deletions | additions       diff --git a/Shot optimization.tex b/Shot optimization.tex  index 3eaede1..52d8452 100644  --- a/Shot optimization.tex  +++ b/Shot optimization.tex 
...
\section{Shot Optimization}  Shot optimization was performed using an asexual genetic algorithm. A genetic algorithm is a stategy for finding an optimal value out of all possible values. The idea is inspired heavily by biology; each iteration of the process produces a new set of values, each with small modifications of the current set, while favoring the optimima.  For the present study, the typically asexual operator, a mutation, was used: a single parent chrosome is altered randomly to produce a new one. With each iteration, or generation, members of a population, in this case the set of possible circle arrangements in the square lattice, are then assessed by a fitness function. The fitness determines which chromosomes are passed on to the next generation, a kind of "selection" function. The selection function can be described as aroulette selection, or a  fitness proportionate selection: let $p$ denote population size, let $m_i$ denote population members, for $1≤i≤p$, and let $f(m)$ denote the fitness function for a chromosome; the total fitness is set by \begin{equation}  F= \sideset{}{}\sum_{1}^{p}f(m_i)  \end{equation}  In other words, the sum of probabilities of a single  chromosome being chosen by the fitness function is equal to total fitness. The fitness level is used to associate a probability of selection with each individual chromosome (in this case, the gene for radius or for position).  $F_{1}$ score. It utilizes two terms - precision and recall. In terms of our mathematical model, precision is the number of tumor cells covered in the shot divided by the total number of cells covered by the shot. Recall, in context, is the number of cancer cells covered in a shot divided by the total number of cancer cells. The F_{1} score in our problem was calculated with this equation: