Signal classification with a point process distance on the space of persistence diagrams

In this paper, we consider the problem of signal classification. First, the signal is translated into a persistence diagram through the use of delay-embedding and persistent homology. Endowing the data space of persistence diagrams with a metric from point processes, we show that it admits statistical structure in the form of Fréchet means and variances and a classification scheme is established. In contrast with the Wasserstein distance, this metric accounts for changes in small persistence and changes in cardinality. The classification results using this distance are benchmarked on both synthetic data and real acoustic signals and it is demonstrated that this classifier outperforms current signal classification techniques.     

REMARKS ON UNIVERSAL COEFFICIENT THEOREMS FOR GENERALIZED HOMOLOGY THEORIES

Abstract

New proofs of universal coefficient theorems for generalized homology theories (cf. ∮ 2, ∮ 3) including L. G. Brown's result, relating Brown-Douglas-Fillmore's Ext (X) with complex K-theory are presented. They are all based on a theorem asserting the existence of a chain functor for a generalized homology theory (cf. ∮ 1), which was originally designed for the construction of strong homology theories on strong shape categories.     

Betti numbers in multidimensional persistent homology are stable functions

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of scalar‐valued filtering functions, persistent homology groups can be studied through their persistent Betti numbers, that is, the dimensions of the images of the homomorphisms induced by the inclusions of lower level sets into each other. Whenever such inclusions exist for lower level sets of vector‐valued filtering functions, we can consider the multidimensional analog of persistent Betti numbers. Varying the lower level sets, we obtain that persistent Betti numbers can be seen as functions taking pairs of vectors to the set of non‐negative integers. In this paper, we prove stability of multidimensional persistent Betti numbers. More precisely, we prove that small changes of the vector‐valued filtering functions imply only small changes of persistent Betti numbers functions. This result can be obtained by assuming the filtering functions to be just continuous. Multidimensional stability opens the way to a stable shape comparison methodology based on multidimensional persistence. In order to obtain our stability theorem, some other new results are proved for continuous filtering functions. They concern the finiteness of persistent Betti numbers for vector‐valued filtering functions and the representation via persistence diagrams of persistent Betti numbers, as well as their stability, in the case of scalar‐valued filtering functions. Finally, from the stability of multidimensional persistent Betti numbers, we obtain a lower bound for the natural pseudo‐distance. Copyright © 2013 John Wiley & Sons, Ltd.     

Testing multivariate normality in incomplete data of small sample size

In longitudinal studies with small samples and incomplete data, multivariate normal-based models continue to be a powerful tool for analysis. This has included a broad scope of biomedical studies. Testing the assumption of multivariate normality (MVN) is critical. Although many methods are available for testing normality in complete data with large samples, a few deal with the testing in small samples. For example, Liang et al. (J. Statist. Planning and Inference 86 (2000) 129) propose a projection procedure for testing MVN for complete-data with small samples where the sample sizes may be close to the dimension. To our knowledge, no statistical methods for testing MVN in incomplete data with small samples are yet available. This article develops a test procedure in such a setting using multiple imputations and the projection test. To utilize the incomplete data structure in multiple imputation, we adopt a noniterative inverse Bayes formulae (IBF) sampling procedure instead of the iterative Gibbs sampling to generate iid samples. Simulations are performed for both complete and incomplete data when the sample size is less than the dimension. The method is illustrated with a real study on an anticancer drug.     

A robust multigrid approach for variational image registration models

Variational registration models are non-rigid and deformable imaging techniques for accurate registration of two images. As with other models for inverse problems using the Tikhonov regularization, they must have a suitably chosen regularization term as well as a data fitting term. One distinct feature of registration models is that their fitting term is always highly nonlinear and this nonlinearity restricts the class of numerical methods that are applicable. This paper first reviews the current state-of-the-art numerical methods for such models and observes that the nonlinear fitting term is mostly ‘avoided’ in developing fast multigrid methods. It then proposes a unified approach for designing fixed point type smoothers for multigrid methods. The diffusion registration model (second-order equations) and a curvature model (fourth-order equations) are used to illustrate our robust methodology. Analysis of the proposed smoothers and comparisons to other methods are given. As expected of a multigrid method, being many orders of magnitude faster than the unilevel gradient descent approach, the proposed numerical approach delivers fast and accurate results for a range of synthetic and real test images.     

A link-free method for testing the significance of predictors

One important step in regression analysis is to identify significant predictors from a pool of candidates so that a parsimonious model can be obtained using these significant predictors only. However, most of the existing methods assume linear relationships between response and predictors, which may be inappropriate in some applications. In this article, we discuss a link-free method that avoids specifying how the response depends on the predictors. Therefore, this method has no problem of model misspecification, and it is suitable for selecting significant predictors at the preliminary stage of data analysis. A test statistic is suggested and its asymptotic distribution is derived. Examples are used to demonstrate the proposed method.     

