-
Notifications
You must be signed in to change notification settings - Fork 0
/
model.tex
79 lines (78 loc) · 3.86 KB
/
model.tex
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
\begin{algorithm}
\caption{Pseudocode representation of \emph{cosmos} model}
\label{alg:model}
\begin{algorithmic}[1]
\State $g \sim \mathbf{HalfNormal}(50)$
\Comment{camera gain}
\State $\sigma^{xy} \sim \mathbf{Exponential}(1)$
\Comment{standard deviation of on-target spot position (pixels)}
\State $\pi \sim \mathbf{Beta}(1/2, 1/2)$
\Comment{mean specific binding probability}
\State $\lambda \sim \mathbf{Exponential}(1)$
\Comment{non-specific binding density}
\ForAll{$\mathsf{AOI}[N+N_\mathsf{c}]$}
\State $\mu^b \sim \mathbf{HalfNormal}(1000)$
\Comment{mean background intensity}
\State $\sigma^b \sim \mathbf{HalfNormal}(100)$
\Comment{standard deviation of background intensity}
\ForAll{$\mathsf{frame}[F]$}
\State $b \sim \mathbf{Gamma}(\mu^b, \sigma^b)$
\Comment{background intensity}
\State $
z \sim
\begin{cases}
\mathbf{Bernoulli}(\pi) & \text{on-target AOI} \\
\mathbf{Bernoulli}(0) & \text{control off-target AOI} \rule{0pt}{4ex}
\end{cases}
$
\Comment{target-specific spot presence}
\State $
\theta \sim
\begin{cases}
\mathbf{Categorical}\left( \begin{bmatrix} 1, 0, \dots, 0 \end{bmatrix} \right) & z = 0 \\
\mathbf{Categorical}\left( \begin{bmatrix} 0, \frac{1}{K}, \dots, \frac{1}{K} \end{bmatrix} \right) & z = 1 \rule{0pt}{4ex}
\end{cases}
$
\Comment{target-specific spot index}
\ForAll{$\mathsf{spot}[K]$}
\State $ m_{\mathsf{spot}(k)} \sim
\begin{cases}
\mathbf{Bernoulli}(1) & \text{$\theta = k$} \\
\mathbf{Bernoulli} \left( \sum_{l=1}^K \dfrac{l \cdot \mathbf{TruncPoisson}(l; \lambda, K)}{K} \right) & \text{$\theta = 0$} \rule{0pt}{4ex} \\
\mathbf{Bernoulli} \left( \sum_{l=1}^{K-1} \dfrac{l \cdot \mathbf{TruncPoisson}(l; \lambda, K-1)}{K-1} \right) & \text{otherwise} \rule{0pt}{4ex}
\end{cases} $
\Comment{spot presence}
\State $h \sim \mathbf{HalfNormal}(10000)$
\Comment{spot intensity}
\State $w \sim \mathbf{Uniform}(0.75, 2.25)$
\Comment{spot width}
\State $ x_{\mathsf{spot}(k)} \sim
\begin{cases}
\mathbf{AffineBeta}\left( 0, \sigma^{xy}, -\dfrac{P+1}{2}, \dfrac{P+1}{2} \right) & \theta = k \\
\mathbf{Uniform}\left(-\dfrac{P+1}{2}, \dfrac{P+1}{2} \right) & \theta \neq k \rule{0pt}{4ex} \end{cases} $
\Comment{$x$-axis center}
\State $ y_{\mathsf{spot}(k)} \sim
\begin{cases}
\mathbf{AffineBeta}\left( 0, \sigma^{xy}, -\dfrac{P+1}{2}, \dfrac{P+1}{2} \right) & \theta = k \\
\mathbf{Uniform}\left(-\dfrac{P+1}{2}, \dfrac{P+1}{2} \right) & \theta \neq k \rule{0pt}{4ex} \end{cases} $
\Comment{$y$-axis center}
\ForAll{$\mathsf{pixelX}[P] \times \mathsf{pixelY}[P]$}
\State $\mu^{S}_{\mathsf{pixelX}(i), \mathsf{pixelY}(j)} =
\dfrac{m \cdot h}{2 \pi w^2} \exp{\left ( -\dfrac{(i-x-x^\mathsf{target})^2 + (j-y-y^\mathsf{target})^2}{2w^2} \right)}$
\Comment{2-D Gaussian spot}
\EndFor
\EndFor
\ForAll{$\mathsf{pixelX}[P] \times \mathsf{pixelY}[P]$}
\State $\delta \sim \mathbf{Empirical}( \delta_\mathsf{samples}, \delta_\mathsf{weights})$
\Comment{offset signal}
\State $\mu^I = b + \sum_{\mathsf{spot}} \mu^S$
\Comment{mean pixel intensity w/o offset}
\State $I \sim \mathbf{Gamma} (\mu^I, \sqrt{\mu^I \cdot g})$
\Comment{pixel intensity w/o offset}
\State $D = \delta + I$
\Comment{observed pixel intensity}
\EndFor
\EndFor
\EndFor
\end{algorithmic}
\end{algorithm}