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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Approximation of a rational function

Suppose a function $f$ has a known form $ f(x) = \dfrac{P(x)}{Q(x)}$ where both $P,Q$ are polynomials of degree at most $d$. Assume $d$ is fairly low, take $d\leq 5$ for example. What is the "...
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Do all moments of a random variable need to be well controlled for a valid 2nd order Taylor approximation, or is the third moment sufficient?

In this post, the accepted answer states that we need certain conditions before a second order Taylor series approximation is robust, due to the fact that the variance does not control higher moments. ...
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Given x or y can and the correlation coefficient can you approximate the other?

Give x or y can and the correlation coefficient can you approximate the other? The definition of correlation coefficient is: $$r=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2(y_i-\...
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Poisson Approximation to Negative Binomial

While r tends to infinity, p tends to 1, and (1-p)r tends to lambda, we obtain Poisson distribution from Negative Binomial Distribution. X in Neg Binom(r,p) is the number of failures; x=0,1,2,3,4.... ...
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36 views

Can I approximate with a normal distribution?

I feel like I should know this (I graduated in physics a couple of years ago), but I'm really unsure about whether or not it's appropriate to use a normal distribution for the following case: I have ...
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The quality of approximation

I have $N$ random values and I initially know that it is not a Normal distribution (it is a discrete one), but it is really close to that. I estimate the expectation and variance using my number set ...
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comparing two statistics with an approximation distribution [duplicate]

I need help starting this homework question. Can anyone explain to me how I can assume these two vaccines are the same. Can I represent the two values as binomial distributions?
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wald test and score test, normal or chi square?

I learnt from section 10.3 of statistical inference that both Wald test statistic $\frac{W_n-\theta_0}{S_n}\approx\frac{W_n-\theta_0}{\sqrt{\hat I_n(W_n)}}$ and score test statistic $\frac{S(\theta_0)}...
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Why isn't the Welch–Satterthwaite equation nonsensical?

As I understand it, the Welch–Satterthwaite equation says that if $s_i^2$ is the sample variance of group $i$ (with $n_i$ samples), then, assuming that the measurements in each group are iid's that ...
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Approximation of chi-squared distribution quantiles by means of the standard normal distribution quantiles

I've been searching for weeks now but I can't find a proof for the following relationship between the quantiles of the chi-squared distribution and the quantiles of the standard normal distribution: $$...
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101 views

exponential approximation at adaboost

I am reading about AdaBoost algorithm, I cannot understand (7). How can it be like this? If you want full document, you can get it at http://cseweb.ucsd.edu/classes/fa01/cse291/AdaBoost.pdf
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Determining under what conditions an exponential function is linear

I'm working through an exercise to determine when an exponential function of the form: y = ae^(bx)+c is approximately or exactly linear (of the form ...
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Is it reasonable to look at the output of simulating from a multivariate distribution as univariate distribution? If yes, what is this called?

Suppose I have $X_{n} \sim MVN(\underline{\mu},\Sigma)$ where $n$ is large (several thousands). However, the $\mu_i's$ and the elements of $\Sigma$ are such that almost every simulation from $X_n$ ...
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Low rank approximation

I'm looking for literature that deals with the following problem (does anybody know any paper related to it). The Low-Rank Approximation problem is well known: $$\min \|X - \hat{X}\|_{F}, \: \text{s.t....
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Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
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Deriving posterior update equation in a Variational Bayes inference

I'm reading a paper (He, et al. 2010) that has used variational Bayesian inference to solve an inverse problem. I have difficulties deriving the relations for updating the variational approximations ...
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Is the set of distribution $\{ X | \max_t |f_X(t) - f_Y(t)| \leq \epsilon \}$ convex, where f is the cdf or inverse cdf?

