%\VignetteIndexEntry{totalcovariance}
\documentclass{article}

\title{Poisson PCA}
\author{Tianshu Huang and Toby Kenney}

\begin{document}

\maketitle

\verb!Poisson_Corrected_PCA(X,k=dim(X)[2],transformation=NA,seqdepth=FALSE)!


Performs PCA estimation with Poisson noise correction. X is the data
matrix of Poisson random variables with means Lambda given by some underlying
multivariate distribution. We want to estimate the principal
components of the distribution of f(Lambda) where f is some transformation.
We may also correct for sequencing depth --- that is, we assume that
the Poisson means in each sample are subject to large multiplicative
noise associated more with the experimental procedure than with the
important signal. This assumption is common in microbiome analysis.

Example:

<<>>=
library(PoissonPCA)
X <- rbind(c(0,1,3,5,2),c(2,1,4,4,0),c(3,2,2,0,5))
Poisson_Corrected_PCA(X)
@ 


\verb!LinearCorrectedVariance(X)!

\verb!LinearCorrectedVarianceSeqDepth(X)!

These functions estimate the latent variance of the Poisson means. The
first function assumes that total count is a potentially important
variable. The second assumes it has high noise, and treats the latent
Poisson means as compositional.



\verb!TransformedVariance(X,g,CVar)!

\verb!TransformedVarianceECV(X,g,ECVar)!

These functions estimate the variance of some function of the Poisson
means. g is an estimator of this function of the Poisson means. CVar
and ECVar are estimators of the conditional variance of g(X) for fixed
latent Lambda. CVar is a single estimator for a given X, while ECVar
estimates the expected value of g(X) over a sample of values with
different values of Lambda.


\verb!make_compositional_variance(Sigma)!

\verb!make_compositional_min_var(Sigma)!

These functions take a variance estimate Sigma and assume that each
entry has had a constant added to it. The first function estimates the
compositional variance matrix that best approximates Sigma, while the
second function subtracts the largest possible constant from all
entries of Sigma, so that the resulting matrix remains non-negative definite.


\verb!get_scores(X,V,d,k,transformation,mu)!

\verb!get_scores_log(X,V,d,k,mu)!

Given the principal components of the transformation of Lambda, these
functions estimate the scores of each observation X. That is, the
projection of the corresponding latent Lambda onto the principal
component space. This essentially gives the projection of the data
onto the latent principal component space. The first function applies
to a general transformation of Lambda, while the second function works
specifically for the logarithm transformation.

\verb!polynomial_transformation(coeffs)!

\verb!makelogtransformation(a,N)!


For many of the methods in this package, a transformation consists of
a list of 4 functions:

\begin{itemize}
  
\item A function \texttt{f} which evaluates the transformation on a particular value.

\item A function \texttt{g} which estimates \texttt{f(Lambda)} from an
  observation \texttt{X}.

\item A function \texttt{solve} which computes \texttt{Lambda} from \texttt{f(Lambda)}.
  
  \item A function \texttt{CVar} which estimates the conditional
    variance of \texttt{g(X)} for the latent value of \texttt{Lambda}
    corresponding to an observed value of \texttt{X}. For some
    transformations, it is more convenient to have a function
    \texttt{ECVar} which computes the average of this conditional
    expectation over a vector of observed values of \texttt{X}.
  
\end{itemize}

Example:

<<>>=
library(PoissonPCA)
X <- rbind(c(0,1,3,5,2),c(2,1,4,4,0),c(3,2,2,0,5))
Poisson_Corrected_PCA(X)
@ 


\end{document}