sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

a pyramid vector quantizer the geometric properties of a memoryless laplacian source are presented and used to establish a source coding theorem motivated by this geometric structure a pyramid vector quantizer pvq is developed for arbitrary vector dimension the pvq is based on the cubic lattice points that lie on the surface of an l dimensional pyramid and has simple encoding and decoding algorithms a product code version of the pvq is developed and generalized to apply to a variety of sources analytical expressions are derived for the pvq mean square error mse and simulation results are presented for pvq encoding of several memoryless sources for large rate and dimension pvq encoding of memoryless laplacian gamma and gaussian sources provides rose improvements of 5 64 8 40 and 2 39 db respectively over the corresponding optimum scalar quantizer although suboptimum in a rate distortion sense because the pvq can encode large dimensional vectors it offers significant reduction in rose distortion compared with the optimum lloyd max scalar quantizer and provides an attractive alternative to currently available vector quantizers

the rank transform method in some two factor designs abstract the use of the rank transform for testing problems in some two factor designs is considered the rank transform procedure consists of replacing the observations by their ranks in the combined sample and performing one of the standard analysis of variance anova procedures on these ranks the asymptotic version of the rank transform is introduced and its usefulness as a means for understanding the nature of the rank transform is examined it is demonstrated that the asymptotic version of the transformation helps identify the testing problems where the rank transform method works and helps suggest which anova procedure should be used on the ranks this approach is applied on the balanced and unbalanced nested model and the balanced and unbalanced two way layout with and without interaction the extension of the rank transform method to some of these models is new the proposed procedures do not share the known simplicity of existing rank transform statistics but they do allow heteroscedasticity

universal multiple description scalar quantization analysis and design this paper introduces a new high rate analysis of the multiple description scalar quantizer mdsq with balanced descriptions the analysis provides insight into the structure of the mdsq suggesting the nonoptimality of uniform central quantizer cell lengths as well as a method to approximate optimal cell lengths for both level constrained and entropy constrained mdsq new upper bounds on the granular distortion for sources with smooth probability density functions pdfs are derived under the mean squared error measure which are 0 4 db lower than previous results based on the insights a universal multiple description scalar quantizer umdsq is proposed which at high rate can achieve nearly the same performance as the fully optimized entropy constrained mdsq ecmdsq without requiring extensive training the proposed umdsq has only two control parameters and a continuum of tradeoff points between the central and side distortions can be achieved as the two parameters are varied

probabilistic index models summary we present a semiparametric statistical model for the probabilistic index which can be defined as p y y a where y and y a are independent random response variables associated with covariate patterns x and x a respectively a link function defines the relationship between the probabilistic index and a linear predictor asymptotic normality of the estimators and consistency of the covariance matrix estimator are established through semiparametric theory the model is illustrated with several examples and the estimation theory is validated in a simulation study

upper bounds to the asymptotic performance of block quantizers upper bounds to the asymptotic performance of block quantizers with difference distortion measures are derived in many eases these upper bounds approach known lower bounds as the block length of the quantizer approaches infinity a condition for the optimal point density function of the output levels is derived it is shown to particularize to a known result of gersho the behavior of the bounds for large block lengths is investigated

probabilistic index an intuitive non parametric approach to measuring the size of treatment effects effect sizes es tell the magnitude of the difference between treatments and ideally should tell clinicians how likely their patients will benefit from the treatment currently used es are expressed in statistical rather than in clinically useful terms and may not give clinicians the appropriate information we restrict our discussion to studies with two groups one with n patients receiving a new treatment and the other with m patients receiving the usual or no treatment the standardized mean difference e g cohen s d is a well known index for continuous outcomes there is some intuitive value to d but measuring improvement in standard deviations sd is a statistical concept that may not help a clinician how much improvement is a half sd a more intuitive and simple to calculate es is the probability that the response of a patient given the new treatment x is better than the one for a randomly chosen patient given the old or no treatment y i e p x y larger values meaning better outcomes this probability has an immediate identity with the area under the curve auc measure in procedures for receiver operator characteristic roc curve comparing responses to two treatments it also can be easily calculated from the mann whitney u wilcoxon or kendall statistics we describe the characteristics of an ideal es we propose p x y as an alternative index summarize its correspondence with well known non parametric statistics compare it to the standardized mean difference index and illustrate with clinical data copyright 2005 john wiley sons ltd

