# Tagged Questions

**1**

vote

**0**answers

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

**0**answers

159 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

**1**answer

124 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 ...

**9**

votes

**3**answers

436 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

**1**answer

427 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

**1**answer

131 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

**0**answers

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

**1**answer

240 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

**4**answers

288 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

**2**answers

277 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

**2**answers

311 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

**1**answer

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

**2**answers

351 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

**2**answers

795 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 ...

**3**

votes

**2**answers

2k 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

**2**answers

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

**2**answers

974 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

**1**answer

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

**1**answer

588 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

**0**answers

275 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

**3**answers

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

**1**answer

685 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

**1**answer

489 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

**3**answers

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

**2**answers

3k 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?