Questions tagged [numerical-integration]

A class of algorithms to approximate definite integrals.

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How can I compute rectangular confidence regions for parameters using R?

Simultaneous confidence regions for multivariate parameters (say, a confidence region for multivariate mean, or for regression parameters) usually find an elliptical region when the parameters' ...
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2 votes
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Bayesian Quadrature to find expectation of unkown function w.r.t. known pdf

I am interested in estimating the integral $\int f(x) P(x) dx$, where $f(x)$ is an expensive function and $P(x)$ is has an analytic form. I would like to evaluate this with as few evaluations of $f(x)$...
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4 votes
1 answer
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Approximate Posterior Predictive Quantiles with Numerical Methods

I have a posterior function which is easy to approximate using numerical methods (the posterior has only 2 parameters, and is approximately Gaussian because of the large sample). However, I need to ...
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Conditional expectation of a normal variable given a lower bound [duplicate]

I saw the identity below but I'm not sure how to derive it. $$E[X\mid X>K] = \mu + \sigma \frac{\phi(z)}{\Phi(-z)} \text{ where } z = \frac{K-\mu} \sigma$$ I'm stuck at the following step: $$E[X\...
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3 votes
1 answer
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Gompertz-Makeham Model Life Expectancy using Numerical Integration and Analytical Solution of Castellares et al. (2020)

I am trying to calculate life expectancies for the Gompertz-Makeham model, but can't replicate the results of the paper which gives the formulas. I use R to ...
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3 votes
2 answers
223 views

How to calculate Quasi-Monte Carlo integration error when sampling with Sobol's sequence?

My understanding is that QMC integration using random sampling will converge with $O(\frac{1}{\sqrt{n}})$, while using Sobol's sampling will converge with $O(\frac{(\log{n})^d}{n})$. However I'm ...
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11 votes
5 answers
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Estimate the Euler–Mascheroni constant ($\gamma$) by Monte Carlo simulations

The Euler–Mascheroni constant is defined simply as the limiting difference between harmonic series and the natural logarithm. $$\gamma =\lim_{n\to \infty}\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\...
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What is the expectation of a random variable satisfying some conditions?

How to find the expectation E[X.I(Y<x,X<x)], where X and Y are independent random variables with respective cumulative distribution functions F(.) and G(.) respectively. x is a positive value. ...
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10 votes
1 answer
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Plain English explanation of Ito's integral?

I'm looking for a plain English explanation of Ito's integral. I don't need an exhaustive proof, derivation, etc. Just a simple ~this is effectively what it does and why it's better than a Riemann sum ...
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Numerically integrate gaussian pdf

Suppose I want to numerically integrate the function $g(\mathbf{x}) = \exp\bigg(-\frac12 \mathbf{x}^\mathsf{T}\mathbf{\Lambda} \mathbf{x}\,\bigg)$ to obtain the normalization constant $$\int_{\mathcal{...
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Kullback–Leibler divergence between multivariate t and the multivariate normal?

I want to calculate the Kullback–Leibler divergence between a multivariate $t$ distribution and a multivariate normal distribution, for different values of the degrees of freedom $\nu$. However, this ...
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Numerical integration of the exact distribution of the product of two random gaussian variables

I am trying to implement this function in R, which is equation (2) in this paper by Cui et al. (2016), which is the exact distribution of the product of two random gaussian variables. ...
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How do mixed models bypass the overflow inside the integration?

In a mixel model, when you try to maximize the likelihood, you have to integrate over the random effects. This means that you cannot take logs, since what would normally be a product of individual ...
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Computing the probability $P(\exists X\in\{X_1,\ldots, X_N\}:X>\text{max}\{Y_1,\ldots, Y_M\})$

I am trying to somewhat generalize a question, which has been asked in one way or another a several times here on StackExchange**. However, I have not managed to find an answer to the below problem. ...
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How to calculate confidence intervals for a generic (non-normal) pdf

In particular, this is a question about distributions with some skew and/or where the mean, median and mode might all be different. For any pdf, it's possible to find multiple different domains within ...
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Different estimates for mixed effects Logistic regression and pwrssUpdate Error message with binomial glmer

Hi this is my first post here. I reported an issue on Github and got valuable help from Dr. Bolker. However, I did some other analyses on the same data and got confused about some of the results. I ...
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Discrete correlation function (sample cross-covariance)

The continuous correlation function for the random variable $A(t)$ at a instant of time $t$ is given by \begin{equation} C_{AA}(\tau) =\frac{1}{T} \int_{0}^T d\bar{t} A(\bar{t})A(\bar{t}+\tau) \end{...
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Positive or negatively bounded CDFs [closed]

If $X\in\mathbb{R}^n$ is a continuous random variable whose cumulative distribution function is ordinarily $$F_X(x) = \int_{-\infty}^{\infty} f_X(x) dx $$ what is the meaning of $$F_X(x) = \int_{0}^{\...
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7 votes
1 answer
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Expressing a marginal probability using copulas

Please correct me if I am wrong and kindly provide me with the correct notations. I have two questions: We know that for the variables $(X,Y,Z)\in \mathbb{R}^3$, the marginal joint density $f(x,y)$ ...
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How to calculate Integral error from MCMC

I've recently started using Markov Chain Monte Carlo to calculate integrals, but I can't seem to find how to calculate the error for such an integral. In the standard case of importance sampling, the ...
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Computational Complexity of Multivariate Normal CDF

I'm going to post this here also as per user suggestions since I feel like the root cause of the issue is more maths related than code related. I'm working on a multivariate cross-entropy minimization ...
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1 vote
1 answer
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Finding confidence interval for unimodal function equivalent to and comparable with standard deviation of normal

I'm trying to characterise an arbitrary, unimodal distribution in a way that is a) easily understandable (to a physics audience) and b) comparable with a normal distribution. My thinking goes like ...
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3 votes
1 answer
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Numerical integration with measurement errors: what scheme to use?