vector quantizer design for memoryless gaussian gamma and laplacian sources locally optimum vector quantizer vq designs are presented for memoryless gaussian gamma and laplacian sources for gaussian sources low 2 6 dimensional vector quantization provides relatively little improvement in mean squared error mse compared to the minimum mean squared error mmse scalar quantizer for laplacian or gamma sources however significant improvement in mse is available with vector quantization the laplacian and gamma 6 bit sixdimensional vector quantizers achieve respectively improvements of 2 and 4 5 db over the corresponding scalar mmse quantizer distortions

the rate distortion region for multiple descriptions without excess rate during recent years there has been strong interest in a certain source coding problem which some authors call the problem of multiple descriptions old and new wringing techniques enable us to establish a single letter characterization of the rate distortion region in the case of no excess rate for the joint description

optimality and suboptimality of multiple description vector quantization with a lattice codebook the asymptotic analysis of multiple description vector quantization mdvq with a lattice codebook for sources with smooth probability density functions pdfs is considered in this correspondence goyal et al 2002 observed that as the side distortion decreases and the central distortion correspondingly increases the quantizer cells farther away from the coarse lattice points shrink in a spatially periodic pattern in this correspondence two special classes of index assignments are used along strategic groupings of central quantizer cells to derive a straightforward asymptotic analysis which provides an analytical explanation for the aforementioned observation mdvq with a lattice codebook was shown earlier to be asymptotically optimal in high dimensions with a curious converging property that the side quantizers achieve the space filling advantage of an n dimensional sphere instead of an n dimensional optimal polytope the analysis presented here explains this behavior readily while central quantizer cells on a uniform lattice are asymptotically optimal in high dimensions the present authors have shown that by using nonuniform rather than uniform central quantizer cells the central side distortion product in an mdsq can be reduced by 0 4 db at asymptotically high rate the asymptotic analysis derived here partially unifies these previous results in the same framework though a complete characterization is still beyond reach

bit error resilient index assignment for multiple description scalar quantizers this paper addresses the problem of increasing the robustness to bit errors for two description scalar quantizers our approach is to start with an m diagonal index assignment and further apply a permutation to the indexes of each description to increase the minimum hamming distance d rm min of the set of valid index pairs in particular we show how to construct linear permutation pairs achieving d rm min geq 3 and establish a lower bound in terms of the description rate r for the highest value of m for which such permutations exist

the f statistic in the two way layout with rank score transformed data abstract the limiting null distribution of the usual f statistic for main effects in the two way layout is shown to have the same limiting distribution when applied to ranks and scores based on ranks as when applied to normal data the limit is taken as the cell size n increases without bound the denominator of the f statistic times j 1 j is shown to provide an unbiased and consistent estimator of the limit as n of n 1 times the variance of the treatment sum of scores when the null hypothesis is true the test based on the scores statistic is demonstrated to be consistent for a wide class of fixed alternative hypotheses

tables of sphere packings and spherical codes the theta function of a sphere packing gives the number of centers at each distance from the origin the theta functions of a number of important packings a n d n e n the leech lattice and others and tables of the first fifty or so of their coefficients are given in this paper

quantization based on a novel sample adaptive product quantizer sapq in this paper we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq this is a structurally constrained vector quantizer that uses unions of product codebooks sapq is based on a concept of adaptive quantization to the varying samples of the source and is very different from traditional adaptation techniques for nonstationary sources sapq quantizes each source sample using a sequence of quantizers even when using scalar quantization in sapq we can achieve performance comparable to vector quantization with the complexity still close to that of scalar quantization we also show that important lattice based vector quantizers can be constructed using scalar quantization in sapq we mathematically analyze sapq and propose a algorithm to implement it we numerically study sapq for independent and identically distributed gaussian and laplacian sources through our numerical study we find that sapq using scalar quantizers achieves typical gains of 13 db in distortion over the lloyd max quantizer we also show that sapq can he used in conjunction with vector quantizers to further improve the gains

