# doZeroMStep: Compute the zero Maximization step. In HCBravoLab/metagenomeSeq: Statistical analysis for sparse high-throughput sequencing

## Description

Performs Maximization step calculation for the mixture components. Uses least squares to fit the parameters of the mean of the logistic distribution. \$\$ pi_j = sum_i^M frac1Mz_ij \$\$ Maximum-likelihood estimates are approximated using the EM algorithm where we treat mixture membership \$delta_ij\$ = 1 if \$y_ij\$ is generated from the zero point mass as latent indicator variables. The density is defined as \$f_zig(y_ij = pi_j(S_j) cdot f_0(y_ij) +(1-pi_j (S_j))cdot f_count(y_ij;mu_i,sigma_i^2)\$. The log-likelihood in this extended model is \$(1-delta_ij) log f_count(y;mu_i,sigma_i^2 )+delta_ij log pi_j(s_j)+(1-delta_ij)log (1-pi_j (sj))\$. The responsibilities are defined as \$z_ij = pr(delta_ij=1 | data)\$.

## Usage

 `1` ```doZeroMStep(z, zeroIndices, mmZero) ```

## Arguments

 `z` Matrix (m x n) of estimate responsibilities (probabilities that a count comes from a spike distribution at 0). `zeroIndices` Index (matrix m x n) of counts that are zero/non-zero. `mmZero` The zero model, the model matrix to account for the change in the number of OTUs observed as a linear effect of the depth of coverage.

## Value

List of the zero fit (zero mean model) coefficients, variance - scale parameter (scalar), and normalized residuals of length sum(zeroIndices).

`fitZig`