a geometric characterization of c optimal designs for heteroscedastic regression we consider the common nonlinear regression model where the variance as well as the mean is a parametric function of the explanatory variables the c optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest a geometric characterization of c optimal designs in this context is presented which generalizes the classical result of elfving 1952 for c optimal designs as in elfving s famous characterization c optimal designs can be described as representations of boundary points of a convex set however in the case where there appear parameters of interest in the variance the structure of the elfving set is dierent roughly speaking the elfving set corresponding to a heteroscedastic regression model is the convex hull of a set of ellipsoids induced by the underlying model and indexed by the design space the c optimal designs are characterized as representations of the points where the line in direction of the vector c intersects the boundary of the new elfving set the theory is illustrated in several examples including pharmacokinetic models with random eects

local differential geometry as a representation of the supersymmetric oscillator the choice of a coordinate chart on an analytical rn rna provides a representation of the n dimensional supersymmetric oscillator the 1 parameter group of dilations provides a euclidean evolution moving the system through a sequence of charts that at each instant supply a hilbert space by cartan s exterior algebra endowed with a suitable scalar product stationary states and coherent states are eigenstates of the lie derivatives generating the dilations and the translations respectively

designing dose finding studies with an active control for exponential families in a recent paper dette et al 2014 introduced optimal design problems for dose finding studies with an active control these authors concentrated on regression models with normal distributed errors with known variance and the problem of determining optimal designs for estimating the smallest dose which achieves the same treatment eect as the active control this paper discusses the problem of designing activecontrolled dose finding studies from a broader perspective in particular we consider a general class of optimality criteria and models arising from an exponential family which are frequently used analyzing count data we investigate under which circumstances optimal designs for dose finding studies including a placebo can be used to obtain optimal designs for studies with an active control optimal designs are constructed for several situations and the dierences arising from dierent distributional assumptions are investigated in detail in particular our results are applicable for constructing optimal experimental designs to analyze active controlled dose finding studies with discrete data and we illustrate the eciency of the new optimal designs with two

supersymmetry and morse theory it is shown that the morse inequalities can be obtained by consideration of a certain supersymmetric quantum mechanics hamiltonian some of the implications of modern ideas in mathematics for supersymmetric theories are discussed

stable local dimensionality reduction approaches dimensionality reduction is a big challenge in many areas a large number of local approaches stemming from statistics or geometry have been developed however in practice these local approaches are often in lack of robustness since in contrast to maximum variance unfolding mvu which explicitly unfolds the manifold they merely characterize local geometry structure moreover the eigenproblems that they encounter are hard to solve we propose a unified framework that explicitly unfolds the manifold and reformulate local approaches as the semi definite programs instead of the above mentioned eigenproblems three well known algorithms locally linear embedding lle laplacian eigenmaps le and local tangent space alignment ltsa are reinterpreted and improved within this framework several experiments are presented to demonstrate the potential of our framework and the improvements of these local algorithms

optimal designs for comparing curves we consider the optimal design problem for a comparison of two regression curves which is used to establish the similarity between the dose response relationships of two groups an optimal pair of designs minimizes the width of the confidence band for the difference between the two regression functions optimal design theory equivalence theorems efficiency bounds is developed for this non standard design problem and for some commonly used dose response models optimal designs are found explicitly the results are illustrated in several examples modeling dose response relationships it is demonstrated that the optimal pair of designs for the comparison of the regression curves is not the pair of the optimal designs for the individual models in particular it is shown that the use of the optimal designs proposed in this paper instead of commonly used non optimal designs yields a reduction of the width of the confidence band by more than 50

a generalization of the bargmann fock representation to supersymmetry in the bargmann fock representation the coordinates act as bosonic creation operators while the partial derivatives act as annihilation operators on holomorphic 0 forms as states of a d dimensional bosonic oscillator also considering p forms and further geometrical objects as the exterior derivative and lie derivatives on a holomorphic we end up with an analogous representation for the d dimensional supersymmetric oscillator in particular the supersymmetry multiplet structure of the hilbert space corresponds to the cohomology of the exterior derivative in addition a 1 complex parameter group emerges naturally and contains both time evolution and a homotopy related to cohomology the emphasis is on calculus

