New statistical control limits using maximum copula entropy

Statistical quality control methods are noteworthy to produced standard production in manufacturing processes. In this regard, there are many classical manners to control the process. Many of them have a global assumption around distributions of the process data. They are supposed to be normal, which is clear that it is not always valid for all processes. Such control charts made some false decisions that waste funds. So, the main question while working with multivariate data set is how to find the multivariate distribution of the data set, which saves the original dependency between variables. Up to our knowledge, a copula function guarantees the dependence on the result function. But it is not enough when there is no other functional information about the statistical society, and we have just a data set. Therefore, we apply the maximum entropy concept to deal with this situation. In this paper, first of all, we find out the joint distribution of a data set, which is from a manufacturing process that needs to be control while running the production process. Then, we get an elliptical control limit via the maximum copula entropy. In the final step, we represent a practical example using the stated method. Average run lengths are calculated for some means and shifts to show the ability of the maximum copula entropy. In the end, two real data examples are presented

A COMPARISON BETWEEN DATA-DRIVEN AND PHYSICS OF FAILURE PHM APPROACHES FOR SOLDER JOINT FATIGUE

Prognostics and systems health management technology is an enabling discipline of technologies and methods with the potential of solving reliability problems that have been manifested due to complexities in design, manufacturing, environmental and operational use conditions, and maintenance. Over the past decade, research has been conducted in PHM to provide benefits such as advance warning of failures, enable forecasted maintenance, improve system qualification, extend system life, and diagnose intermittent failures that can lead to field failure returns exhibiting no-fault-found symptoms. While there are various methods to perform prognostics, including model-based and data-driven methods, these methods have some key disadvantages. This thesis presents a fusion prognostics approach, which combines or ―fuses together‖ the model based and data-driven approaches, to enable increasingly better estimates of remaining useful life. A case study using an electronics system to illustrate a step by step implementation of the fusion approach is also presented. The various benefits of the fusion approach and suggestions for future work are included

Sublinear Computation Paradigm

This open access book gives an overview of cutting-edge work on a new paradigm called the “sublinear computation paradigm,” which was proposed in the large multiyear academic research project “Foundations of Innovative Algorithms for Big Data.” That project ran from October 2014 to March 2020, in Japan. To handle the unprecedented explosion of big data sets in research, industry, and other areas of society, there is an urgent need to develop novel methods and approaches for big data analysis. To meet this need, innovative changes in algorithm theory for big data are being pursued. For example, polynomial-time algorithms have thus far been regarded as “fast,” but if a quadratic-time algorithm is applied to a petabyte-scale or larger big data set, problems are encountered in terms of computational resources or running time. To deal with this critical computational and algorithmic bottleneck, linear, sublinear, and constant time algorithms are required. The sublinear computation paradigm is proposed here in order to support innovation in the big data era. A foundation of innovative algorithms has been created by developing computational procedures, data structures, and modelling techniques for big data. The project is organized into three teams that focus on sublinear algorithms, sublinear data structures, and sublinear modelling. The work has provided high-level academic research results of strong computational and algorithmic interest, which are presented in this book. The book consists of five parts: Part I, which consists of a single chapter on the concept of the sublinear computation paradigm; Parts II, III, and IV review results on sublinear algorithms, sublinear data structures, and sublinear modelling, respectively; Part V presents application results. The information presented here will inspire the researchers who work in the field of modern algorithms

