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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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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4 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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19 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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20 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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22 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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22 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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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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27 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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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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25 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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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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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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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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67 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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31 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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63 views

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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35 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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56 views

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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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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88 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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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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138 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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144 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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108 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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1answer
110 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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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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28 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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51 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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124 views

How to get better approximation than Central Limit Theorem

This is continuation of my problem Calculate variance of sum random variables Suppose random variable $X$ takes 3 values $1, 2, 3$ with probability $\frac{1}{2}$, $\frac{1}{3}$ and $\frac{1}{6}$. ...
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74 views

How to approximate the distribution of the sum of multiple multinomial random variables?

Say we have $T$ independent Multinomial random variables $X_1,X_2\dots X_T$, with $p(X_t=m)=p_{t,m},m\in\{1,2,...M\}$. What would be the distribution of $X_1+X_2+...+X_T$? If there is no closed-form, ...
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Approximation for the sampling error of the number of positive cases in a Bernoulli trial

Reading the book "Energy for Future Presidents" I found a way of approximating the binomial proportion sampling error which I never saw before, and I would like to know if my derivation is correct. ...
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362 views

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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Order of continuity of an ANN approximation dependent on the activation functions used?

If I have understood this correctly, a result from Hornik et al.'s Universal Approximation of an Unknown Mapping and Its Derivatives Using Multilayer Feedforward Networks essentially states that, if ...
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Definition of Mean Squared Value Error with respect to action-value functions in Reinforcement Learning algorithms

I am referring to page 199 of Sutton and Barto book on Reinforcement Learning available here: book There the Mean Squared Value Error for an vector-parameterized function approximation $\hat{v}(s,\...
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Approximate a density function from sampled data

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ and $\mathcal E_0\subseteq\left.\mathcal E\right|_{E_0}$ be finite and disjoint with $$E_0=\biguplus\...
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Asymptotic approximation of log-probability using first four moments

Consider a random variable $X \sim p_{n,\theta}$ where the first four moments are given by known functions: $$\begin{matrix} \ \ \ \ \ \ \mathbb{E}(X) \equiv \mu(n,\theta) & & & \ \ \ \ \ ...
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59 views

Analytical solution to the multivariate CDF given multivariate pdf

Is there any way of approximating or analytically solving the below CDF (let's say even for $n\to\infty$)? I am trying to find the below probability: \begin{align} &P\left[X_{2}-X_{1} \leq 0,...
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96 views

Expectation Value of a Product of Many IID variables

First of all, I apologize for not being rigorous, but I am not a statistitian by background. Imagine you have $N$ i.i.d. positive random variables $X_1...X_N$ and you are trying to compute a ...
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33 views

“At least” approximation calculation

I have a vector of different probabilities to get 1, for example probs = [0.1, 0.5, 0.2, 0.9, 0.25, 0.55] I have to calculate the probability of having at least ...
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153 views

Back-Transformation for Ln(X+1) of zero rich data

I have seen and read several similar questions, but mine pertains specifically to zero rich data. I will be back transforming my data based on a first order Taylor series approximation. As outlined ...

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