flexible symmetric multiple description lattice vector quantizer with l geq 3 descriptions in the previous work on multiple description lattice vector quantizers mdlvqs with l geq 3 descriptions once the central and side lattice codebooks are fixed the decoding quality is determined for all numbers k of received descriptions therefore it is not possible to achieve tradeoffs between the quality of reconstruction for different values of k 1 leq k leq l 1 this paper proposes a flexible mdlvq capable of overcoming the above drawback for this a different reconstruction method is employed and a heuristic index assignment ia algorithm which uses l 2 parameters to control the distortions for 2 leq k leq l 1 is developed experimental results show that the proposed mdlvq in addition to achieving the desired tradeoffs significantly outperforms the classic md scheme based on unequal erasure protection the second contribution of this paper is a structured ia for the case of l 3 and the derivation of the corresponding expressions of the distortions at high resolution the proposed ia has a simple mechanism for controlling the tradeoff between the reconstruction quality for k 1 2 the ia is able to achieve a wide range of distortion values while keeping the product of the distortions for k 1 2 the same as in the prior work

k sample rank tests for umbrella alternatives abstract distribution free test procedures are proposed for the k sample problem where the alternatives of interest are of the form f 1 x fl x fk x for all x with at least one strict inequality these are referred to as umbrella alternatives and both the l point of umbrella known and l unknown settings are considered small sample null distributions are discussed and the results of a monte carlo power study are presented a detailed example illustrating the use of the unknown umbrella point procedure is also given

multiple description coding with many channels an achievable region for the l channel multiple description coding problem is presented this region generalizes two channel results of el gamal and cover 1982 and of zhang and berger 1987 it further generalizes three channel results of gray and wyner 1974 and of zhang and berger a source that is successively refinable on chains is shown to be successively refinable on trees a new outer bound on the rate distortion rd region for memoryless gaussian sources with mean squared error distortion is also derived the achievable region meets this outer bound for certain symmetric cases

lattice based robust distributed source coding for three correlated sources the problem of robust distributed source coding for three correlated sources is studied in this work a lattice based scheme is proposed and the analysis of its performance is provided in the high resolution regime special attention is paid to the degenerate case where the three sources are gaussian and identical in this case our scheme is shown to achieve within an asymptotic gap of 0 069 bits in terms of rate per description from the information theoretic limit of quadratic symmetric gaussian multiple description coding with central and individual decoders when the side distortion and the ratio between the central and side distortions both approach 0

n channel entropy constrained multiple description lattice vector quantization in this paper we derive analytical expressions for the central and side quantizers which under high resolution assumptions minimize the expected distortion of a symmetric multiple description lattice vector quantization md lvq system subject to entropy constraints on the side descriptions for given packet loss probabilities we consider a special case of the general n channel symmetric multiple description problem where only a single parameter controls the redundancy tradeoffs between the central and the side distortions previous work on two channel md lvq showed that the distortions of the side quantizers can be expressed through the normalized second moment of a sphere we show here that this is also the case for three channel md lvq furthermore we conjecture that this is true for the general n channel md lvq for given source target rate and packet loss probabilities we find the optimal number of descriptions and construct the md lvq system that minimizes the expected distortion we verify theoretical expressions by numerical simulations and show in a practical setup that significant performance improvements can be achieved over state of the art two channel md lvq by using three channel md lvq

exact and asymptotically robust permutation tests given independent samples from p and q two sample permutation tests allow one to construct exact level tests when the null hypothesis is p q on the other hand when comparing or testing particular parameters of p and q such as their means or medians permutation tests need not be level or even approximately level in large samples under very weak assumptions for comparing estimators we provide a general test procedure whereby the asymptotic validity of the permutation test holds while retaining the exact rejection probability in finite samples when the underlying distributions are identical a quite general theory is possible based on a coupling construction as well as a key contiguity argument for the binomial and hypergeometric distributions the ideas are broadly applicable and special attention is given to a nonparametric k sample behrens fisher problem whereby a permutation test is constructed which is exact level under the hypothesis of identical distributions but has asymptotic rejection probability under the more general null hypothesis of equality of means a monte carlo simulation study is performed 2010 msc subject classifications primary 62e20 secondary 62g10