On scalable inference and learning in spike-and-slab sparse coding

Sparse coding is a widely applied latent variable analysis technique. The standard formulation of sparse coding assumes Laplace as a prior distribution for modeling the activations of latent components. In this work we study sparse coding with spike-and-slab distribution as a prior for latent activity. A spike-and-slab distribution has its probability mass distributed across a ’spike’ at zero and a ’slab’ spreading over a continuous range. For its capacity to induce exact zeros with a higher likelihood, a spike-and-slab prior distribution constitutes a more accurate model of sparse coding. The distribution as a prior also allows for the sparseness of latent activity to be directly inferred from observed data, which essentially makes spike-and-slab sparse coding more flexible and self-adaptive to a wide range of data distributions. By modeling the slab with a Gaussian distribution, we furthermore show that in contrast to the standard approach to sparse coding, we can indeed derive closed-form analytical expressions for exact inference and learning in linear spike-and-slab sparse coding. However, as the posterior landscape of a spike-and-slab prior turns out to be highly multi-modal with a prohibitive exploration cost, in addition to the exact method, we also develop subspace and Gibbs sampling based approximate inference techniques for scalable applications of the linear model. We contrast our approximation methods with variational approximation for scalable posterior inference in linear spike-and-slab sparse coding. We further combine the Gaussian spike-and-slab prior with a nonlinear generative model, which assumes a point-wise maximum combination rule for the generation of observed data. We analyze the model as a precise encoder of low-level features such as edges and their occlusions in visual data. We again combine subspace selection with Gibbs sampling to overcome the analytical intractability of performing exact inference in the model. We numerically analyze our methods on both synthetic and real data for their verification and comparison with other approaches. We assess the linear spike-and-slab approach on source separation and image denoising benchmarks. In most experiments we obtain competitive or state-of-the-art results, while we find that spike-and-slab sparse coding overall outperforms other comparable approaches. By extracting thousands of latent components from a large amount of training data we further demonstrate that our subspace Gibbs sampler is among the most scalable posterior inference methods for a linear sparse coding approach. For the nonlinear model we experiment with artificial and real images to demonstrate that the components learned by the model lie closer to the ground-truth and are easily interpretable as the underlying generative causes of the input. We find that in comparison to standard sparse coding, the nonlinear spike-and-slab approach can compressively encode images using naturally sparse and discernible compositions of latent components. We also demonstrate that the components inferred by the model from natural image patches are statistically more consistent with respect to their structure and distribution to the response patterns of simple cells in the primary visual cortex of the brain. This work thereby contributes novel methods for sophisticated inference and learning in spike-and-slab sparse coding, while it also empirically showcases their functional efficacy through a variety of applications.Sparse Coding ist eine weit verbreitete Technik der latenten Variablenanalyse. Die Standardformulierung von Sparse Coding setzt a priori eine Laplace-Verteilung zur Modellierung der Aktivierung von latenten Komponenten voraus. In dieser Arbeit untersuchen wir Sparse Coding mit einer a priori Spike-and-Slab-Verteilung für latente Aktivität. Eine Spike-and-Slab-Verteilung verteilt ihre Wahrscheinlichkeitsmasse um ein Aktionspotential (“Spike”) um Null und eine dicke Verteilung (“slab”) über einen kontinuierlichen Wertebereich. Durch die Induktion von exakten Nullen mit einer höheren Wahrscheinlichkeit erzeugt eine Apriori-Spike-and-Slab-Verteilung ein genaueres Modell von Sparse Coding. Als A-priori-Verteilung erlaubt sie es uns die Seltenheit von latenten Komponenten direkt von Daten abzuleiten, sodass ein Spike-and-Slab-getriebenes Modell von Sparse Coding sich besser verschiedensten Verteilungen von Daten anpasst. Durch das Modellieren des Slab mittels einer Gauß-Verteilung zeigen wir, dass – im Gegensatz zur Standardformulierung von Sparse Coding – wir in der Tat geschlossene analytische Ausdrücke ableiten können, um eine exakte Ableitung und das Lernen eines linearen Spike-and-Slab-Sparse-Coding-Modell durchzuführen. Weil eine Spike-and-Slab-A-priori-Verteilung zu einer hoch multimodalen A-posteriori-Landschaft mit viel zu hohen Suchkosten führt, entwickeln wir zusätzlich zur exakten Methode Näherungslösungen basierend auf einem Teilraum und Gibbs-Sampling für skalierbare Anwendungen des Modells. Wir vergleichen unseren Ansatz der näherungsweisen Inferenz mit näherungsweiser Variationsrechnung des linearen Spike-and-Slab-Sparse Coding. Des Weiteren kombinieren wir die Spike-and-Slab-A-priori-Verteilung mit einem nicht-linearen Sparse-Coding-Modell, das eine punktweise Maximum-Kombinationsregel zur Datengenerierung voraussetzt. Wir analysieren das Modell als genauen Kodierer von untergeordneten Merkmalen in Bildern wie z.B. Kanten und deren Okklusionen. Wir lösen die analytische Ausweglosigkeit, eine Ableitung von multimodalen A-posteriori-Verteilungen im Modell durchzuführen, durch die Kombination von Gibbs-Sampling und der Auswahl eines Teilraums, um eine skalierbare Prozedur für die approximative Inferenz des Modells zu entwickeln. Wir analysieren unsere Methode numerisch durch synthetische und wirkliche Daten zum Nachweis und Vergleich mit anderen Ansätzen. Wir bewerten den linearen Spike-and-Slab-Ansatz mittels Maßstäben für die Quellentrennung und zur Rauschunterdrückung in Bildern. In den meisten Experimenten erhalten wir vergleichsweise oder die beste Resultate. Gleichzeitig finden wir, dass Spike-and-Slab-Sparse-Coding insgesamt andere vergleichbare Ansätze übertrifft. Durch die Extraktion von Tausenden von latenten Komponenten aus einer riesigen Menge an Trainingsdaten zeigen wir des Weiteren, dass unserer Teilraum Gibbs-Sampler zu den skalierbarsten Inferenzmethoden der linearen Sparse-Coding-Modelle gehört. Für das nichtlineare Modell experimentieren wir mit künstlichen und echten Bildern zur Demonstration, dass die von dem Modell gelernten Komponenten näher an der “Ground Truth” liegen und leichter zu interpretieren sind als die zugrundeliegenden generierenden Einflüsse der Eingabe. Wir finden, dass – im Vergleich zu Standard-Sparse-Coding – der nichtlineare Spike-and-Slab-Ansatz Bilder komprimierend kodieren kann durch natürliche dünnbesetzte und klar erkennbare Kompositionen von latenten Komponenten. Wir zeigen auch, dass die vom Modell abgeleiteten Komponenten von natürlichen Bildern statistisch konsistenter sind in ihrer Struktur und Verteilung mit dem Antwortmuster von einfachen Zellen im primären visuellen Kortex. Diese Arbeit leistet durch neue Methoden zur komplexen Inferenz und zum Erlernen ivvon Spike-and-Slab-Sparse-Coding einen Beitrag und demonstriert deren praktikable Wirksamkeit durch einen Vielzahl von Anwendungen