Questions on Monte Carlo methods, methods that require the repeated generation of (pseudo-, quasi-)random numbers for computing their results.

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53 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
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1answer
24 views

What is the average minimum distance between two Sobol points?

Having the first n points of a d-dimensional Sobol sequence, what is the average Euclidean distance from one arbitrarily point to its nearest neighbour?
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2answers
304 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
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2answers
62 views

Probabilities seem to be growing exponentially

We have instituted random drug testing at our company. I was charged with writing the code to generate the weekly random list of employees. We've gotten some complaints because of people getting ...
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1answer
107 views

Histogram normalization

I need to generate random numbers from Gaussian distribution and to draw an equalized histogram. I've generated them in Matlab using Box-Muller transformation. Since I wasn't sure how to equalize the ...
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59 views

Standard deviation of the mean through bootstrap resampling of dependent samples

I'm trying to do a Monte Carlo approximation of an integral where the samples are not independent (how much so can be tuned by a parameter giving how often I sample). Therefore the regular expression ...
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94 views

Monte Carlo sampling and Kolmogorov–Smirnov test in practice

I have two deterministic algorithms, Algorithm 1 and Algorithm 2. The first has $m$ inputs and one output, and the second has $n$ inputs and one output. The distributions of the inputs of the ...
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96 views

Importance Sampling of 2D constant piecewise function convertible to 1D?

So I have a constant piecewise 2D function (luminance values of pixels of an image) that I am writing an importance sampling algorithm for. I was going to write my algorithm by first sampling the 1D ...
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19 views

LDA with fixed topics?

Suppose I have a collection of "topic" probability distributions $\{\phi^{z}\}$ for LDA (Latent Dirichlet Allocation) that I have found via alternate methods; is there a closed form MLE for the ...
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1answer
103 views

Double integral estimation with Monte Carlo method by importance

I have this integral $\int_{0}^{\infty}\int_{0}^{\infty} e^{-x-y-xy}dxdy$ I have to estimate the value of it using the Monte Carlo method, using as importance function the exponential PDF. Does ...
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62 views

Monte Carlo simulations, accuracy of mean vs variance of answers

I am working on a Monte Carlo simulation where two inputs are being used, $N$ is the amount of simulations I use, and $M$ controls the detail of each simulation (specifically the amount of time step ...
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1answer
39 views

Looking for Method to evaluate the optimal node rate vs number of simulation rate in a Monte Carlo simulation

I am currently working on evaluating an American Option using a Monte Carlo simulation, and I am getting answers but they vary quite a bit. The two variables that I can alter are number of simulations ...
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49 views

Mixing time for metropolis chain on graph coloring

I'm reading the Markov Chains and Mixing Times by David Levin et al.. In section 5.4 page 71 a proof is given for a bound of mixing time for the Metropolis Chain on graph coloring. In the proof, such ...
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19 views

Additive model and simulation of the time series

It will be great if somebody could help me with following task.. Im trying to solve it about 3 days and read a lot of papers according this issue, but it doesnt work My task is to simulate a time ...
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1answer
20 views

A question about importance sample and Metropolis Algorithm

I am reading this paper by Beichl, I., & Sullivan, F. (2000) on Metropolis algorithm. I understand rejection sample. In the section "The Rejection Sample", I can understand the equation: ...
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23 views

How to choose a proper numerical optimisation method

Given a problem in numerical analysis in finance/econometrics, how to decide whther to choose Monte Carlo, Newton Raphson , Finite Difference , Gradient descent? I had this silly misconception that ...
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2answers
238 views

How to use Monte Carlo method to find volume?

So the question asks to use the Monte Carlo method to find the volume of an irregular figure defined as: ...
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1answer
86 views

Applying MCMC Metropolis algorithm

I'm interested in all possible paths (on the grid $\mathbb{N}^2 $) that goes from $ (0,0) $ to $ (n, n) $. At each step there are two possibilities: go right or go up. The path is a sequence $ ...
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1answer
75 views

Random sampling using Metropolis vs. Accept-Reject

I'm working on a project comparing the accept-reject (von Neumann) and Metropolis methods of sampling. I'm generating a sample of size $N$ from a normal distribution $N(1,(\frac{1}{2})^2)$. What I ...
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2answers
170 views

Are irreducible, positiv-definite Markov chains aperiodic?

