zoukankan      html  css  js  c++  java
  • Introduction To Monte Carlo Methods

    Introduction To Monte Carlo Methods

    I’m going to keep this tutorial light on math, because the goal is just to give a general understanding.

    The idea of Monte Carlo methods is this—generate some random samples for some random variable of interest, then use these samples to compute values you’re interested in.

    I know, super broad. The truth is Monte Carlo has a ton of different applications. It’s used in product design, to simulate variability in manufacturing. It’s used in physics, biology and chemistry, to do a whole host of things that I only partially understand. It can be used in AI for games, for example the chinese game Go. And finally, in finance, to evaluate financial derivatives or option pricing [1]. In short—it’s used everywhere.

    The methods we use today originated from the Manhattan Project, as a way to simulate the distance neutrons would travel through through various materials [1]. Ideas using sampling had been around for a little while, but they took off in the making of the atomic bomb, and have since appeared in lots of other fields.

    The big advantage with Monte Carlo methods is that they inject randomness and real-world complexity into the model. They are also more robust to adjustments such as applying a distribution to the random variable you are considering. The justification for a Monte Carlo method lies in the law of large numbers. I’ll elaborate in the first example.

    The examples I give are considered simple Monte Carlo. In this kind of problem we want to know the expected value of some random variable. We generate a bunch of these random variables and take their average. The random variable will often have a probability distribution.

    Estimating Pi

    We can use something called the random darts method, a Monte Carlo simulation, to estimate pi. Here is my R code for this example.

    The logic goes as follows—

    If we inscribe a circle in a square, where one side length of the square equals the diameter of the circle we can easily calculate the ratio of circle area to square area.

    Screen Shot 2015-07-25 at 6.53.12 PM

    Now if we could estimate this value, we would be able to estimate pi.

    We can do this by randomly sampling points in the square, and calculating the proportion of those inside the circle to the total points. So I just calculate red points over total points, and multiply by 4.

    CirclePic

    Now as the number of points increases, the closer our value will get to pi.

    Pi_Error

    This is a very simple example of a Monte Carlo method at work.

    Simulating Traffic

    Here is a more useful example. We can simulate traffic using the Nagel–Schreckenberg model. In this model, we have a road, which is made up by cells or spaces and contains a speed limit, and then a certain number of cars. We iterate through the cars and update their velocity based on the four following rules. Note – a car’s velocity = v.

    1. Cars not at the maximum velocity will increase their velocity by one unit.
    2. We then assess the distance d between a car and the car in front of it. If the car’s velocity is greater than or equal to the distance, we adjust it’s velocity to d-1.
    3. Now we add some randomization. This is the step that makes it Monte Carlo-esque. With a probability p, we will reduce the cars velocity by 1.
    4. Then, we move the car up v units.

    This model is simple, but it does a pretty good job of simulating traffic behavior. It doesn’t deal with accidents or bad drivers; it’s purpose is to assess those times when traffic just appears and vanishes without any apparent reason. More sophisticated models exist, but many of them are based on this model.

    View my code for getting the simulated data here, and for visualizing it in Processing here.

    MonteCarloRoad

    I love how some of the “cars” are right on the bumper of the one in front of it, and others are just chilling out taking their time. Haha

     

    Challenges with Monte Carlo Methods

    The first big challenge for Monte Carlo is how to come up with independent samples for whatever distribution your dealing with. This is a harder than you might think. In my code I just called R or Python’s built in random functions, but sampling can become much more sophisticated. That is a lot of what you will read about from more academic sources.

    Here is a link on how R’s built in uniform sampling distribution works.

    Another problem is getting the error to converge. Notice with the pi example how the error stopped converging quickly for the latter part of the graph. Most Monte Carlo applications just use really large samples due to low computing costs to compensate.

    Monte Carlo methods are an awesome topic to explore, and I hope this post popularizes them even a little bit more (outside of finance and physics, that is).

    Sources

    1. Wikipedia

    2. Art Owen’s textbook on the subject. My favorite resource so far.

    3. Kevin Murphy’s textbook.

  • 相关阅读:
    mysql 数据库的简单操作 2
    mysql简单操作,增删查改.
    当mysq启动时出现错误1067时应如何解决
    JS中 逻辑或 || 逻辑与 && 的使用方法总结
    编写一段程序,运行时向用户提问“你考了多少分?(0~100)”,接受输入后判断其等级并显示出来。判断依据如下:等级={优 (90~100分);良 (80~89分);中 (60~69分);差 (0~59分);}
    IF的使用
    第一个输出程序 Console.WriteLine
    Day2_and_Day3 文件操作
    linux安装VLAN,系统怎么划分VLAN打标签上交换机
    Python3.5+selenium(11)脚本模块化&参数化
  • 原文地址:https://www.cnblogs.com/yymn/p/4681076.html
Copyright © 2011-2022 走看看