Regression for non-Euclidean data using distance matrices

Regression methods for common data types such as measured, count and categorical variables are well understood but increasingly statisticians need ways to model relationships between variable types such as shapes, curves, trees, correlation matrices and images that do not fit into the standard framework. Data types that lie in metric spaces but not in vector spaces are difficult to use within the usual regression setting, either as the response and/or a predictor. We represent the information in these variables using distance matrices which requires only the specification of a distance function. A low-dimensional representation of such distance matrices can be obtained using methods such as multidimensional scaling. Once these variables have been represented as scores, an internal model linking the predictors and the responses can be developed using standard methods. We call scoring as the transformation from a new observation to a score, whereas backscoring is a method to represent a score as an observation in the data space. Both methods are essential for prediction and explanation. We illustrate the methodology for shape data, unregistered curve data and correlation matrices using motion capture data from an experiment to study the motion of children with cleft lip.     

On the measurement and analysis of asymmetry with applications to facial modelling

Summary.  Asymmetry is a feature of shape which is of particular interest in a variety of applications. With landmark data, the essential information on asymmetry is contained in the degree to which there is a mismatch between a landmark configuration and its relabelled and matched reflection. This idea is explored in the context of a study of facial shape in infants, where particular interest lies in identifying changes over time and in assessing residual deformity in children who have had corrective surgery for a cleft lip or cleft lip and palate. Interest lies not in whether the mean shape is asymmetric but in comparing the degrees of asymmetry in different populations. A decomposition of the asymmetry score into components that are attributable to particular features of the face is proposed. A further decomposition allows different sources of asymmetry due to position, orientation or intrinsic asymmetry to be identified for each feature. The methods are also extended to data representing anatomical curves across the face.     

Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum–Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented.     

基于B-样条基底展开的曲线聚类方法

随着大数据时代的来临,近年来函数型数据分析方法成为研究的热点问题,针对曲线的聚类分析方法引起了学界的关注.给出一种曲线聚类的方法:以L2距离作为亲疏程度的度量,在B样条基底函数展开表述下,将曲线本身信息、曲线变化信息引入聚类算法构建,并实现了曲线聚类与传统多元统计聚类方法的对接.作为应用,以城乡收入函数聚类实例验证了该曲线聚类方法,结果表明,在引入曲线变化信息的情况下,比仅考虑曲线本身信息能够取得更好的聚类效果.     

On Random and Gibbsian Particle Motions for Point Processes Evolving in Space and Time

This article presents an analysis of space-time interdependencies of spatial point processes considering random and deterministic Gibbsian point motions caused by repulsion effects between particles. Two deterministic models of Gibbsian motions are considered by formulating a constant (i.e., Strauss-like) and a linear interaction motion functions. Given that theoretical development of continuous space-time stochastic processes are mathematically intractable, we have mainly based our analysis on numerical simulations. Our results suggest that to fully understand such complex dynamics, the analysis of purely spatial patterns should be combined with their interactions in the space-time domain. Otherwise, analysis of pure spacial patterns may not fully explain the real mechanism generating such dynamical configurations. We highlight that adding movement to sedentary points opens new areas of application and research to study biological phenomena, where particles not only evolve through time but also can change spatial positions in terms of their neighbor locations.     

Fitting Curves to Human Respiratory Data

Records of gas flow during breathing are cyclical, with the cycles varying in duration. The shape of these cycles may change with the intensity of respiratory stimulation or the development of respiratory disease, but currently research is hampered by the lack of a fully satisfactory technique for determining the shape of a typical cycle. The approach adopted here is to replace the time series by a 'phase diagram', plotting the time integral of flow against flow itself. Principal curves are then fitted. These are curves `through the middle of the data' which were introduced by Hastie and Stuetzle. The shapes of these curves are compared, either directly or after reconstructing an average cycle corresponding to each fitted curve. This has the advantage that the shape of the waveform is separated from the amplitude, and from the duration of the breath. A disadvantage is that periods of zero flow are lost, and the reconstructed average cycle may show irregularities at points near zero flow as a result. In practice, the methodology showed clear differences in shape between the protocols, gave reasonable average cycles and ordered the waveform shapes according to the hardness of breathing induced by the protocols.     

