A vast area which includes generating results from computer models.

learn more… | top users | synonyms

4
votes
0answers
45 views

Shaking a box of rocks (Optimal Packing)

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...
4
votes
0answers
153 views

How to compute this triple integral?

I am trying to do this triple integral $$\int_{0}^{\infty }\int_{0}^{\infty }\int_{0}^{\infty }(u+w)e^{-\frac{(u+w)^2}{2}}(v+w)e^{-\frac{(v+w)^2}{2}}(u+v)e^{-\frac{(u+v)^2}{2}}e^{-(\mu +\lambda ...
3
votes
0answers
31 views

Simulating a SDE

I have a question about simulating a SDE. I want to simulate $dS=\alpha(K-S)dt+\sigma S dZ$ with use of a Euler-marayama scheme. The numerical scheme becomes: $S_{i+1}=S_{i}+\alpha(K-S_{i})dt+\sigma ...
3
votes
0answers
71 views

Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with ...
3
votes
0answers
182 views

Simulation and Stochastic Processes

Supposing we want to take a sample from the distribution $p(x)=cp^*(x)$ where $c$ is the normalization constant and $p^*(x)$ is given by $$p^*(x)=0.5\exp(-(x-\mu_1)^2)+0.5\exp(-(x-\mu_2)^2).$$ ...
2
votes
0answers
32 views

Test a law-of-iterated-logarithm-like result, with numerical simulation

I have a non-standard random walk $S_n$ for which the increments are not exactly independent (I could describe it, but it would be a totally different long and complex topic). I expect it to have ...
2
votes
0answers
25 views

Calculating integral with antithetic variables

Use simulation with antithetic variables and find $$\int_{-\infty}^\infty \int_0^\infty \sin(x+y)e^{-x^2+4x-y} \, dx \, dy.$$ so, my question and doubt is how struggle with the infinite limit ? It ...
2
votes
0answers
27 views

Likelihood that two markov chains are derived from the same transition matrix

Forgive me for my weak statistic background, hopefully what I'm asking makes sense. So some quick background, I have one markov chain from a data set and many additional chains that I'm producing from ...
2
votes
0answers
70 views

Solving Kepler's Equation

I've been working on simulating orbits. I've found that, when solving Kepler's equation, $M = E - \varepsilon\sin{E}$, I'm unsure about the solution to use. For a true anomaly $< \pi$, using the ...
2
votes
0answers
33 views

Are these two approaches to calculating return rate mathematically consistent?

I have coded two C# programs, which use two different approaches to evaluate the outcome of a certain casino-style game (casino-style in the sense that the user pays points to take a turn, and ...
2
votes
0answers
24 views

orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
2
votes
0answers
125 views

Positive eigenvalues in differential-algebraic equations not appearing in time-domain simulation

I am solving a system of equations derived from power system applications. It consists of index-1 differential and algebraic equations in the form: $$\dot{x}=f(x,y) \\ 0=g(x,y)$$ To get the ...
2
votes
0answers
143 views

Simulating first passage times

I have a Brownian motion $X_t = \mu t+\sigma W_t$, where $W_t$ standard Brownian motion. I know that the first passage time $\tau = \min\{t|X_t\leq\alpha\}$, is Inverse Gaussian distributed i.e., ...
1
vote
0answers
36 views

Simulate a non-homogeneous poisson process

I was trying to simulate a non-homogeneous poisson process with hazard rate function $$ \lambda(t)=3+\sin(2\pi t) $$ I tried to use the property that given $N=n$, arrivals in $[0,T]$ are distributed ...
1
vote
0answers
29 views

Why does this cellular automaton generate circular patterns?

