  In this paper, we study the problem of computing a Principal Component
Analysis of data affected by Poisson noise. We assume samples are drawn from
independent Poisson distributions. We want to estimate principle components of
a fixed transformation of the latent Poisson means. Our motivating example is
microbiome data, though the methods apply to many other situations. We develop
a semiparametric approach to correct the bias of variance estimators, both for
untransformed and transformed (with particular attention to log-transformation)
Poisson means. Furthermore, we incorporate methods for correcting different
exposure or sequencing depth in the data. In addition to identifying the
principal components, we also address the non-trivial problem of computing the
principal scores in this semiparametric framework. Most previous approaches
tend to take a more parametric line. For example the Poisson-log-normal (PLN)
model, approach. We compare our method with the PLN approach and find that our
method is better at identifying the main principal components of the latent
log-transformed Poisson means, and as a further major advantage, takes far less
time to compute. Comparing methods on real data, we see that our method also
appears to be more robust to outliers than the parametric method.
