Questions tagged [numerical-integration]
A class of algorithms to approximate definite integrals.
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questions
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Convergence of Diffusion Process Monte-Carlo
Let $X_t$ be a $d$-dimensional diffusion process initialized at $x \in \mathbb{R}^d$; given as the strong solution to the SDE
$$
X_t = x + \int_0^t a(t,X_t)dt + \int_0^t b(t,X_t)dW_t;
$$
where $a$ and ...
0
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0answers
13 views
How to approximate expectation and variance of an integral from a discrete Time series financial dataset?
I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this.
...
2
votes
2answers
92 views
Mean and variance of a non-standard pdf
I have tried to compute the variance and the mean for $\mu=0.5$ of the following PDF using Wolfram cloud but I failed
$$ F(z,\mu,\sigma)=\frac{2 (z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2
\...
0
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1answer
26 views
is it a good idea to take the derivative or integral of some features and add them as new features in machine learning?
I'm learning how to do feature Engineering and come across some ideas in my head that's why I want to ask if I had some dataset with some features let's say 2 features and I have a timestamp column ...
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0answers
17 views
More efficient way to calculate convolution of Weibull distribution
Hi I was trying to compute the convolution of Weibull distribution as follows with parameter t, lambda(scale parameter) and k:
...
0
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1answer
43 views
Expectation of continuous rv X^2
I ran the following algorithm to find the expected value of X^2 for a random variable X with pdf:
exp(-abs(x)^3/3)
This is what I did and my results:
...
7
votes
2answers
101 views
What is better in Monte Carlo integration: product of means or mean of products?
Let $X$ and $Y$ be two independent continuous random variables with pdfs $f_X$ and $f_Y$, respectively. Let $\varphi_1$ and $\varphi_2$ be two continuous functions from ${\mathbb R}$ to ${\mathbb R}$. ...
2
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0answers
15 views
Double Integral involving Beta Functions (about Pareto Distribution)?
I have tried evaluate $(m_i,m_j)$th product moment of $X_{(i)}$ and $X_{(j)}$ order statistics of Pareto Distribution, that is $E[X_{(i)},X_{(j)}]$, where $i\le j$ , $X_1,X_2,...,X_n$ i.i.d. from ...
12
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2answers
188 views
Why is pseudo-random sampling applicable for Monte Carlo integration, even though it does not satisfy the CLT requirements?
Assume we have a function $f\left(x\right)$ defined on $\left[0, 1\right]$ that we want to integrate and estimate the error using Monte Carlo method. We generate realizations of uniformly distributed ...
7
votes
4answers
398 views
Expectation of $\ln(1 + e^x)$, where $x$ is normally distributed
I need to evaluate the following integral:
$$\int_{-\infty}^\infty\mathrm d x \exp\left(-\frac{(x-\mu)^2}{2\nu}\right) \ln(1+e^x)$$
where $\mu$ is a finite real number and $\nu > 0$. This is just ...
0
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1answer
41 views
Understanding Gauss-Hermite Weights
I routinely use Gauss-Hermite as a tool for approximating complex integrals. While I am proficient in its applications, I am not proficient in its development. I am working to understand the weights ...
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0answers
38 views
quickly finding the moments of a numerically-defined PDF
I have a two-dimensional continuous PDF which is numerically defined. From this PDF I would like to extract the second central moment (variance) of a "slice" of this distribution.
The "slice" is to ...
2
votes
0answers
8 views
Best methods to integrate summed probability and uncertainty from multiple sources?
I am trying to calculate risk of coastal flooding. In our model, we have 4 major sources of uncertainty:
Elevation at location (height + gaussian error)
Sea level rise (value at any given year + ...
2
votes
1answer
113 views
Why should there be two solutions for each parameter of likelihood ratio equation for Weibull-distribution?
I try to calculate the confidence ínterval of a Weibull distribution by means of the Likelihood method as described in ReliaWiki. In order to find the confidence intervals, I have to solve the ...
4
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0answers
33 views
Integration by sampling from truncated distribution
I'm reading the book Ben Lambert's Bayesian Statistics: problems and answers, which by the way I like.
There is a group of problems in "Integration by Sampling" chapter 12.
The first integral is
$$...
2
votes
1answer
39 views
How to calculate mean and variance from very small function proportional to the density?
Problem
Say I have the following function $g(x)$, which is proportional to the density function $f_\theta(\theta)$ of random variable $\theta$, i.e. $g(\theta) \propto f(\theta)$, such that
$$
\...
1
vote
0answers
29 views
Error bars of Monte Carlo expectation with correlated samples
I will try to phrase the question in a general way, then give my specific case as an example.