fully nonparametric hypotheses for factorial designs i multivariate repeated measures designs abstract we introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models such as the hypotheses of no main effects no interaction effects and no factor effects these natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations we concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses these statistics are the rank transform rt versions of the classical statistics for testing hypotheses in repeated measures designs we emphasize that even though recent research has demonstrated the inappropriateness of the rt method for many parametric hypotheses the rt procedure is always valid for testing our nonparametric hypotheses we show that the rank statistics converge in distribution to central chi squared distributions under their respective nonparametric null hypotheses the noncentrality paramet

optimal index assignment for multiple description scalar quantization with translated lattice codebooks we design a k description scalar quantizer whose construction is based on a structure of translated scalar lattices and a lattice in k 1 dimensional space the use of translated lattices provides a performance advantage by exploiting a so called staggering gain the use of the k 1 dimensional lattice facilitates analytic insight into the performance and significantly speeds up the computation of the index assignment compared to state of the art methods using a common decoding method the proposed index assignment is proven to be optimal for the k description case it is shown that the optimal index assignment is not unique this is illustrated for the two description case where a periodic index assignment is selected from possible optimal assignments and described in detail the performance of the proposed quantizer accurately matches theoretic analysis over the full range of operational redundancies moreover the quantizer outperforms the state of the art md scheme as the redundancy among the description increases

generalized multiple description coding with correlating transforms multiple description md coding is source coding in which several descriptions of the source are produced such that various reconstruction qualities are obtained from different subsets of the descriptions unlike multiresolution or layered source coding there is no hierarchy of descriptions thus md coding is suitable for packet erasure channels or networks without priority provisions generalizing work by orchard wang vaishampayan and reibman see proc ieee int conf image processing vol i santa barbara ca p 608 11 1997 a transform based approach is developed for producing m descriptions of an n tuple source m spl les n the descriptions are sets of transform coefficients and the transform coefficients of different descriptions are correlated so that missing coefficients can be estimated several transform optimization results are presented for memoryless gaussian sources including a complete solution of the n 2 m 2 case with arbitrary weighting of the descriptions the technique is effective only when independent components of the source have differing variances numerical studies show that this method performs well at low redundancies as compared to uniform md scalar quantization

multiple description vector quantization with a coarse lattice a multiple description md lattice vector quantization technique for two descriptions was previously introduced in which fine and coarse codebooks are both lattices the encoding begins with quantization to the nearest point in the fine lattice this encoding is an inherent optimization for the decoder that receives both descriptions performance can be improved with little increase in complexity by considering all decoders in the initial encoding step the altered encoding relies only on the symmetries of the coarse lattice this allows us to further improve performance without a significant increase in complexity by replacing the fine lattice codebook with a nonlattice codebook that respects many of the symmetries of the coarse lattice examples constructed with the two dimensional 2 d hexagonal lattice demonstrate large improvement over time sharing between previously known quantizers

the covering radius of the leech lattice we investigate the points in 24 dimensional space that are at maximal distance from the leech lattice i e the deep holes in that lattice the distance of such a point from the leech lattice is 1 sqrt 2 times the minimal distance between the lattice points furthermore there are 23 inequivalent types of deep hole one for each of the 23 even unimodular 24 dimensional lattices found by niemeier

computation of volume and surface body moments abstract the paper presents explicit formulae for calculation of surface moments for polytopes in r n we describe also efficient algorithms for calculation of 3d body volume and surface moments the algorithms are based on explicit formulae for moment calculation and take advantages of shape polygonal representation they use only coordinates of the body vertices and faces orientation

the root lattices as low bit rate vector quantizers vector quantizers based on the a and d lattices are explored using the mean square error mse fidelity criterion for independent identically distributed sources with normal laplacian and gamma probability density functions code books are studied for uniform coding rates of up to 2 b sample and two and four dimensions the expected mse performance of the quantizers is better than that of optimal nonuniform lloyd max quantizers while the design algorithm presented offers only modest improvements in mse performance for a normally distributed source improvement gains of up to several decibels can be achieved when coding laplacian and gamma sources