Persistence Terrace for Topological Inference of Point Cloud Data

Topological data analysis (TDA) is a rapidly developing collection of methods for studying the shape of point cloud and other data types. One popular approach, designed to be robust to noise and outliers, is to first use a smoothing function to convert the point cloud into a manifold and then apply persistent homology to a Morse filtration. A significant challenge is that this smoothing process involves the choice of a parameter and persistent homology is highly sensitive to that choice; moreover, important scale information is lost. We propose a novel topological summary plot, called a persistence terrace, that incorporates a wide range of smoothing parameters and is robust, multi-scale, and parameter-free. This plot allows one to isolate distinct topological signals that may have merged for any fixed value of the smoothing parameter, and it also allows one to infer the size and point density of the topological features. We illustrate our method in some simple settings where noise is a serious issue for existing frameworks and then we apply it to a real dataset by counting muscle fibers in a cross-sectional image. Supplementary material for this article is available online.     

The relationship between the diagonal and the bar constructions on a bisimplicial set

The aim of this paper is to prove that the homotopy type of any bisimplicial set X is modelled by the simplicial set , the bar construction on X. We stress the interest of this result by showing two relevant theorems which now become simple instances of it; namely, the Homotopy colimit theorem of Thomason, for diagrams of small categories, and the generalized Eilenberg-Zilber theorem of Dold-Puppe for bisimplicial Abelian groups. Among other applications, we give an algebraic model for the homotopy theory of (not necessarily path-connected) spaces whose homotopy groups vanish in degree 4 and higher.     

Evaluation of reproducibility for paired functional data

Evaluation of reproducibility is important in assessing whether a new method or instrument can reproduce the results from a traditional gold standard approach. In this paper, we propose a measure to assess measurement agreement for functional data which are frequently encountered in medical research and many other research fields. Formulae to compute the standard error of the proposed estimator and confidence intervals for the proposed measure are derived. The estimators and the coverage probabilities of the confidence intervals are empirically tested for small-to-moderate sample sizes via Monte Carlo simulations. A real data example in physiology study is used to illustrate the proposed statistical inference procedures.     

A geometric decomposition of spaces into cells of different types II: Homology theory

We develop the homology theory of CW(A)-complexes, generalizing the classical cellular homology theory for CW-complexes. A CW(A)-complex is a topological space which is built up out of cells of a certain core A.     

Homology of generalized partition posets

We define a family of posets of partitions associated to an operad. We prove that the operad is Koszul if and only if the posets are Cohen-Macaulay. On the one hand, this characterization allows us to compute completely the homology of the posets. The homology groups are isomorphic to the Koszul dual cooperad. On the other hand, we get new methods for proving that an operad is Koszul.     

The Accumulated Persistence Function,a New Useful Functional Summary Statistic for Topological Data Analysis,With a View to Brain Artery Trees and Spatial Point Process Applications

We start with a simple introduction to topological data analysis where the most popular tool is called a persistence diagram. Briefly, a persistence diagram is a multiset of points in the plane describing the persistence of topological features of a compact set when a scale parameter varies. Since statistical methods are difficult to apply directly on persistence diagrams, various alternative functional summary statistics have been suggested, but either they do not contain the full information of the persistence diagram or they are two-dimensional functions. We suggest a new functional summary statistic that is one-dimensional and hence easier to handle, and which under mild conditions contains the full information of the persistence diagram. Its usefulness is illustrated in statistical settings concerned with point clouds and brain artery trees. The supplementary materials include additional methods and examples, technical details, and the R code used for all examples.     

Torsion in the matching complex and chessboard complex

Topological properties of the matching complex were first studied by Bouc in connection with Quillen complexes, and topological properties of the chessboard complex were first studied by Garst in connection with Tits coset complexes. Björner, Lovász, Vre?ica and ?ivaljevi? established bounds on the connectivity of these complexes and conjectured that these bounds are sharp. In this paper we show that the conjecture is true by establishing the nonvanishing of integral homology in the degrees given by these bounds. Moreover, we show that for sufficiently large n, the bottom nonvanishing homology of the matching complex Mn is an elementary 3-group, improving a result of Bouc, and that the bottom nonvanishing homology of the chessboard complex Mn,n is a 3-group of exponent at most 9. When , the bottom nonvanishing homology of Mn,n is shown to be Z₃. Our proofs rely on computer calculations, long exact sequences, representation theory, and tableau combinatorics.     

The covariance structure and likelihood function for multivariate dyadic data

This paper extends the results in Li and Loken [A unified theory of statistical analysis and inference for variance component models for dyadic data, Statist. Sinica 12 (2002) 519-535] on the statistical analysis of measurements taken on dyads to the situations in which more than one attribute are measured on each dyad. Starting from the covariance structure for the univariate case obtained in Li and Loken (2002), the covariance structure for the multivariate case is derived based on the group symmetry induced by the assumed exchangeability in the units. Our primary objective is to document the Gaussian likelihood and the sufficient statistics for multivariate dyadic data in closed form, so that they can be referenced by researchers as they analyze those data. The derivation carried out can also serve as an example of multivariate extension of univariate models based on exchangeability.     