If $M$ is the transition matrix of a discrete Markov chain, and $M$ is both irreducible, symmetric and positiv-definite, is the resulting Markov chain necessarily aperiodic? In my intuition, ...
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20 views

How to rewrite function for squared uniform distribution

The question is as follows: I am evaluating the following integral: $$\int_o^1\frac{\exp(\sqrt{1-x^2})}{\sqrt{x}}dx$$ by assuming it equals $E[f(U)]$ for a uniform distribution. I worked it out via ...
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1answer
20 views

Logic with increasing monte carlo possible output

I am working on Monte Carlo algorithm : Given that you have an experiment MC which has a p-correct of 75%, which means it gives you the right answer 75% of the time. You run MC3 which repeats MC 3 ...
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17 views

SEnsitivity Indices are non zero

I am trying to compute the sensitivity indices (SI) of a function using Monte Carlo simulation. I had written a matlab code that perform the computation directly and just return the final answer of my ...
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1answer
29 views

Does the perimeter of a 2-D object “count” toward its area?

I'm writing a quick Monte-Carlo simulation for a class in Matlab in order to estimate the value of pi as demonstrated in this gif: http://en.wikipedia.org/wiki/File:Pi_30K.gif However, I'm not sure ...
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32 views

Is there a such thing as a quasi-random shuffle?

I've recently experimented with Quasi-random numbers in monte-carlo applications. Is there a way to construct a quasi-random shuffle? By that I mean can I take a sequence $Q$ and shuffle it to produce ...
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14 views

Monte Carlo Markov Chain Simulation Issues

The Markov Chain is uniformly distributed across all $50$x$50$ matrices of entries $0$ and $$1 with no neighboring $1's$. I am supposed to run a MC simulation to check the probability that the ...
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36 views

Monte Carlo Markov Chain simulation

I am going to post the python code logic we used however I want someone to look at the number that are printing out. The Markov chain is uniformly distributed across all $50x50$ matrices with entries ...
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1answer
55 views

Compute the Std. Deviation of Multiple Monte Carlo Estimation of $\pi$

For a school programming assignment, I am trying to compute the value of $\pi$ via the classic Monte Carlo estimation of $\pi$. In the experiment, we throw a variable number of darts at a circle that ...
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25 views

How to sample points from a bounded polytope?

I have a bounded polytope $C \subset \mathbb{R}^n$ characterized by the following restraints: $$ x \in C \Leftrightarrow \sum_{i=1}^n x_i = 1 \text{ and } Ax \leq b$$ for some matrix $A \in ...
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1answer
38 views

Approximation and Monte Carlo simulation.

I am a bit up over my head here, I will present an argument and then I hope you guys will say if my reasoning is correct or what should be changed, ultimately I am hoping to say something qualified ...
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91 views

Correlated variables from Latin Hypercube

Say I have a vector $\mathbf{Y}$ of $n$ normally distributed random variables. I have its mean vector $\mu$ and covariance matrix $\Sigma$. Normally if I were to generate a sample, I would decompose ...
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53 views

How can I prove that the error of the Monte Carlo method for finding $\pi$ decreases as $N$, the number of samples, increases?

