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Questions tagged [density-estimation]

Estimation of probability density functions, whether by kernel density estimation, log-spline estimation or other methods.

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How to identify hot spots in one-dimension

I am looking to identify stretches of a road along which a notably high number of accidents occur. My data can be represented as a two column table in which each row represents one accident, and the ...
Josh O'Brien's user avatar
4 votes
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How to accurately estimate the probability of a rare event in a large dataset?

I have a dataset of 30,155 names and out of curiosity I verified that the longest name has 68 characters, which is quite big considering the mean and SD were 24.78 and 5.64, respectively. Based on ...
WordP's user avatar
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estimation of multivariate probability

Let $(X_{1}, \dots, X_{n})$ be a multivariate distribution and I can generate the sample from it. Next, assume that I have to compute $$ P(X_{1}\in A_{1}, \dots, X_{n}\in A_{n}), $$ where $A_{1}, \...
ABK's user avatar
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MLE of marginal distribution for continuous random variable

Let $\mathcal{F}$ be a family of multivariate probability densities such that for a sufficiently large data sample, there always exists a unique MLE. Assume also that all marginal and conditional ...
12345's user avatar
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How to show $\sup_{x\in [a,b]}|f_n(x)-f(x)|=O_p(\sqrt{\frac{\log n}{nh}}+h^2)$ when the kernel $K(\cdot) $ is of bounded variation?

Consider the kernel estimate $f_n$ of a real univariate density defined by $$f_n(x)=\sum_{i=1}^{n}(nh)^{-1}K\left\{h^{-1}(x-X_i)\right\}$$ where $X_1,...,X_n$ are independent and identically ...
Kevin's user avatar
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Scaling of different kernels when estimating densities in R

The implementation of the density function in R says that the kernels are scaled so that the bandwidth becomes the standard deviation of the smoothing kernel. For the Gaussian kernel, it is ...
shani's user avatar
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Density estimation vs estimation

Given a statistical model $(\mathcal{X},\Sigma,\mathcal{P})$, where $\mathcal{P}$ is a collection of probability measures on $\mathcal{X}$, and given a random sample with values in $\mathcal{X}$, we ...
user124910's user avatar
3 votes
1 answer
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Is this a known or valid divergence between two densities?

I am testing various metrics for learning a density estimate. Specifically, I have a sample of data from a distribution $p$, and am learning a function $f$ to estimate $p$ by minimizing a distance or ...
Travis L's user avatar
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Reference datasets for conditional density estimation

[In case you feel inclined to close this question because I'm asking for a dataset - I'm looking for solutions in the spirit of point 2 (on-topic) in the accepted answer to this question about asking ...
Scriddie's user avatar
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Developing a Confidence Interval of Density Functions for Uniform Periods in Seasonal Time Series Data

Suppose I have a set of observational data as a time series where the observations are collected at uniform interval over the course of several years. The data exhibits seasonality over the course of ...
mtp's user avatar
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Measuring the Distance Between KDE Distributions with Different Bin Counts

I have two KDE distributions, each with a different number of bins. I'd like to compare them effectively, and I'm wondering if there's a recommended technique for this. Should I unify the number of ...
Adham Enaya's user avatar
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Is there a method to estimate the distribution of error term in linear model?

Consider the linear model where $A$ is not known $$ y = Ax + \epsilon $$ where we want to estimate the distribution $\epsilon$ from a set of samples. To prevent over-fitting, we want to impose some ...
Ma Joad's user avatar
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At what circumstances will the difficulty for the tasks of density evaluation and sampling be different?

In this tutorial video of normalizing flow, the presenter mentioned that for the original autoregressive flow, the density evaluation is fast and the sampling is slow. In contrast, for the inverse ...
8cold8hot's user avatar
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Kernel density estimation for noisy samples with known non-iid noise

I'm interested in the following variant of the usual one-dimensional density-estimation problem: I wish to estimate some unknown density $\rho$. There are iid samples $Y_{1},\ldots,Y_{n} \sim \rho$, ...
l2c's user avatar
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Kernel Density Estimation on a Log-Scale: Log Transformation vs. Geometric Space

I’m working on a project where I need to plot a Kernel Density Estimation (KDE) on a log-scale x-axis. I’ve come across two different methods and I’m unsure which one would be more appropriate for my ...
Karesple's user avatar
4 votes
1 answer
668 views

Difference between KDE, MLE and EM for density estimation

I'm reviewing kernel density estimation (KDE), maximum likelihood estimation (MLE) and expectation maximization (EM) algorithm for density estimation and struggling to differentiate what each ...
Amith Adiraju's user avatar
1 vote
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Why is histogram density estimation nonparametric?

