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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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at what probability will the probability we start considering the data?

For example I have this problem, Do Americans tend to vote for the taller of the two candidates in a presidential election? In 30 presidential elections since 1856, 18 of the winners were taller than ...
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1answer
34 views

Sum of product as product of sums

Assuming I have two random non-independent vectors $A,B$ which are within [-1,1]. I want to approximate their sum of product by product of sums (everything is a dot product), i.e. $\sum_{i=1}^NA_iB_i ...
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10 views

Sum of multivariate lognormals

Is it possible to approximate the sum of multivariate lognormals using Wilkinson approximation? Any reference?
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22 views

Simple arithmetic approximations to categorical analyses

Suppose I have a two by two table: $$ \begin{array}{c|ccc} & Y & \neg Y & \\ \hline X & a & b& &\\ \neg X & c & d& &\\ \end{array} $$ and I am interested ...
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Estimation of function using Spline Interpolation

My problem is the following: Estimate the function from given data (below) and show that the estimated function has the following properties: (i) $f(0)=0$ (ii) $f(x)>0, x>0$ and $f(x)<0, x<...
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Approximate prediction interval in linear regression

Suppose we have a linear regression model of the following format : $$ y(x) = \beta_0 + \beta_1 x_1+ \beta_2x_2+\beta_3x_3+\epsilon$$ We know that the prediction interval associated with a level $\...
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65 views

Can a Bernoulli distribution be approximated by a Normal distribution?

$$\sum_{i=1}^n bernoulli(p) = binomial(n,p) \approx \mathcal N(np, np(1-p)) = \sum_{i=1}^n \mathcal N(p, p(1-p))$$ Can I conclude that $\mathcal N(p, p(1-p))$ could represent an approximation of $...
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CLT approximation - how large should sample be so probability is equal to 0.95? [duplicate]

We have a measurement which has mean $\mu$ and variance $\sigma^2$ = 25. Let $\bar{X}$ be average of $\textit{n}$ such independent measurements. How large should $\textit{n}$ be in so that $P(|\bar{...
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Fast Approximate Sampling from Multivariate Normal Parameterized by Precision Matrix

I want to efficiently sample $x \sim N(\mu, \Omega)$ where $\Omega$ is a precision matrix (e.g., the inverse of the covariance. The challenge is that the dimension of $x$ is massive (~ 100K to 10M) ...
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28 views

Is there a universal approximation theorem for monotone functions?

The universal approximation theorem basically states that a feed-forward network with a single hidden layer containing a finite number of neurons can approximate continuous functions on compact ...
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1answer
30 views

Approximation of the critical value for $\alpha$ of $\Gamma(n-1,1)$

Say I have the critical region for a test statistic $T$ and some constant $c$, as follows, $$ n(T - 1)^2 \ge c $$ where $nT \sim \Gamma(n-1, 1)$ and the probability of rejection is $\alpha = P(n(T - ...
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Approximation of copulas

I'm studying copulas, finished the Introduction to Copulas by Nelsen. I'm interested in the latest/best known/etc approaches for approximating any Copula, or some families of copulas, so would be ...
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Approximating the first moment of $h(x)$ where $x$ ~${\rm log\,normal}(\mu, \sigma)$

What is the best way to approximate $E(h(X))$, where $X$ ~ Lognomal($\mu, \sigma$)? So far, I can think of Monte Carlo Methods and Gaussian Hermite quadrature as below: \begin{align} E(h(X)) &= ...
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1answer
37 views

Compute Mean of Normal Distribution where x% of Values are over y

I am looking for a way to determine the mean of a normal distribution (with given variance), where e.g. $z = 0,37 = 37\% $ of values should be above a certain value $a$ (e.g. 0,2)? My first idea was ...
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Approximation error on standard and sparse PCA

I am trying to understand the approximation error of pca explained in this tutorial. $\sum_{n=1}^{N}||x_n-\tilde{x}_n||=N\sum_{i=M+1}^{D}\lambda_i$ where $||x_n-\tilde{x}_n||=\sum_{i=M+1}^{D}\left(\...
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1answer
79 views

How state aggregation in reinforcement learning works?

