Reduced-rank approximations to spectroscopic and compositional data: A universal framework based on log-ratios and counting statistics

Abstract

Although not immediately apparent, acquisition of all spectroscopic and all compositional data involves counting or a similar process. Therefore, these data are subject to counting errors. In this contribution, the performance of Reduced Rank Approximation (RRA) methods was investigated which accommodate for counting errors to a different extent. Our premise is that count data can be treated as compositional quantities and should therefore be modelled in a suitable metric space. In this contribution that is the centred log-ratio (clr) space.Although the clr space is scale-invariant, error propagation showed that this does not apply to the associated counting error covariance matrix. Whereas its structure is determined by the location of the data in clr space, its magnitude reflects the sample size which is either directly (e.g., the number of counted objects) or indirectly (e.g., the measurement time) controlled by the data-acquisition protocol. It was also shown that (i) the counting error correlation structure in clr space is identical for data originating from Poisson and multinomial distributions, and (ii) the counting errors in clr space become independent only in the trivial case where the data set comprises multiple realizations from the same population and the number of variables goes to infinity.In a simulation experiment, the performance of unweighted RRA (i.e., RRA using a limited number of Principal Components) was compared with those of three alternative methods which make successively less restrictive assumptions about the error covariance structure. The alternative methods are weighted RRA (w-RRA), diagonal Maximum Likelihood RRA (d-MLRRA) and full Maximum Likelihood RRA (f-MLRRA). Their performance was tested by comparing data of known rank with RRAs to the same data set with added noise. As expected, f-MLRRA always gives the most accurate RRA. However, less computationally intensive methods give equally satisfying results in case the number of variables is large. A set of selection rules defined an algorithm which yields optimal reduced-rank approximations to noisy count data, i.e. Optimal Scale-Invariant Reduced-rank Approximation (OSIRA).