My understanding of histogram density estimation: For $k$ predefined equal-width bins $(b_0, b_1], (b_1, b_2], ..., (b_{k-1}, b_k]$ and $n$ observations $x_1,...,x_n \in (b_0,b_k]$, we estimate ...
fin's user avatar
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0 votes
2 answers
74 views

Flexmix maxima are not where they are expected to be

For my dataset I have plotted the density with ggplot. As the data's density is multimodal (a total of 6 destinct modi) I tried to gain insight on the normal distributions associated to each modus. ...
Lukas's user avatar
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Estimating the distribution of a sum of two random variables if the family of one of the variables is known

Assume I have a random variable $Y=X_1+X_2$. I want to estimate the distribution $f$ of $Y$ given a sample $y_1,\ldots,y_N$. If this was all that is known about $Y$ the best way would probably be to ...
LiKao's user avatar
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3 answers
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Parametric models for mixed discrete/continuous data

I'm curious if there are any common parametric distribution models for mixed discrete/continuous data. For illustration, suppose I have two random vectors, $X_c,X_d$, where $X_c$ is continuous and $...
icurays1's user avatar
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2 votes
1 answer
131 views

Loss function for estimating the conditional variance by fitting $y_i^2$

I'm trying to detect anomolies in a dataset $i \in \{1,2,...,N\}$ where a random variable $y_i$ is expected to be drawn from a normal distribution with mean $\mu_i=0$ and variance $\sigma_i^2 (X_i)$ ...
JoseOrtiz3's user avatar
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85 views

Expected value (and variance) of a Dirichlet Process

Suppose I have a measure $G$ that follows a Dirichlet Process, $$G \sim DP(H_0,\alpha)$$ where $H_0$ is some base measure. Is there a closed form solution for the expected value of $G$?
dogs4ever's user avatar
1 vote
1 answer
154 views

kernel density estimation on 2D data with rotational symmetry

My question is: what is the appropriate way to apply a kernel density estimator (KDE) to a 2D dataset that has a rotational symmetry? Specifically, I have the points ($x_i$, $y_i$) and want the ...
ElectronsAndStuff's user avatar
5 votes
2 answers
549 views

Is density estimation the same as parameter estimation?

I was studying parameter estimation from Sheldon Ross' probability and statistics book. Here the task of parameter estimation is described as follows: Is this task the same of density estimation in ...
tail's user avatar
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120 views

How to transform histogram to kernel density?

I have data aggregated as a histogram $$ (m_1, c_1), (m_2, c_2), \dots, (m_k, c_k) $$ where $m_1 < m_2 < \dots < m_k$ are the midpoints of the histogram bins and $c_i$ are the counts that sum ...
Tim's user avatar
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1 vote
0 answers
58 views

Why is Rectangular density kernel not cut off at tails?

When we create kernel densities we could use different kernels. Here I create an example with Gaussian, Rectangular and Triangular kernel: When we check the start and end points of the distributions ...
Quinten's user avatar
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1 vote
0 answers
70 views

Computation of conditional expectation [closed]

Suppose that we have a random vector $X \in \Bbb R^n$ and a random variable $Y \in \Bbb R$, and that the joint density $f(x, y)$ is known. For a given $x \in \Bbb R^n$, what is the most efficient way ...
John D's user avatar
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Package for Multidimensional Density Estimation

I may be missing something obvious, but is there a python package that can reliably do density estimation of a PDF in high dimensions (e.g. 512)? I know of scipy's ...
user102938's user avatar
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45 views

Sample from one distribution such that it’s PDF matches another distribution

Problem: I have a set of samples from a continuous distribution (multivariate), call this set $W$. I have another set of samples from a different distribution $X$. I want to sample from $W$ (with ...
user102938's user avatar
1 vote
1 answer
58 views

Online Estimation of a Joint Distribution from batches of data

I want to implement an algorithm for the online estimation of a joint probability distribution from a sequence of mini batches sampled from the real distribution. The distribution is discrete and non ...
Bach05's user avatar
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0 answers
98 views

Storing a probability distribution without saving single values

I saw this question "Storing a probability distribution without saving single values" on stackexchange and thought it deserved a statistical answer. Example Scenario I could see this problem ...
Alex's user avatar
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1 vote
0 answers
188 views

How to quantitatively compare parametric density fit and kernel density (KDE) fit of a multivariate data?

