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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27 views

How to approximate a function using Fast Fourier Transform? [closed]

I'm trying to approximate a complex function (I have millions of points to sample) using Fourier transform. I used Legendre and Chebyshev polynomials to approximate it before, and it worked pretty ...
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Science practice: Where to introduce approximations?

In my work, I am using an algorithm which relies on estimates of the gradient of the log-posterior at a collection of Monte Carlo samples. Since this gradient is not available in closed form, I must ...
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Approximating mean/covariance of truncated/folded/censored normal distribution

Given a normally distributed $X$, what is the best way to approximate the covariance matrix and mean vector of $\tilde{X} = \max(0, X)$? I am interested in the censored distribution, but the truncated ...
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Why can't we approximate the General TSP while we can approximate the Euclidean TSP? [closed]

Euclidean TSP is approximatable, whereby the triangle inequality is obeyed. However, what is the exact reason which does not allow us to approximate General TSP?
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Approximate PDF function from “how many in each range” data

I have the following data which represent how many graduates (out of 578) have an average grade in each range: $58$ with average grade in the range $[5, 5.99]$ $336$ with average grade in the range $[...
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42 views

Composite priors in bayesian linear regression?

I'm not certain that "composite" is the right word for this; I've seen blogs tutorials and books that seem to link prior beliefs together. Consider MTCARS data, where miles per gallon (mpg) ...
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Is it possible to go back to initial point from kth iterated point in a Newton Raphson method?

I am trying to find preimage of a kth iterated point under Newton method. Is it possible to find an initial point from which the kth iterated is derived?
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Approximating Bayesian Posterior with Weighted Likelihood Bootstrap (WLB)

$\DeclareMathOperator*{\argmax}{arg\,max}$Given a set of $N$ i.i.d. observations $X=\left\{x_1, \ldots, x_N\right\}$, we train a model $p(x|\boldsymbol{\theta})$ by maximizing marginal log-likelihood $...
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314 views

Why do we use parametric distributions instead of empirical distributions?

The probability density function (pdf) is the first derivative of the cumulative distribution (cdf) for a continuous random variable. I take it that this only applies to well-defined distributions ...
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Approximating k-dimensional lipschitz function

I have a known function $f:[0,1]^K \rightarrow [0,1]^K$ which is L-Lipschitz (w.r.t to $L_1$ but can also be w.r.t to $L_2$ if the results differ). Denote the input vector by $\theta$. Each entry in $\...
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Weighted sum of negative binomial distributions - approximate fast parameter calculation

Let's suppose we have a convolution (weighted sum) of three negative binomials (parameterised as mean and overdispersion). ...
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Quantifying the universal approximation theorem

Let $m\geq 1$ be an integer and $F\in \mathbb{R}[x_1, \dots, x_m]$ be a polynomial. I want to approximate $F$ on the unit hypercube $[0, 1]^m$ by a (possibly multilayer) feedforward neural network. ...
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Understanding additive function approximation or Understanding matching pursuit

I am trying to read Greedy function approximation: A gradient boosting machine. On page 4 (it is marked as page 1192) under 3. Finite data the author tells how the function approximation approach ...
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Approximate or exact distribution of the sum of inverse gamma variables

The random variable ${\left| {H\left( {n,m} \right)} \right|^{ - 2}} \sim Inv - Gamma\left( {{\omega },\frac{\Omega }{{\omega }}} \right)$and independent of each other. What distribution does its sum ...
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Error analysis in sampling multivariate distribution

Consider a discrete joint distribution $p(x_1, x_2, x_3)$ over variables $x_1,x_2,x_3 \in \{0,1\}$. By the chain rule of probability, the following algorithm samples correctly from the joint ...
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Analytical Approximation for Conditional Moments

Say I have a function of a latent variable: $F(X_{t+1})$. $F(X_{t+1})=-log(\sum\limits_{\substack{k \neq j}}\alpha^{k}_{j}\frac{S^{k}_{t+1}}{S^{j}_{t+1}})$ The term in brackets is $X_{t+1}$. I know ...
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205 views