Suppose I want to evaluate $Q = \mathbb E \left[ f\left(X, Y \right) \right]$ where $X$ and $Y$ are ...
1
vote
1answer
198 views
Estimating the population median from a kernel density estimator
I have a 1-d kernel density estimate in the form of two vectors:
x_grid is a vector of x-values at which the density function was sampled
...
1
vote
0answers
35 views
Monte Carlo integration for Bayesian parameter estimation
I want to determine the credible interval of a quantity $\theta_1$. I want to make this estimate using observed data by assuming a certain model which depends on $\theta_1$ as well as about n=15 ...
0
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0answers
11 views
What kind of algorithms are appropriate for this sort of medium-dimensional integration problem?
I'm trying to model a situation in which an agent must select one of several choices (not more than ten). Each choice is associated with a vector, known to the agent, representing its effectiveness in ...
6
votes
2answers
210 views
Numerically/approximately integrating over independent gamma variables
Problem Statement
For a problem in biology, I am testing out a joint distribution of the form:
$$
X \sim Multinomial(\frac{\theta_1}{\sum \theta_i}, ...,\frac{\theta_n}{\sum{\theta_i}}) \\
\theta_i \...
1
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0answers
38 views
How to perform MCMC integration when no prior over the integrated function is available? [closed]
As far as I can tell, MCMC integration (e.g. VEGAS) is performed by sampling from a distribution proportional to $f(x)$ using MCMC, then building a density estimator $g(x)$ using these samples (for ...
2
votes
1answer
47 views
Evaluating problematic function when cdf is close to one?
Let $F(x;\theta)$ be a cumulative distribution function and $\beta>0$. I need to evaluate
$$\rho=\frac{F(x;\theta)^\beta}{F(x;\theta)-F(x;\theta)^{\beta+1}},$$
but, for some values of $\theta$, R ...
2
votes
1answer
51 views
Calculating Integral Using MCMC
Consider the integral $\int_{\Theta}f(\theta|\mathbf{x}) \Pi(\theta)d\theta$,where $\theta$ is a univariate parameter and $\Theta$ is the support of $\Pi(\theta)$. I need to evaluate the value of this ...
2
votes
1answer
85 views
Issues with qbeta and pbeta [closed]
I'm trying to get samples from truncated beta. I have written the following function
based on inverse transform sampling method to do so, however, it seems that when CDF converges to 1 fast, pbeta ...
4
votes
2answers
127 views
MCMC combined with numerical integration towards more efficient Bayesian inference
I am quite new to Bayesian statistics so the question can be a bit naive.
My question is on how to deal with a model with individual coefficients. Simple versions of a task and a model I deal with is ...
2
votes
0answers
190 views
Calculating expected loss of posterior distribution
I'm working with 2 posterior distributions from AB tests. For the sake of simplicity let's assume:
$$ A\sim Beta(10, 20) $$
$$ B\sim Beta(5, 25) $$
I want to calculate the posterior expected loss of ...
2
votes
1answer
625 views
What is the difference between monte carlo integration and gibbs sampling?
I am aware that both are methods of sampling from the posterior. MC integration replaces the integral by a sample MC sample.
Is this sample independent?
Gibbs sampling is a class of MCMC ...
0
votes
1answer
105 views
Approximate a CDF
Suppose we have $n$ equations with an integral of the form $\int_0^{x_i} F(z)dz = c_i,\ i=1,\ldots,n$ where $F(y)=\mathbb{P}(X \le y)$ is an unknown cumulative distribution function of a non-negative ...
5
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1answer
246 views
Constant of Laplace approximation
I'm reading Example 3.16 of Robert & Casella's Monte Carlo Statistical Methods. It uses a Laplace approximation for approximating an integral related with the Gamma distribution namely $$\int_a^b\...
6
votes
1answer
783 views
How to choose best proposal distribution for importance sampling
From Robert & Casella p95, we know that the choice of proposal distribution $g(x)$ with minimal variance is the $g$ proportional to $|h(x)|f(x)$. If we restrict our proposal distribution to cetain ...
1
vote
2answers
185 views
Expected value of $X$ which follows a normal distribution, between a certain interval [duplicate]
What is the process of finding the expected value of $X$ in a normal distribution between a certain interval?
In particular I want to find: $E(X | a \le X \le b)$.
For example, if $X$ has $\mu=0$ ...
3
votes
1answer
304 views
Calculating integral by using Monte Carlo method
I am asked to calculate the value of the following integral by using Monte Carlo method. $$I=\int_{\mathbb{R^{10}}}(2\pi)^{-10/2}\exp\left(-\sum_{i=1}^{10}\frac{x_i^2}{2}+\sin\left(\sum_{i=1}^{10}x_i\...
3
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0answers
384 views
Estimating standard error of Monte Carlo integration, non-MCMC version
Let us suppose that we're to evaluate the expectation of a random variable $h$ with respect to some distribution $\pi$, $\text{E}_{\pi}[h]$. The standard Monte Carlo estimate, using a sample of $X_1, ...
1
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1answer
54 views
Efficient estimation of conditional means from pdf, CDF, & quantile function supplied numerically
Suppose I have a a probability distribution that I know to have a continuous differentiable unimodal pdf, with pdf(x) strictly greater than zero for all x in the positive half-plane. In addition, I ...
0
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1answer
2k views
How to calculate joint distribution by integrating over all possible values of model parameters and observations
I was wondering how to calculate the following joint distribution by assuming that $x_i$'s are continuous observations from normal distirbution $N(\mu, \sigma)$ with mean $\mu$ and variance $\sigma$ ...
3
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0answers
194 views
Low-valued multivariate integral in R [closed]
I'm trying to calculate multivariate normal integrals in R, with relatively high dimensions. I've tried the openMX and cubature packages. The problem is that the integrals become too small and I get ...
0
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1answer
295 views
Metropolis-Hastings Algorithm for Numerical Integration [duplicate]
I'm attempting to implement a Metropolis-Hastings Algorithm to evaluate integrals of the following form:
$$I =\frac{1}{\sqrt\pi}\int_{-\infty}^{\infty} {f(x)\exp(-x^2)} \text{d}x$$
Now we can ...
2
votes
1answer
510 views
What does it mean that high dimensional integration is difficult?
When we say numerical integration is difficult for high dimensional problems, what do we mean by high dimensional?
For example, in the Bayesian framework, the marginal normalizing constant can be ...
2
votes
2answers
178 views
Savitzky-Golay … integrator?
Background:
Savitzky-Golay filters (yes they have other names) are robust estimators of slope. Where a small noise can substantially damage the slope estimate of textbook finite difference methods, ...
asked Jul 20 '17 at 13:54
EngrStudent - Reinstate Monica
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0
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1answer
269 views
Rollmean vs. Integrate.xy for computing Integrals
I have a density function in R that reflects an underlying null distribution, for example:
density_null=density(rnorm(100))
I want to integrate between 0 and ...
1
vote
0answers
195 views
Advantages of monte carlo over numerical quadrature for integration in low dimension MLE?
I am doing a parameter estimation for a two-level model via maximum likelihood estimation. My MLE optimization procedure requires multiple numerical integrations over a 3 dimensional space at each ...
1
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0answers
56 views
Proving an equivalence relation by using numerical methods such as Gaussian quadrature
The background is residual useful life prediction. The following is my problem description.
Degradation signal path: $r(t)=\phi+\theta t$ , where $\phi$ is assumed to be the same for all units，and $\...
2
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0answers
53 views
Numerical methods for one-dimensional Bayesian inference
I am doing inference using a Bayesian model that has only one variable, which boils down to computing a (one-dimensional) cumulative distribution and a quantile function given the log of a probability ...
2
votes
0answers
145 views
Maximum likelihood estimation using a random number generator
Imagine that I have a random number generator $X$ where it is impossible (or mathematicaly intractable) to calculate its probability density function
- this can happen when we compose several simpler ...
0
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1answer
472 views
Having integration in Bayesian model: what package to use?
I have a bit experience on Bayesian analysis and am a JAGS user. Recently I have to run a more complicated model that contains integration (I use adaptIntegrate from cubature package in the original ...
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1answer
84 views
I have three probabilities of a value falling within the following ranges +/- 5%, +/- 10% and +/- 20% of the mean value
I have the mean value and my question is: is it possible to calculate the standard deviation assuming a normal distribution? Looking at previous questions which have been asked, this question is ...
2
votes
1answer
105 views
The median of the absolute value of the difference of two dependent log normal random variables
Assume X and Y have a bivariate lognormal distribution (x,y>0) that is:
$f_{X,Y}(x,y)$=$$\frac{1}{2p\sqrt{1-r^2}xy\sigma_1\sigma_2}exp\{\frac{-1}{2(1-r^2)}[(\frac{ln(x)-\mu_1}{\sigma_1})^2-2r(\frac{...
1
vote
1answer
160 views
Linear regression of B-splines with terms inside an integral?
I have encountered a problem that the literature suggests linear regression is able to solve, but I am at a loss.
I have a function $F$ that I want to estimate. This function obeys $N$ equations of ...
6
votes
1answer
625 views
Use Importance Sampling and Monte carlo for estimating a summation
Maybe my question is pretty basic and dumb. I'm studying computer science. In one problem i have to use Monte Carlo method and Importance sampling in order to estimate a big sum. I've seen Monte Carlo ...