computing exact d optimal designs by mixed integer second order cone programming i 1 wiaia t i where ai i 1 s are known matrices with m rows in this paper we show that the criterion of d optimality is secondorder cone representable as a result the method of second order cone programming can be used to compute an approximate d optimal design with any system of linear constraints on the vector of weights more importantly the proposed characterization allows us to compute an exact d optimal design which is possible thanks to highquality branch and cut solvers specialized to solve mixed integer second order cone programming problems our results extend to the case of the criterion of dk optimality which measures the quality of w for the estimation of a linear parameter subsystem defined by a full rank coefficient matrixk we prove that some other widely used criteria are also secondorder cone representable for instance the criteria of a ak gand i optimality we present several numerical examples demonstrating the efficiency and general applicability of the proposed method we show that in many cases the mixed integer second order cone programming approach allows us to find a provably optimal exact design while the standard heuristics systematically miss the optimum

robust optimum designs for transformation of the responses in a multivariate chemical kinetic model transformation of both sides of a nonlinear model is often necessary to obtain tractable error distributions but can have a dramatic effect on optimum designs for the parameters of the model this article develops methods for the design of experiments when the value of the power transformation parameter is not precisely known optimum designs for particular transformations together with the efficiency of these designs as the transformation varies are studied for a multivariate model it is shown that by a suitable choice of prior bayesian designs can be found that are robust to the correct transformation a typical equispaced design is evaluated and the properties of designs are studied under various departures from the assumptions

alignment of overlapping locally scaled patches for multidimensional scaling and dimensionality reduction data observations that lie on a manifold can be approximated by a collection of overlapping local patches the alignment of which in a low dimensional euclidean space provides an embedding of the data this paper describes an embedding method using classical multidimensional scaling as a local model based on the fact that a manifold locally resembles a euclidean space a set of overlapping neighborhoods are chosen by a greedy approximation algorithm of minimum set cover local patches derived from the set of overlapping neighborhoods by classical multidimensional scaling are aligned in order to minimize a residual measure which has a quadratic form of the resulting global coordinates and can be minimized analytically by solving an eigenvalue problem this method requires only distances within each neighborhood and provides locally isometric embedding results the size of the eigenvalue problem scales with the number of overlapping neighborhoods rather than the number of data points experiments on both synthetic and real world data sets demonstrate the effectiveness of this method extensions and variations of the method are discussed

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

optimal weights for experimental designs on linearly independent support points an explicit formula is derived to compute the a optimal design weights on linearly independent regression vectors for the mean parameters in a linear model with homoscedastic variances the formula emerges as a special case of a general result which holds for a wide class of optimality criteria there are close links to iterative algorithms for computing optimal weights

a geometric characterization of optimal designs for regression models with correlated observations we consider the problem of optimal design of experiments for random effects models especially population models where a small number of correlated observations can be taken on each individual whereas the observations corresponding to different individuals are assumed to be uncorrelated we focus on c optimal design problems and show that the classical equivalence theorem and the famous geometric characterization of elfving from the case of uncorrelated data can be adapted to the problem of selecting optimal sets of observations for the n individual patients the theory is demonstrated by finding optimal designs for a linear model with correlated observations and a non linear random effects population model which is commonly used in pharmaco kinetics

