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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Best way to approximate head point having only face keypoints

I'm using the BlazeFace model from TensorFlow which only has this few keypoints: I need those keypoints plus a head keypoint, like this one: My question is, which would be the best way to ...
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LSA and cosine similarity approximation with large matrix

This question is the non-r related version of this one I have a tdm with 16k x 350k dimension, for which I am trying to get the document-document similarity (cosine type). With RSpectra I have found a ...
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Distribution of the approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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Universal approximation of Gaussians

Can gaussian kernels reproduce non continuous L2 integrable functions? ( Do non continuous L2 integrable functions lie in the RKHS constructed by a Gaussian Kernel?) Edit: I think my question is being ...
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Simplifying the Kullback-Leibler divergence for a sum of distributions

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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Universal Approximation Capabilities of Mixture of Weibulls

Can a mixture of $N$ Weibull distributions approximate any continuous density with non-negative support, if $N$ is sufficiently large? (If so, a reference to the proof would be greatly appreciated). (...
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Is there a way to correct for degrees of freedom when using a generalized linear model with a Poisson distribution featuring random effects?

I am running a generalized linear mixed effect model with a Poisson distribution to analyse count data. The model has a random effect that takes into account multiple observation obtained by the same ...
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Conservative coverage probability when an biased estimator is used for the variance

Suppose that $X_n\sim N(\mu, \sigma^2_n)$. Thus, to construct a 95% CI for $\mu$, we can use $X_n\pm 1.96 \sigma_n$. The coverage probability, $P(\mu\in [X_n\pm 1.96 \sigma_n])$, is equal to 95%. ...
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Confidence interval for multiple samples of ratios of counts (in R)

Data and objective I have count data from two groups, A and B, from across multiple samples. I want to estimate the average ratio of A to B across all samples, along with a confidence interval. Issues ...
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Mean squared value error only useful for policy evaluation?

The mean squared value error is defined as: $\overline{VE}(\mathbf{w}) \equiv \sum_{s \in \mathcal{S}} \mu (s) \left[v_{\pi}(s)- \hat{v}(s,\mathbf{w})\right]^2$ Just by looking at this metric, it ...
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What is known, in principle, about the possibility of approximating the random discrepancy between a statistical estimate and its parameter?

The difference between the value of a statistical estimate and its parameter's value is almost never exactly $0$. For example, $r - \rho$, for a unique sample $r$, is likely to be some non-zero ...
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A nondeterministic covariance-stationary process approximated by an ARMA process

We know that the Wold Decomposition Theorem says that any purely nondeterministic covariance-stationary process, $x = [x_t : t \in \mathbb{Z}]$, can be written as a linear combination of lagged values ...
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Fast likelihood evaluation for Gaussian distribution with diagonal plus low rank covariance

Let's assume the likelihood $$ y \sim\mathcal N_p(0, \Sigma + \Lambda\Lambda^\top) $$ where $\Sigma$ is diagonal and $\Lambda$ is a $p \times d$ matrix with $d \ll p$. What is the fastest way to ...
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Number of points a one hidden layer neural-network can interpolate

We am trying to understand the number of points that a neural network of a particular size can interpolate. I think this may be isomorphic to its degree of freedom? We are not interested in whether ...
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Connection between mean field inference and mean field theory (physics)

In variational inference, the mean-field family of probability distributions is the set of distributions that factors over its terms (i.e. each component is independent of all others). This allows us ...
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Is there a closed form approximation for the composition of the Gamma CDF with the inverse Normal CDF?

Given $k$, $\theta$ fixed shape and scale parameters for some Gamma distribution which has a CDF $F$. Let $G^{-1}$ be the inverse CDF of the standard Normal distribution. Consider the composition $H(x)...
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Approximation of a polynomial via histogram

Note: I originally tried to pose this question generally, without discussing the specific type of stochastic process. I hope that this can still be an interesting question generally. Assume that we ...
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How to solve negative binomial regression? [closed]

I want to estimate negative binomial regression for from scratch i.e. I want to write a script that will maximize maximum likelihood obtaining optimal parameters. To do so we can calculate derivatives ...
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Approximate Posterior Predictive Quantiles with Numerical Methods

I have a posterior function which is easy to approximate using numerical methods (the posterior has only 2 parameters, and is approximately Gaussian because of the large sample). However, I need to ...
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Unscented Transform - Combination of multiple Sets of Sigma Points?

