Merge pull request #9 from pim-n/dev

update docs

docs/prefab_roads.ipynb

@@ -23,7 +23,11 @@
     "N = \\frac{L}{\\Delta s}.\n",
     "$$\n",
     "\n",
-    "Let $\\left( p_1, p_2, \\dots, p_K \\right)$ represent the proportion of $N$ that each prefab will be assigned, where $\\sum p_i = 1$. One useful distribution here is the [Dirichlet distribution](https://en.wikipedia.org/wiki/Dirichlet_distribution), which is parametrized by a vector $\\mathbf{\\alpha} = \\left(\\alpha_1, \\alpha_2, \\dots, \\alpha_K \\right)$, known as the *concentration factor*. Setting a uniform $\\alpha$ across the entire parameter space makes the distribution symmetric, meaning all components are equal.\n",
+    "Let $\\left( p_1, p_2, \\dots, p_K \\right)$ represent the proportion of $N$ that each prefab will be assigned, where $\\sum p_i = 1$. One useful distribution here is the [Dirichlet distribution](https://en.wikipedia.org/wiki/Dirichlet_distribution), which is parametrized by a vector $\\mathbf{\\alpha} = \\left(\\alpha_1, \\alpha_2, \\dots, \\alpha_K \\right)$. The special case where all $\\alpha_i$, the scalar parameter $\\alpha$ is called a *concentration parameter*. Setting the same $\\alpha$ across the entire parameter space makes the distribution symmetric, meaning no prior assumptions are made regarding the proportion of $N$ that will be assigned to each segment. $\\alpha = 1$ leads to what is known as a flat Dirichlet distribution, whereas higher values lead to more dense and evenly distributed $\\left( p_1, p_2, \\dots, p_K \\right)$. On the other hand, keeping $\\alpha \\leq 1$ gives a sparser distribution which can lead to larger variance in apportioned number of waypoints to $\\left( p_1, p_2, \\dots, p_K \\right)$.\n",
+    "\n",
+    "#### Expectation value and variance of Dirichlet distribution\n",
+    "\n",
+    "Suppose we draw our samples for proportion of length from the Dirichlet distribution\n",
     "\n",
     "$$\n",
     "(p_1, p_2, \\ldots, p_K) \\sim \\text{Dirichlet}(\\alpha, \\alpha, \\ldots, \\alpha)\n",
@@ -35,7 +39,7 @@
     "\\operatorname {E} [p_{i}]={\\frac {\\alpha _{i}}{\\alpha _{0}}}, \\; \\operatorname {Var} [p_{i}]={\\frac {\\alpha _{i}(\\alpha _{0}-\\alpha _{i})}{\\alpha _{0}^{2}(\\alpha _{0}+1)}}.\n",
     "$$\n",
     "\n",
-    "But, since $\\alpha_i$ is the same for all $i$,\n",
+    "If $\\alpha$ is a scalar, then $\\alpha _{0}= K \\alpha$ and the above simplifies to\n",
     "\n",
     "$$\n",
     "\\operatorname {E} [p_{i}]={\\frac {\\alpha}{K \\alpha}}={\\frac {1}{K}}, \\; \\operatorname {Var} [p_{i}]={\\frac {\\alpha(K \\alpha -\\alpha)}{(K \\alpha)^{2}(K \\alpha +1)}}.\n",
@@ -47,7 +51,7 @@
     "(N \\cdot p_1, N \\cdot p_2, \\ldots, N \\cdot p_K)\n",
     "$$\n",
     "\n",
-    "to get the randomly assigned number of waypoints for each prefab. We now have a distribution which can give randomly assigned lengths to a given list of prefabs, with a parameter to control the degree of randomness. With very high concentration factor $\\alpha$, the distribution of lengths will be uniform, with each prefab getting $N \\cdot \\operatorname {E} [p_{i}]={\\frac {N}{K}}$ waypoints assigned to it.\n",
+    "to get the randomly assigned number of waypoints for each prefab. We now have a distribution which can give randomly assigned lengths to a given list of prefabs, with a parameter to control the degree of randomness. With a large concentration parameter $\\alpha$, the distribution of lengths will be more uniform, with each prefab getting $N \\cdot \\operatorname {E} [p_{i}]={\\frac {N}{K}}$ waypoints assigned to it. Likewise, keeping $\\alpha$ low increases variance and allows for a more random assignment of proportions of waypoints to each prefab segment.\n",
     "\n",
     "#### Random angles\n",
     "\n",