Jolly and Hampton (1990) documentation updates

docs/_config.yml

@@ -47,7 +47,7 @@ sphinx:
   extra_extensions: [
     'sphinx_automodapi.automodapi',
     'numpydoc',
-    # 'sphinx.ext.autodoc',
+    'sphinx.ext.autodoc',
     'sphinxcontrib.bibtex',
     'sphinx.ext.intersphinx',
     'sphinx.ext.mathjax',

docs/theory/stratification.md

@@ -78,42 +78,123 @@ $$
 \sum_{s,\ell,\alpha} L^i_{s,\alpha,\ell} = 1
 $$
 
+## Jolly-Hampton (1990) stratified sampling 
+This analysis provides a coefficient of variation ($\textit{CV}$) for the entire survey by reweighting biomass estimates for $k$ based on the length of each transect $t$ stratified by $i$ to derive estimates of the mean and variance. The first step is summing biomass estimates for each $k$ within each $t$:
 
+$$ 
+B^t =
+    \sum_{k} B^{k,t}
+\label{eq:biomass_transect} \tag{1}
+$$
+
+Total transect distance $d_{x,y}^t$ for each $t$ is similarly computed:
+
+$$ 
+d_{x,y}^t =
+    \sum_{k} d_{x,y}^{k,t}
+\label{eq:distance_transect} \tag{2}
+$$
+
+These transect distances are then summed for each $i$:
+
+$$ 
+D_{x,y} =
+    \sum_{t} d_{x,y}^{t}
+\label{eq:distance_stratum} \tag{3}
+$$
+
+Values of $B^t$ $\eqref{eq:biomass_transect}$, $d_{x,y}^t$ $\eqref{eq:distance_transect}$, and $D_{x,y}$ $\eqref{eq:distance_stratum}$ are then all used to compute the mean transect-length-weighted biomass estimates for each $i$:
+
+$$ 
+\tilde{\rho} = 
+    \frac{\sum\limits_{t} B^t d_{x,y}^t}{D_{x,y}}
+\label{eq:mean_estimate} \tag{4}
+$$
+
+Next $B^t$ $\eqref{eq:biomass_transect}$ for each $t$ is standardized to provide a mean biomass-per-distance estimate:
+
+$$ 
+\tilde{\rho}^{~t} = 
+    \frac{B^t}{d_{x,y}^{t}}
+\label{eq:transect_estimate} \tag{5}
+$$
+
+which can then be used to compute the squared deviation from the mean along with $\tilde{\rho}$ $\eqref{eq:mean_estimate}$ for each $t$ within $i$:
+
+$$
+\tilde{s}^{~t} =
+    (\tilde{\rho}^{~t} - \tilde{\rho})^2
+\label{eq:squared_deviation} \tag{6}
+$$
+
+This can then be used to calculate the sum of weighted squared deviations:
 
-<!-- ## Jolly-Hampton (1990) stratified sampling 
+$$
+\tilde{s} =
+    \sum\limits_{t} w_t^{2} \tilde{s}^t
+\label{eq:summed_squared_deviation} \tag{7}
+$$
+
+where the stratified weights ($w_t$) for each $t$ are:
+
+$$
+w_t = 
+    \frac{d_{x,y}^t}{\bar{D}_{x,y}}
+\label{eq:stratified_weights} \tag{8}
+$$
+
+The variance ($\tilde{\sigma}$) for each $i$ is then calculated:
+
+$$
+\tilde{\sigma} =
+    \frac{\tilde{s}}{\nu}
+\label{eq:variance} \tag{9}
+$$
 
-Mean density for stratum $i$:
+where the $\nu$ represents the degrees of freedom:
 
 $$
-\hat{ \rho }_{A,B}^{ i } = 
-    \frac{1}{ n^{ i } }
-    \sum\limits_{i=0}^{n^{i} } w^{i,j} \hat{ \rho }_{A,B}^{ i,j,k}
+\begin{equation}
+\nu =
+    \begin{cases}
+        n^t(n^t-1), & \text{if } n^t > 1 \\
+        (n^t)^2, & \text{if } n^t = 1
+    \end{cases}
+\tag{10} \label{eq:dof} 
+\end{equation}
 $$
 
-where $w^{i,j}$ is the transect weight calculated via:
+Variance estimates for each $i$ are then weighted by the total area ($A$) for $i$ to compute the overall (weighted) survey variance:
 
 $$
-w^{i,j} = \frac{
-    d(x,y)^{i,j}
-}{
-    \frac{1}{n^{i}}
-    \sum\limits_{j=1}^{n^{i}} d(x,y)^{i,j}
-}
+\hat{\sigma} = 
+    \sum\tilde{\sigma} A^2
+\label{eq:weighted_variance} \tag{11}
 $$
 
-where $d(x,y)^{i,j}$ is the transect length of $n^{i}$ transects within each stratum. 
-This procedure is then repeated by using the different areas ($A^{i}$) of each stratum to
-weight the final $\hat{ \rho }_{A,B}$ estimate:
+where:
 
 $$
-\hat{ \rho }_{A,B} =
-    \frac{
-        \sum\limits_{i} A_{i} \hat{ \rho_{A,B}^{ i } }
-    }{
-        \sum\limits_{i} A_{i}
-    }
-$$ -->
+A =
+    \sum\limits_k A^k
+\label{eq:transect_area} \tag{12}
+$$
+
+Similar to the weighted variance estimates $\eqref{eq:weighted_variance}$, biomass estimates for each $i$ were also weighted by $A$ to calculate the overall (weighted) survey mean:
 
+$$
+\hat{\mu} =
+    \sum \tilde{\rho} A
+\label{eq:weighted_mean} \tag{13}
+$$
+
+The overall survey variance $\eqref{eq:weighted_variance}$ and mean $\eqref{eq:weighted_mean}$ are then both used to calculate $\textit{CV}$:
+
+$$
+\textit{CV} =
+    \frac{\sqrt{\hat{\sigma}}}{\hat{\mu}}
+\label{eq:cv} \tag{14}
+$$
 
 ## Stratification schemes used in the hake survey
 For Pacific hake, two types of stratifications are used: