Outlier rejection with mixture models in alignment

Abstract

Outliers in measurement often interfere with alignment. They are caused by sudden damages in the alignment mark, and existence of particles, resist damages and so on. In a conventional way to identify outliers, the observations that have larger residual than previously determined threshold are identified as outlier. It works well only with the operator’s labor of adjusting the threshold according to the deviation of <i>ordinaries</i> (non-outliers). However, labor is a problem especially in Small-Quantity Large-Variation fabrication such as for ASIC, System-LSI and so on. A novel method for elimination of the labor has been developed. It utilizes normal mixture models whose number of components is determined based on the Maximum Penalized Likelihood (MPL) method. It can be regarded as an identification method that determines threshold adaptively using <i>ordinaries’ </i>deviation. Simulation results show that the penalty coefficient, the only parameter of the method, can be a constant in the variation of <i>ordinarie's</i> deviation. It also shows that in the absence of outliers, the accuracy of the method is comparable with the maximum likelihood estimation that is commonly considered to be the best method when the observations follow the normal distribution. The method performs better than conventional ones when there are a sufficient number of observations (no less than ten) in the standard Enhanced Global Alignment (EGA). Superiority of the adaptive method is dependent upon the probability of outlier occurrence, variation of the <i>situation</i>, the number of observations and the complexity of the model fitted to the observations.