Equal Covariance Research Articles

Longitudinal registration of thoracic CT images with radiation-induced lung diseases: A divide-and-conquer approach based on component structure wise registration using coherent point drift

Background and ObjectiveRegistration of pulmonary computed tomography (CT) images with radiation-induced lung diseases (RILD) was essential to investigate the voxel-wise relationship between the formation of RILD and the radiation dose received by different tissues. Although various approaches had been developed for the registration of lung CTs, their performances remained clinically unsatisfactory for registration of lung CT images with RILD. The main difficulties arose from the longitudinal change in lung parenchyma, including RILD and volumetric change of lung cancers, after radiation therapy, leading to inaccurate registration and artifacts caused by erroneous matching of the RILD tissues. MethodsTo overcome the influence of the parenchymal changes, a divide-and-conquer approach rooted in the coherent point drift (CPD) paradigm was proposed. The proposed method was based on two kernel ideas. One was the idea of component structure wise registration. Specifically, the proposed method relaxed the intrinsic assumption of equal isotropic covariances in CPD by decomposing a lung and its surrounding tissues into component structures and independently registering the component structures pairwise by CPD. The other was the idea of defining a vascular subtree centered at a matched branch point as a component structure. This idea could not only provide a sufficient number of matched feature points within a parenchyma, but avoid being corrupted by the false feature points resided in the RILD tissues due to globally and indiscriminately sampling using mathematical operators. The overall deformation model was built by using the Thin Plate Spline based on all matched points. ResultsThis study recruited 30 pairs of lung CT images with RILD, 15 of which were used for internal validation (leave-one-out cross-validation) and the other 15 for external validation. The experimental results showed that the proposed algorithm achieved a mean and a mean of maximum 1 % of average surface distances <2 and 8 mm, respectively, and a mean and a maximum target registration error <2 mm and 5 mm on both internal and external validation datasets. The paired two-sample t-tests corroborated that the proposed algorithm outperformed a recent method, the Stavropoulou's method, on the external validation dataset (p < 0.05). ConclusionsThe proposed algorithm effectively reduced the influence of parenchymal changes, resulting in a reasonably accurate and artifact-free registration.

Testing hypotheses about correlation matrices in general MANOVA designs

Correlation matrices are an essential tool for investigating the dependency structures of random vectors or comparing them. We introduce an approach for testing a variety of null hypotheses that can be formulated based upon the correlation matrix. Examples cover MANOVA-type hypothesis of equal correlation matrices as well as testing for special correlation structures such as sphericity. Apart from existing fourth moments, our approach requires no other assumptions, allowing applications in various settings. To improve the small sample performance, a bootstrap technique is proposed and theoretically justified. Based on this, we also present a procedure to simultaneously test the hypotheses of equal correlation and equal covariance matrices. The performance of all new test statistics is compared with existing procedures through extensive simulations.

Are principal components with estimated parameters feasible in multivariate process control?

AbstractPrincipal components have been used in conjunction with Hotelling's T2 chart to monitor multivariate processes. It is known that prohibitively large sample sizes are needed to estimate the process parameters with enough precision to deploy the chart. We investigate whether principal components can be used to reduce the dimensionality of a process so that multivariate process control can be performed using estimated parameters. The chart based on the first k principal components, which we will refer to as the chart, is investigated. Specifically, we explore three research questions in this paper: (1) can the chart with estimated parameters be applied with moderate preliminary sample sizes?, (2) are there situations where the charts are able to detect a shift in the mean vector quicker than the T2 charts?, and (3) is it possible to exploit assumptions about the covariance matrix, such as equal covariances, to improve the performance of the chart. Using simulation, we find that for high dimensions, the chart with estimated parameters can be used to detect shifts in the direction of the first principal component. Otherwise, the chart requires very large preliminary sample sizes. When it is reasonable to assume equal covariances, the number of parameters to be estimated is substantially reduced and the performance of the chart is improved.

