parallel tree contraction part 2 further applications this paper applies the parallel tree contraction techniques developed in miller and reif s paper randomness and computation vol 5 s micali ed jai press 1989 pp 47 72 to a number of fundamental graph problems the paper presents an o log n time and n log n processor a 0 sided randomized algorithm for testing the isomorphism of trees and an o log n time n algorithm for maximal subtree isomorphism and for common subexpression elimination an o log n time n processor algorithm for computing the canonical forms of trees and subtrees is given an olog n time algorithm for computing the tree of 3 connected components of a graph an o log 2 n time algorithm for computing an explicit planar embedding of a planar graph and an o log 3 n time algorithm for computing a canonical form for a planar graph are also given all these latter algorithms use only n o 1 processors on a parallel random access machine pram model with concurrent writes and concurrent reads

entropy based tests of uniformity abstract the power properties of an entropy based test are investigated when used for testing uniformity on 0 1 percentage points and power against seven alternatives are reported compared with other tests of uniformity the entropy based test possesses good power properties for many alternatives some asymptotic null and alternative distributions are derived for sample sizes up to 100 the table of percentage points provides a practical guide for using this test a theory of entropy based tests of distributional hypotheses other than uniformity is outlined

singular value decomposition based alternative splicing detection altered alternative splicing has been identified as an important factor in tumorigenesis the affymetrix exon tiling array is designed for detecting alternative splicing events in a transcriptome wide fashion however there are currently few analysis tools that have been well studied for effective detection of alternative splicing events we propose a new screening procedure based on singular value decomposition svd of the residual matrix from a robust additive model fit to probe selection region psr data with this approach we analyze the exon tiling array data from a brain cancer study conducted at the m d anderson cancer center and show that the proposed svd based approach is able to better accommodate outlying measures and capitalize on the multidimensional group by psr gene expression profiles for more effective detection of group specific alternative splicing events as well as the psrs most likely associated with the alternative splicing lab validation confirmed some of our findings but

dynamic modeling of gene expression data we describe the time evolution of gene expression levels by using a time translational matrix to predict future expression levels of genes based on their expression levels at some initial time we deduce the time translational matrix for previously published dna microarray gene expression data sets by modeling them within a linear framework by using the characteristic modes obtained by singular value decomposition the resulting time translation matrix provides a measure of the relationships among the modes and governs their time evolution we show that a truncated matrix linking just a few modes is a good approximation of the full time translation matrix this finding suggests that the number of essential connections among the genes is small

binary regression models for contaminated data on etudie la nature des observations aberrantes dans le contexte des donnees de regression binaires on propose un modele simple qui permet qu un petit nombre de reponses binaires soit mal enregistre

deciding context unification in first order term unification variables represent well formed terms over a given signature and we are to solve equations built using function symbols from the signature and such variables this problem is well known to be decidable in linear time in second order term unification the variables take arguments i e other terms and a substitution uses those arguments an arbitrary number of times for instance an equation f x c x c e x f c c has a solution x e b where b is a special symbol denoting the place in which the argument is substituted under this substitution both sides evaluate to f c c there are other solutions for instance x e f b b which evaluates both sides tof f c c f c c in general a solution that evaluates both sides to full binary tree of arbitrary height is easy to construct second order unification is in general undecidable context unification is a natural problem in between first and second order unification we deal with equations over terms the variables take arguments but we restrict the set of solutions the argument is used exactly once formally contexts are terms with exactly one occurrence of the special symbol b and in context unification we are given an equation over terms with variables representing contexts and ask about the satisfiability of this equation for instance when the aforementioned equation f x c x c e x f c c is treated as a context unification problem then it has exactly one solution x e b other substitutions that are solutions of it as an instance of the second unification problem say x e f b b are not valid as b is used more than once context unification also generalizes satisfiability of word equations which is decidable in pspace the decidability status of context unification remained unknown for almost two decades in this article we show that context unification is in pspace in exptime when tree regular constraints are also allowed those results are obtained by extending the recently developed recompression technique which was previously defined for strings and used to obtain a new pspace algorithm for satisfiability of word equations in this article the technique is generalized to trees and the corresponding algorithm is generalized from word equations to context unification the idea of recompression is to apply simple compression rules replacing pairs of neighboring function symbols to the solution of the context equation to this end we appropriately modify the equation without the knowledge of the actual solution so compressing the solution can be simulated by compressing parts of the equation it is shown that if the compression operations are appropriately chosen then the size of the instance is polynomial during the whole algorithm thus giving a pspace upper bound