I am currently playing around with the Monte Carlo method of finding $\pi$. The idea is pretty simple, I work with a unit square of length one on each side, with the coordinates of $(0,0) , (1,0), ...
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50 views

Linear Filtering Problem (Keynman Fac/Particle Model)

$lienar Filtering Problem $$X_n^1 = X_{n-1}^1 + \epsilon_n *W_n $$ $$X_n^2 = (1-\alpha* \delta) X_{n-1}^2 + \beta*\delta X_n^1 $$ $$X_n^3 = X_{n-1}^3 + \delta*X_n^2$$ above is $$\approx$$ $$dX_n^1 ...
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39 views

Normalisation of Monte Carlo overlap

Background: In quantum mechanics you are sometimes required to compute the overlap (inner product) of two wave functions (square integrable complex functions) $\Psi(x)$ and $\Phi_i(x)$ as $$ ...
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347 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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1answer
33 views

Find Monte Carlo Variance When Expected Value is not Known

I'm working on a problem that can be approached in two different ways. Both are Monte Carlo algorithms--but it's a hard problem, so I am unsure whether the expected values are indeed the same. I ...
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2answers
56 views

Sampling from the diamond: $|x_1|+\ldots+|x_n| \le 1$?

Let $\left(x_1, \ldots, x_n \right)$ be a point in $\mathbb R^n$. Sample uniformly at random from the diamond $$ |x_1|+\ldots+|x_n| \le 1. $$ In $\mathbb R^2$, one way is to sample the square, then ...
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51 views

Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.) Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal ...
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0answers
35 views

Finding Volume of Monte Carlo Integration

Suppose $\mathbf{X}\in\mathrm{R}^n$ is an $n-$ dimensional random vector having joint Gaussian distribution i.e. $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol\mu,\boldsymbol\Sigma\right)$, where, ...
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120 views

Convergence of Monte Carlo simulation

I am not sure if this is a valid question but here goes. For the monte carlo method I know that estimation of the mean is also a random quantity and follows a normal distribution. The standard error ...
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1answer
34 views

Metropolis Monte Carlo with modified acceptance

What happens, if I change the acceptance criterion in a Metropolis Monte Carlo algorithm? I do know the classic proof of detailed balance, which for symmetric transition matrices gives a set of ...
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1answer
116 views

Forecasting future revenue and expensces

I am trying to forecast future revenue and expenses in a company. In the past I used moving average method but later I am more inclined to try to do that by using monte carlo simulation. I am ...
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1answer
63 views

Numerical CDF estimation for complicated random variable

Given a combination $U$ of several random variables $X,Y,Z...$ with known distributions, what is an efficient numerical algorithm to estimate PDF or CDF of $U$, if its CDF, PDF, characteristic ...
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130 views

What does “sequence is equidistributed in [0, 2]” mean?

I was reading an article in which they are mentioning this sentence: "sequence is equidistributed in [0, 2]" where the sequence in question, is a sequence of real number (the article in question is ...
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2answers
160 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
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1answer
25 views

Monte Carlo Methods for non-orthogonal functions

I'm trying to approximate a function using a set of piecewise polynomials. For example, perhaps I'd like to uniformly split the domain [-1,+1] 20 times and place Wendland RBF, Gaussian, or maybe a ...
3
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1answer
168 views

Using Monte Carlo method to estimate the value of $\int_0^1\int_0^1e^{xy} \ dx \ dy$

Monte Carlo, estimate the value of $$\int_0^1\int_0^1e^{xy} \ dx \ dy.$$ I am using matlab to solve this problem. My code is the following: ...
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1answer
196 views

Double Integrals & Expected Value Monte Carlo Method

Tell me if I'm wrong Let $\Omega = [a,b]\times[c,d]\subseteq\mathbb{R}^2$, then $$ \iint_\Omega ...
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1answer
1k views

Evaluating double integrals using monte carlo methods in matlab.

I used the monte carlo method to integrate $\int_{0}^{1}x^2dx$ in matlab. My matlab code was simply the following: A=1; N=10000; s=0; for i=1:N x=rand; y=rand; if y<= x^2; ...
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1answer
31 views

Understanding claim in Newman and Barkema's Monte Carlo book

In Newman and Barkema's Monte Carlo Methods in mathematical physics, on page 23-24, the following claim is made: "Assume we have a function f(x) and the integral $I(x)=\int_0^xf(x')dx'$. Then pick a ...