Questions tagged [numerical-integration]
A class of algorithms to approximate definite integrals.
147
questions
0
votes
0
answers
16
views
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' ...
2
votes
1
answer
51
views
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 $f(x)$...
4
votes
1
answer
56
views
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 ...
1
vote
0
answers
58
views
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:
$$E[X\...
3
votes
1
answer
74
views
Gompertz-Makeham Model Life Expectancy using Numerical Integration and Analytical Solution of Castellares et al. (2020)
I am trying to calculate life expectancies for the Gompertz-Makeham model, but can't replicate the results of the paper which gives the formulas. I use R to ...
3
votes
2
answers
223
views
How to calculate Quasi-Monte Carlo integration error when sampling with Sobol's sequence?
My understanding is that QMC integration using random sampling will converge with $O(\frac{1}{\sqrt{n}})$, while using Sobol's sampling will converge with $O(\frac{(\log{n})^d}{n})$. However I'm ...
11
votes
5
answers
782
views
Estimate the Euler–Mascheroni constant ($\gamma$) by Monte Carlo simulations
The Euler–Mascheroni constant is defined simply as the limiting difference between harmonic series and the natural logarithm.
$$\gamma =\lim_{n\to \infty}\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln n\...
1
vote
0
answers
22
views
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. ...
10
votes
1
answer
330
views
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 ...
2
votes
1
answer
65
views
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{...
2
votes
1
answer
336
views
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 ...
2
votes
0
answers
88
views
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.
...
1
vote
0
answers
17
views
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 ...
1
vote
1
answer
79
views
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.
...
2
votes
1
answer
215
views
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 ...
0
votes
1
answer
477
views
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 ...
0
votes
0
answers
32
views
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{...
-3
votes
1
answer
70
views
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}^{\...
7
votes
1
answer
229
views
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 $f(x,y)$ ...
1
vote
0
answers
29
views
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 ...
0
votes
1
answer
171
views
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 ...
1
vote
1
answer
75
views
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 ...
3
votes
1
answer
122
views
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 ($(...
1
vote
1
answer
168
views
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 $...
0
votes
1
answer
47
views
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 ...
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, ...
3
votes
1
answer
83
views
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 ...
1
vote
1
answer
19
views
how do you find the vector that minimizes [closed]
How do you find the vector that minimizes $\|A\mathbf{x}-\mathbf{b}\|_2^2$ ?
3
votes
0
answers
178
views
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 ...
0
votes
1
answer
87
views
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:
$$\int f(x\mid \theta)\pi(\...
0
votes
0
answers
199
views
Minimize asymptotic variance of fintely many estimates
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded function with integrable Euclidean norm on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\...
2
votes
0
answers
23
views
Examples for integration estimator
suppose I'm interested in estimating $C=\int_{a}^{b}g(x)dx$, where $a$ and $b$ are known, and $g(x)=E(Y|X=x)$ is an unknown function of $x$. The data I have is $\{Y_{i},X_{i}\}_{i=1}^{n}$, then a ...
1
vote
0
answers
107
views
Compute which of a finite number of integrals is minimal (not interested in the actual value of the integral)
Let
$(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space;
$f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
0
votes
0
answers
23
views
multi-dim monte carlo integration -- some complicated integration
I want to integrate
$$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}...
0
votes
0
answers
120
views
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 $$\int_{\{\:p\:>\:0\:\}}\frac{|f-\...
1
vote
0
answers
88
views
Numerical Integration with respect to a mixture of Normals [closed]
I have a likelihood function that contains an integral of a latent parameter. I would like to numerically integrate it using Monte Carlo, as in, $L = \prod_{i=1}^N \int f(X, \tilde{\theta}; \beta) d ...
1
vote
0
answers
68
views
Negative values returned from pmvnorm in R, for multivariate normal integration
I'm using pmvnorm to obtain triple integration for multivariate normal density function in R. The integration range is from negative infinity to finite numbers for all three variables. However, ...
2
votes
1
answer
111
views
Nested Uniform Distributions in Monte Carlo Integration
In terms of importance sampling for numerical Monte Carlo integration we can proceed as follows:
\begin{align}
\int_{\Omega} p(\mathbf{x}) d\mathbf{x} &= \int_{\Omega} p(\mathbf{x}) \frac{q(\...
0
votes
0
answers
21
views
Convergence of Diffusion Process Monte-Carlo
Let $X_t$ be a $d$-dimensional diffusion process initialized at $x \in \mathbb{R}^d$; given as the strong solution to the SDE
$$
X_t = x + \int_0^t a(t,X_t)dt + \int_0^t b(t,X_t)dW_t;
$$
where $a$ and ...
1
vote
0
answers
66
views
How to approximate expectation and variance of an integral from a discrete Time series financial dataset?
I have discrete time series financial data, with time($u$), price($S$) and someVariable($q$) which looks something like this.
...
2
votes
2
answers
145
views
Mean and variance of a non-standard pdf
I have tried to compute the variance and the mean for $\mu=0.5$ of the following PDF using Wolfram cloud but I failed
$$ F(z,\mu,\sigma)=\frac{2 (z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2
\...
0
votes
1
answer
265
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 ...
0
votes
1
answer
56
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
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 ${\mathbb R}$. ...
2
votes
0
answers
23
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 ...
13
votes
2
answers
246
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
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 ...
0
votes
1
answer
134
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 ...
0
votes
0
answers
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 ...
2
votes
0
answers
16
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 + ...