complete neighborhood preserving embedding for face recognition neighborhood preserving embedding npe is a linear approximation to the locally linear embedding algorithm which can preserve the local neighborhood structure on the data manifold however in typical face recognition where the number of data samples is smaller than the dimension of data space it is difficult to directly apply npe to high dimensional matrices because of computational complexity moreover in such case npe often suffers from the singularity problem of eigenmatrix which makes the direct implementation of the npe algorithm almost impossible in practice principal component analysis or singular value decomposition is applied as a preprocessing step to attack these problems nevertheless this strategy may discard dimensions that contain important discriminative information and the eigensystem computation of npe could be unstable towards a practical dimensionality reduction method for face data we develop a new scheme in this paper namely the complete neighborhood preserving embedding cnpe cnpe transforms the singular generalized eigensystem computation of npe into two eigenvalue decomposition problems moreover a feasible and effective procedure is proposed to alleviate the computational burden of high dimensional matrix for typical face image data experimental results on the orl face database and the yale face database show that the proposed cnpe algorithm achieves better performance than other feature extraction methods such as eigenfaces fisherfaces and npe etc

riemannian manifold learning recently manifold learning has been widely exploited in pattern recognition data analysis and machine learning this paper presents a novel framework called riemannian manifold learning rml based on the assumption that the input high dimensional data lie on an intrinsically low dimensional riemannian manifold the main idea is to formulate the dimensionality reduction problem as a classical problem in riemannian geometry that is how to construct coordinate charts for a given riemannian manifold we implement the riemannian normal coordinate chart which has been the most widely used in riemannian geometry for a set of unorganized data points first two input parameters the neighborhood size k and the intrinsic dimension d are estimated based on an efficient simplicial reconstruction of the underlying manifold then the normal coordinates are computed to map the input high dimensional data into a low dimensional space experiments on synthetic data as well as real world images demonstrate that our algorithm can learn intrinsic geometric structures of the data preserve radial geodesic distances and yield regular embeddings

optimal designs for active controlled dose finding trials with efficacy toxicity outcomes author s schorning kirsten dette holger kettelhake katrin wong weng kee bretz frank abstract we derive optimal designs to estimate efficacy and toxicity in active controlled dose finding trials whenthe bivariate continuous outcomes are described using nonlinear regression models we determine upperbounds on the required number of different doses and provide conditions under which the boundary pointsof the design space are included in the optimal design we provide an analytical description of minimallysupported optimal designs and show that they do not depend on the correlation between the bivariateoutcomes

locally linear discriminant embedding an efficient method for face recognition in this paper an efficient feature extraction method named as locally linear discriminant embedding llde is proposed for face recognition it is well known that a point can be linearly reconstructed by its neighbors and the reconstruction weights are under the sum to one constraint in the classical locally linear embedding lle so the constrained weights obey an important symmetry for any particular data point they are invariant to rotations rescalings and translations the latter two are introduced to the proposed method to strengthen the classification ability of the original lle the data with different class labels are translated by the corresponding vectors and those belonging to the same class are translated by the same vector in order to cluster the data with the same label closer they are also rescaled to some extent so after translation and rescaling the discriminability of the data will be improved significantly the proposed method is compared with some related feature extraction methods such as maximum margin criterion mmc as well as other supervised manifold learning based approaches for example ensemble unified lle and linear discriminant analysis en ullelda locally linear discriminant analysis llda experimental results on yale and cmu pie face databases convince us that the proposed method provides a better representation of the class information and obtains much higher recognition accuracies

the usefulness of optimum experimental designs optimum experimental designs were originally developed by kiefer mainly for response surface models this survey of recent developments emphasizes potential or actual usefulness for linear models the construction of exact designs particularly over irregular design regions is stressed as is the blocking of response surface designs other important areas include systematic designs that are robust against trend and designs for mixtures with irregular design regions several industrial examples are mentioned both d and c optimum designs are found for a non linear model of the economic response of cereal production to fertilizer level the c optimum design being for the conditions of maximum economic return locally optimum and bayesian designs are both described similar results for generalized linear models lead to designs for the ld 95 in a logistic model in which male and female subjects respond differently designs with structure in the variance suggest alternatives to the potentially wasteful product designs of taguchi designs for sequential clinical trials to include random balance are presented the last section outlines some applications including life testing and models in which time is a factor