Familial spinal shape

It has been proposed that there is a familial relationship in the shape of the spine. This paper describes a pilot study investigating familial shape in the sagittal plane (side view), using three data sets of normal Leeds schoolchildren. The study is exploratory in nature, because only small samples were available. Data acquisition was by means of the Quantec system, which obtains surface shape measurements and extracts a line representing the spinal curve. The coordinates of the spine line in the sagittal plane are then used to investigate familial correlations of spinal shape. The spine lines first undergo some preprocessing, including Procrustes rotations to remove location, rotation and size effects. Smoothed principal component analysis of the curves provides suitable shape variables, and familial correlations between curves are then investigated. The covariates of sex and height are also investigated in the analysis. It does appear that there could be some evidence for familial correlations in sagittal spinal shape, although a further large-scale study is required. Finally, a discussion of the approach and other alternatives is considered.     

Parametric fractional imputation for nonignorable missing data

Parameter estimation with missing data is a frequently encountered problem in statistics. Imputation is often used to facilitate the parameter estimation by simply applying the complete-sample estimators to the imputed dataset.In this article, we consider the problem of parameter estimation with nonignorable missing data using the approach of parametric fractional imputation proposed by Kim (2011). Using the fractional weights, the E-step of the EM algorithm can be approximated by the weighted mean of the imputed data likelihood where the fractional weights are computed from the current value of the parameter estimates. Calibration fractional imputation is also considered as a way for improving the Monte Carlo approximation in the fractional imputation. Variance estimation is also discussed. Results from two simulation studies are presented to compare the proposed method with the existing methods. A real data example from the Korea Labor and Income Panel Survey (KLIPS) is also presented.     

Finite sample distributions of statistics in change point analysis

This paper is concerned with derivation of finite sampling distributions of some statistics which appear frequently in change point analysis. The exact distribution of cusum test statistic is approximated by two methods. Approximations are presented and their accuracies are measured. We first consider the change point in mean problem and we study the exact distribution of change point estimator. Finally, we consider the change point in variance case.     

Asymptotic Methods of Enumeration and Applications to Markov Chain Models

Abstract

The purpose of this article is to present analytic methods for determining the asymptotic behaviour of the coefficents of power series that can be applied to homogeneous discrete quasi death and birth processes. It turns that there are in principle only three types for the asymptotic behaviour. The process either converges to the stationary distribution or it can be approximated in terms of a reflected Brownian motion or by a Brownian motion. In terms of Markov chains these cases correspond to positive recurrence, to null recurrence, and to non recurrence. The same results hold for the continuous case, too.     

Classification of biomedical signals for differential diagnosis of Raynaud's phenomenon

This paper discusses a supervised classification approach for the differential diagnosis of Raynaud's phenomenon (RP). The classification of data from healthy subjects and from patients suffering for primary and secondary RP is obtained by means of a set of classifiers derived within the framework of linear discriminant analysis. A set of functional variables and shape measures extracted from rewarming/reperfusion curves are proposed as discriminant features. Since the prediction of group membership is based on a large number of these features, the high dimension/small sample size problem is considered to overcome the singularity problem of the within-group covariance matrix. Results on a data set of 72 subjects demonstrate that a satisfactory classification of the subjects can be achieved through the proposed methodology.     

Adaptive threshold method for peak detection of surface electromyography signal from around shoulder muscles

This paper illustrates the accurate identification of the surface electromyography signal obtained from the shoulder muscles (Teres, Trapezius and Pectoralis) of amputee subjects with three different arm motions (elevation, protraction and retraction). During the acquisition of the signal, a variety of variations (amplitude, frequency and noise) were introduced into the acquired signal which will misguide in the prediction of motion of the shoulder. Therefore, a novel approach has been aimed to adaptively adjust the threshold of Teager energy operator in order to filter the unwanted peaks in the pre-processing stage of the surface electromyography (SEMG) signal. Results show that the proposed approach is accurate and effective in the analysis of biomedical signal where peaks are important to detect without the knowledge of the shape of the waveform. As clinical research continues, these algorithms helps us to process SEMG signal and the identified signal would be used to design more accurate and efficient controllers for the upper-limb amputee.     

Local functional data model for characterizing shape variation of similar curves

“Similar curves” in the present article refers to a family of curves whose major shape are similar, but who have variation coming from curve-specific sources. The goal here is to develop a general methodology to describe small changes among similar curves. Previous methods mainly focus on dimension reduction through FPCA, which are not appropriate for quantifying local variation. Here, we consider a local functional data model which divides data into segments adaptively and models each segment with a shape invariant model. Such model has great flexibility in characterizing local variation of curves, as illustrated by simulation and real data examples.     