multiple description quantization via gram schmidt orthogonalization the multiple description md problem has received considerable attention as a model of information transmission over unreliable channels a general framework for designing efficient md quantization schemes is proposed in this paper we provide a systematic treatment of the el gamal cover egc achievable md rate distortion region and show it can be decomposed into a simplified egc segc region and a superimposed refinement operation furthermore any point in the segc region can be achieved via a successive quantization scheme along with quantization splitting for the quadratic gaussian case the proposed scheme has an intrinsic connection with the gram schmidt orthogonalization which implies that the whole gaussian md rate distortion region is achievable with a sequential dithered lattice based quantization scheme as the dimension of the optimal lattice quantizers becomes large moreover this scheme is shown to be universal for all independent and identically distributed i i d smooth sources with performance no worse than that for an i i d gaussian source with the same variance and asymptotically optimal at high resolution a class of md scalar quantizers in the proposed general framework is also constructed and is illustrated geometrically the performance is analyzed in the high resolution regime which exhibits a noticeable improvement over the existing md scalar quantization schemes

lattices for distributed source coding jointly gaussian sources and reconstruction of a linear function consider a pair of correlated gaussian sources x 1 x 2 two separate encoders observe the two components and communicate compressed versions of their observations to a common decoder the decoder is interested in reconstructing a linear combination of x 1 and x 2 to within a mean square distortion of d we obtain an inner bound to the optimal rate distortion region for this problem a portion of this inner bound is achieved by a scheme that reconstructs the linear function directly rather than reconstructing the individual components x 1 and x 2 first this results in a better rate region for certain parameter values our coding scheme relies on lattice coding techniques in contrast to more prevalent random coding arguments used to demonstrate achievable rate regions in information theory we then consider the case of linear reconstruction of k sources and provide an inner bound to the optimal rate distortion region some parts of the inner bound are achieved using the following coding structure lattice vector quantization followed by ldquocorrelatedrdquo lattice structured binning

achievable rates for multiple descriptions consider a sequence of independent identically distributed i i d random variables x l x 2 cdots x n and a distortion measure d x i x i on the estimates x i of x i two descriptions i x in 1 2 cdots 2 nr 1 and j x in 1 2 cdots 2 nr 2 are given of the sequence x x 1 x 2 cdots x n from these two descriptions three estimates i x x2 j x and hat x o i x j x are formed with resulting expected distortions e frac 1 n sum n k 1 d x k hat x mk d m m 0 1 2 we find that the distortion constraints d 0 d 1 d 2 are achievable if there exists a probability mass distribution p x p hat x 1 hat x 2 hat x 0 x with ed x hat x m leq d m such that r 1 i x hat x 1 r 2 i x hat x 2 where i cdot denotes shannon mutual information these rates are shown to be optimal for deterministic distortion measures

on convergence properties of pocket algorithm the problem of finding optimal weights for a single threshold neuron starting from a general training set is considered among the variety of possible learning techniques the pocket algorithm has a proper convergence theorem which asserts its optimality however the original proof ensures the asymptotic achievement of an optimal weight vector only if the inputs in the training set are integer or rational this limitation is overcome in this paper by introducing a different approach that leads to the general result furthermore a modified version of the learning method considered called pocket algorithm with ratchet is shown to obtain an optimal configuration within a finite number of iterations independently of the given training set

learning in feedforward layered networks the tiling algorithm the authors propose a new algorithm which builds a feedforward layered network in order to learn any boolean function of n boolean units the number of layers and the number of hidden units in each layer are not prescribed in advance they are outputs of the algorithm it is an algorithm for growth of the network which adds layers and units inside a layer at will until convergence the convergence is guaranteed and numerical tests of this strategy look promising

voronoi regions of lattices second moments of polytopes and quantization if a point is picked at random inside a regular simplex octahedron 600 cell or other polytope what is its average squared distance from the centroid in n dimensional space what is the average squared distance of a random point from the closest point of the lattice a n or d n e n a n ast or d n ast the answers are given here together with a description of the voronoi or nearest neighbor regions of these lattices the results have applications to quantization and to the design of signals for the gaussian channel for example a quantizer based on the eight dimensional lattice e8 has a mean squared error per symbol of 0 0717 cdots when applied to uniformly distributed data compared with 0 08333 cdots for the best one dimensional quantizer