I made a kind of cellular automaton game with the following rules. Each cell in a rectangular grid has a "water level" (a 32-bit floating-point number). In the next generation, water "flows" from each ...
1
vote
0answers
29 views

Quantile lines of stationary process

Quantile lines of any stationary process are parallel and constant. But for different procesess I've obtained different behavior of quantile lines. First case was process after Lamperti transormation ...
1
vote
0answers
44 views

Self learning complex systems modeling

I am currently a senior studying mathematics. My program is heavily focused on pure maths, however, my interests lie in the area of applied mathematics. Specifically, I am interested in the study of ...
1
vote
0answers
70 views

interarrival time

I am trying to model the arrival rate (delay between the arrivals) of the cars in my city. I have some real data with the resolution of one minute. For example, Number of counted cars at position x at ...
1
vote
0answers
30 views

Determanistically skipping through time of a time homogeneous Markov chain

Suppose I have an infinite number of time steps $X_0,\ldots,X_i,\ldots$, where each $X_i$ is an infinite dimensional random vector consisting of 0's and 1's. I now specify $P(X_i|X_{(i-1)})$ and an ...
1
vote
0answers
19 views

How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
1
vote
0answers
164 views

distribution of the length for a random walk on an infinite 2D grid

In connection with the flatland paradox, consider a 2D-random walk $(X_n)$ on $\mathbb{Z}^2$: the four moves of length one to W,E,N, and S are equaly likely at each time. For a fixed number of moves ...
1
vote
0answers
58 views

Simulating r.v.'s from a joint density by rejection sampling in R. Continued

I wish to sample variables $v$ and $w$ from the joint density $$(v+w)e^{-\frac{(v+w)^{2}}{2x_{0}}-2\mu v-(\mu -\lambda )w},$$ where $x_0$, $\mu$ and $\lambda$ can be seen as positive constant. Since ...
1
vote
0answers
94 views

Personal Experiences with Probability Simulation

Simulations methods are increasingly used in theoretical and (especially) applied probability. Personally, I have used simulation for purposes that range from recreational Q&A to applications of ...
1
vote
0answers
49 views

Is Principal Component Analysis applicable to this type of situation?

I'm trying to model the response of ant populations to pheromones in this way: The ants are simulated as Self Propelled Particles with internal energy. They undergo acceleration due to their internal ...
1
vote
0answers
165 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 ...
1
vote
0answers
125 views

Statistics Marble Simulation

Given the jar containing 20 red, 5 white, and 10 blue marbles, Joey thinks a more interesting problem would be to find the number of marbles you would have to draw, without replacing them in the jar, ...
1
vote
0answers
161 views

Backward Euler method with a cross-product.

I want to solve the following differential equation with the backward Euler method ...
1
vote
0answers
20 views

Stability of simulation of brownian noise

As I understand, Brownian noise can be simulated by the process $$x_{n+1}=x_n+R_n$$ where $R\sim U[-a,a]$. The expected value for $x_n$ is then $x_0$. But $\text{Var} x_n\to\infty$ as $n\to\infty$ ...
1
vote
0answers
50 views

Distribution of phone calls during 24h

I would like to model the amount of phone calls at each time of the day. The phone calls should follow a poisson distribution and at 12:00 there should be the peak. So, semantically what I would like ...
1
vote
0answers
72 views

Forest fire simulation; analytically constructing a function for tree residual after fire

Consider a Cellular Automaton with an $n \times n$ grid, where each cell corresponds either to a tree or dirt. We assign a tree to cell $(i,j)$ by probability $p$. Next, we initiate a fire in some ...
1
vote
0answers
107 views

Kalman Filter application to non-linear system.

I want to use the Kalman filter to have a better estimate of the state of a system which I know its equations of motion: ...
1
vote
0answers
79 views

simulate hitting time of Brownian motion

let's say I have a brownian motion $W_t$, and I know the value of $W_1$. Is there a way to simulate the hitting time of $W_t$ and a given function $f(t)$ ? For instance I know that if $f(t)$ is a ...
1
vote
0answers
52 views

Random graphs simulation

Reading the article "Emergence of scaling in random network, by Barabasi and Albert" I faced a lot of results obtained by simulations of the A-B random graph model. I always wanted to do such ...
1
vote
0answers
69 views

Samples from the Dirichlet measure

In Ferguson, 1973, Definition 2, he defines a sample of size $n$ from a random probability measure $G$ on $(\mathcal{X}, \mathcal{B})$ as: $$ P(X_1 \in C_1, \cdots, X_n \in C_n | G(A_1), \cdots, ...
1
vote
0answers
34 views

