1
vote
0answers
83 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
4
votes
0answers
161 views

Pair of two-variable polynomial equations of high order

I have the following pair of equations to be solved for two variables $\rho$ and $D$ resulting from a certain Maximum Likelihood Estimation for a time series $X_n > 0$, $n=0, \ldots, N+1$ with $N ...
1
vote
1answer
131 views

Understanding the rationale behind “batch means” estimation

Hello all, I am implementing an MCMC algorithm for my work, and I've come upon something in the literature which I just can't understand. Specifically, I am attempting to estimate the amount of ...
10
votes
3answers
444 views

Rapid evaluation of multivariate normal integral

I'm implementing a model that requires me to numerically evaluate a multivariate normal integral of the following form $$\int_{-\infty}^\infty \phi(z)\displaystyle\prod_{i=1}^N \Phi(a_iz+b_i) \, ...
15
votes
1answer
434 views

The Chow & Robbins game ≈ 0.79295350640: improvements could come from simple statistics, or from a continuous version of the game

This question seeks help with improving a numerical estimate of the value of the Chow and Robbins game. Much about this game is unknown, such as whether its value is rational, but there are two routes ...
2
votes
1answer
133 views

Optimization problem

I'm trying to solve a very practical optimization problem and I think I hit a dead-end. There are $N$ products ($N \sim 50$). Each product can have a price $p_i$ in range between 1 and 40 dollars. ...
2
votes
0answers
196 views

Convergence rate of iterated nonlinear equations?

For $i=1, \dots, n$ ($n$ could be large) we have variables $x_i$ and $y_i$ relating to probability bounds s.t. $x_i, y_i \geq 0, x_i+y_i \leq 1 \; \forall i$. Each $i$ has a constant $\theta_i$, and ...
1
vote
1answer
258 views

Stieltjes convolution with white noise

I'm looking for a reference that would discuss a Stieltjes convolution between a wiener process and a function of bounded variation. Additionally, I had a question about this sort of convolution. Is ...
2
votes
4answers
293 views

Avoiding overfitting by averaging polynomials fit to part of the data?

I was thinking about the problem of overfitting data. Suppose you have a hundred data points sampled from an unknown function (call this the training set). You could try fitting a ...
4
votes
2answers
282 views

Convolutive noise removal

I have the time domain signal $$ u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t) $$ where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...
2
votes
2answers
317 views

My overdetermined linear system gives both bad and good estimates. Why ?

Hello to everyone. What the question means is that different ways of expressing the same relation between the data and unknown variables produce really weird fit results: The problem: I have the ...
3
votes
1answer
107 views

exact simulation of a large sample histogram

Say I want to create a histogram of $N$ samples from some simple compactly supported distribution on $\mathbb{R}$, where $N$ is very large, say $N = 10^{30}$. The histogram has $K$ disjoint bins, ...
0
votes
2answers
364 views

Is there a method to find (fit) a function with four (4) independent variables?

I have a system with 4 sensors (say $s_1..s_4$) which I want to combine into a single signal. I have logged the 4 outputs as well as a "control" sensor ($s_c$) which has the desired ouput signal. ...
5
votes
2answers
826 views

Solving for Moore Penrose pseudo inverse

I have a system to solve, set up as : $$Ax = b$$ with a square rank deficient matrix $A$. The paper suggests to use a Moore Penrose pseudo inverse, which in my case can be computed using the ...
4
votes
2answers
3k views

Numerically most robust way to compute sum of products (standard deviation) in floating-point?

I stumbled across a paper by Welford (1962), where he proclaims a method that should compute the standard deviation numerically more robust than the naive algorithms ...
5
votes
2answers
424 views

Inverting products of matrices

I need to compute a large number of inverses of the following form: $(A \Lambda_k A^\top)^{-1}$ where $A \in \mathbb{R}^{m \times n}$, $n > m$ and $\Lambda_k = \text{diag}(\lambda_1, ..., ...
1
vote
2answers
1k views

Generating samples of a multivariate cauchy distribution

The question is very simple: Do you know an efficient algorithm to generate samples of the multivariate cauchy distribution ...
1
vote
1answer
2k views

Inverting Hessian matrix

I need to invert a Hessian matrix to calculate the covariance matrix. The matrices are fairly large, typical sizes are (300x300), or values of that order. In general, the Hessian is very ...
4
votes
1answer
616 views

how to deal with bad-scaled covariance matrix?

Hi, When Aetkin linear model is used, problem holder has to provide weight matrix which is defined as $\Sigma^{-1/2}$. As far as the covariance matrix is always positive-defined the raising to the ...
0
votes
0answers
276 views

Estimating a multinomial sum

I have the following sum \begin{equation} \sum_{r_1=q+1}^{\tau}\dots\sum_{r_\lambda=q+1}^{\tau}{\tau\choose r_1,\dots,r_\lambda,\tau-r_1-\dots -r_\lambda} (\Lambda-\lambda)^{\tau-r_1-\dots-r_\lambda} ...
1
vote
3answers
1k views

Polynomial Regression/Least Squares

I have a couple questions on the same topic: 1) Does ordinary linear least squares have a generally defined confidence interval. I know that you can determine the confidence interval of each term, ...
6
votes
1answer
686 views

Random, Linear, Homogeneous Difference Equations and Time Integration Methods for ODEs

Most methods (that I know of) of numerically approximating the solution of ODEs are "general linear methods". For this type of method, the so-called 'linear stability' is examined by applying the ...
4
votes
1answer
492 views

O(n^2) algorithm to approximate the sum of the log of the singular values of a matrix

Given an $M \times N$ matrix of rank $N$ ($M \ge N$) with $i^{th}$ singular value $\sigma_i$, does their exist an $O(M^2)$ algorithm for approximating the sum $ H =\sum_{i=1}^N \log(\sigma_i)$ with ...
5
votes
3answers
3k views

measure of quality of curve fit

I am interested in a measure for the quality of fit to a curve which would distinguish the two cases shown in the following image (without addressing the fact that incidentally the right one has more ...
4
votes
2answers
4k views

What's an efficient way to calculate covariance for a large data set?

What is the best algorithm for computing covariance that would be accurate for a large number of values like 100,000 or more?