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

quantitative analysis of nonlinear embedding a lot of nonlinear embedding techniques have been developed to recover the intrinsic low dimensional manifolds embedded in the high dimensional space however the quantitative evaluation criteria are less studied in literature the embedding quality is usually evaluated by visualization which is subjective and qualitative the few existing evaluation methods to estimate the embedding quality neighboring preservation rate for example are not widely applicable in this paper we propose several novel criteria for quantitative evaluation by considering the global smoothness and co directional consistence of the nonlinear embedding algorithms the proposed criteria are geometrically intuitive simple and easy to implement with a low computational cost experiments show that our criteria capture some new geometrical properties of the nonlinear embedding algorithms and can be used as a guidance to deal with the embedding of the out of samples

face recognition using laplacianfaces we propose an appearance based face recognition method called the laplacianface approach by using locality preserving projections lpp the face images are mapped into a face subspace for analysis different from principal component analysis pca and linear discriminant analysis lda which effectively see only the euclidean structure of face space lpp finds an embedding that preserves local information and obtains a face subspace that best detects the essential face manifold structure the laplacianfaces are the optimal linear approximations to the eigenfunctions of the laplace beltrami operator on the face manifold in this way the unwanted variations resulting from changes in lighting facial expression and pose may be eliminated or reduced theoretical analysis shows that pca lda and lpp can be obtained from different graph models we compare the proposed laplacianface approach with eigenface and fisherface methods on three different face data sets experimental results suggest that the proposed laplacianface approach provides a better representation and achieves lower error rates in face recognition

efficient rounding of approximate designs summary discretization methods to round an approximate design into an exact design for a given sample size n are discussed there is a unique method called efficient rounding which has the smallest loss of efficiency under a wide family of optimality criteria the efficient rounding method is a multiplier method of apportionment which otherwise is known as the method of john quincy adams or the method of smallest divisors

efficient design of experiments in the monod model summary estimation and experimental design in a non linear regression model that is used in microbiology are studied the monod model is defined implicitly by a differential equation and has numerous applications in microbial growth kinetics water research pharmacokinetics and plant physiology it is proved that least squares estimates are asymptotically unbiased and normally distributed the asymptotic covariance matrix of the estimator is the basis for the construction of efficient designs of experiments in particular locally d e and c optimal designs are determined and their properties are studied theoretically and by simulation if certain intervals for the non linear parameters can be specified locally optimal designs can be constructed which are robust with respect to a misspecification of the initial parameters and which allow efficient parameter estimation parameter variances can be decreased by a factor of 2 by simply sampling at optimal times during the experiment

finding representative landmarks of data on manifolds data driven non parametric models such as manifold learning algorithms are promising data analysis tools however to fit an off training set data point in a learned model one must first locate the point in the training set this query has a time cost proportional to the problem size which limits the model s scalability in this paper we address the problem of selecting a subset of data points as the landmarks helping locate the novel points on the data manifolds we propose a new category of landmarks defined with the following property the way the landmarks represent the data in the ambient euclidean space should resemble the way they represent the data on the manifold given the data points and the subset of landmarks we provide procedures to test whether the proposed property presents for the choice of landmarks if the data points are organized with a neighbourhood graph as it is often conducted in practice we interpret the proposed property in terms of the graph topology we also discuss the extent to which the topology is preserved for landmark set passing our test procedure another contribution of this work is to develop an optimization based scheme to adjust an existing landmark set which can improve the reliability for representing the manifold data experiments on the synthetic data and the natural data have been done the results support the proposed properties and algorithms

