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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31 views

How well can an AR(p) process model any given stationary time series?

Are there any theorems which tell us how well AR(p) models are able to approximate any stationary finite time series? If so, what are the relevant results?
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1answer
223 views

natural log approximation

I've got an equation that contains x^p - 1 x is any positive number, such as 2, and p is a small positive number close to 0 such as 0.001 For some reason (that ...
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1answer
74 views

How to deal with increasing action space in TD learning using linear function approximation

I am working on an application of reinforcement learning in an environment in which the number of possible actions increasing throughout an episode. The first step has only 8 possible actions, which ...
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1answer
638 views

Lower Bound on the Total Variation Distance between two Binomials

Let $X= B(n,1/2)$, $Y=B(n,1/2 + \delta)$, for a small $\delta >0$ be two Binomial Distributions. Question 1. I am looking for a lower bound on the Total Variation Distance the two Binomials ...
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1answer
72 views

What are some of the common techniques for density estimation?

I'm trying to estimate the probability density function of a real random variable given its iid realizations. What are some of the standard techniques to do this? One method I have heard of is the ...
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3answers
121 views

How to calculate or approximate the integral $\int_{-\infty}^{\infty}\sigma(x)(1-\sigma(x))\mathcal{N}(x|0,1)dx$

$\sigma(x)$ is the sigmoid function, that is, $\sigma(x)=\frac{1}{1+e^{-x}}$. And $x$ is from a Normal distribution of 0 mean and 1 variance. Now I'd like to calculate the expectation of the ...
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1answer
58 views

Is approximate normality limited to the exponential family?

A GLM looks like $$g(\mu) = X\beta,\ \ \mu = EY_i$$ where $Y_i$ is an exponential family. It is commonly assumed for a decent sample size that $\beta$ is approximately normal with mean $\hat{\beta}$ ...
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48 views

Is vcov the information matrix /n from the theory?

If I fit a model m <- glm/gam/gamm/lme/whatever(y ~ x + z, family = some exponential family) and extract coef(m) vcov(m) ...
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1answer
37 views

Simultaneously finding least squares predictors of every feature based on all others

Given set of $d$ datapoints and $f$ features in a $X_{f,d}$ data matrix, I'm trying to fit a least squares predictor of every feature based on every other feature. In other words for every feature $i$...
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1answer
238 views

Kahn Pseudo Normal distribution

I read about a so-called "Kahn Pseudo Normal" distribution in a forum post. It says when $U$ is uniformly distributed over $[0,1]$, then $\log(1/U-1)$ is approximately normal distributed, that is ...
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1answer
177 views

A Bound based on Jensen's Inequality

Consider: $$X \sim \text{Gamma}(\alpha, \beta)$$ $$Y = \frac{1}{X+c}, \ c > 0$$ I am interested in $E(Y)$, which I'm pretty sure is intractable... $$E(Y) = \frac{\beta^\alpha}{\Gamma(\alpha)}\...
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338 views

Multivariable piece-wise linear approximation in R?

I am trying to run linear approximation on a function of 2 or 4 variables. The approx function in R is a very nice, optimized time saver, but it only works for 1 dimension functions I think. Anyone ...
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1answer
34 views

How to determine recurring use numbers for my app

A few months ago, I wrote an Android App. I am getting two key pieces of data from this app regarding its usage. The first metric, is number of downloads per day. The second metric, is basically, ...
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1answer
32 views

Algorithm to Fit Intuitively Important Data Points

It is hard to formalize what I am asking without knowing the answer, but in experimental mathematics it is very common to see a graph of points hinting at limiting behavior. The graph below makes me ...
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3answers
1k views

regression with constraints

I have some domain knowledge I want to use in a regression problem. Problem statement The dependent variable $y$ is continuous. The independent variables are $x_1$ and $x_2$. Variable $x_1$ is ...
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25 views

Are there barriers to sampling derivatives of $\log(p)$?

Suppose I have a probability distribution $p_\theta$ on $\Bbb R^n$ dependant on some parameters $\theta$. A natural problem is to evaluate the derivative of some expectation by the parameters: $$d_\...
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1answer
526 views

Finding the optimal number of hidden nodes and training epochs for function approximation [duplicate]

In an attempt to find the optimal number of hidden nodes and number of training epochs (to obtain optimal performance while not over-fitting) in a single-hidden layer neural network, I generated the ...
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3answers
2k views

What is the CDF of the sum of weighted Bernoulli random variables?

