Gerhard Pöppel

Towards unique solutions of non-negative matrix factorization problems by a determinant criterion

We propose a determinant criterion to constrain the solutions of non-negative matrix factorization problems and achieve unique and optimal solutions in a general setting, provided an exact solution exists. We demonstrate how optimal solutions are obtained by a heuristic named detNMF in an illustrative example and discuss the difference to sparsity constraints. Furthermore, an intuitive explanation of multi-layer techniques is discussed also.

A Nonnegative Blind Source Separation Model for Binary Test Data

A novel method called binNMF is introduced which aimed to extract hidden information from multivariate binary data sets. The method treats the problem in the spirit of blind source separation: The data are assumed to be generated by a superposition of several simultaneously acting sources or elementary causes which are not observable directly. The superposition process is based on a minimum of assumptions and reversed to identify the underlying sources. The method is motivated, developed, and demonstrated in the context of binary wafer test data which evolve during microchip fabrication.

Projection pursuit for analyzing data from semiconductor environments

Complex environments with complex productions lead to complex data; this certainly applies to the work on semiconductors. Acquiring the data offers a challenge, and the challenge only grows when attempting to analyze the data structures and dependencies. We put forward the well-known Projection Pursuit method; it has apparently not yet been used so far in the production of semiconductors. Extensions of this method are introduced which include robust index functions, new optimizing strategies, and an approach to analyzing the dependencies between different groups of variables.

Binary Nonnegative Matrix Factorization Applied to Semi-conductor Wafer Test Sets

A method of forming a fiber reinforced synthetic resin rod-like molding including a rod-like core portion having at least in the outer portion thereof a reinforcing fiber bundle integrally adhered together by a thermosetting resin and a thermoplastic resin layer coating the core portion is disclosed. The outer surface portion of the core portion and the inner surface portion of the thermoplastic resin layer are integrally adhered together due to an anchor effect generated by contacting between the thermosetting resin and the thermoplastic resin in a semifluid state under pressure.

Bayesian extensions of non-negative matrix factorization

Although non-negative matrix factorization has become a popular data analysis tool for non-negative data sets, there are still some issues remaining partly unsolved. We investigate the potential of Bayesian techniques towards the solution of two important open questions concerning uniqueness and actual number of sources underlying the data. We derive a general Bayesian optimality condition for NMF solutions and elaborate on the criterion for the Gaussian likelihood case. We further derive a variational Bayes NMF algorithm for the Gaussian likelihood using rectified Gaussian prior distributions and study its ability to estimate the true number of sources in a toy data set.

Nonnegative Matrix Factorization for Binary Data to Extract Elementary Failure Maps from Wafer Test Images

We introduce a probabilistic variant of nonnegative matrix factorization (NMF) applied to binary datasets. Hence we consider binary coded images as a probabilistic superposition of underlying continuous-valued basic patterns. An extension of the well-known NMF procedure to binary-valued datasets is provided to solve the related optimization problem with nonnegativity constraints. We demonstrate the performance of our method by applying it to the detection and characterization of hidden causes for failures during wafer processing. Therefore, we decompose binary coded (pass/fail) wafer test data into underlying elementary failure patterns and study their influence on the quality of single wafers.

