The Stack Overflow podcast is back! Listen to an interview with our new CEO.

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.

Filter by
Sorted by
Tagged with
0
votes
1answer
15 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 ...
0
votes
1answer
34 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 ...
1
vote
0answers
24 views

Penalize the MSE of half of predicted values more than the other half

I'm using MSE loss for an multi-layer perceptron that learns to approximate the target feature vector $[\hat{x_1}, \dots, \hat{x_N}, \hat{y_1}, \dots, \hat{y_N}]^\intercal$. The catch here is that I'd ...
2
votes
1answer
65 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 $\frac{1}{6}$. ...
0
votes
1answer
29 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, ...
0
votes
0answers
11 views

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. ...
1
vote
1answer
69 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 ...
0
votes
0answers
8 views

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 ...
1
vote
0answers
16 views

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,\...
0
votes
0answers
24 views

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\...
1
vote
0answers
22 views

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) & & & \ \ \ \ \ ...
1
vote
1answer
36 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,...
0
votes
0answers
114 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 $$\sum_i^n{\max(0, -y_i\mathbf{w}^T\mathbf{x}_i)}$$ ...
2
votes
1answer
69 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
29 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
17 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
60 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
27 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
11 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
36 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
8 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
43 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
207 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
45 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
43 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^{*} \...
5
votes
1answer
208 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
28 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
49 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
15 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
45 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
80 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
220 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
64 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) ...
5
votes
1answer
110 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
34 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
47 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
97 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
38 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 ...
1
vote
1answer
301 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
109 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
256 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
1answer
109 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 ...
4
votes
1answer
153 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
372 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
269 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 ...
1
vote
2answers
63 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 ...
6
votes
1answer
199 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 $I = \mathop {\arg \max }\limits_{i = 1,n} {X_i}$. $I$ ...