Let's say we have a random variable $Y$ defined as the sum of $N$ Bernoulli variables $X_i$, each with a different, success probability $p_i$ and a different (fixed) weight $w_i$. The weights are ...
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1answer
1k views

Function approximation using multilayer perceptron (neural network)

I've been asked to solve a problem for a project. I'm working on Python or R. I need to approximate a function with multiplayer perceptron (neural network). The function is: $y= 2\text{cos}(x)+4$ on ...
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38 views

Data approximation with unequal intervals

For some sampled function of real variable, defined on [a, b], how can [a, b] be split into subintervals of unequal width, so that for each interval one can say, "On this interval, function is ...
6
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1answer
552 views

Half-normal probability plot

To construct the half-normal probability plot, plot the absolute values in a certain statistical diagnostic (residual, leverage, Cook distance and others) versus $z_i$ where: $\displaystyle z_{i} = \...
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89 views

Estimate mean and variance of pdf from truncated taylor expansion of logarithm of pdf

In a maximum likelihood fit, one estimates the parameter with the mode of the likelihood $L$, and the variance of this estimator with the second derivative of $\log(L)$: $$ \bar\theta = \mathrm{Mode}[...
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1answer
1k views

Do Gaussian process (regression) have the universal approximation property?

Can any continuous function on [a, b], where a and b are real numbers, be approximated or arbitrarily close to the function (in some norm) by Gaussian Processes (Regression)?
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420 views

Approximating Uniform Distribution with Mixture of Gaussians

Let $T$ be a compact, connected, proper subset of $\mathbb{R}^3:\quad T \subset \mathbb{R}^3$. Further let $\left\{ \boldsymbol{\mu}_i \right\}_{i=1}^n$ be a given finite set of $n$ point in $T$: $$ \...
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1answer
402 views

Approximating a joint distribution from marginals of sums of variables

Suppose I have a set of random variables ${X_1, X_2, ..., X_n}$. For each of the variables, I have the marginal distribution. Furthermore, I have the marginal distribution for various sums of subsets ...
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1answer
37 views

Using clusters to estimate model variance

I am working with a blackbox prediction model which takes known inputs and outputs a single mean response. I know this model's residuals to be heteroskedastic, but also can assume the error term of ...
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5answers
995 views

Approximation error of confidence interval for the mean when $n \geq 30$

Let $\{X_i\}_{i=1}^n$ be a family of i.i.d. random variables taking values in $[0,1]$, having a mean $\mu$ and variance $\sigma^2$. A simple confidence interval for the mean, using $\sigma$ whenever ...
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1answer
70 views

Calculate probabilities from binomial or normal distribution

I have the following experiment: Two balls are let fall from a certain high (could be the same or different) Then for each of them, I get a random number from 0 to 1 with a uniform distribution and ...
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2answers
271 views

Claim that a distribution is almost Gaussian

I prove a some theorem under the assumption that some random variable X is Gaussian. Now in practice, in my experiments section I have real-world samples from ...
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2answers
189 views

Show that $1-\Phi(x)$ is approximately $\varphi(x)/x$ for large $x$ (standard-normal random variable) [duplicate]

Demonstrate that, for a standard normal random variable: $$1-\Phi(x) \approx \frac{\varphi(x)}x$$ for large values of $x$.
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26 views

When is it possible to estimate the non-linearity error when approximating data with a linear model?

The most common form of linear regression estimates the best values of $\vec{\beta}$ and $\sigma^2$ assuming that data is sampled from a model $y = \vec{\beta} \cdot \vec{x} + \vec{\epsilon}$ where $\...
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1answer
78 views

Difference between contrastive divergence and importance sampling

The terms Contrastive Divergence and Importance Sampling are sometimes used interchangeably. I understand that both are used to approximate partition functions (normalization terms for probabilities)...
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110 views

approximate a nonparametric CDF in R

I have two vectors of same length. The first vector is a collection of realization from an unknown random variable. The second vector is the distribution computed at each particular realization. A ...
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37 views

Can the data from two resources be mixed?