Information Fusion Strategies for Collaborative Inertial Radio SLAM

Information fusion strategies for vehicles navigating while aiding their inertial navigation systems (INSs) with terrestrial signals of opportunity (SOPs) are developed and studied. The following problem is considered. Multiple navigating vehicles with access to global navigation satellite system (GNSS) signals are aiding their on-board INSs with GNSS pseudoranges. While navigating, vehicle-mounted receivers draw pseudorange measurements from terrestrial SOPs (e.g., AM/FM radio, digital television, cellular) with unknown emitter positions and unknown and unsynchronized clocks. The vehicles share INS data and SOP pseudoranges to collaboratively estimate the SOPs’ states through an extended Kalman filter using tight coupling. After some time, GNSS signals become unavailable, at which point the navigating vehicles use shared INS and SOP information to continue navigating in a collaborative inertial radio simultaneous localization and mapping (CIRSLAM) framework. This paper develops such CIRSLAM framework and synthesizes what SOP and INS information should be shared between collaborators. Two information fusion strategies are compared: 1) sharing time-of-arrival (TOA) measurements from SOPs; 2) sharing time-difference-of-arrival (TDOA) measurements taken with reference to an SOP. Next, a strategy to efficiently share INS information along with SOP information is discussed. Monte Carlo simulation results are presented that support the analytical findings that vehicles navigating in a CIRSLAM framework, while sharing and fusing SOP TOA measurements, produce a smaller or equal estimation error covariance compared to fusing SOP TDOA measurements. Experimental results are presented demonstrating two unmanned aerial vehicles (UAVs) navigating in a CIRSLAM framework with SOP TOA measurements from terrestrial cellular towers. The final UAVs’ localization error after 30 seconds of GPS unavailability were reduced compared to using an INS alone from around 55 m to around 6 m.

Effective multiview registration of point clouds based on Student’s-t mixture model

Recently, the expectation maximization (EM) algorithm has been used to address the multiview registration of point clouds problem. Most approaches suppose that each data point is generated from the Gaussian mixture model, which notably has difficulty handling outliers and heavy-tail noise. Consequently, this study proposes a novel multiview registration approach based on Student’s-t mixture model (SMM). Specifically, we assume that each data point is generated from a unique SMM, where its nearest neighbors (NNs) in other point clouds are SMM centroids with fixed degrees of freedom, equal covariances, and membership probabilities. Therefore, the multiview registration problem is formulated as the maximization of the likelihood function. Subsequently, the EM algorithm is employed to optimize the rigid transformations used for multiview registration and the only Student’s t-distribution covariance. Because only a few model parameters are optimized, our algorithm achieves promising registration performance. Additionally, all SMM centroids are obtained using the NN search algorithm, which is exceedingly efficient. Moreover, the Student’s t-distribution renders our algorithm inherently robust to outliers and heavy-tail noise. The results of tests on benchmark datasets demonstrate our algorithm’s superior registration performance in terms of accuracy and robustness, compared with state-of-the-art algorithms.

Likelihood Maximization and Moment Matching in Low SNR Gaussian Mixture Models

AbstractWe derive an asymptotic expansion for the log‐likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal‐to‐noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation‐Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone towards the analysis of EM and maximum likelihood estimation in a wide range of models. A motivating application for the study of low SNR mixture models is cryo‐electron microscopy data, which can be modeled as a GMM with algebraic constraints imposed on the mixture centers. We discuss the application of our expansion to algebraically constrained GMMs, among other example models of interest. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