multiple description distribution preserving quantization the notion of a multiple description quantizer mdq includes providing multiple distortion levels if a human observer is involved the design of mdq requires a suitable distortion measure to achieve a graceful quality degradation in the case of description losses while the mean squared error is a ubiquitous distortion measure for many classic mdq schemes it is known to be perceptually relevant only at low distortions we propose a new mdq designed according to an unconventional distortion criterion that combines the mean squared error with a constraint on the probability distribution of the reconstructed signal the performance of the new mdq is shown to approach that of the classic mdq asymptotically as rate increases however once applied in the context of transform audio coding the new mdq significantly outperforms a classic mdq in perceptual tests the new scheme is suitable for a wide range of distortions and renders a seamless transition between coding that preserves signal features and coding of a waveform

linear codes interpolation from noisy patterns by means of a vector quantization process abstract an algorithm inferring a boolean linear code from noisy patterns received by a noisy channel under the assumption of uniform occurrence distribution over the codewords and an upper bound to the amount of data are presented a vector quantizer is designed from the noisy patterns choosing the obtained codebook as code approximation it is shown both theoretically and experimentally that when the data are affected by independent random errors this strategy requires a small number of patterns to obtain a good identification with high probability of the code from the noisy data

nonparametric models and methods for nonlinear analysis of covariance summary a fully nonparametric model for nonlinear analysis of covariance is proposed the term nonlinear means that the covariate influences the response in a possibly nonlinear and nonpolynomial fashion while the term fully nonparametric implies that the distributions for each factor level combination and covariate value are not restricted to comply with any parametric or semiparametric model the possibility of different shapes of covariate effect in different factor level combinations is also allowed this generality is useful whenever modelling assumptions such as proportional odds or linearity and homoscedasticity appear suspect in the context of this nonparametric model hypotheses of no main effect no interaction and no simple effect which adjust for the covariate values are defined and test statistics are developed both the response and the covariate are allowed to be ordinal the test statistics are based on averages over the covariate values of certain nadarayawatson type nonparametric regression quantities and asymptotically they have under their respective null hypotheses a central z2 distribution simulation results show that the statistics have good power properties the procedures are demonstrated on two real datasets

the hexagon theorem we wish to int s z f z 2 s the unit square over all choices of f z which take just n values for large n it is shown that f z approaches that function which on each face of a hexagonal mesh has the value of the centroid of that face this problem in the design of quantizers for a pulse code modulation pcm system arose in which pairs of adjacent samples are quantized in an effort to capitalize on the correlation between samples and on the geometry of the plane

a regression framework for rank tests based on the probabilistic index model we demonstrate how many classical rank tests such as the wilcoxon mann whitney kruskal wallis and friedman test can be embedded in a statistical modeling framework and how the method can be used to construct new rank tests in addition to hypothesis testing the method allows for estimating effect sizes with an informative interpretation resulting in a better understanding of the data supplementary materials for this article are available online

nonparametric hypotheses and rank statistics for unbalanced factorial designs abstract factorial designs are studied with independent observations fixed number of levels and possibly unequal number of observations per factor level combination in this context the nonparametric null hypotheses introduced by akritas and arnold are considered new rank statistics are derived for testing the nonparametric hypotheses of no main effects no interaction and no factor effects in unbalanced crossed classifications the formulation of all results includes tied observations extensions of these procedures to higher way layouts are given and the efficacies of the test statistics against nonparametric alternatives are derived a modification of the test statistics and approximations to their finite sample distributions are also given the small sample performance of the procedures for two factors is examined in a simulation study as an illustration a real dataset with ordinal data is analyzed

multiple description vector quantization with lattice codebooks design and analysis the problem of designing a multiple description vector quantizer with lattice codebook spl lambda is considered a general solution is given to a labeling problem which plays a crucial role in the design of such quantizers numerical performance results are obtained for quantizers based on the lattices a sub 2 and z sup i i 1 2 4 8 that make use of this labeling algorithm the high rate squared error distortions for this family of l dimensional vector quantizers are then analyzed for a memoryless source with probability density function pdf p and differential entropy h p spl infin for any a spl epsiv 0 1 and rate pair r r it is shown that the two channel distortion d sub o and the channel 1 or channel 2 distortion d sub s satisfy lim sub r spl rarr spl infin d sub o 2 sup 2r 1 a 1 4 g spl lambda 2 sup 2h p and lim sub r spl rarr spl infin d sub s 2 sup 2r 1 a g s sub l 2 sup 2h p where g spl lambda is the normalized second moment of a voronoi cell of the lattice spl lambda and g s sub l is the normalized second moment of a sphere in l dimensions