Given an initial state distribution $x \sim N(m_x, S_x) \in R^{n_x}$ and transition function $y = f(x)$ one can use the unscented transform to approximate the distribution $p(y) \approx N(m_y, S_y)$ ...
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Distribution of sum of independent but not i.i.d. lognormal variables?

I am trying to find the distribution of the following variable Z: $X_i$ are each independent with Lognormal distribution ($\mu_i, \sigma^2_i$), $X_i \in L^2$ forall $\forall i$ Z = $\sum_i cX_i$ where ...
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approximate fisher information for intractable likelihoods

Suppose I have a data set $X_1, \ldots, X_n$, and from that I compute a statistic $T(X_1, \ldots, X_n) := T$. I want to assess how reactive/sensitive this calculation is to changes in parameter values....
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How to approximate the expression to $\sum x_i$

How to approximate the expression on the left hand side to $\sum_{i=1}^Nx_i$ as $n\to \infty$ $$ \frac{\sum\limits_{i=1}^{N}x_i^2}{n-2\frac{\sum\limits_{i=1}^{N}x_i}{N}} \left(\sqrt{1+\frac{Nn\left(...
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If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$

Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$. However, can they be said to be approximately equal? If ...
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Normal approximation to Bernoulli variable

I'm looking for a normal approximation for a Bernoulli variable (so I can later sum multiple correlated approximated variables) The trivial approximation is taking the mean and variance of the ...
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Correlation of Financial Returns using Period-End vs. Period-Average Values

I have two time series of financial returns for assets $A$ and $B$ defined below for $n$ periods. The return $a_i$ is the percent growth in the asset price of $A$ using period-end values for $i-1$ and ...
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How to prove that the two equations for the slope of the line of best fit are equivalent? [duplicate]

I was reading this great article on deriving the equation for the line of best fit (https://www.neelocean.com/simple-ols-estimators/), and got confused when I came across: Rearranging: $$\hat \beta = \...
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2 votes
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How do I approximate a multivariable polynomial equation using Neural Networks?

I've been trying to experiment and test the extents to which a neural network works. I was only able to make something with broad categorical variables function in an acceptable amount of time and in ...
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Polynomials that converges pointwise to a simple function on (-1,1) and bounded by $e^{|x|}$?

I am trying to prove a theorem related to the moment generating function. I will need a sequence of polynomial that converges to a simple function $K_{(-1,1)}(x)$ pointwise on the real line while ...
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Can I use the delta method with a function that depends on n to approximate the distribution of a function of the sum of iid random variables?

Let $X_1, X_2,...$ be i.i.d. random variables with finite mean $\mu$ and finite variance $\sigma^2$. From the Central Limit Theorem, we know that $\sqrt{n}(\bar{X_n}-\mu)$ tends in distribution to $N(...
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Can a neural net approximate any conditional density asymptotically?

Assume that the conditional density of $ y \vert x $ is a Beta distribution for all values of x. Can a Beta distribution with parameters computed by a neural net, i.e. Beta($\hat{\alpha}$, $\hat{\beta}...
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Why does non-parametric approach break down when the joint distribution is estimated by a finite data sample?

I am currently reading the paper on Gradient Boosting Machines - J. H. Friedman, “Greedy function approximation: A gradient boosting machine,” Ann. Stat., vol. 29, no. 5, pp. 1189–1232, 2001, doi: 10....
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Predicting Repurchase Curves next value based on usual functional form

Some definitions first: Acquired customers: Customers placing an order for their first time. Cohort: Group of customers that have been acquired during the same time period. Repurchase: An order placed ...
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2 votes
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CLT theorem and Berry–Esseen bounds for this special case of sampling

Consider a finite set $S=\{s_1,s_2,..s_n\}$, where $a \leq s_i\leq b$ are integers. Each element in $S$ can be chosen to a subset $S'$ in probability $p$. We consider $n$ to be very large. My question:...
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What are the variations of Expectation Maximization?