I'm trying to figure out if the set is convex, where the maximum difference between cdf(or inverse cdf) of X and a reference distribution Y is smaller than $\epsilon$. 1. Let $f_X(t)$ denote the cdf ...
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Inverse-normal CDF approximation in Excel, Python or R

I read that the implementations of Inverse-normal cumulative distribution function (CDF) /quantile / ppf in R, Python (scipy) and Excel give similar results. However, I can't find the very formulae ...
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Constructing a continuous distribution to match $m$ moments

Suppose I have a large sample drawn from a continuous distribution, size $n$, and $2 < m\ll n$ moments from that sample. Alternatively, suppose I have been given those moments by an angel, ...
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1answer
174 views

Universal approximation of probability distribution with latent variable model

I want to show that under certain circumstances this form can approximate any probability distribution. For that, I came up with the following argument. Consider a directed graphical model of the ...
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117 views

Validity of approximating a covariance matrix by making use of a probability limit?

I want to know can we approximate the covariance matrix of a random vector by making use of a probability limit. Define the linear regression model in matrix form as $$ \mathbf{Y} = \mathbf{X} \beta + ...
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How to show that normal distribution is a second order approximation to any distribution around the mode?

How can I show that normal distribution is a second order approximation to any distribution around the mode?
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Numerically validating rates of convergence of approximations of expectations?

In applied mathematics it is standard practise to often validate theoretical approximations using numerical simulations. Since these simulations typically use numerical methods that convergence very ...
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174 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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Expected value of a “logistic uniform” multivariate

Let $\mathbf{a}_1,\ldots,\mathbf{a}_n \in \mathbb R^d$ and $b_1,\ldots,b_n \in \mathbb R$ be fixed. For $\mathbf{x} \sim \mathcal U([0,1]^d)$ and $j \in \{1,\ldots,n\}$, consider the real variable ...
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232 views

The distribution of the product of a multivariate normal and a lognormal distribution

If $$X=\left(\begin{array}{c} X_{1}\\ X_{2} \end{array}\right)\sim N\left[\left(\begin{array}{c} \mu_{X_{1}}\\ \mu_{X_{2}} \end{array}\right),\left(\begin{array}{cc} \sigma_{X_{1}}\\ \sigma_{X_{1}X_{2}...
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Trying to approximate $E[f(X)]$ - Woflram Alpha gives $E[f(X)] \approx \frac{1}{\sqrt{3}}$ but I get $E[f(X)] \approx 0$?

Let $X \sim \mathcal{N}(\mu_X,\sigma_X^2) = \mathcal{N}(0,1)$. Let $f(x) = e^{-x^2}$. I want to approximate $E[f(X)]$. Wolfram Alpha gives \begin{align} E[f(X)] \approx \frac{1}{\sqrt{3}}. \end{align} ...
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Express standard deviation of a sequence in matrix form

I am working improving an existing program that does everything in matrices. So if I can express below concept in matrix that would make my life a bit easier. We all know that for matrix ...
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17 views

Approximate / Standardize value in certain range

I have table with numeric values like ...
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1k views

Distribution of the Levenshtein distance between two random strings

The Levenshtein or edit distance between two strings is the minimum number of edits (adding a letter, removing a letter or changing a letter) required to transform one into the other. Assume that we ...
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1answer
340 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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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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54 views

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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Approximation of logarithm of standard normal CDF for x<0

Does anyone know of an approximation for the logarithm of the standard normal CDF for x<0? I need to implement an algorithm that very quickly calculates it. The straightforward way, of course, is ...
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Difference of two i.i.d. lognormal random variables

Let $X_1$ and $X_2$ be 2 i.i.d. r.v.'s where $\log(X_1),\log(X_2) \sim N(\mu,\sigma)$. I'd like to know the distribution for $X_1 - X_2$. The best I can do is to take the Taylor series of both and ...
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589 views

Tail distribution of sum of Weibull distributed variables

Is there a bound on the tail distribution of a finite sum of i.i.d Weibull random variables?
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51 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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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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1answer
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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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Analytical approximation of probability of one beta-distributed var being greater than another?

The answer to What is the probability P(X > Y) given X ~ Be(a1, b1), and Y ~ Be(a2, b2), and X and Y are independent? provides an analytical solution for this, but is there a less computationally ...
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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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32 views

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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