## UlamMethod.jl

A package for discretizing trajectory data into a transition probability matrix using Ulam's method.
Author 70Gage70
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January 2023

# UlamMethod.jl

## Introduction

This package is an implementation of Ulam's method 1 2 (see also Galerkin projection 3) for the discretization of a stochastic operator using pure Julia. Given a set of two-dimensional, one-step trajectories

$$(x_{0, 1}, y_{0, 1}) \to (x_{T, 1}, y_{T, 1}), (x_{0, 2}, y_{0, 2}) \to (x_{T, 2}, y_{T, 2}) \dots$$

defined in a domain, the essential goal of Ulam's method is to partition the domain into a series of non-intersecting regions and construct a transition probability matrix $P$ on these regions. In UlamMethod.jl, this is accomplished in two main steps

1. The user provides a polygon containing the data, and covering of the the domain is generated by polygons according to one of several different [binning algorithms](@ref binning).
2. The number of trajectories beginning in polygon $i$ and ending in polygon $j$ is used to create the entry $P_{i, j}$ of $P$ such that the edges of the domain are handled by a [stochasticization algorithm](@ref stoc).

The polygons which form the covering and the transition probability matrix are the main outputs.

## Installation

In the Julia REPL, run the following code and follow the prompts:

using Pkg
Pkg.add("UlamMethod")

Make the functions in this package available to use in your code by including the following line:

using UlamMethod

## Quickstart

The core functionality is provided by

ulam_method(traj::UlamTrajectories, domain::UlamDomain)

where traj contains information about trajectories and domain contains information about the domain and covering. Here are 10000 random trajectories in the domain $[0, 10]^2$

using UlamMethod

n_data = 10000
x0, y0, xT, yT = 10*rand(n_data), 10*rand(n_data), 10*rand(n_data), 10*rand(n_data)
traj = UlamTrajectories(x0 = x0, y0 = y0, xT = xT, yT = yT)

We will take our domain to be the rectangular subset $[3, 5] \times [4, 8]$ and generate a covering with 40 rectangles.

xmin, xmax, ymin, ymax = 3, 5, 4, 8
domain = UlamDomain(xmin, xmax, ymin, ymax, poly_type = "rec", poly_number = 40)

Run Ulam's method.

ulam = ulam_method(traj, domain)    # the main calculation

ulam.P_closed                       # the transition matrix
pt = PolyTable(ulam.polys)          # PolyTable makes a simple list of nodes and edges
pt.nodes                            # |x|y| table of polygon nodes
pt.edges[:,3]                       # the index of the polygon that the i'th node belongs to

## Footnotes

1. Ulam, Stanislaw M. A collection of mathematical problems. No. 8. Interscience Publishers, 1960.

2. Li, Tien-Yien. "Finite approximation for the Frobenius-Perron operator. A solution to Ulam's conjecture." Journal of Approximation theory 17.2 (1976): 177-186.

3. Reddy, Junuthula Narasimha. Introduction to the finite element method. McGraw-Hill Education, 2019.

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