estimation of expression indexes for oligonucleotide arrays using the singular value decomposition multiprobe oligonucleotide arrays are a widely used type of expression microarray with the attractive feature that numerous probes are used to represent each transcript an expression index is a statistic used to represent expression level for a particular gene that is estimated from the probe hybridization intensities we show that a popular model based expression index proposed by li and wong has an interpretation as a component of the singular value decomposition svd of the probe intensity matrix following this observation we propose a new svd model that accounts for differing variance structure across probes we also propose that nonlinearity in intensity response to expression can be corrected to some extent through a data transformation which is guided by an svd entropy measure the methods are demonstrated with simulation and applications to two real datasets

the expressibility of languages and relations by word equations classically several properties and relations of words such as being a power of the same word can be expressed by using word equations this paper is devoted to a general study of the expressive power of word equations as main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations in particular the primitiveness and the equal length are such properties as well as being any word over a proper subalphabet

analyzing multiple probe microarray estimation and application of gene expression indexes gene expression index estimation is an essential step in analyzing multiple probe microarray data various modeling methods have been proposed in this area amidst all a popular method proposed in li and wong 2001 is based on a multiplicative model which is similar to the additive model discussed in irizarry et al 2003a at the logarithm scale along this line hu et al 2006 proposed data transformation to improve expression index estimation based on an ad hoc entropy criteria and naive grid search approach in this work we re examined this problem using a new profile likelihood based transformation estimation approach that is more statistically elegant and computationally efficient we demonstrate the applicability of the proposed method using a benchmark affymetrix u95a spiked in experiment moreover we introduced a new multivariate expression index and used the empirical study to shows its promise in terms of improving model fitting and power of detecting differential expression over the commonly used univariate expression index as the other important content of the work we discussed two generally encountered practical issues in application of gene expression index normalization and summary statistic used for detecting differential expression our empirical study shows somewhat different findings from the maqc project maqc 2006

complexity of makanin s algorithm the exponent of periodicity is an important factor in estimates of complexity of word unification algorithms we prove that the exponent of periodicity of a minimal solution of a word equation is of order 2 1 07d where d is the length of the equation we also give a lower bound 2 0 29d so our upper bound is almost optimal and exponentially better than the original bound 6d 22d4 2 consequently our result implies an exponential improvement of known upper bounds on complexity of word unification algorithms

the undecidability of the second order unification problem abstract it is shown that there is no effective procedure for determining whether or not two terms of the language of second order logic have a common instance

approximation of grammar based compression via recompression in this paper we present a simple linear time algorithm constructing a context free grammar of size o g log n g for the input string where n is the size of the input string and g the size of the optimal grammar generating this string the algorithm works for arbitrary size alphabets but the running time is linear assuming that the alphabet of the input string can be identified with numbers from 1 n c for some constant c otherwise additional cost of o n log is needed algorithms with such an approximation guarantee and running time are known the novelty of this paper is a particular simplicity of the algorithm as well as the analysis of the algorithm which uses a general technique of recompression recently introduced by the author furthermore contrary to the previous results this work does not use the lz representation of the input string in the construction nor in the analysis

