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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Polynomials that converges pointwise to a simple function on (-1,1) and bounded by $e^{|x|}$?

I am trying to prove a theorem related to the moment generating function. I will need a sequence of polynomial that converges to a simple function $K_{(-1,1)}(x)$ pointwise on the real line while ...
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48 views

Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
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41 views

Can a neural net approximate any conditional density asymptotically?

Assume that the conditional density of $ y \vert x $ is a Beta distribution for all values of x. Can a Beta distribution with parameters computed by a neural net, i.e. Beta($\hat{\alpha}$, $\hat{\beta}...
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Why does non-parametric approach break down when the joint distribution is estimated by a finite data sample?

I am currently reading the paper on Gradient Boosting Machines - J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Ann. Stat., vol. 29, no. 5, pp. 1189–1232, 2001, doi: 10....
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Predicting Repurchase Curves next value based on usual functional form

Some definitions first: Acquired customers: Customers placing an order for their first time. Cohort: Group of customers that have been acquired during the same time period. Repurchase: An order placed ...
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CLT theorem and Berry–Esseen bounds for this special case of sampling

Consider a finite set $S=\{s_1,s_2,..s_n\}$, where $a \leq s_i\leq b$ are integers. Each element in $S$ can be chosen to a subset $S'$ in probability $p$. We consider $n$ to be very large. My question:...
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What are the variations of Expectation Maximization?

To explain my question better, I will use this analogy: In the case of the Gradient-Descent method, we have multiple variations/expansions for the main algorithm, like stochastic gradient descent (SGD)...
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label set of data points

I have a set of data points in a 3-dimensional space. Points are approximately in two rows, like so(3 in secondary row and 7 in primary row): I need to label both rows separately. For example, the ...
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41 views

Derivation of confidence interval of incidence rate ratio

I am trying to understand the confidence interval equation for a Incidence Rate Ratio (IRR) given several places: $ 95\text{% CL(IRR)} = \exp(\log(\text{IRR}) \pm 1.96\times \text{SE(log(IRR))})$, ...
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Local quadratic approximation

I am busy working my way through a paper by Guo et al. (pairwise variable selection for high dimensional model-based clustering) and I am just completely stuck. In the paper they use the EM algorithm ...
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Nonlinear Iterative Partial Least Squares algorithm can calculate accurately all Principal Components?

I wanted to demonstrate a small example in order to understand better the $\textbf{Nonlinear Iterative Partial Least}$ $\textbf{Squares algorithm}$. My goal is to calculate all the Principal ...
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Universal Approximation: how does a neural network handle a ratio of inputs

Related questions/background info: Universal Approximation Theorem — Neural Networks Does the universal approximation theorem for neural networks hold for any activation function? A universal ...
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How can i find out closest lognormal distribution parameters from a GEV distributed data in R

The question is a bit weird so i'll open it up. So i have a table of return periods for different amounts of rain. The table has been made using GEV distribution on known data and then the mean and ...
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Approximate covariance matrix while batch sampling

I have a fallowing problem: In each iteration I take samples $x^k_t, x^k_{t+1} ... x^k_{t+\tau}$ from a process from $t$ to $t+\tau$. Then I construct $k \;x\; k$ covariance matrix. Then I do it once ...
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Approximation of a probability distribution

I have a continuous random variable $X$ that can easily be sampled. I don't have any other assumption on $X$. Let's say I have sampled $X$ and I have constructed the set $S$. We can assume that $S$ is ...
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Approximate a distribution as mixture of $K$ other (known, fixed) distributions

I'd like to draw samples from some "target" probability density function $f(x)$. However, I don't have a way to do that -- instead I just have access to $N$ samples, each drawn from one of $...
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Sampling marginal distribution from joint density

Suppose we know that random vectors $x, y$ have joint density $p(x, y) \propto \exp(-U(x_1, \ldots, x_m, y_1, \ldots, y_n))$, and we want to draw a random sample from the marginal $p(x)$ (i.e. we want ...
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Why does Kullback–Leibler divergence measure information loss when approximating a probability distribution?

I've encountered a sentence: In information theory, Kullback–Leibler divergence is regarded as a measure of the information lost when probability distribution Q is used to approximate a true ...
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Why does log of incidence rate is approximately normal?

Say I have $n$ events over $T$ person years for some disease. Assume $n$ is small. $\textbf{Q:}$ Why does incidence rate is approximately normally distributed? What is justification for such ...
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Graphical construction of normal approximation to histogram

In Statistics by Freedman et al. it is described how to construct a normal approximation for a histogram, as follows: Calculate mean and SD for the histogram (in the image $63.5$ and $3$ inches). ...
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Approximation for Multivariate Gaussian

I am reading a paper, where they say they approximate a 2D multivariate Gaussian distribution by its second moment. The corresponding formula is the following: $\displaystyle d(u) = \frac{1}{1 + (u - \...
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Predicting a value based solely on Correlation Coefficient

Let me set the stage. We are dealing with two variables; $A$ and $B$. We can easily obtain $A(x)$ for a specific data point $x$. $B(x)$, on the other hand, is very difficult to know. We know Pearson'...
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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 descent 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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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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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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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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55 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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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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100 views

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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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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37 views

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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35 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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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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158 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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130 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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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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Approximate / Standardize value in certain range

I have table with numeric values like ...
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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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27 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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160 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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61 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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