I have an unknown 'smooth' function that is sampled at some $n$ equally spaced locations $x_i$ between $0$ and $1$, with some measurement values $f(x_i)=y_i$, but each has some Gaussian uncertainty ($(...
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Calculating variance of MC estimator of an integral

Let $$ f(x) = \sqrt{1-x^2}, \quad I = \int \limits_0^1 f(x) dx. $$ We consider the following estimator of I: $$ \hat{I}_n = \frac{1}{n} \sum \limits_{i=1}^n f(X_i), $$ where $X_1, X_2, ...$ are iid $...
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Why are my sampled values are non Gaussian?

I just have a quick question regarding Importance Sampling Monte Carlo integration. If I sample from some pdf, $p(x,y)$, to calculate an integral. I.e., $I = \int f(x,y) \ dx\ dy$ It can be ...
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1 vote
1 answer
322 views

Integration with accept reject sampling Monte Carlo

I've got a quick question with regards to accept-reject Monte Carlo integration that I can't solve. Suppose I want to integrate some function, $f(x,y)$, with samples of $x, y$ from $p(x,y)$. Now, ...
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3 votes
1 answer
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How to calculate individual moments of a 2-dimensional distribution via Monte Carlo Integration

I've recently been using Monte Carlo integration to calculate a particular integrals which I can do fine but I've hit a problem where I can't calculate the individual moments of my distribution for ...
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1 vote
1 answer
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how do you find the vector that minimizes [closed]

How do you find the vector that minimizes $\|A\mathbf{x}-\mathbf{b}\|_2^2$ ?
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3 votes
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Uncertainty propagation in ODEs

I want to see the effect of parameter uncertainty in the Euler method for ODEs. For a differential equation: $dx/dt=f$ with initial condition $x(0)=xo$ and a function $f$ (that has uncertain ...
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Relatively fast approximations to the marginal likelihood?

Let $\theta\in{\mathbb R}^d$ be a multidimensional parameters, where $d$ can be large (e.g. $d=100$ or more). What approximations can I use for the marginal likelihood: $$\int f(x\mid \theta)\pi(\...
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199 views

Minimize asymptotic variance of fintely many estimates

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded function with integrable Euclidean norm on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\...
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2 votes
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Examples for integration estimator

suppose I'm interested in estimating $C=\int_{a}^{b}g(x)dx$, where $a$ and $b$ are known, and $g(x)=E(Y|X=x)$ is an unknown function of $x$. The data I have is $\{Y_{i},X_{i}\}_{i=1}^{n}$, then a ...
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Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
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multi-dim monte carlo integration -- some complicated integration

I want to integrate $$I= \int^1_0 \int^1_0 \frac{g(x,y)}{\int^1_0 h(x,y)\,dx} \,dx\,dy. $$ For now, I sampled $n \times n$ simulation samples from $unif[0,1]$ and estimate $$\hat{I} = n^{-2}\sum_{j}...
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Estimate $\lambda\frac{|f-\lambda f|^2}p$ without looping twice

Let $(E,\mathcal E,\lambda)$ be a measure space, $p$ be a probability density on $(E,\mathcal E,\lambda)$ and $f\in\mathcal L^2(\lambda)$. Say I want to estimate $$\int_{\{\:p\:>\:0\:\}}\frac{|f-\...
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1 vote
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Numerical Integration with respect to a mixture of Normals [closed]

I have a likelihood function that contains an integral of a latent parameter. I would like to numerically integrate it using Monte Carlo, as in, $L = \prod_{i=1}^N \int f(X, \tilde{\theta}; \beta) d ...
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1 vote
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Negative values returned from pmvnorm in R, for multivariate normal integration

I'm using pmvnorm to obtain triple integration for multivariate normal density function in R. The integration range is from negative infinity to finite numbers for all three variables. However, ...
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2 votes
1 answer
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Nested Uniform Distributions in Monte Carlo Integration

In terms of importance sampling for numerical Monte Carlo integration we can proceed as follows: \begin{align} \int_{\Omega} p(\mathbf{x}) d\mathbf{x} &= \int_{\Omega} p(\mathbf{x}) \frac{q(\...
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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 ...
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1 vote
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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. ...
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2 votes
2 answers
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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 \...
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1 answer
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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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1 answer
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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: ...
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7 votes
2 answers
131 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 ${\mathbb R}$. ...
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2 votes
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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 ...
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13 votes
2 answers
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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 ...
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7 votes
4 answers
878 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 ...
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0 votes
1 answer
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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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126 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 ...
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2 votes
0 answers
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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 + ...
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