Kernel approximation with Nystroem method and usage in scikit-learn

I am planning to use the Nystroem method to approximate a Gram matrix induced by any kernel function. I found the Nystroem implementation in scikit-learn. As far as I understood, the full Gram Matrix ...
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Maximum-likelihood histogram from noisy data

Given a sequence of noisy observations $\{x_k\in\mathbb{R}\}$ and a set of thresholds $\{t_i\in\mathbb{R}\}$ we can bin the observations using the thresholds to create a histogram. However, since we ...
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Approximating a distribution with an integer histogram

Given a distribution $f:[0,a)\rightarrow\mathbb{R}$, is there a simple algorithm by which to find a sequence $\{h_i\in\mathbb{N_0}\}$ such that $f(x)$ is approximated by $h_{floor(x)}$ as a histogram ...
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How do I combine the weights of two predictor in a regression model with GRNN?

I am trying to build an algorithm that uses GRNN for regression, a model based on the formula: My csv files are looks like: ...
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Good list of references and books on statistical approximation, simulation and computational methods?

I am looking for books and resources that cover simulation and approximation techniques so that we do not have to follow the strict assumptions held by the many statistical models. With how fast ...
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Inferring an approximate distribution for noising of data given 300,000 samples of human noising [closed]

I'm trying to find a statistical way to get an approximate distribution of all human noising. I have a dataset of over 300,000 samples of people noising words. I took basic Statistics and I would know ...
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Poissonization use for sampling size

I'm interested in using the Poissonization trick to solve the following problem, which I made up: Suppose I have a categorical random variable $X$ taking values $1$, $2$, and $3$, with probability ...
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Different forms of Stirling's approximation

I have Stirling's approximation in the form: Please could someone explain how this is equivalent to the form: log(n!) = nlog(n) - n for large n?
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Proof of theorem on Poisson distribution [duplicate]

Can someone help prove this theorem? Many thanks! If $p\to0$ and $n\to\infty$ in such a way that $\lim np = \lambda > 0$, then for $k=0, 1,\dots$: $$\lim_{n\to\infty}\binom nkp^k (1-p)^{n-k}=\...
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Approximate the mean area of 2D Voronoi cell

Consider a random uniform distribution of $N$ points in $2D$ space bounded by $[0, 1]$ in both dimensions. Example: If I want to estimate the mean area of their Voronoi cells, I have to obtain the ...
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34 views

Using Gumbel distribution to approximate distribution of sample maximum — formulae for the parameters?

Suppose you have an observable sample $X_1,...,X_n \sim \text{IID } F_X$ which has a right-tail that decreases sufficiently rapidly to apply the extreme-value theorem (e.g., a normal distribution) to ...
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70 views

The “correct” way to approximate $\text{var}(f(X))$ via Taylor expansion

tl;dr: There are two commonly reported formulas for approximating $\text{var}(f(X))$, but one is notably better than the other. Since it isn't the "standard" Taylor expansion, where does it come from, ...
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29 views

When is the quadratic approximation of a log-likelihood function used?

In the notes I'm working through, we're told that we can make a quadratic approximation to the log-likelihood function. Why would one need to(or want to) do this? At the moment, I don't see the ...
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27 views

Approximation of chained functions

Say we want to approximate a set of $n$ continuous functions $f_n(g(x))=y$ where $x \in \mathbb{R}^d, y \in \mathbb{R}, g(x) \in \mathbb{R}^m$ by fitting them to $n$ different datasets $(X, Y)_n$ ...
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76 views

Expected value with dependent samples

It is well known that the expected value of a function can be approximated with i.i.d. samples: $$ E_X[f(X)] = \frac{1}{n}\sum_{i=1}^n f(x_i),\quad x_i\sim_{i.i.d.} X $$ What methods exist to ...
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37 views

Approximating distribution by moment matching?