Topology of matching,chessboard, and general bounded degree graph complexes

We survey results and techniques in the topological study of simplicial complexes of (di-, multi-, hyper-)graphs whose node degrees are bounded from above. These complexes have arisen in a variety of contexts in the literature. The most well-known examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.In memory of Gian-Carlo Rota     

Categorification of Persistent Homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category. A set of such diagrams has an interleaving distance, which we show generalizes the previously studied bottleneck distance. To illustrate the utility of this approach, we generalize previous stability results for persistence, extended persistence, and kernel, image, and cokernel persistence. We give a natural construction of a category of ε-interleavings of $\mathbf {(\mathbb {R},\leq)}$ -indexed diagrams in some target category and show that if the target category is abelian, so is this category of interleavings.     

Theory and inference for regression models with missing responses and covariates

In this paper, we carry out an in-depth theoretical investigation for inference with missing response and covariate data for general regression models. We assume that the missing data are missing at random (MAR) or missing completely at random (MCAR) throughout. Previous theoretical investigations in the literature have focused only on missing covariates or missing responses, but not both. Here, we consider theoretical properties of the estimates under three different estimation settings: complete case (CC) analysis, a complete response (CR) analysis that involves an analysis of those subjects with only completely observed responses, and the all case (AC) analysis, which is an analysis based on all of the cases. Under each scenario, we derive general expressions for the likelihood and devise estimation schemes based on the EM algorithm. We carry out a theoretical investigation of the three estimation methods in the normal linear model and analytically characterize the loss of information for each method, as well as derive and compare the asymptotic variances for each method assuming the missing data are MAR or MCAR. In addition, a theoretical investigation of bias for the CC method is also carried out. A simulation study and real dataset are given to illustrate the methodology.     

Portfolio value at risk based on independent component analysis

Risk management technology applied to high-dimensional portfolios needs simple and fast methods for calculation of value at risk (VaR). The multivariate normal framework provides a simple off-the-shelf methodology but lacks the heavy-tailed distributional properties that are observed in data. A principle component-based method (tied closely to the elliptical structure of the distribution) is therefore expected to be unsatisfactory. Here, we propose and analyze a technology that is based on independent component analysis (ICA). We study the proposed ICVaR methodology in an extensive simulation study and apply it to a high-dimensional portfolio situation. Our analysis yields very accurate VaRs.     

Hierarchical Bayes variable selection and microarray experiments

Hierarchical and empirical Bayes approaches to inference are attractive for data arising from microarray gene expression studies because of their ability to borrow strength across genes in making inferences. Here we focus on the simplest case where we have data from replicated two colour arrays which compare two samples and where we wish to decide which genes are differentially expressed and obtain estimates of operating characteristics such as false discovery rates. The purpose of this paper is to examine the frequentist performance of Bayesian variable selection approaches to this problem for different prior specifications and to examine the effect on inference of commonly used empirical Bayes approximations to hierarchical Bayes procedures. The paper makes three main contributions. First, we describe how the log odds of differential expression can usually be computed analytically in the case where a double tailed exponential prior is used for gene effects rather than a normal prior, which gives an alternative to the commonly used B-statistic for ranking genes in simple comparative experiments. The second contribution of the paper is to compare empirical Bayes procedures for detecting differential expression with hierarchical Bayes methods which account for uncertainty in prior hyperparameters to examine how much is lost in using the commonly employed empirical Bayes approximations. Third, we describe an efficient MCMC scheme for carrying out the computations required for the hierarchical Bayes procedures. Comparisons are made via simulation studies where the simulated data are obtained by fitting models to some real microarray data sets. The results have implications for analysis of microarray data using parametric hierarchical and empirical Bayes methods for more complex experimental designs: generally we find that the empirical Bayes methods work well, which supports their use in the analysis of more complex experiments when a full hierarchical Bayes analysis would impose heavy computational demands.     

Parameter estimation under ambiguity and contamination with the spurious model

Recently, we proposed variants as a statistical model for treating ambiguity. If data are extracted from an object with a machine then it might not be able to give a unique safe answer due to ambiguity about the correct interpretation of the object. On the other hand, the machine is often able to produce a finite number of alternative feature sets (of the same object) that contain the desired one. We call these feature sets variants of the object. Data sets that contain variants may be analyzed by means of statistical methods and all chapters of multivariate analysis can be seen in the light of variants. In this communication, we focus on point estimation in the presence of variants and outliers. Besides robust parameter estimation, this task requires also selecting the regular objects and their valid feature sets (regular variants). We determine the mixed MAP-ML estimator for a model with spurious variants and outliers as well as estimators based on the integrated likelihood. We also prove asymptotic results which show that the estimators are nearly consistent.The problem of variant selection turns out to be computationally hard; therefore, we also design algorithms for efficient approximation. We finally demonstrate their efficacy with a simulated data set and a real data set from genetics.