vector gaussian multiple description with individual and central receivers t multiple descriptions of a vector gaussian source for individual and central receivers are investigated the sum rate of the descriptions with covariance distortion measure constraints in a positive semidefinite ordering is exactly characterized for two descriptions the entire rate region is characterized the key component of the solution is a novel information theoretic inequality that is used to lower bound the achievable multiple description rates jointly gaussian descriptions are optimal in achieving the limiting rates we also show the robustness of this description scheme the distortions achieved are no larger when used to describe any non gaussian source with the same covariance matrix

on the use of a friedman type statistic in balanced and unbalanced block designs distribution free tests and multiple comparison procedures are considered for randomized block designs a general friedman type statistic is defined that is usable in almost any block design with arbitrary missing data structure for balanced incomplete block designs a multiple comparison procedure and exact tables are given for a statistic equivalent to one proposed by durbin 1951 finally for designs with missing observations on only one treatment we present exact tables for the test statistic and we consider a multiple comparison procedure

multiple description source coding with no excess marginal rate multiple description source coding concerns situations in which the transmission of the source information is distributed over two data streams at rates r sub 1 and r sub 2 respectively when both data streams are received the decoder uses the combined data at rate r sub 1 r sub 2 to reconstruct the source information with average distortion d sub 0 if a communication breakdown prevents one of the data streams from reaching the receiver the decoder has to base its reconstruction solely on the available data at rate either r sub 1 or r sub 2 this results in a higher distortion of either d sub 1 or d sub 2 respectively the region spl rscr of all achievable quintuples r sub 1 r sub 2 d sub 0 d sub 1 d sub 2 has been determined in the so called no excess rate sum case defined by imposing the requirement r sub 1 r sub 2 r d sub 0 where r spl middot is the rate distortion function of the source the case with excess rate sum characterized by r sub 1 r sub 2 r d sub 0 is challenging we study in this paper a special case of it in which the requirements r sub t r d sub t t 1 2 are imposed we refer to this as the no excess marginal rate case the lower and upper bounds on d sub 0 we obtain are separated by only a tiny gap when evaluated for a binary equiprobable source and the hamming distortion measure

a study of vector quantization for noisy channels several issues related to vector quantization for noisy channels are discussed an algorithm based on simulated annealing is developed for assigning binary codewords to the vector quantizer code vectors it is shown that this algorithm could result in dramatic performance improvements as compared to randomly selected codewords a modification of the simulated annealing algorithm for binary codeword assignment is developed for the case where the bits in the codeword are subjected to unequal error probabilities resulting from unequal levels of error protection an algorithm for the design of an optimal vector quantizer for a noisy channel is briefly discussed and its robustness under channel mismatch conditions is studied numerical results for a stationary first order gauss markov source and a binary symmetric channel are provided it is concluded that the channel optimized vector quantizer design algorithm if used carefully can result in a fairly robust system with no additional delay the case in which the communication channel is nonstationary as in mobile radio channels is studied and some preliminary ideas for quantizer design are presented

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

local algorithms for pattern recognition and dependencies estimation in previous publications bottou and vapnik 1992 vapnik 1992 we described local learning algorithms which result in performance improvements for real problems we present here the theoretical framework on which these algorithms are based first we present a new statement of certain learning problems namely the local risk minimization we review the basic results of the uniform convergence theory of learning and extend these results to local risk minimization we also extend the structural risk minimization principle for both pattern recognition problems and regression problems this extended induction principle is the basis for a new class of algorithms

