Inter-event Times Statistic in Stationary Processes: Nonlinear ARMA Modeling of Wind Speed Time Series

The random sequence of inter-event times of a level-crossing is a statistical tool that can be used to investigate time series from complex phenomena. Typical features of observed series as the skewed distribution and long range correlations are modeled using non linear transformations applied to Gaussian ARMA processes. We investigate the distribution of the inter-event times of the level-crossing events in ARMA processes in function of the probability corresponding to the level. For Gaussian ARMA processes we establish a representation of this indicator, prove its symmetry and that it is invariant with respect to the application of a non linear monotonic transformation. Using simulated series we provide evidence that the symmetry disappears if a non monotonic transformation is applied to an ARMA process. We estimate this indicator in wind speed time series obtained from three different databases. Data analysis provides evidence that the indicator is non symmetric, suggesting that only highly non linear transformations of ARMA processes can be used in modeling. We discuss the possible use of the inter-event times in the prediction task.

Asymptotics and Quotient Spaces of Solutions of Operator-Difference Equations and Differential Equations with Small Delay

Formerly, in order to conduct the in-depth study of differential equations with delay, the author proposed the method of splitting the solution space reducing such equations to the systems of operator-difference equations. Using this method, the author assumed new conditions, i.e. the absolute domains for coefficients sufficient for the existence of special (slowly changing) solutions, and proved the presence of approximating and asymptotically approximating properties in them, as well as the asymptotic one-dimensional space of solutions of the initial problems for linear scalar differential equations with insignificantly retarded argument and the corresponding operator-difference equation systems (special solutions correspond, to the solutions with a slowly changing first component and a relatively small second component). For the purposes of the single-point representation of the obtained results and other data related to the theory of dynamic systems (the distance between the solution values tends to zero alongside the unlimited increase in argument), throughout this research paper the author uses the concept of the asymptotic equivalence of solutions for dynamic systems, as it was introduced by the author in their previous research. In order to shape the new mathematical objects, the concept of asymptotic Hausdorff equivalence of solutions for dynamic systems is introduced (the distance between solution values tends to zero with unlimited increase in argument of one solution and monotonic transformation of argument of another solution).

Subspace clustering using ensembles of K-subspaces

Subspace clustering is the unsupervised grouping of points lying near a union of low-dimensional linear subspaces. Algorithms based directly on geometric properties of such data tend to either provide poor empirical performance, lack theoretical guarantees or depend heavily on their initialization. We present a novel geometric approach to the subspace clustering problem that leverages ensembles of the $K$-subspace (KSS) algorithm via the evidence accumulation clustering framework. Our algorithm, referred to as ensemble $K$-subspaces (EKSSs), forms a co-association matrix whose $(i,j)$th entry is the number of times points $i$ and $j$ are clustered together by several runs of KSS with random initializations. We prove general recovery guarantees for any algorithm that forms an affinity matrix with entries close to a monotonic transformation of pairwise absolute inner products. We then show that a specific instance of EKSS results in an affinity matrix with entries of this form, and hence our proposed algorithm can provably recover subspaces under similar conditions to state-of-the-art algorithms. The finding is, to the best of our knowledge, the first recovery guarantee for evidence accumulation clustering and for KSS variants. We show on synthetic data that our method performs well in the traditionally challenging settings of subspaces with large intersection, subspaces with small principal angles and noisy data. Finally, we evaluate our algorithm on six common benchmark datasets and show that unlike existing methods, EKSS achieves excellent empirical performance when there are both a small and large number of points per subspace.

Extracting Job Title Hierarchy from Career Trajectories: A Bayesian Perspective

A job title usually implies the responsibility and the rank of a job position. While traditional job title analysis has been focused on studying the responsibilities of job titles, this paper attempts to reveal the rank of job titles. Specifically, we propose to extract job title hierarchy from employees' career trajectories. Along this line, we first quantify the Difficulty of Promotion (DOP) from one job title to another by a monotonic transformation of the length of tenure based on the assumption that a longer tenure usually implies a greater difficulty to be promoted. Then, the difference of two job title ranks is defined as a mapping of the DOP observed from job transitions. A Gaussian Bayesian Network (GBN) is adopted to model the joint distribution of the job title ranks and the DOPs in a career trajectory. Furthermore, a stochastic algorithm is developed for inferring the posterior job title rank by a given collection of DOPs in the GBN. Finally, experiments on more than 20 million job trajectories show that the job title hierarchy can be extracted precisely by the proposed method.

Unified Least Squares Methods for the Evaluation of Diagnostic Tests With the Gold Standard

The article proposes a unified least squares method to estimate the receiver operating characteristic (ROC) parameters for continuous and ordinal diagnostic tests, such as cancer biomarkers. The method is based on a linear model framework using the empirically estimated sensitivities and specificities as input “data.” It gives consistent estimates for regression and accuracy parameters when the underlying continuous test results are normally distributed after some monotonic transformation. The key difference between the proposed method and the method of Tang and Zhou lies in the response variable. The response variable in the latter is transformed empirical ROC curves at different thresholds. It takes on many values for continuous test results, but few values for ordinal test results. The limited number of values for the response variable makes it impractical for ordinal data. However, the response variable in the proposed method takes on many more distinct values so that the method yields valid estimates for ordinal data. Extensive simulation studies are conducted to investigate and compare the finite sample performance of the proposed method with an existing method, and the method is then used to analyze 2 real cancer diagnostic example as an illustration.

Surface Inspection Using Computer Vision and Gradient Spectrum

This paper is concerned with the problem of automatic inspection of hot-rolled plate surface using computer vision. An automated visual inspection system has been developed to take images of external hot-rolled plate surfaces and an intelligent surface defect detection paradigm based on gradient spectrum technique is presented. Gradient spectrum characterizes the spatial configuration of local image texture and is robust against any monotonic transformation of the gray scale. Texture features based on gradient spectrum are extracted from ROI in hot-rolled plate surface images and integrated into a feature vector which uniquely differentiates the abnormal regions from normal surface. Classification accuracies using the gradient spectrum and gradient-based method are compared. The results indicate that gradient spectrum performs well in classifying the samples with the lowest classification error.

Investigating Local Dependence With Conditional Covariance Functions

The local dependence of item pairs is investigated via a conditional covariance function estimation procedure. The conditioning variable used in the procedure is obtained by a monotonic transformation of total score on the remaining items. Intuitively, the conditioning variable corresponds to the unidimensional latent ability that is best measured by the test. The conditional covariance functions are estimated using kernel smoothing, and a standardization to adjust for the confounding effect of item difficulty is introduced. The particular standardization chosen is an adaptation of Yule’s coefficient of colligation. Several models of local dependence are discussed to explain special situations, such as speededness and latent space multidimensionality, in which the assumptions of unidimensionality and local independence are violated.

On the Sensitivity of a Regression Coefficient to Monotonic Transformations

This note presents a procedure which enables the user to check whether a monotonic transformation can change the sign of a regression coefficient. One possible use of this procedure is to examine the robustness of key regression coefficients.