There are several sources of temperature-dependent viscosity data, which goes as following: |:--------------------------------| | 273.15 | 320.5 | 330.25 | |---------------------------------| | 1....
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406 views

Quantile approximation using Cornish-Fisher expansion

I am trying to approximate a set of quantiles from the estimated mean, variance, skewness and kurtosis of a random variable with unknown distribution. I tried to apply the Cornish-Fisher expansion of ...
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0answers
28 views

Regression without known covariates: approximating the likelihood

I would like to perform (linear) regression for an outcome $y$ but do not have access to the covariates $x$. Instead, each record $i$ belongs to one of $K$ groups and I know the distribution of $x_i$ ...
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0answers
179 views

Gaussian Exponential Sum Approximation

Let $x_j,\ j\in\{1,\cdots,n\}$ be independent standard Gaussian random variables and $a_j,\ j\in\{1,\cdots,n\}$ be constants. What would a function $f$ and a standard Gaussian random variable $x_0$ ...
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3answers
1k views

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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1answer
1k views

Probability of candidate winning majority vote

I was trying to work out probability of a candidate winning the majority vote but I got a crazy z value, which i'm not sure is correct or not but it sure as hell seems crazy. Bear in the mind, that I'...
4
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1answer
33 views

Approximating the posterior

I would like to determine the MAP estimate $\hat\theta$ for a posterior distribution $P(\theta|x)$, where $x$ is data. The exact posterior is difficult to evaluate but I can approximate by $Q(\theta)$ ...
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1answer
392 views

Why spectral norm low-rank approximation error is stronger than frobenius norm?

Suppose a rank $k$ matrix $Z$ with orthonormal columns $z_1,..., z_k$ satisfies Frobenius Norm Error: $$\|A − ZZ^TA_k\|_F ≤ (1 + \epsilon)\|A − A_k\|_F -- (1)$$ It is said that Frobenius norm error ...
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57 views

Learning samples from similarity matrix

Having a set of points $X\in\mathbb{R}^{n\times d}$ and a similarity matrix $S\in \mathbb{R}^{n\times m}$, I am interested in an approximation of the second set of points $U\in \mathbb{R}^{m\times d}$....
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0answers
137 views

Distribution that has shape of normal and uniform distribution

I have a pdf of a random variable that looks like image below (the support is $[-1,1]$). It is actually function of another random variable, let's say $T = aX + bY + cZ$. When I changed parameters $(a,...
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1answer
643 views

Find a polyline best approximating gps points

I don't have much experience in data analysis, so big sorry for anything stupid below. I have thousands of gps points taken from a public transport vehicle, you can see them below. I would like to ...
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1answer
40 views

Generating data with given median from empirical data

I have an empirical distribution of household income based on the US census from wikipedia and I'd like to generate a plausible distribution for the income distribution in a region given its median ...
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2answers
4k views

Why use the normal approximation to the binomial?

In school, I was taught about the normal approximation to the binomial, and it was suggested that I could use it effectively under some conditions, because it can be 'easier to calculate'. I ...
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0answers
592 views

Standard normal quantile approximation

In the book Asymptotic theory of statistics and probability by Anirban DasGupta (2008, Springer Science & Business Media) in page 109 Example 8.13 I found the following approximation $$\Phi^{-1}\...
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0answers
153 views

Moment/cumulant series expansion of probability distribution?

Let $\mu_n = \langle x^n \rangle$ and $\kappa_n$ be the moments and cumulants of a probability density function $p(x)$. Given a finite number of the moments, $\mu_1, \dots, \mu_N$, or equivalently ...
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1answer
47 views

Frequency of N Independent Events

I'm attempting to implement a numerical approximation to assess the frequency of outcomes given $N$ independent random variables. There is a slight twist however, each $N$ has a weight, Each variable ...
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1answer
149 views

Approximated relation between the estimated coefficient of a regression using and not using log transformed outcomes

Consider the two regression equations: \begin{align} Y &= \alpha+\beta X+\varepsilon \\ \log Y &= \gamma+\delta X+\eta \end{align} Is there a "simple" approximated relation between the ...