A Robust Comparison Powers of Four Multivariate Analysis of Variance Tests

This paper compared the powers of four test statistics of Multivariate Analysis of Variance and test statistics include Lawley-Hotelling ,Pillai's trace, Roy’s largest root and Wilks' lambda. The R Statistics was employed to simulate the data used to compare the four test statistics under the Multivariate Gamma and Multivariate Normal distributions. The sample sizes utilized were 10, 20, 30, 40, 100, 200, 300, 400, 600, 700, 800 and 1000); Number of variables (p = 2, 3, and 4) equal and unequal samples and variance co-variance matrix were also used. The comparison were done at two levels of significance (α = 0.01 and 0.05) using power of the test. The results obtained indicated that the Roy’s largest Root test statistic is better than all other test statistic considered when p = g = 2 because it has the least powers. The result of the analysis further showed that when p = g = 3 and p = g = 4 the Wilks' lambda proved better than all other test statistics both for small and large sample sizes. The results equally showed that when the data are multivariate normal and Gamma with g =2 and p = 2 the power of the four test statistics from best to least is Roy’s largest root, followed Lawley’s trace = Pillae’s trace and the least is Wilks’ Lambda at significant levels of 0.01 and 0.05 for equal and unequal samples. More so, when data are multivariate normal as well as gamma and p= g = 3 and p=g=4, the power of the four statistics ranked from best to least is Wilks’ Lambda , Pillae’s trace = Lawley’s trace and Roy’s largest root respectively . From the findings, it is evidently obvious that Roy’s largest root should be applied when p=g=2 while Wilks’ Lambda be used when p=g=3 and p=g=4. This study will help researchers to plan studies with controlled probabilities of detecting a meaningful effect thereby giving conclusive results with maximum efficiency.

On Testing Homogeneity of Covariance Matrices with Box’s M and the Approximate Tests for Multivariate Data

Homogeneity of covariance matrices, or equal covariance matrices across groups, is one of the most important assumptions in multivariate analysis of variance (MANOVA) and in discriminant analysis. Box’s M test, an exact test, is a generally accepted method used to check the violation of this assumption. The Box’s statistic M can be transformed to the statistics serving as the approximate tests based on chi-squared and F distributions. This study aims to assess the performance of Box’s M test compared with the approximate test using chi-squared distribution for normal data. When the data are non-normal, the performance of the approximate test and the nonparametric test using the bootstrap method are examined. The results showed that under normality with equal sample sizes and certain conditions to conduct the exact test, the Box’s M and the approximate chi-squared tests perform well whereas the performance of the approximate test is slightly better for detecting heterogeneity of covariance matrices. In the case of unequal sample sizes, in which the data do not meet the conditions to conduct Box’s M test, the approximate chi-squared test also performs well so it is applicable for such a case. When dealing with non-normal data, such as multivariate t-distributed data, the performance of the approximate test as well as the nonparametric test using the bootstrap method is unsatisfactory in the situations studied so it needs to be further developed.

Asymptotic approximation of misclassification probabilities in linear discriminant analysis with repeated measurements

We propose asymptotic approximations for the probabilities of misclassification in linear discriminant analysis when the group means follow a growth curve structure. The discriminant function can classify a new observation vector of p repeated measurements into one of several multivariate normal populations with equal covariance matrix. We derive certain relations of the statistics under consideration in order to obtain asymptotic approximation of misclassification errors for the two group case. Finally, we perform Monte Carlo simulations to evaluate the reliability of the proposed results.

مقارنة بين أسلوبي C-H والامکان الأعظم لتصنيف البيانات وذلک بالتطبيق علي مرضي الفشل الکلوي