Oscillation analysis using optical flow: Application to an EUV movie of the solar corona

Movies of the solar corona in Extreme UltraViolet (EUV) bandpasses exhibit complex patterns of magnetically structured plasma features surrounding the solar photosphere. Among the various phenomena to be observed in the EUV movies, coronal oscillations are an essential process for determining physical parameters of the plasma. In this paper we demonstrate the ability of our motion estimation algorithm to explore and analyse the oscillating motions of coronal loops present in EUV image sequences. The motion fields of each image pair in the sequence are estimated; selected features are tracked using the motion estimation to form trajectories. The oscillating features are then selected from the Morlet wavelet analysis of the trajectories that provides parameters such as local oscillation period. The proposed method will be particularly useful to process datasets expected from new solar missions.     

Checking the censored two-sample accelerated life model using integrated cumulative hazard difference

In this paper, new statistical tests for the censored two-sample accelerated life model are discussed. From the estimating functions using integrated cumulative hazard difference, stochastic processes are constructed. They can be described by martingale residuals, and, given the data, conditional distributions can be approximated by zero mean Gaussian processes. The new methods, based on these processes, provide asymptotically consistent tests against a general departure from the model. A graphical method is also discussed. In various numerical studies, the new tests performed better than the existing method, especially when the hazard curves cross. The proposed procedures are illustrated with two real data sets.     

On limiting likelihood ratio processes of some change-point type statistical models

Different change-point type models encountered in parametric statistical inference give rise to different limiting likelihood ratio processes. In this paper we consider two such likelihood ratios. The first one is an exponential functional of a two-sided Poisson process driven by some parameter, while the second one is an exponential functional of a two-sided Brownian motion. We establish that for sufficiently small values of the parameter, the Poisson type likelihood ratio can be approximated by the Brownian type one. As a consequence, several statistically interesting quantities (such as limiting variances of different estimators) related to the first likelihood ratio can also be approximated by those related to the second one. Finally, we discuss the asymptotics for large values of the parameter and illustrate the results by numerical simulations.     

Bayesian regression under combinations of constraints

A Bayesian method for regression under several types of constraints is proposed. The constraints can be range-restricted and include shape restrictions, constraints on the value of the regression function, smoothness conditions and combinations of these types of constraints. The support of the prior distribution is included in the set of piecewise linear functions. It is shown that the proposed prior can be arbitrarily close to the distribution induced by the addition of a polynomial plus an (m−1)-fold integrated Brownian motion. Hence, despite its piecewise linearity, the regression function behaves (approximately) like an m−1 times continuously differentiable random function. Furthermore, thanks to the piecewise linear property, many combinations of constraints can easily be considered. The regression function is estimated by the posterior mode computed by a simulated annealing algorithm. The constraints on the shape and the values of the regression function are taken into account thanks to the proposal distribution, while the smoothness condition is handled by the acceptation step. Simulations from the posterior distribution are obtained by a Gibbs sampling algorithm.     

Approximations for weighted Kolmogorov–Smirnov distributions via boundary crossing probabilities

A statistical application to Gene Set Enrichment Analysis implies calculating the distribution of the maximum of a certain Gaussian process, which is a modification of the standard Brownian bridge. Using the transformation into a boundary crossing problem for the Brownian motion and a piecewise linear boundary, it is proved that the desired distribution can be approximated by an n-dimensional Gaussian integral. Fast approximations are defined and validated by Monte Carlo simulation. The performance of the method for the genomics application is discussed.     

Testing for brain anomalies: A hippocampus study

A mathematical classification method is presented to show how numerical tests for abnormal anatomical shape change can be used to study geometrical shape changes of the hippocampus in relation to the occurrence of schizophrenia. The method uses the well-known best Bayesian decision rule for two simple hypotheses. Furthermore, the technique is illustrated by applying the hypothesis testing method to some preliminary hippocampal data. The data pool available for the experiment consisted of 10 subjects, five of whom were diagnosed with schizophrenia and five of whom were not schizophrenics. Even though the information used in the experiment is limited and the number of subjects is relatively small, we are confident that the mathematical classification method presented is of significance and can be used successfully, given proper data, as a diagnostic tool.