Single evaluation for using exponential sampling until past a point

I am trying to improve an algorithm that looks like the following (and am getting stumped): I am provided with a starting time, rate, and a target time. I then use an exponential distribution to ...
1
vote
0answers
52 views

Simulating of GBM

I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form $dS(t) = \mu S(t) dt + ...
1
vote
0answers
130 views

Approximation of SDE

I have been struggling with the following problem: If you want to find a numerical result by simulating the paths of a stochastic differential equation, in particular a geometric brownian motion I ...
1
vote
0answers
119 views

Maximizing noisy unknown function

I'm interested in maximizing a function $f(\mathbf \theta)$, where $\theta \in \mathbb R^p$. The problem is that I don't know the analytic form of the function, or of its derivatives. The only thing ...
1
vote
0answers
191 views

Expected value involving a conditional multinomial distribution

$X|z$ has a multinomial distribution $MN(m, \mathbf{q}(z))$ where $z$ is either 0 or 1 with probability $1/2$. I need to find: $E_X[\max\{\Pr(z=1|X), \Pr(z=0|X\}]$. Is there an analytical form to ...
1
vote
0answers
274 views

Semi implicit integration - stability issues

I am trying to decide whether to use semi-implicit integration vs. explicit integration (particularly Position Verlet over Semi implicit Euler). Although the Verlet approach is widely used and is ...
1
vote
0answers
200 views

Optimization via Simulation

I want to minimize and objective function $\hat{B_i}$ $i\in l$, which can be computed by a matlab code (assume $\operatorname{findB}(a, b, c)$ returns $B$. I have the following optimization problem: ...
1
vote
0answers
172 views

Markov Chains - Using Gibbs & Metropolis algorithm.

Suppose $f_x,_y$ is bivariate normal distribution. I was given the parameters $(μ_1, μ_2, σ_1^2, σ_2^2)$ and $ρ=0.95$ the correlation coefficient. I want to generate $(x_1,y_1), ...
0
votes
0answers
15 views

Is there a simple distribution/function that behaves like fermi distribution but with a $\exp(-x^2)$ tail?

I have a data file with the following data (see picture). I try to find a simple function/distribution that follows the same trend : A behaviour like a Fermi-Dirac distribution A behaviour like a ...
0
votes
0answers
19 views

Analytical Case Differentiation

is there a analytical way for case differentiation? In my case a MonteCarlo Simulation calculates a system of equations. Parameters can randomly change so that the underlied mathematical condition ...
0
votes
0answers
10 views

Proving weak simulation

I want to prove something but I am not sure if it is the right way to do it. I have two LTS that define different semantics. A=($Q_a,Λ,\to)$, and B=$(Q_b,Λ\cup\{\beta\},\leadsto)$, where $\beta$ is ...
0
votes
0answers
34 views

Adequacy of Monte-Carlo simulations

Suppose we have a number of independent random variables of the form $X_1 \sim U[a_1,b_1], X_2 \sim U[a_2,b_2], X_3 \sim U[a_3,b_3]$. Now, suppose we generate a random variable $Y$ as follows: $$Y = ...
0
votes
0answers
16 views

Preventing cars from leaving a crossroad in a cellular automata traffic simulation

I'm writing a traffic simulation using cellular automata based on this paper It states that the rule in the middle of the crossroad always stays the same (184), but that the cell after the crossroad ...
0
votes
0answers
19 views

Plotting the fundamental diagram of traffic flow

I have a traffic simulation and I don't understand how I can plot the fundamental diagram (flow rate vs density). I simulate the traffic as follows: I have a matrix that has as many columns as the ...
0
votes
0answers
23 views

lemma proof for alias method for generating discrete random variables

I'm looking to prove the lemma written in chapter 11, page 274 of Sheldon M. Ross's Simulation, regarding the alias method for random variable generation. As a prelude to presenting the method for ...
0
votes
0answers
18 views

Using acceptance rejection method with two variables

I'm having trouble using Acceptance rejection method to simulate the following r.v $$ f(x,y)=Ke^{-x^2-y^2+x\sin(y)} $$ where $K$ is just the normalizing constant. Most specifically, any ideas on what ...