ملخص استهدفت تلک الدراسة المقارنة بين طريقتين من طرق التحليل التمييزي الأول هو أسلوب التوليفة الخطية (C-H classifier) Chung and Hanوهو عبارة عن دمج دالتين تمايز خطية واحدة للمشاهدات المکتملة والاخري للمشاهدات غير المکتملة, وأسلوب تصنيف الإمکان الأعظم MLE classifier وهو مشتق من دالة التمييز الخطي التقليدية ولکن اعتمد علي تقدير المعالم باستخدام دالة الامکان الأعظم . وکلا الأسلوبين يتطلبا نمط خاص للبيانات وهو أن تحتوي البيانات على مشاهدات مفقودة علي وتيرة واحدة ويجب ان يکون للمجتمعين نفس النمط. کما تهدف الدراسة إلي التنبؤ بمتجه مشاهدات جديدة ذو بعد إلي إحدي المجتمعين قيد الدراسة, و قامت الدراسه لتقييم کفاءة الأسلوبين باستخدام معيارمقدر البوتستراب المعلمي لفرق معدل الخطأ المتوقع(a parametric bootstrap estimator of Expected Error Rate Difference) لمعرفة أيهما أفضل في التصنيف, ولقد تمت الدراسة التطبيقية علي مجموعة من بيانات مرضي الفشل الکلوي المتاحة بوحدة أمراض الکلي والغسيل الکلوي بمستشفي الأطفال الجامعي (جامعة المنصورة) وقد توصلت الدراسه إلى أن أسلوب C-H هو الأفضل . This study aims to comparing two discriminant analysis methods, namely the linear combination classifier of Chung and Hun (C-H classifier), which is a linear combination of two discriminant functions, one based on the complete observations and the other based on the incomplete observations. Also, the Maximum Likelihood Estimation substitution classifier is the general rule of discrimination based on parameters estimators via MLE estimator. It will be assumed that there are two populations are multivariate normal with equal covariance matrix; one of them is with the same monotone pattern. We consider the problem of classifying a observation into one of two population. We examine the two classifiers to know which is better in the classification by using a parametric bootstrap estimator of the Expected Error Rate Differences, The applied study was done on a set of data of patients with kidney failure available in the Kidney Diseases and Kidney Dialysis Unit at the University Children's Hospital (Mansoura University) the result shows that the C-H classifier is more efficiency to the MLE classifier when the proportion of observations with missing data is substantial.

A Novel Model Using Virtual State Variables and Bayesian Discriminant Analysis to Classify Surrounding Rock Stability

To accurately classify the stability of surrounding rock masses, a novel method (VSV-BDA) based on virtual state variables (VSVs) and Bayesian discriminant analysis (BDA) is proposed. The factors influencing stability are mapped by an artificial neural network (ANN) capable of recognizing the model of rock mass classification, and the obtained output vector is treated as VSVs, which are verified as obeying a multinormal distribution with equal covariance matrixes by normal distribution testing and constructed statistics. The prediction variance ratio test method is introduced to determine the optimal dimension of the VSVs. The VSV-BDA model is constructed through the use of VSVs and the optimal dimension on the basis of the training samples, which are divided from the collected samples into three situations having different numbers. ANN and BDA models are also constructed based on the same training samples. The predictions by the three models for the testing samples are compared; the results show that the proposed VSV-BDA model has high prediction accuracy and can be applied in practical engineering.

Quantitative Decision Under Unequal Covariances and Post-Treatment Variances: A Kidney Disease Application

Proof-of-concept (PoC) trials enable sponsors decide whether to continue or discontinue a compound’s development based on preliminary evidence of safety and efficacy. Many accounts exist on using quantitative approaches for this decision. Still, these accounts are devoid of quantitative decisions under unequal covariances and post-treatment variances for continuous endpoints. Filling this void is important because randomization to treatment groups only guarantees equal variances before treatment initiation. In this article, I show how unequal covariances and post-treatment variances affect the consider zone when using a three-outcome frequentist decision framework. Results herein suggest that the consider zone decreases with increase in correlation between baseline and post-baseline measurements for any sample size. Compared to decrease in consider zone under equal covariances and post-treatment variances, decrease in consider zone was greater under unequal covariances and post-treatment variances. The results suggest that ignoring a baseline covariate correlated with an outcome robs the study team of an opportunity to effectively minimize uncertain decision in PoC trial scoped to support quantitative decisions. Used for illustration is design of a PoC trial in kidney disease.

Residual risk stratification of Taiwanese breast cancers following curative therapies with the extended concurrent genes signature.