I am watching Prediction with linear approximation video course in the RL class by prof. Sutton. He presented state aggregation approach on a random walk problem. It seems that this approach just ...
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Approximation or closed form equation for summation of logistic function [closed]

The spread of epidemics follows a logistic growth, given in the equation below $I(t) = \frac{N}{1+(N-1)exp^{-rNt}}$ where, N is the population size, r is infection rate, t is time , I(t) is ...
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2answers
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The Universal Approximation Theorem vs. The No Free Lunch Theorem: What's the caveat?

The universal approximation theorem: A neural network with 3 layers and suitably chosen activation functions can any approximate continuous function on compact subsets of $R^n$. The no free ...
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Approximate reinforcement learning vs approximate dynamic programmin?

I know that dynamic programming uses the model of the environment while many RL methods are model-free. However, I am willing to know the difference between ADP and ARL and I would be thankful if ...
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34 views

How do Neural Networks use weight and bias to approximate complex functions?

I've been using This well put together article to understand the concept of the Universal Theorem. The problem is that I still cannot understand exactly how a Neural Network's weights and bias work ...
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Compute p-value for very high z scores [duplicate]

For a statistical test I need to compute the p-value given a z score. I am using the Python method: scipy.stats.norm.cdf the problem is that for z score > 8, ...
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1answer
66 views

Approximating the error of maximum likelihood estimation

I have a log likelihood function of a model and I want to find $\mu$ and $\sigma^2$ which maximize the log likelihood. Since the log lik function is quite complex, I decided to use Nelder-Mead ...
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222 views

Estimate wear distribution based on smal amount of samples

This is a task where I think bayesian statistics can help, but as I only know the basics about it and the question is rather complex I have troubles to get started... Assume a machine where some ...
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1answer
126 views

In exactly what sense do MCMC draws approximate the target?

Background We want to sample from some intractable density $\pi(\theta)$. Using an MCMC algorithm, we generate a sample of draws $\{\theta_i\}_{i=1}^N$ from a Markov chain that has $\pi(\theta)$ as ...
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Optimization: Approximate function - Which points to evaluate next?

I am looking for a statistical method (and a link to a nice R package would be cool too!) which allows me to find which point to evaluate next for a given function. I have a non-stochastic function z ...
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1answer
208 views

Normal approximation to Poisson: With Continuity Correction the Approximation Seems Worse

This is Exercise 3 in Section 6.3 of Probability and Statistics, 4th edition, by DeGroot and Schervish: Suppose that the distribution of the number of defects on any given bolt of cloth is the ...
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1answer
199 views

Analytically solving sampling with or without replacement after Poisson/Negative binomial

Short version I am trying to analytically solve/approximate the composite likelihood that results from independent Poisson draws and further sampling with or without replacement (I don't really care ...
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2answers
50 views

Compression of 18000 curves

I have over $18000$ curves that I need to compress to save $\geq 50\%$ of space. Each curve is described by points $f(1), f(2), ..., f(96)$, each $f(x)$ is 8-bit long. The curves in compressed form ...
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Convergence of $(X_i, Y_i)$ when each converges in distribution to normal.

I'm trying to prove the following statement. Let $X_i$ and $Y_j$ are independent $\forall i, j=1,2,\ldots$ If $$ X_n \overset{d}{\to} N(\mu_1, \sigma_1^2), Y_n \overset{d}{\to} N(\mu_2, \sigma_2^2)...
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1answer
113 views

Approximating the mathematical expectation of the argmax of a Gaussian random vector

Let $X = \left( {{X_1},...,{X_n}} \right) \sim \mathcal{N}\left( {{\mathbf{\mu }},{\mathbf{\Sigma }}} \right)$ be a Gaussian random vector and $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...
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1answer
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CLT for random sums: Anscombe's Theorem vs. “classical” version

Given a compound Poisson distribution $$S(t):=\sum_{k=1}^{N(t)} X_{k}$$ with $N(t)\in\mathbb{N},\,t\geq0$ a Poisson process with rate $\lambda.$ $X_{k}$ are non-negative iid random variables such ...
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187 views

Continuous approximation to binomial distribution

Consider an integer variable $k$ that follows a binomial distribution, $$\binom{N}{k}p^{k}\left(1-p\right)^{N-k}$$ with total draws $N$ and probability of success $p$. I am interested in the ...
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39 views

What's a good approximation to the count distribution of people for days of birth?

I have data on $n$ people's dates of birth and let's ignore the years and look only at the $k$ = 366 days of the year (including Feb 29). Assuming that dates of birth are uniformly and independently ...
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1answer
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Difference vs log-difference: do they behave similarly?