Robuste Indizes für Projection Pursuit

Zusammenfassung. Die moderne und hoch komplexe Fertigung, z.B. in der Halbleiterindustrie, erfordert für die dabei anfallenden Messdaten der Anlagen- bzw. Prozessparameter multivariate Analysemethoden. Eine dieser möglichen Analysemethoden ist Projection Pursuit (PP). Dieses Verfahren ist durch geschickte Wahl des so genannten Projektionsindex in der Lage, verschiedenste Datencharakteristika zu detektieren und diese auf anschauliche Weise zu visualisieren. Bei den Projektionsindizes handelt es sich um Funktionen, die eine Projektion auf unterschiedliche Merkmale hin bewerten und dabei jeder Projektion einen Funktionswert zuweisen. Dieser Funktionswert spiegelt die Aussagekraft der Projektion (in Abhängigkeit vom verwendeten Index) wider. Die Auswahl bzw. der Aufbau dieser Indizes ist hierbei von entscheidender Bedeutung. Alle hier vorgestellten Indizes zeichnen sich vor allem durch ihren robusten Charakter gegenüber durch Ausreißer kontaminierte Daten, wie sie in realen Datenszenarien vorkommen, aus. Die durch Anwendung dieser Indizes gewonnenen Einblicke in die komplexe Struktur der Daten ermöglichen es, hochdimensionale Parameterabweichungen und komplexe Zusammenhänge innerhalb der Daten zu finden.Abstract. Modern and highly complex production environments e.g. in the semiconductor industry require multivariate analysis methods for the huge amount of equipment and processing data. One of these multivariate methods is Projection Pursuit (PP). This method in combination with a suitable choice of a so-called projection index is able to detect and visualize various characteristics of the data. Projection indices are functions which rate projections of different characteristics and thereby give them a functional value. This functional value reflects the information content of the projection (dependent on the projection index). The choice, or the construction of these indices is therefore of decisive importance. The indices which are presented here are distinguished by their robust characteristics in dealing with outliers, the latter being very common in real datasets. The insight into the complex structure of the underlying data through the usage of these indices makes it possible to find high-dimensional parameter divergence and complex connections within the data.

A Nonnegative Tensor Factorization Approach for Three-Dimensional Binary Wafer-Test Data

We introduce a new Blind Source Separation Approach called binNTF which operates on tensor-valued binary datasets. Assuming that several simultaneously acting sources or elementary causes are generating the observed data, the objective of our approach is to uncover the underlying sources as well as their individual contribution to each observation with a minimum number of assumptions in an unsupervised fashion. We motivate, develop and demonstrate our method in the context of binary wafer test data which evolve during microchip fabrication. In this application, we also have to deal with incomplete datasets which can occur due to the commonly used stop-on-first-fail testing procedure or result from the aggregation of several distinct tests into BIN categories.

From Binary NMF to Variational Bayes NMF: A Probabilistic Approach

A survey of our recent work on probabilistic NMF is provided. All variants discussed here are illustrated by their application to the analysis of failure patterns emerging from manufacturing and processing silicon wafers. It starts with binNMF, a variant developed to apply NMF to binary data sets. The latter are modeled as a probabilistic superposition of a finite number of intrinsic continuous-valued failure patterns characteristic for the manufacturing process. We further discuss related theoretical work on a semi-non-negative matrix factorization based on the logistic function, which we called logistic NMF. While addressing uniqueness issues, we propose a Bayesian Optimality Criterion for NMF and a determinant criterion to geometrically constrain the solutions of NMF problems, leading to detNMF. This approach also provides an intuitive explanation for the often used multilayer approach. Finally, we present a Variational Bayes NMF (VBNMF) algorithm which represents a generalization of the famous Lee–Seung method. We also demonstrate its ability to estimate the intrinsic dimension (model order) of the NMF method.

Localization of temperature sensitive areas on analog circuits

Abstract We introduce a novel easy to apply method to detect critical temperature sensitive areas on analog circuits. Our method is based on heat diffusion on a silicon micro-chip: the corners of a temperature sensitive micro-chip are heated up directly by ESD diodes or infrared laser light. This heat stimulus at the corners results in an inhomogeneous temperature distribution. Thus, the temperature is a function in time and space. The elapsed time to change the chip status from “fail” to “pass” as a reaction to the heat stimulus correlates with the distance to the heat source. This correlation is extracted from COMSOL simulations and experimental results. A numerical program based on that correlation succeeded in localization of the temperature sensitive chip module. Micro-chips affected by corner MOSFETs in the subthreshold regime are used to demonstrate our method.