approximation of smallest linear tree grammar a simple linear time algorithm for constructing a linear context free tree grammar of size o r g r g log n r g for a given input tree t of size n is presented where g is the size of a minimal linear context free tree grammar for t and r is the maximal rank of symbols in t which is a constant in many applications this is the first example of a grammar based tree compression algorithm with a good i e logarithmic in terms of the size of the input tree approximation ratio the analysis of the algorithm uses an extension of the recompression technique from strings to trees

finding all solutions of equations in free groups and monoids with involution this paper presents a pspace algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids this became possible due to the recent recompression technique while this technique was successfully applied for pure word equations without involution or rational constraints it could not be used as a black box for free groups actually the presence of an involution and rational constraints complicates the situation and some additional analysis is necessary still the technique is general enough to accommodate both extensions in the end it simplifies proofs that satisfiability of word equations is in pspace and the corresponding result for equations in free groups with rational constraints as a byproduct we can decide in pspace whether the solution set is finite

decidability of bounded second order unification it is well known that first order unification is decidable whereas second order and higher order unification is undecidable bounded second order unification is second order unification under the restriction that only a bounded number of bound variables in the instantiating terms for second order variables is permitted however the size of the instantiation is not restricted in this paper a decision algorithm for bounded second order unification is described this is the first nontrivial decidability result for second order unification where the signature is not restricted and there are no restrictions on the occurrences of variables this supports the claim that bounded second order unification is easier than context unification whose decidability is currently an open question

regression modeling of ordinal data with nonzero baselines summary this paper develops regression models for ordinal data with nonzero control response probabilities the models are especially useful in dose response studies where the spontaneous or natural response rate is nonnegligible and the dosage is logarithmic these models generalize abbott s formula which has been commonly used to model binary data with nonzero background observations we describe a biologically plausible latent structure and develop an em algorithm for fitting the models the em algorithm can be implemented using standard software for ordinal regression a toxicology data set where the proposed model fits the data but a more conventional model fails is used to illustrate the methodology

the complexity of monadic second order unification monadic second order unification is second order unification where all function constants occurring in the equations are unary here we prove that the problem of deciding whether a set of monadic equations has a unifier is np complete where we use the technique of compressing solutions using singleton context free grammars we prove that monadic second order matching is also np complete

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

recompression a simple and powerful technique for word equations in this article we present an application of a simple technique of local recompression previously developed by the author in the context algorithms for compressed strings jez 2014a 2015b 2015a to word equations the technique is based on local modification of variables replacing x by ax or xa and iterative replacement of pairs of letters occurring in the equation by a fresh letter which can be seen as a bottom up compression of the solution of the given word equation or to be more specific building a straight line programme for the solution of the word equation using this technique we give new independent and self contained proofs of many known results for word equations to be more specific the presented nondeterministic algorithm runs in o n log n space and in time polynomial in n and log n where n is the size of the input equation and n the size of the length minimal solution of the word equation furthermore for o 1 variables the bound on the space consumption is in fact linear that is o m where m is the size of the space used by the input this yields that for each k the set of satisfiable word equations with k variables is context sensitive the presented algorithm can be easily generalised to a generator of all solutions of the given word equation without increasing the space usage furthermore a further analysis of the algorithm yields an independent proof of doubly exponential upper bound on the size of the length minimal solution the presented algorithm does not use exponential bound on the exponent of periodicity conversely the analysis of the algorithm yields an independent proof of the exponential bound on exponent of periodicity

on robustness in the logistic regression model summary we investigate robustness in the logistic regression model copas has studied two forms of robust estimator a robust resistant estimate of pregibon and an estimate based on a misclassification model he concluded that robust resistant estimates are much more biased in small samples than the usual logistic estimate is and recommends a bias corrected version of the misclassification estimate we show that there are other versions of robust resistant estimates which have bias often approximately the same as and sometimes even less than the logistic estimate these estimates belong to the mallows class in addition the corrected misclassification estimate is inconsistent at the logistic model we develop a simple consistent modification the modified estimate is a member of the mallows class but unlike most robust estimates it has an interpretable tuning constant the results are illustrated on data sets featuring different kinds of outliers