The aim of the study was to perform digital RNA counting to validate a gene expression signature for operable breast cancers initially treated with curative intention, and the risk of recurrence, distant metastasis, and mortality was predicted. Candidate genes were initially discovered from the coherent genomic and transcriptional alternations from microarrays, and the extended concurrent genes were used to build a risk stratification model from archived formalin-fixed paraffin-embedded (FFPE) tissues with the NanoString nCounter. The extended concurrent genes signature was prognostic in 144 Taiwanese breast cancers (5-year relapse-free survival: 89.8 and 69.4% for low- and high-risk group, log-rank test: P = 0.004). Cross-platform comparability was evidenced from significant and positive correlations for most genes as well as equal covariance matrix across 64 patients assayed for both microarray and digital RNA counting. Archived FFPE samples could be successfully assayed by the NanoString nCounter. The purposed signature was prognostic stratifying breast cancer patients into groups with distinct survival patterns, and clinical applicability of the residual risk model was proved.

A comprehensive treatment of quadratic-form-based inference in repeated measures designs under diverse asymptotics

Split-Plot or Repeated Measures Designs with multiple groups occur naturally in sciences. Their analysis is usually based on the classical Repeated Measures ANOVA. Roughly speaking, the latter can be shown to be asymptotically valid for large sample sizes ni assuming a fixed number of groups a and time points d. However, for high-dimensional settings with d>ni, this argument breaks down and statistical tests are often based on (standardized) quadratic forms. Furthermore, analysis of their limit behaviour is usually based on certain assumptions on how d converges to ∞ with respect to ni. As this may be hard to argue in practice, we do not want to make such restrictions. Moreover, sometimes also the number of groups a may be large compared to d or ni. To also have an impression about the behaviour of (standardized) quadratic forms as test statistic, we analyze their asymptotics under diverse settings on a, d and ni. In fact, we combine all kinds of combinations, where they diverge or are bounded in a unified framework. To this aim, we assume equal covariance matrices between all groups. Studying the limit distributions in detail, we follow Sattler and Pauly (2018) and propose an approximation to obtain critical values. The resulting test and its approximation approach are investigated in an extensive simulation study focusing on the exceptional asymptotic frameworks that are the main focus of this work.

L1-norm unsupervised Fukunaga-Koontz transform

The Fukunaga-Koontz transform (FKT) is a powerful supervised feature extraction method used in two-class recognition problems, particularly when the classes have equal mean vectors but different covariance matrices. The present work proves that it is also possible to perform the FKT in an unsupervised manner, sparing the need for labeled data, by using a variant of L1-norm Principal Component Analysis (L1-PCA) that minimizes the L1-norm in the feature space. Rigorous proof is given in the case of data drawn from a mixture of Gaussians. A working iterative algorithm based on gradient-descent in the Stiefel manifold is put forward to perform L1-norm minimization with orthogonal constraints. A number of numerical experiments on synthetic and real data confirm the theoretical findings and the good convergence characteristics of the proposed algorithm.

Annotations on the Relationship Among Discriminant Functions

Different forms of discriminant functions and the essence of their appearances were considered in this study. Various forms of classification problems were also considered, and in each of the cases mentioned, classification from simple functions of the observational vector rather than complicated regions in the higher-dimensional space of the original vector were made. Violation of condition of equal variance covariance matrix for Linear Discriminant Function (LDF) results to Quadratic Discriminant Function (QDF). The relationships among the classification statistics examined were established: The Anderson’s (W) and Rao’s (R) statistics are equivalent when the two sample sizes are equal, and when a constant is equal to 1, W, R and John-Kudo’s (Z) classification statistics are asymptotically comparable. A linear relationship is also established between W and Z classification statistics.

Registration of multi-view point sets under the perspective of expectation-maximization.