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

robust locally linear embedding in the past few years some nonlinear dimensionality reduction nldr or nonlinear manifold learning methods have aroused a great deal of interest in the machine learning community these methods are promising in that they can automatically discover the low dimensional nonlinear manifold in a high dimensional data space and then embed the data points into a low dimensional embedding space using tractable linear algebraic techniques that are easy to implement and are not prone to local minima despite their appealing properties these nldr methods are not robust against outliers in the data yet so far very little has been done to address the robustness problem in this paper we address this problem in the context of an nldr method called locally linear embedding lle based on robust estimation techniques we propose an approach to make lle more robust we refer to this approach as robust locally linear embedding rlle we also present several specific methods for realizing this general rlle approach experimental results on both synthetic and real world data show that rlle is very robust against outliers

locally optimal designs for errors in variables models we consider the construction of optimal designs for nonlinear regression models when there are measurement errors in the covariates corresponding approximate design theory is developed for maximum likelihood and least squares estimation with the latter leading to nonconcave optimization problems analytical characterizations of the locally d optimal saturated designs are provided for the michaelis menten e rm max and exponential regression models through concrete applications we illustrate how measurement errors in the covariates affect the optimal choice of design and show that the locally d optimal saturated designs are highly efficient for relatively small misspecifications of the parameter values

locality preserved maximum information projection dimensionality reduction is usually involved in the domains of artificial intelligence and machine learning linear projection of features is of particular interest for dimensionality reduction since it is simple to calculate and analytically analyze in this paper we propose an essentially linear projection technique called locality preserved maximum information projection lpmip to identify the underlying manifold structure of a data set lpmip considers both the within locality and the between locality in the processing of manifold learning equivalently the goal of lpmip is to preserve the local structure while maximize the out of locality global information of the samples simultaneously different from principal component analysis pca that aims to preserve the global information and locality preserving projections lpps that is in favor of preserving the local structure of the data set lpmip seeks a tradeoff between the global and local structures which is adjusted by a parameter alpha so as to find a sub space that detects the intrinsic manifold structure for classification tasks computationally by constructing the adjacency matrix lpmip is formulated as an eigenvalue problem lpmip yields orthogonal basis functions and completely avoids the singularity problem as it exists in lpp further we develop an efficient and stable lpmip qr algorithm for implementing lpmip especially on high dimensional data set theoretical analysis shows that conventional linear projection methods such as weighted pca maximum margin criterion mmc linear discriminant analysis lda and lpp could be derived from the lpmip framework by setting different graph models and constraints extensive experiments on face digit and facial expression recognition show the effectiveness of the proposed lpmip method

optimal designs for dose finding studies understanding and properly characterizing the dose response relationship is a fundamental step in the investigation of a new compound be it a herbicide or fertilizer a molecular entity an environmental toxin or an industrial chemical in this article we investigate the problem of deriving efficient designs for the estimation of target doses in the context of clinical dose finding we propose methods to determine the appropriate number and actual levels of the doses to be administered to patients as well as their relative sample size allocations more specifically we derive local optimal designs that minimize the asymptotic variance of the minimum effective dose estimate under a particular dose response model we investigate the small sample properties of these designs together with their sensitivity to a misspecification of the true parameter values and of the underlying dose response model through simulation finally we demonstrate that the designs derived for a fixed model are rather sensitive

the equivalence of constrained and weighted designs in multiple objective design problems abstract several competing objectives may be relevant in the design of an experiment the competing objectives may not be easy to characterize in a single optimality criterion one approach to these design problems has been to weight each criterion and find the design that optimizes the weighted average of the criteria an alternative approach has been to optimize one criterion subject to constraints on the other criteria an equivalence theorem is presented for the bayesian constrained design problem equivalence theorems are essential in verifying optimality of proposed designs especially when as in most nonlinear design problems numerical optimization is required this theorem is used to show that the results of cook and wong on the equivalence of the weighted and constrained problems apply much more generally the results are applied to bayesian nonlinear design problems with several objectives

