will not assume that these systems are given by linear contractions generalising in this way some previous work of dutkay jorgensen and pedersen to the non linear setting as a byproduct we are able to provide a fourier basis also for such linear fractals like the middle third cantor set which have been left out by previous approaches

geometrical validity of curvilinear finite elements in this paper we describe a way to compute accurate bounds on jacobian determinants of curvilinear polynomial finite elements our condition enables to guarantee that an element is geometrically valid i e that its jacobian determinant is strictly positive everywhere in its reference domain it also provides an efficient way to measure the distortion of curvilinear elements the key feature of the method is to expand the jacobian determinant using a polynomial basis built using bezier functions that has both properties of boundedness and positivity numerical results show the sharpness of our estimates