To explain my question better, I will use this analogy: In the case of the Gradient-Descent method, we have multiple variations/expansions for the main algorithm, like stochastic gradient descent (SGD)...
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label set of data points

I have a set of data points in a 3-dimensional space. Points are approximately in two rows, like so(3 in secondary row and 7 in primary row): I need to label both rows separately. For example, the ...
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Derivation of confidence interval of incidence rate ratio

I am trying to understand the confidence interval equation for a Incidence Rate Ratio (IRR) given several places: $ 95\text{% CL(IRR)} = \exp(\log(\text{IRR}) \pm 1.96\times \text{SE(log(IRR))})$, ...
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Local quadratic approximation

I am busy working my way through a paper by Guo et al. (pairwise variable selection for high dimensional model-based clustering) and I am just completely stuck. In the paper they use the EM algorithm ...
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Nonlinear Iterative Partial Least Squares algorithm can calculate accurately all Principal Components?

I wanted to demonstrate a small example in order to understand better the $\textbf{Nonlinear Iterative Partial Least}$ $\textbf{Squares algorithm}$. My goal is to calculate all the Principal ...
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Universal Approximation: how does a neural network handle a ratio of inputs

Related questions/background info: Universal Approximation Theorem — Neural Networks Does the universal approximation theorem for neural networks hold for any activation function? A universal ...
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1 vote
1 answer
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How can i find out closest lognormal distribution parameters from a GEV distributed data in R

The question is a bit weird so i'll open it up. So i have a table of return periods for different amounts of rain. The table has been made using GEV distribution on known data and then the mean and ...
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Approximate covariance matrix while batch sampling

I have a fallowing problem: In each iteration I take samples $x^k_t, x^k_{t+1} ... x^k_{t+\tau}$ from a process from $t$ to $t+\tau$. Then I construct $k \;x\; k$ covariance matrix. Then I do it once ...
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8 votes
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Approximation of a probability distribution

I have a continuous random variable $X$ that can easily be sampled. I don't have any other assumption on $X$. Let's say I have sampled $X$ and I have constructed the set $S$. We can assume that $S$ is ...
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Approximate a distribution as mixture of $K$ other (known, fixed) distributions

I'd like to draw samples from some "target" probability density function $f(x)$. However, I don't have a way to do that -- instead I just have access to $N$ samples, each drawn from one of $...
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Sampling marginal distribution from joint density

Suppose we know that random vectors $x, y$ have joint density $p(x, y) \propto \exp(-U(x_1, \ldots, x_m, y_1, \ldots, y_n))$, and we want to draw a random sample from the marginal $p(x)$ (i.e. we want ...
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Why does Kullback–Leibler divergence measure information loss when approximating a probability distribution?

I've encountered a sentence: In information theory, Kullback–Leibler divergence is regarded as a measure of the information lost when probability distribution Q is used to approximate a true ...
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Graphical construction of normal approximation to histogram

In Statistics by Freedman et al. it is described how to construct a normal approximation for a histogram, as follows: Calculate mean and SD for the histogram (in the image $63.5$ and $3$ inches). ...
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Predicting a value based solely on Correlation Coefficient

Let me set the stage. We are dealing with two variables; $A$ and $B$. We can easily obtain $A(x)$ for a specific data point $x$. $B(x)$, on the other hand, is very difficult to know. We know Pearson'...
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Gradient based optimization of step function w.r.t number of steps

I am trying to optimize the parameter b in the following simple function using gradient descent in PyTorch: $$ y = \frac{\lfloor{xb} \rfloor + 0.5}{b} $$ x is in $[0,1]$ and b is continuous and in $[5,...
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3 votes
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Choosing training data inputs to optimize approximation

Suppose you have a smooth function $f^*:D_1 \times D_2\rightarrow\mathbb{R}$ that you observe with error as $f$ such that $$f(x,y)=f^*(x,y)+\epsilon$$ where $\epsilon$ has zero expectation (you can ...
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