Compute the value of the count density function from the count model residuals.
Calculate density values from a normal: $f(x) = 1/(sqrt (2 pi ) sigma ) e^-((x - mu )^2/(2 sigma^2))$. Maximum-likelihood estimates are approximated using the EM algorithm where we treat mixture membership $deta_ij$ = 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 $z_ij = pr(delta_ij=1 | data)$.
Residuals from the count model.
Whether or not we are calculating from a log-normal distribution.
Density values from the count model residuals.
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