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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32 views

Gradient based optimization of step function w.r.t number of steps

I am trying to optimize the parameter b in the following simple function using gradient descend in PyTorch: $$ y = \frac{\lfloor{xb} \rfloor + 0.5}{b} $$ x is in $[0,1]$ and b is continuous and in $[5,...
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Improving accuracy of regression equation by using another equation [closed]

Is it possible to improve the accuracy of the regression equation by using another equation? Let me make a point more clear. Suppose we measure depth (H) and density of rocks ($ \rho $) in a certain ...
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Approximating a predictor with a kernel

Assume that some predictor $f$ is a kernel machine, but the kernel function $K(\cdot, \cdot)$ is unknown. Is there a way to recover the kernel $K(\cdot, \cdot)$ that "best approximates" $f$? ...
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30 views

Choosing training data inputs to optimize approximation

Suppose you have a smooth function $f^*:D_1 \times D_2\rightarrow\mathbb{R}$ that you observe with error as $f$ such that $$f(x,y)=f^*(x,y)+\epsilon$$ where $\epsilon$ has zero expectation (you can ...
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Why KL divergence fails to approximate the means of distributions? [closed]

We have two distributions, $P$ and $Q$ such that $P$ is our input distribution and $Q$ is our target distribution. The formulation of $KL = \mathbb{E}_{P}\left[\log\frac{P}{Q}\right]$ allows us to ...
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1answer
34 views

Optimal way to rank candidates - concrete statement

I'd like to have some statistical/probabilistic formalisations (solutions..) of the following concrete case I have heard : "Imagine you have a set of candidates to be interviewed for a job. You ...
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1answer
37 views

When is the ratio of two normals approximately normal?

Suppose that $X \sim N(\mu_1,\sigma_1)$ and $Y \sim N(\mu_2,\sigma_2)$ are two independent normal random variables. Define $Z = X/Y$. I noticed that there are some cases where the distribution of $Z$ ...
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Random subsampling to approximate distribution

Suppose I have an uniform grid $A = [ a_1, a_2, a_3, \dots, a_n ]$ of $n$ points on the interval $[-c, +c]$. As an example, consider $c=10$ and $n=10^4$ so that $$A = [-10, -9.9979998, -9.9959996, \...
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How to judge the approximated density appropriate to the original one?

I want to approximate type 1 extreme value distribution (standard Gumbel distribution ) to mixture Gaussian distribution using R software. I did it using "fminsearch" function in "...
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16 views

Approximation of a rational function [closed]

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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1answer
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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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37 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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1answer
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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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32 views

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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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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1answer
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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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1answer
101 views

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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1answer
121 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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2answers
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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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1answer
17 views

Approximate / Standardize value in certain range

I have table with numeric values like ...
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26 views

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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19 views

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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1answer
69 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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1answer
52 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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1answer
300 views

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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1answer
424 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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12 views

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
42 views

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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82 views

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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17 views

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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47 views

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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5 views

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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28 views

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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1answer
380 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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34 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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31 views

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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1answer
17 views

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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19 views

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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1answer
32 views

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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0answers
14 views

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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18 views

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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1answer
63 views

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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0answers
20 views

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 ...
2
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1answer
71 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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