distribution of the length of the longest significance run on a bernoulli net and its applications we consider the length of the longest significance run in a two dimensional bernoulli net and derive its asymptotic limit distribution our results can be considered as generalizations of known theorems in significance runs we give three types of theoretical results 1 reliability style lower and upper bounds 2 erdos renyi law and 3 the asymptotic limit distribution to understand the rate of convergence to the asymptotic distributions we carry out numerical simulations the convergence rates in a variety of situations are presented to understand the relation between the length of the longest significance run s and the success probability p we propose a dynamic programming algorithm to implement simultaneous simulations insights from numerical studies are important for choosing the values of design parameters in a particular application which motivates this article the distribution of the length of the longest significance run in a bernoulli net is critical in applying a multiscale metho

semiparametric linear transformation models effect measures estimators and applications semiparametric linear transformation models form a versatile class of regression models with the cox proportional hazards model being the most well known member these models are well studied for right censored outcomes and are typically used in survival analysis we consider transformation models as a tool for situations with uncensored continuous outcomes where linear regression is not appropriate we introduce the probabilistic index as a uniform effect measure for the class of transformation models we discuss and compare three estimators using a working cox regression model the partial likelihood estimator an estimator based on binary generalized linear models and one based on probabilistic index model estimating equations the latter has a superior performance in terms of bias and variance when the working model is misspecified for the purpose of illustration we analyze data that were collected at an urban alcohol and drug detoxification unit

uniform and piecewise uniform lattice vector quantization for memoryless gaussian and laplacian sources lattice vector quantizer design procedures for nonuniform sources are presented the procedures yield lattice vector quantizers with excellent performance and retaining the structure required for fast quantization analytical methods for truncating and scaling lattices to be used in vector quantizations are given and their utility is demonstrated for independent and identically distributed i i d gaussian and laplacian sources an analytical technique for piecewise linear multidimensional compandor designs is evaluated for i i d gaussian and laplacian sources by comparing its performance to that of the other vector quantizers

optimization of the index assignments for multiple description vector quantizers the optimization criterion and a practically feasible new algorithm is stated for the optimization of the index assignments of a multiple description unconstrained vector quantizer with an arbitrary number of descriptions in the simulations the index optimized multiple description vector quantizer achieves significant gains in source signal to noise ratio over scalar multiple description schemes

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

geometric piecewise uniform lattice vector quantization of the memoryless gaussian source the aim of this paper is to find a quantization technique that has low implementation complexity and asymptotic performance arbitrarily close to the optimum more specifically it is of interest to develop a new vector quantizer design procedure for a memoryless gaussian source that yields vector quantizers with excellent performance and the structure required for fast quantization to achieve this we combined a fast lattice encoding algorithm with a geometric approach to generate a model of a geometric piecewise uniform lattice vector quantizer expressions for granular distortion and the optimal number of outputs points in each region were derived both exact and approximative asymptotic analyses were carried out during this process the constant probability density function of the input signal vector was kept inside the whole region the analysis demonstrated the existence of piecewise constant approximations to the input vector probability density function which is optimal for the proposed geometric piecewise uniform vector quantizer the considered quantization technique is near optimal for a memoryless gaussian source in other words this paper proposes a method for a near optimum low complex vector quantizer design based on probability density function discretization the presented methodology gives a signal to quantization noise ratio that in some cases differs from the optimum by 0 1db or less improvements of the considered model in performance and complexity over some of the existing techniques are also demonstrated

design of multiple description scalar quantizers the design of scalar quantizers for communication systems that use diversity to overcome channel impairments is considered the design problem is posed as an optimization problem and necessary conditions for optimality are derived a design algorithm a generalization of s p lloyd s 1962 algorithm for quantizer design is developed unlike a single channel scalar quantizer the performance of a multiple description scalar quantizer is dependent on the index assignment the problem of index assignment is addressed good index assignments performance results and sample quantizer designs are presented for a memoryless gaussian source comparisons are made with rate distortion bounds for the multiple description problem

fast quantizing and decoding and algorithms for lattice quantizers and codes for each of the lattices a n n geq 1 d n n geq 2 e 6 e 7 e 8 and their duals a very fast algorithm is given for finding the closest lattice point to an arbitrary point if these lattices are used for vector quantizing of uniformly distributed data the algorithm finds the minimum distortion lattice point if the lattices are used as codes for a gaussian channel the algorithm performs maximum likelihood decoding