I am going to compare distributions by moment-matching (expected value, standard deviation, skewness, kurtosis etc). The question is simple: As the moment-matching relates to Taylor expansion, would ...
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Approximating the expected value of a random variable as a function of the prior mean of a parameter

I have a parameterised statistical model and I'm trying to calculate the expected value of a random variable. I know that the expected value is a function of the value of the unknown parameter. So I ...
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43 views

kl divergence of two different distributions of subsets

Suppose there is a set $S=\{1, 2, 3, ..., n\}$, then I need a distribution of its subsets with fixed size k, which can be denoted as $A=\{x_1, x_2, ..., x_k\}$ where $x_1$ to $x_k$ are from 1 to n. ...
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Approximate known non-linear function using linear regression

Consider the following model: $$ y_{i}=f\left(\boldsymbol{x}_{i};\theta\right)+\varepsilon_{i} $$ where $y_{i}$ is the dependent variable, $\boldsymbol{x}_{i}$ is a vector of explanatory variables, $...
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162 views

Maximum sample size for one-way ANOVA?

Lists of requirements for one-way ANOVA include the following: Samples should be mutually independent Samples should be from a population with a normal distribution Samples should have the same ...
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94 views

Estimating Quartiles with Moments

The Wikipedia article on Skewness indicates that the median of a distribution can be estimated from the mean, standard deviation, and skeweness with an error term that goes as $O(skewness^2)$. ...
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30 views

Binary Matrix Low Rank Matrix Factorization

Low Rank Matrix Factorization is a pretty popular problem in data mining. We need to find 2 matrices, $W, H$ such as $F = W \cdot H$. I know that this approximation is NPC problem, so we won't find ...
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187 views

Sampling distribution is not normal. How is that possible?

As central limit theorem suggests, sampling distribution is approaching normal on the large sample sizes regardless of the initial distribution of the variable. And it's always been true for me until ...
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Nystrom approximation with inexact/stochastic kernel evaluation

Suppose we have several data points $x_1,\ldots,x_m\in\mathbb R^n$ as well as a positive definite kernel $K(x,y):\mathbb R^n\times\mathbb R^n\to\mathbb R$ that can be written in the form $$K(x,y)=\...
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35 views

Robustness of a model to learnt parameters

There is a recent push to study how sensitive a model is to small changes in its input. This has also been studied from an adversarial point of view: e.g what is the smallest input perturbation that ...
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181 views

What is the difference between approximate bayesian computation vs approximate bayesian inference?

What are the main differences between approximate bayesian computation vs approximate bayesian inference? Are they essentially the same? Do they refer to the same of different family of models? My ...
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Approximate inverse of a Gaussian Process

I'm using a GP in order to learn the transition function of a continuous Markov Decision Process, i.e. P(s'|s,a). This works reasonably well, but I'm now also ...
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109 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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How to transform $P[k_1\leq (x_i-\mu - \sigma\cdot Z)^2 \leq k_2]$ to $P[k_1\leq \frac{(x_i-\mu)^2}{\sigma^2}+e \leq k_2]$?

Taste estimation As an example consider an experiment conducted to determine the optimal concentration of salt in popcorn. The concentration of salt in sample $i$ is denoted by ${x_i}$. The subject ...
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139 views

Multivariate Gaussian FItting

When trying to approximate a distribution of random vectors D by using multivariate gaussian what properties must we ensure that D has ie; what distributions can be estimated by Multivariate gaussian ...
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Computing KL Divergence for distributions over sets

I have a distribution over a set of (hundreds of) discrete terms, and I'd like to describe the difference between I see a couple of options, and none seems really attractive: Take the KL divergence ...
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248 views

what is the probability of detecting departure from H0?

The desired percentage of SiO$_2$ in a certain type of aluminous cement is 5.5. To test whether the true average percentage is 5.5 for a particular production facility, 16 independently obtained ...
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30 views

Poisson distribution question

An airline has found that the number of people booked on flights who do not arrive at the airport follows a Poisson distribution at the rate of 2% per flight.For a flight with 146 seats ,150 are sold ...
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72 views

How can I apply the Poisson ($\mu$) distribution to two series of random draws?

I have the following question: A box contains 1000 balls, of which 2 are black and the rest are white. If two series of 1000 draws are made at random from this box, what approximately, is the chance ...

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