Questions tagged [approximation]

Approximations to distributions, functions, or other mathematical objects. To approximate something means to find some representation of it which is simpler in some respect, but not exact.

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341 views

Hessian for Laplace Approximation in Uncertainty Propagation

This is possibly a silly conceptual question, ... but anyway: Imagine I have a function: $f = F(\mathbf{x}) = F(x_1,x_2) = ax_1^2 + bx_2^3,$ where $x_1,x_2 \sim N(0,1)$ for example. For a naive ...
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Is there a way to approximately convert a standard deviation of log returns to a standard deviation of simple returns?

If I have a standard deviation of log returns, what do I need to know to convert it to a standard deviation of simple returns? I have got the following R functions which seem to give consistent ...
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24 views

A close-form solution to a low dimensional curve fitting

I have a very short vector of intensities (let's say 5 elements per vector). The intensities are a discrete sampling of a curve around its maxima (the maxima is not discrete though). What I'd like to ...
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Approximate computation of a linear function

I am not sure that this is the right stackexchange site for this, but here goes: Let $d_1,\ldots,d_n$ be known real numbers, some positive, some negative. Let $v_1,\ldots,v_n$ be non-negative numbers ...
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code Welch's t-test in C++ using approximations for student's t values

I need to code a Welch's t-test between two populations in C++ without using external libraries like, for example, boost. I know that given my two populations of size $N_1$ and $N_2$, I can calculate ...
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108 views

Ways to approximate multiple samples of same function in R

Example dataset (simplified): ...
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235 views

Interpretting Confidence Interval Questions

In a survey conducted by a mail order company a random sample of $200$ customers yielded $172$ who indicated that they were highly satisfied with the delivery time of their orders. Calculate an ...
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119 views

Computing the partition-function of an exponential family member

I am working on a Monte Carlo Expectation Propagation problem. In that context I got the following property: $ I = \sum\limits_i w^{(i)} \log p_\eta(x^{(i)}) $ where $\{w^{(i)}\}_i$ are weights, $...
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42 views

Ratio in a random subset without replacement

This function returns a random proportion. It appears normally distributed with mean $a/(a+b)$. What is its approximate variance? ...
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Why does the reconstruction error of truncated SVD equal the sum of squared singular values?

I saw this formula in a textbook: squared Frobenius norm of the original matrix $\mathbf X$ minus its truncated SVD $\mathbf X_k$ (which can be seen as the approximation error) equals the sum of ...
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1answer
42 views

Other types of mean error

Let $\tilde f$ be an approximation of the function $f(x) = \arccos(x)$. I'm using MATLAB to figure out how good this approximation is by calculating a mean error. My first idea was to use this formula ...
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Digits that are insignificant by virtue of representing extremely small quantities within a much larger quantity: what are they called?

In computer science, sometimes we can measure run-times with high accuracy. As a hypothetical scenario, suppose a computation takes a week to run and we can measure the run-time accurate to the ...
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389 views

68–95–99.7 rule for binomial distribution?

What I want to calculate is the probability that the size $k'$ of an intersection of two subgroups with sizes $n$ and $K$ that have been randomly selected from a group with size $N$ that is smaller ...
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normal approximation to the binomial distribution: why np>5?

Nearly every text book which discusses the normal approximation to the binomial distribution mentions the rule of thumb that the approximation can be used if $np\geq5$ and $n(1-p)\geq 5$. Some books ...
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1answer
704 views

Dirac Delta function Notation

I am trying to understand the delta function notation used to be express a monte carlo approximation of a probability distribution. The notation used in this (p10) is $\pi(x_{1:n}) = \frac{1}{N}\sum^...
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Training Restricted Boltzmann Machines according to the Likelihood Function

Is it possible to chose the parameters of a RBM to maximize the likelihood of the observed data? (I follow the notation of the deeplearning tutorial ). Denote the observable data by $x$, hidden data ...
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2answers
450 views