Consider two positively-valued time series, $\{X(t),Y(t)>0|t\geq0\}$. Now consider two transformations: $$ U(t) = Y(t) - \beta X(t),\\ V(t) = \ln{[Y(t)]} - \ln{[\beta X(t)]}, $$ with $\beta>0$ ...
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Approximation of cumulative time series data

I have a time-series data in the form of a histogram, it is a cumulative one. I have a cumulative time series data of a particular feature and non-cumulative time series data of that same feature (...
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1answer
108 views

Estimate correlation between data and data-fit model for variance reduction in Monte Carlo estimate

Say that I want to estimate the mean of a function $f$, $\mathbb{E}[f(X)]$, given some input distribution $x\sim P(x)$. I don't know anython about the form of $f$ except that it is smooth and ...
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1answer
95 views

Estimating function with Gaussian Procceses

I do not have strong math background, but I am trying to understand Gaussian Processes by example using the lecture Machine learning - Introduction to Gaussian processes by Nando de Freitas. Here is ...
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1answer
70 views

Evaluate approximation of PCA from randomized algorithms

I have been comparing different PCA implementations (some via explicit calculation of the covariance matrix, some with randomized/truncated SVD) in terms of speed, and now wanted to compare how good ...
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1answer
38 views

Contraction or Reduction of a Neural Network

Circuit size reduction is common practice in Theoretical Computer science. It is very common to approximate a circuit with a smaller circuit (Or a polynomial and so on). Are there any such techniques ...
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34 views

Better skill test for RPGs - Conditional probability given 2 independent parameters

I am trying to find a better way (theoretically, not practically speaking) to roll the dice for a skill test in RPGs. In the d20 system, the Game Master choose a Difficulty Level for the skill test, ...
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1answer
99 views

How good an approximation is sampling with replacement to sampling without replacement?

I'm learning about probability with Feller's book and he states that, when the population size $n$ is big in comparison with the sample size $r$, then $n_r$, which is a shorthand for $\frac{ n!}{(n-r)!...
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9 views

How to approximate table-defined function using non-linear least squares

I read least squares method and haven`t found a good example of using non-linear least squares. Problen: I have an arbitrary values for x = 1, 2, 3, 4, 5, therefore, I have a table-defined function. ...
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30 views

Approximation of fractional function that has real-power numerator

I have the function $f(x)=\frac{(1+x)^k}{1+ax}$, where $x>0, 0<a<k<1$. The function has only one maximum at $x_0=\frac{a-k}{a(k-1)}$, increases on the left of $x_0$ and decreases on the ...
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22 views

Reducing a logistic model used for prediction

I'm developing a logistic regression used for prediction. I have pre-selected, based on prev. literature, 15 candidate predictors (fitting my ~200 events). Now, I want a reduced/more parsimonious ...
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156 views

Approximation of an infinite sum in R [closed]

I'm calculating the probability mass function for a count variable and the normalization term is an infinite sum of the form $\sum_{n = 0}^{\infty} f(n)$. I'm looking for a function in R that ...
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41 views

Associated eigenvalue problem to spline penalization

In Natural Spline functions, their associated eigenvalue problem by F. Utreras they prove a result. I am interested to know if a more generalized form exists and where to find it. Consider the space ...
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1answer
33 views

Regularization for approximation in neural networks

In the case of approximation tasks using neural networks, should we standardize the data, as in the classification ?
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145 views

Testing a low rank estimator of a covariance matrix

I am exploring ways to reduce the noise of a covariance matrix estimator when the number of variables is greater than the number of observations, i.e. $n > t$. First, I tried using a low rank ...
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1answer
86 views

Find $\mathbb{E} \bigg[ \frac{\textbf{h}^{H} \textbf{y}\textbf{y}^{H} \textbf{h}}{ \| \textbf{y} \|^{4} } \bigg]$ with Mathematica? [duplicate]

Considering the following complex random vectors (Complex Gaussian random variables): \begin{align} \textbf{h} &= [h_{1}, h_{2}, \ldots, h_{M}]^{T}\ \ \sim \mathcal{CN}(\textbf{0}_{M},d\textbf{I}_{...
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Approximation of the hessian matrix

What circumstances have to be avaible that the approximation of the hessian matrix leads to wrong standard errors?