local coordinates alignment with global preservation for dimensionality reduction dimensionality reduction is vital in many fields and alignment based methods for nonlinear dimensionality reduction have become popular recently because they can map the high dimensional data into a low dimensional subspace with the property of local isometry however the relationships between patches in original high dimensional space cannot be ensured to be fully preserved during the alignment process in this paper we propose a novel method for nonlinear dimensionality reduction called local coordinates alignment with global preservation we first introduce a reasonable definition of topology preserving landmarks tpls which not only contribute to preserving the global structure of datasets and constructing a collection of overlapping linear patches but they also ensure that the right landmark is allocated to the new test point then an existing method for dimensionality reduction that has good performance in preserving the global structure is used to derive the low dimensional coordinates of tpls local coordinates of each patch are derived using tangent space of the manifold at the corresponding landmark and then these local coordinates are aligned into a global coordinate space with the set of landmarks in low dimensional space as reference points the proposed alignment method called landmarks based alignment can produce a closed form solution without any constraints while most previous alignment based methods impose the unit covariance constraint which will result in the deficiency of global metrics and undesired rescaling of the manifold experiments on both synthetic and real world datasets demonstrate the effectiveness of the proposed algorithm

principal manifolds and nonlinear dimensionality reduction via tangent space alignment we present a new algorithm for manifold learning and nonlinear dimensionality reduction based on a set of unorganized data points sampled with noise from a parameterized manifold the local geometry of the manifold is learned by constructing an approximation for the tangent space at each data point and those tangent spaces are then aligned to give the global coordinates of the data points with respect to the underlying manifold we also present an error analysis of our algorithm showing that reconstruction errors can be quite small in some cases we illustrate our algorithm using curves and surfaces both in two dimensional three dimensional 2d 3d euclidean spaces and in higher dimensional euclidean spaces we also address several theoretical and algorithmic issues for further research and improvements

optimal designs for three dimensional shape analysis with spherical harmonic descriptors we determine optimal designs for some regression models which are frequently used for describing three dimensional shapes these models are based on a fourier expansion of a function defined on the unit sphere in terms of spherical harmonic basis functions in particular it is demonstrated that the uniform distribution on the sphere is optimal with respect to all p criteria proposed by kiefer in 1974 and also optimal with respect to a criterion which maximizes a p mean of the r smallest eigenvalues of the variance covariance matrix this criterion is related to principal component analysis which is the common tool for analyzing this type of image data moreover discrete designs on the sphere are derived which yield the same information matrix in the spherical harmonic regression model as the uniform distribution and are therefore directly implementable in practice it is demonstrated that the new designs are substantially more efficient than the commonly used designs in three dimensional shape analysis

orthogonal neighborhood preserving projections a projection based dimensionality reduction technique this paper considers the problem of dimensionality reduction by orthogonal projection techniques the main feature of the proposed techniques is that they attempt to preserve both the intrinsic neighborhood geometry of the data samples and the global geometry in particular we propose a method named orthogonal neighborhood preserving projections which works by first building an affinity graph for the data in a way that is similar to the method of locally linear embedding lle however in contrast with the standard lle where the mapping between the input and the reduced spaces is implicit onpp employs an explicit linear mapping between the two as a result handling new data samples becomes straightforward as this amounts to a simple linear transformation we show how we can define kernel variants of onpp as well as how we can apply the method in a supervised setting numerical experiments are reported to illustrate the performance of onpp and to compare it with a few competing methods

robust t optimal discriminating designs this paper considers the problem of constructing optimal discriminating experimental designs for competing regression models on the basis of the t optimality criterion introduced by atkinson and fedorov 1975a t optimal designs depend on unknown model parameters and it is demonstrated that these designs are sensitive with respect to misspecification as a solution of this problem we propose a bayesian and standardized maximin approach to construct robust and efficient discriminating designs on the basis of the t optimality criterion it is shown that the corresponding bayesian and standardized maximin optimality criteria are closely related to linear optimality criteria for the problem of discriminating between two polynomial regression models which differ in the degree by two the robust t optimal discriminating designs can be found explicitly the results are illustrated in several examples ams subject classification 62k05