Adding idea about how to sample more efficiently

.vscode/settings.json

@@ -70,5 +70,11 @@
     "cfenv": "cpp",
     "cinttypes": "cpp"
   },
-  "intel-corporation.oneapi-analysis-configurator.binary-path": "/home/george/Documents/development/epiworld/tests/01c.o"
+  "intel-corporation.oneapi-analysis-configurator.binary-path": "/home/george/Documents/development/epiworld/tests/01c.o",
+  "grammarly.selectors": [
+    {
+      "language": "quarto",
+      "scheme": "file"
+    }
+  ]
 }

paper/mixing.md

@@ -0,0 +1,54 @@
+# Mixing probabilities in connected model
+George G. Vega Yon, Ph.D.
+2024-04-25
+
+We will look into the probability of drawing infected individuals to
+simplify the algorithm. There are $I$ infected individuals at any time
+in the simulation; thus, instead of drawing from $Bern(c/N, N)$, we will
+be drawing from $Bern(c/N, I)$. The next step is to check which infected
+individuals should be drawn. Let’s compare the distributions using the
+hypergeometric as an example:
+
+``` r
+set.seed(132)
+nsims <- 1e5
+N <- 400
+rate <- 2
+p <- rate/N
+I <- 10
+
+sim_complex <- parallel::mclapply(1:nsims, \(i) {
+  nsamples <- rbinom(N, N, p)
+  sum(rhyper(N, m = I, n = N, k = nsamples) > 0)
+}, mc.cores = 4L) |> unlist()
+
+sim_simple <- parallel::mclapply(1:nsims, \(i) {
+  sum(rbinom(N, I, p) > 0) 
+}, mc.cores = 4L) |> unlist()
+
+
+op <- par(mfrow = c(1,2))
+MASS::truehist(sim_complex)
+MASS::truehist(sim_simple)
+```
+
+![](mixing_files/figure-commonmark/Simulation-1.png)
+
+``` r
+par(op)
+
+quantile(sim_complex)
+```
+
+      0%  25%  50%  75% 100% 
+       3   16   19   22   40 
+
+``` r
+quantile(sim_simple)
+```
+
+      0%  25%  50%  75% 100% 
+       3   17   19   22   40 
+
+These two approaches are equivalent, but the second one is more
+efficient from the computational perspective.

paper/mixing.qmd

@@ -0,0 +1,39 @@
+---
+format: gfm
+title: Mixing probabilities in connected model
+author: George G. Vega Yon, Ph.D.
+date: 2024-04-25
+---
+
+We will look into the probability of drawing infected individuals to simplify the algorithm. There are $I$ infected individuals at any time in the simulation; thus, instead of drawing from $Bern(c/N, N)$, we will be drawing from $Bern(c/N, I)$. The next step is to check which infected individuals should be drawn. Let's compare the distributions using the hypergeometric as an example:
+
+
+```{r}
+#| label: Simulation
+set.seed(132)
+nsims <- 1e5
+N <- 400
+rate <- 2
+p <- rate/N
+I <- 10
+
+sim_complex <- parallel::mclapply(1:nsims, \(i) {
+  nsamples <- rbinom(N, N, p)
+  sum(rhyper(N, m = I, n = N, k = nsamples) > 0)
+}, mc.cores = 4L) |> unlist()
+
+sim_simple <- parallel::mclapply(1:nsims, \(i) {
+  sum(rbinom(N, I, p) > 0) 
+}, mc.cores = 4L) |> unlist()
+
+
+op <- par(mfrow = c(1,2))
+MASS::truehist(sim_complex)
+MASS::truehist(sim_simple)
+par(op)
+
+quantile(sim_complex)
+quantile(sim_simple)
+```
+
+These two approaches are equivalent, but the second one is more efficient from the computational perspective.