Bayes Factor approximation

A brute force method to approximate the Bayes Factor (the ratio of the denominators (normalizing constants) in the Bayes formula) is to do the following for the two models of interest: repeat ...
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228 views

mixture of Gaussians vs mixture of quadratic denominators (Cauchy)

It is known that mixture of Gaussians are dense in the set of all distribution functions. A 1-dimensional Gaussian has the following density: $$ \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{(\omega-\beta)^...
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Asymptotic distribution of a weighted sum of chi squared variables beyond CLT? [duplicate]

I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
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bad fit - nomenclature for breeds

Question: What is it called when one uses a basis, like the pure line instead of the sigmoid/logistic, in a manner that grossly departs from the "physics" of the problem? There should be a word for ...
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How to understand the geometric intuition of the inner workings of neural networks?

I've been studying the theory behind ANNs lately and I wanted to understand the 'magic' behind their capability of non-linear multi-class classification. This led me to this website which does a good ...
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1answer
345 views

glasso- Assumptions of Meinhausen-Buhlmann approximation?

First off, This is not an R question - it is a conceptual question, so there is no need to perform any code. Problem: I'm trying to invert a large dimensional covariance matrix of p features. For ...
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How can I estimate the approximation quality of the odds ratio based on a sample or how can I ignore infrequent items?

I am analyzing URLs found in tweets of the Twitter stream. I have two samples of tweets from that stream: a random sample of tweets all tweets of a subset of users I want to find URLs which are ...
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Are there any fast approximations to generalized linear mixed models?

Are there any recommended methods or approximations that would help speed up the estimation of fixed effects components of a generalized linear mixed effects model? Specifically, my dataset includes ...
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1answer
648 views

Bound for weighted sum of Poisson random variables

Suppose I have some independent Poisson-distributed random variables $X_1 \ldots X_N$ with parameters $\lambda_1 \ldots \lambda_N$. These can be thought of as processes where each arrival/event ...
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1answer
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Confidence interval from R's prop.test() differs from hand calculation and result from SAS

I'm wondering if anyone has insight into how prop.test() in R calculates its confidence intervals. Although it doesn't state it explicitly in its documentation, my ...
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1answer
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Finding a “skew normal distribution” for given data

I am given a set of $n$ pairs $(x_i, y_i)$, where the $x$-coordinates can be interpreted as the measured values of a random variable $X$ and the $y$-coordinates can be interpreted as some "scaled" ...
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1answer
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Statistical problem related to product lifespan

I design electronics, and while I have some basic education in the principles of statistics, I don't believe I'm qualified to determine whether this problem is tractable as it was posed to me. So I ...
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381 views

If Machine learning is about making accurate predictions, isn't it a catch-22 situation with the no free lunch theorem?

If Statistics is about making approximate summaries while Machine Learning is about making predictions as accurate as possible, doesn't it create a paradox due to the no free-lunch theorem?
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Is there any paper about the distribution of difference of log-normal variable?

I am working on the problem relating to the difference of log-normal distribution. I have found several papers about this topic, however, none of them gives me the answer I want. More specifically, ...
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Approximating $\log( E(X))$

I was casually reading an article (in economics) which had the following approximation for $\log(E(X))$: $\log(E(X)) \approx E(\log(X))+0.5 \mathrm{var}(\log(X))$, which the author says is exact if ...
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1answer
408 views

Approximating P(A,B,C) using P(A,B), P(A,C), P(B,C), and P(A), P(B), P(C)

For some events $A$, $B$, $C$ I know the occurrence probabilities $P(A),\: P(B),\: P(C)$ I also know the pairwise co-occurance probabilities $P(A,B),\: P(A,C),\: P(B,C)$ I want to approximate the ...
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1answer
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Can I approximate the variance of a ratio to the ratio of two variances?