Multi-view registration plays a critical role in 3D model reconstruction. To solve this problem, most previous methods align point sets by either partially exploring available information or blindly utilizing unnecessary information, which may lead to undesired results or extra computation complexity. Accordingly, we propose a novel solution for the multi-view registration under the perspective of Expectation-Maximization (EM). The proposed method assumes that each data point is generated from one unique Gaussian Mixture Model (GMM), where its corresponding points in other point sets are regarded as Gaussian centroids with equal covariance and membership probabilities. As it is difficult to obtain real corresponding points in the registration problem, they are approximated by the nearest neighbor in each other aligned point sets. Based on this assumption, it is reasonable to define the likelihood function including all rigid transformations, which require to be estimated for multi-view registration. Subsequently, the EM algorithm is derived to estimate rigid transformations with one Gaussian covariance by maximizing the likelihood function. Since the GMM component number is automatically determined by the number of point sets, there is no trade-off between registration accuracy and efficiency in the proposed method. Finally, the proposed method is tested on several benchmark data sets and compared with state-of-the-art algorithms. Experimental results demonstrate its superior performance on the accuracy, efficiency, and robustness for multi-view registration.

Annotations on the Relationship Among Discriminant Functions

Different forms of discriminant functions and the essence of their appearances were considered in this study. Various forms of classification problems were also considered, and in each of the cases mentioned, classification from simple functions of the observational vector rather than complicated regions in the higher-dimensional space of the original vector were made. Ever since the emergence of the Linear Discriminant Function (LDF) by Fisher, several other classification statistics have emerged and violation of condition of equal variance covariance matrix for Linear Discriminant Function (LDF) results to Quadratic Discriminant Function (QDF). While the Best Linear Discriminant Function (BLDF) is referred to Best Sample Discriminant Function (BSDF) when the parameters are estimated from a sample and also optimal in the same sense as Quadratic Discriminant Function (QDF), Rao statistic is best for discriminating between options that are close each other. The relationships among the classification statistics examined were established: Among the methods of classification statistics considered, Anderson’s (W) and Rao’s (R) statistics are equivalent when the two sample sizes n₁ and n₂ are equal, and when a constant is equal to 1, W, R and John-Kudo’s (Z) classification statistics are asymptotically comparable. A linear relationship is also established between W and Z classification.

TRANSFORMATION BASED ALGORITHMS FOR CHANGE DETECTION IN FULL POLARIMETRIC REMOTE SENSING IMAGES

Abstract. Thanks to the recent advances in the development of polarimetric synthetic aperture radar (SAR) sensors, this remote sensing field attracts many applications. Among the different applications of these data, change detection is one of the most important applications. PolSAR images, due to interactions between electromagnetic waves and the target, could be used to study changes in the Earth's surface. This paper is a type of transformation-based method for polarimetric change detection (CD) purpose. For this purpose, we use full polarimetry imaging radar and extracted 138 features based on decomposition. The CD methods are the principal component analysis (PCA), the Multivariate Alteration Detection (MAD), the Iteratively Reweighted Multivariate Alteration Detection (IR-MAD), the Covariance Equalization (CE), and the Cross-Covariance (CRC). Assessment of the incorporated methods performed using most common criteria as quantity and quality assessment, such as overall accuracy (OA), kappa coefficient, and as visual analysis. The results of the experiments show that CC has better performance compared with other algorithms.

Performance of Change Detection Algorithms Using Heterogeneous Images and Extended Multi-attribute Profiles (EMAPs)

We present detection performance of ten change detection algorithms with and without the use of Extended Multi-Attribute Profiles (EMAPs). Heterogeneous image pairs (also known as multimodal image pairs), which are acquired by different imagers, are used as the pre-event and post-event images in the investigations. The objective of this work is to examine if the use of EMAP, which generates synthetic bands, can improve the detection performances of these change detection algorithms. Extensive experiments using five heterogeneous image pairs and ten change detection algorithms were carried out. It was observed that in 34 out of 50 cases, change detection performance was improved with EMAP. A consistent detection performance boost in all five datasets was observed with EMAP for Homogeneous Pixel Transformation (HPT), Chronochrome (CC), and Covariance Equalization (CE) change detection algorithms.