Extensions of non-negative matrix factorization and their application to the analysis of wafer test data
R. Schachtner · University of Regensburg Publication Server (University of Regensburg) · 2010
This PhD thesis investigates the data analysis technique named non-negative matrix factorization (NMF) and its applicability for failure analysis in microchip production. A blind source separation approach is pursued in which the observable wafer test data is interpreted as the result of a superposition of several simultaneously acting and not directly observable failure causes. This thesis was created during a three year cooperation with the Infineon Technologies AG Regensburg company. The main scientific contributions of this thesis are general theoretic findings and extensions of NMF whose applications are not only restricted to the field of semiconductor industry covered here. In an introductory chapter the basic principle of the NMF method is explained. A data matrix consisting of non-negative entries is approximated by the product of a non-negative basis matrix and a non-negative coefficient matrix. An important, partly unsolved shortcoming of NMF is the existence of several alternative solutions to the same problem. Geometric considerations explain the problem and offer a possible solution to the uniqueness problem. The here suggested determinant criterion renders the set of (normalized) basis vectors to be optimal which span a minimal volume. Simulations by means of a new developed algorithm support this statement and point out the difference between the determinant criterion and common sparsity constraints. Decompositions of non-negative wafer test data by a conventional NMF algorithm are first shown in this thesis. It is discussed that, in a first approximation, the NMF model applies for this kind of data. While usual NMF algorithms aim the decomposition of non-negative data sets, here also an extension of NMF for binary data is presented which finds direkt application in semiconductor production. The probability that a certain chip passes or fails some special test is modelled by the superposition of several possible failure causes. A fast optimization procedure is develloped which estimates the non-negative basis and coefficient matrices in a NMF-like fashion. The last part of the thesis applies Bayesian techniques to answer the two mayor problems of NMF: the optimal solution given the number of basis components and the actual number of underlying basis components. The answers are gained under prior assumptions by integration over all coefficents, and over coefficients and basis components respectively. The first is exactly solvable in a special case and the result confirms the determinant criterion. The latter integral is approximated by a variational calculation. The herefore developed algorithm is a direct extension of a popular multiplicative NMF algorithm. Simulations demonstrate that the new algorithm is able to detect the correct number of components in artificial datasets even if the assumed form of the reconstruction error does not coincide with the actual one. Finally, potential future developments based on this work are proposed.