equations.Rmd

# Key equations

### Proportion of taxon $i$: Ratio of taxon $i$ to total.

Actual
\begin{align}
  \text{prop}_{i}(s) &= \frac{\text{density}_{i}(s)}{\text{total density}(s)}
\end{align}

Estimator definition 
\begin{align}
  \hat{\text{prop}}_{i}(s) &= \frac{\text{reads}_{i}(s)}{\text{total reads}(s)}
\end{align}

Estimator error 
\begin{align}
  \hat{\text{prop}}_{i}(s) 
  &= \text{prop}_{i}(s) \cdot \frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)}
\end{align}

### Ratio of taxon $i$ to taxon $j$:

actual
\begin{align}
  \text{density}_{i/j}(s) = \frac{\text{density}_{i}(s)}{\text{density}_{j}(s)}
\end{align}

estimate
\begin{align}
  \hat{\text{density}}_{i/j}(s) = \frac{\text{reads}_{i}(s)}{\text{reads}_{j}(s)} = \frac{\hat{\text{prop}}_{i}(s)}{\hat{\text{prop}}_{j}(s)}
\end{align}

error
\begin{align}
  \hat{\text{density}}_{i/j}(s) = \frac{\text{density}_{i}(s)}{\text{density}_{j}(s)} \cdot
  \frac{\text{efficiency}_{i}(s)}{\text{efficiency}_{j}(s)}
\end{align}

### Proportion-based density estimate

Estimator
\begin{align}
  \hat{\text{density}}_{i}(s) 
  = \hat{\text{prop}}_{i}(s) \cdot \hat{\text{total density}}(s)
\end{align}

Error in proportion-based density estimate assuming accurate total
\begin{align}
  \hat{\text{density}}_{i}(s) 
  % &= \hat{\text{prop}}_{i}(s) \cdot \text{total density}(s) \\[0.5ex]
  % &= \text{prop}_{i}(s) \cdot \frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)} \cdot \text{total density}(s) \\[0.5ex]
  &= \text{density}_{i}(s) \cdot \frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)}
\end{align}

Error in proportion-based density estimate
\begin{align}
  \hat{\text{density}}_{i}(s) 
  &= \text{density}_{i}(s) \cdot \frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)} \cdot \text{error in }\hat{\text{total density}}(s)
\end{align}

### Ratio-based density estimate

Estimator
\begin{align}
  \hat{\text{density}}_{i}(s) 
  &= \frac{\text{reads}_{i}(s)}{\text{reads}_{r}(s)} \cdot \hat{\text{density}}_{r}(s)
\end{align}

Error
\begin{align}
  \hat{\text{density}}_{i}(s) 
  &= \text{density}_{i}(s) \cdot \frac{\text{efficiency}_{i}}{\text{efficiency}_{r}} \cdot \text{error in } \hat{\text{density}}_{r}(s)
\end{align}

## Table

<table style="width:100%">
  <tr>
    <td>Quantity</td>
    <td>Actual</td>
    <td>Estimate</td>
    <td>Error</td>
  </tr>
  <tr>
    <td>Proportion</td>
    <td>$\text{prop}_{i}(s) \equiv \frac{\text{density}_{i}(s)}{\text{total density}(s)}$</td>
    <td>$\frac{\text{reads}_{i}(s)}{\text{total reads}(s)}$</td>
    <td>$\frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)}$</td>
  </tr>
  <tr>
    <td>Ratio</td>
    <td>$\text{density}_{i/j}(s) \equiv \frac{\text{density}_{i}(s)}{\text{density}_{j}(s)}$</td>
    <td>$\frac{\text{reads}_{i}(s)}{\text{reads}_{j}(s)} = \frac{\hat{\text{prop}}_{i}(s)}{\hat{\text{prop}}_{j}(s)}$</td>
    <td>$\frac{\text{efficiency}_{i}(s)}{\text{efficiency}_{j}(s)}$</td>
  </tr>
  <tr>
    <td>Density from proportion</td>
    <td>$\text{density}_{i}(s)$</td>
    <td>$\hat{\text{prop}}_{i}(s) \cdot \hat{\text{total density}}(s)$</td>
    <td>$\frac{\text{efficiency}_{i}}{\text{mean efficiency}(s)} \cdot \text{error in }\hat{\text{total density}}(s)$</td>
    <td></td>
  </tr>
  <tr>
    <td>Density from ratio</td>
    <td>$\text{density}_{i}(s)$</td>
    <td>$\frac{\text{reads}_{i}(s)}{\text{reads}_{r}(s)} \cdot \hat{\text{density}}_{r}(s)$</td>
    <td>$\frac{\text{efficiency}_{i}}{\text{efficiency}_{r}} \cdot \text{error in } \hat{\text{density}}_{r}(s)$</td>
  </tr>
</table>