Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
76 views

Can we minimize counting cost function for perceptron algorithm?

In perceptron algorithm (the following analysis might apply to other classification algorithms), a smooth approximation of perceptron cost function $$\sum_i^n{\max(0, -y_i\mathbf{w}^T\mathbf{x}_i)}$$ ...
1
vote
0answers
49 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 ...
0
votes
1answer
24 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 ...
0
votes
0answers
10 views

Query complexity of ANN based on Hamming distance

Ilya R provides the following query complexity of ANN for Hamming distance based on coordinate sampling https://www.ilyaraz.org/static/class_2018/files/20181023.pdf When he says ...
0
votes
0answers
40 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 ...
0
votes
1answer
22 views

Approximation of non linear function with multiple linear functions

How can a non-linear function be approximated by an appropriate amount of linear functions? In the picture below, it would be quite easy to draw 10-15 linear functions to describe all data points ...
0
votes
0answers
10 views

approximation the erf function for finding Z [duplicate]

I am looking for the following probability where X follows normal distrubition; P(X$\leq$x)= ($\frac{1}{2}$(1+erf($\frac{x}{\sqrt2}$)) I have a constraint in my mathematical model like P(X≤x) $\leq$...
1
vote
1answer
31 views

An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
1
vote
0answers
7 views

Learn to mimic function adaptively

Assume I have a function $F: R^n \to R$ that is slow to evaluate, which I, therefore, would like to approximate with something faster by using machine learning. I have seen some work proceeding by ...
3
votes
1answer
40 views

Why is one of the two approximations in the bootstrap worse than the other?

My statistics text has the following diagram: $$\mathbb{V}_F(T_n) \overbrace{\approx}^{ \text{not so small} } \mathbb{V}_{\hat{F}_n}(T_n) \overbrace{\approx}^{ \text{small} } v_{\text{boot}}$$ ...
1
vote
2answers
196 views

Double integral involving the normal CDF

I need to compute (or best approximate?) the following integral $$\int_0^\infty \int_0^\infty (1 + \alpha u)^{-1}(1 + v)^{-1} \Phi\left(\frac{\beta}{\sqrt{\gamma + uv}}\right) \text{d}u \text{d}v,\...
0
votes
1answer
43 views

Approximate the data to a single curve

The question might be simple, but I am not able to find the answer. Hence I am asking here. I did search google but didn't get an answer. I have a continuous stream of data coming from an API in the ...
1
vote
0answers
36 views

Finding mode of posterior using Newton method in R

I am attempting to approximate the posterior $\tilde{\pi_{G}}(z|\theta,Y)$ which is the Gaussian approximation to the full conditional of $z$, and in order to do this I need to find the mode $z^{*} \...
4
votes
1answer
108 views

Variance of Normal Order Statistics

Suppose we have $X_1, \cdots, X_n \overset{\textrm{i.i.d.}}{\sim} \mathcal{N}(0, 1)$ with $n > 50$, and let $X_{(1)}, \cdots, X_{(n)}$ be the associated order statistics. Are there any references ...
1
vote
0answers
27 views

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 ...
0
votes
1answer
38 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 ...
0
votes
0answers
12 views

Sum of multivariate lognormals

Is it possible to approximate the sum of multivariate lognormals using Wilkinson approximation? Any reference?
1
vote
1answer
30 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 ...
0
votes
0answers
35 views

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<...
0
votes
0answers
61 views

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 $\...
3
votes
2answers
124 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 $...
0
votes
0answers
25 views

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{...
0
votes
0answers
51 views

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) ...
1
vote
0answers
38 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 ...
1
vote
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 - ...
2
votes
1answer
43 views

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 ...
5
votes
0answers
96 views

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)) &= ...
0
votes
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 ...
0
votes
1answer
203 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 ...
1
vote
0answers
74 views

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 ...
1
vote
2answers
180 views

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 ...
0
votes
0answers
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 ...
0
votes
0answers
11 views

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, ...
0
votes
1answer
93 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 ...
0
votes
0answers
347 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 ...
4
votes
1answer
141 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 ...
1
vote
0answers
60 views

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 ...
2
votes
1answer
326 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 ...
8
votes
1answer
238 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 ...
0
votes
2answers
56 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 ...
0
votes
0answers
13 views

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)...
5
votes
1answer
127 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 $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...
4
votes
1answer
111 views

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 ...
-2
votes
2answers
444 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 ...
0
votes
0answers
45 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 ...
3
votes
1answer
41 views

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 $\beta>0$ ...
0
votes
0answers
29 views

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 (...
2
votes
1answer
117 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 ...
3
votes
1answer
98 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 ...
5
votes
1answer
83 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 ...