I've been slowly working my way through problems in Casella and Burger's Statistical Inference and I stumbled upon this problem: I haven't been able to get past (a). I've tried using the ...
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1answer
744 views

Assessments of “Approximately Normal” for t-tests

I am testing equality of means using Welch's t-test. The underlying distribution is far from normal (more skewed than the example in a related discussion here). I can obtain more data but would like ...
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3answers
179 views

Algorithm for approximating a density by a mixture density

Given a density $f(x)$ (e.g. the log-normal distribution or log-$t_{\nu=3}$ distribution), I was wondering what algorithm are known/typically used to find a mixture of distributions $g_r(x)$ from ...
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Express $E(x^{\alpha})$ in terms of $E(e^{-\zeta x})$? to a 1st or second order?

I have a random variable, $X$, and am able to find $\mathbf{E}(e^{-\zeta X})$ for many $\zeta$ (through the Laplace transform solving an ODE as this actually evolves over time) Is there any way I can ...
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1answer
428 views

Distribution over the product of three, or $n$, independent Beta random variables

This is a re-post of a question on the Mathematica stack exchange, as per the advice of another user (see here). I am pursuing a computational solution there, but thought it might be worth looking for ...
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131 views

Theoretical properties of Gaussian Process Emulator

I am studying Guassian Prcess Emulator (GPE) to approximate computationally expensive computer models. Basically, we suppose the computer model, or simulator, is denoted by $f(x)$, where $x$ is the ...
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The probability that several normally distributed random variables have a particular order

Let's say we have n random, independent variables $X_1,\ldots,X_n$ with normal distributions. Is there a reasonable way how to compute the probability that they have a particular order, for example $$...
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1answer
273 views

How to approximate Bayes Factor?

I am searching for a computationally simple way to approximate a Bayes Factor. Currently, I'm using an approach which seems pretty logical to me but I would still be interested to know if this is ...
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115 views

Is there any objective criterion to determine the adequate degree of polynomial approximations?

I carry out a polynomial function approximation to a load duration curve (monotone decreasing function). I can approximate the load duration curve by using polynomial degrees from 4 to 12. I need an ...
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107 views

Approximation of quadratic form of normal variances, not necessarily non-negative

I am interested in approximating $$ \sum_{i=1}^{n}{w_i u_i^2} $$ where $u_i \sim N(0,1)$ i.i.d. I have seen a few papers on approximating this sum where $w_i \ge 0$, but not in the general case. Why ...
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108 views

Comparing nonlinear models using the Chi-squared asymptotic approximation

I am attempting to identify the best model fit for a nonlinear mixed effects model using the asymptotic approximation to the Chi-squared test. When calculating the appropriate number of degrees of ...
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When do Taylor series approximations to expectations of (entire) functions converge?

Take an expectation of the form $E(f(X))$ for some univariate random variable $X$ and an entire function $f(\cdot)$ (i.e., the interval of convergence is the whole real line) I have a moment ...
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2answers
588 views

Low rank approximation of binary valued matrix

How does one get the low rank approximation of binary matrix? Is the low rank approximation also a binary matrix? Note - Here binary matrix just means that any entry of the matrix can either be 0 or ...
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257 views

Classification and regression tree (CART) on large data set

I am trying to approximate a multivariate function $y = f(x_1, ...x_n)$, which I have reason to believe will be well approximated by a classification and regression tree. Some of the variables are ...
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1answer
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Approximating the distribution of a linear combination of beta-distributed independent random variables

This question is related with these other two questions in Cross Validated, which has been already answered: Approximate the distribution of the sum of ind. Beta r.v Central limit theorem when the ...
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1answer
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Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
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1answer
936 views

Number of Gaussian mixture components needed to approximate any distribution

I remember reading an actual proven number of components, that can approximate any distribution. Somehow I think it was 18. Can someone point me to a book/article stating something of the sort? Might'...
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1answer
252 views

Dvoretzky–Kiefer–Wolfowitz inequality hold for discrete distributions?

I am wondering whether Dvoretzky–Kiefer–Wolfowitz inequality holds for discrete distributions? Any comments or references would be greatly appreciated.

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