Maximum likelihood principal component analysis of time-of-flight secondary ion mass spectrometry spectral images

Many modern surface analytical instruments are able to acquire huge amounts of data in the form of spectral images. Time-of-flight secondary ion mass spectrometry (TOF-SIMS), for instance, can easily generate a complete mass spectrum at each point in a two-dimensional or three-dimensional spatial array. The challenge for the data analyst, then, is to garner the analytically useful information from the overwhelming quantity of raw spectral data. Factor analysis techniques such as principal component analysis (PCA) have proven quite useful in this endeavor. Standard PCA, however, assumes that noise in the data is uniform, that is, that it does not depend on the magnitude of signal. This is clearly not correct for methods that rely on particle counting where the noise is governed by Poisson statistics. In this case, properly accounting for heteroscedasticity is essential to extracting the chemical information into a minimum number of factors while optimally excluding noise. Maximum likelihood PCA (MLPCA) is one approach to addressing this issue. MLPCA can, in principle, incorporate a separate uncertainty estimate for each individual observation in a data set. This article will present a MLPCA analysis of a simple and intuitive TOF-SIMS spectral image. The results show that there is a trade-off between the number of uncertainty parameters included in the model and the quality of each and, in fact, using poor estimates may be worse than ignoring the noise characteristics altogether. The best results were obtained by using a low-rank approximation to the noise rather than individual estimates. MLPCA will also be compared with an optimal scaling approach. For the particular example given, the added benefits of MLPCA do not outweigh the greatly increased computational demands of the technique.