this is for holding javascript data
Mihir Paithane added Poop.tex
over 9 years ago
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A random walk is a mathematical formalization of a path that consists of a succession of random steps [CITE]. Random walks describe a system in which a point is equally likely to shift left or right on a one-dimensional grid. consist of a set of an infinite number of independent, random variables $Z_{1}...Z_{n}$, each corresponding to a value of either $1$ or $-1$ (left or right) and equally likely to occur ($p=0.5$). This means that the system is equally likely to proceed one direction
After $n$ steps, the output of the function is defined as the sum of all previous movements. Thus, the position $S_{n}$ of the random walk after $n$ steps is described by the following equation:
\begin{equation}
S_{n} = \sideset{}{}\sum_{j=1}^{n}e^{i\theta j}
\end{equation}
This equation is $1$-dimensional in that it only accounts for one axis of motion. To model tumors as $2$-dimensional, we used a random walk to model both axes of motion. As a result, the "tumors" were both randomly configured and contiguous.