I am working on modeling the joint distribution of given multivariate data. I can fit some parametric distributions on the data and evaluate the fitted models by LogLiklihood and AIC values. However, ...
Krishna's user avatar
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1 vote
0 answers
253 views

Bias of kernel density estimator of pdf $f$, where $f$ has bounded first derivative $f'$

Let's say the kernel density estimator is given by $$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{X_i-x}{h_n}\right),$$ where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability ...
Phil's user avatar
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0 votes
0 answers
40 views

Kernel Density Estimator: Misunderstanding in Taylor Series and the bias of KDE [duplicate]

Let's say the kernel density estimator is given by $\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K(\frac{X_i-x}{h_n})$, where $h_n \to 0$, $nh_n \to \infty$, $K$ a symmetric probability distribution ...
Phil's user avatar
  • 636
1 vote
1 answer
73 views

Density Estimation of a Matrix-valued Random Variable?

It seems like the density estimation of a multivariate vector-valued random variable has been well studied, but what if one would like to estimate the probability density of a matrix-valued random ...
sdasqxxz's user avatar
1 vote
0 answers
24 views

Normalized Density vs Unormalized Density [duplicate]

Edited: I have been researching about density function estimation from a sample of data, and I noticed that there are a lot of researches the estimate the density with the normalizing factor and ...
Lazag's user avatar
  • 63
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0 answers
42 views

How are probability density functions, that are computed from real-world datasets, stored and represented by computational software?

In probability and statistics, density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function. The unobservable density ...
CausalQuestions's user avatar
2 votes
0 answers
19 views

Is it correct to say 'estimate of probability density function'?

This question is about terminology: I have a stochastic process from which I get a sample. Ideally I want to know the probability density function (pdf) associated with the process, but from the data ...
user1420303's user avatar
1 vote
0 answers
57 views

Density Estimation of High-dimensional Data

I would like to estimate the probability density function of a data set with a very large number of samples (50,000+) and a large number of continuous variables (2,048). Compute efficiency is somewhat ...
Sebastian Berns's user avatar
4 votes
0 answers
165 views

How to fit a copula when zeros abound?

I am modelling a joint distribution for two random variables: $F(x,y)$. I observe $n$ data points $(x^{}_{i},y^{}_{i})^{N}_{i=1}$. I would like to model $F$ as the product of its marginals and a ...
lasoon's user avatar
  • 103
2 votes
1 answer
658 views

Fitting a copula vs. directly fitting a multivariate distribution

I understand that the joint density of two random variables $f(x,y)$ can be decomposed as the product of its marginals and a copula: $f(x,y) = g(x)k(y) \times c(G(x),K(y))$. Alternatively this may be ...
lasoon's user avatar
  • 103
1 vote
1 answer
149 views

How to estimate the conditional probability p(y|x) if y and x are both continuous but y is discrete given x?

For example, $P(Y=f_1(x)|X=x)=g_1(x)$, $P(Y=f_2(x)|X=x)=1-g_1(x)$. (The functions f1,f2 are unknown and need to be learned.) How can I estimate such a conditional probability? I guess that kernel ...
Ranger Chu's user avatar
0 votes
0 answers
50 views

How to prove symmetry of a Uniform kernel?

I am trying to prove this kernel is valid, $$ K(x) = \frac{1}{2}I(-1 < x < 1) $$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is $\ge$ 0 for ...
user359211's user avatar
4 votes
1 answer
592 views

Estimate parameters of an unknown negative binomial distribution based on known distribution

The PDF of a known NBD given in Equation (1). The parameter a and r are function of $μ$ = sample mean, and $s^2$ = sample variance, as given in Equation (2) and (3) respectively. $r$ = number of ...
vp_050's user avatar
  • 261
1 vote
0 answers
55 views

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). (...
zen_of_python's user avatar
1 vote
0 answers
46 views

Kernel Density: How do the terms 'global' and 'pilot' translate?

I nearly most of the articles on kernel smoothing or concepts that use kernel density estimations, authors speak of 'pilot' and 'global'. https://link.springer.com/article/10.1023/A:1008925425102 &...
four-eyes's user avatar
  • 141
1 vote
0 answers
147 views

How to understand the density in machine learning?

We can calculate the conditional density using Eq.1[3]. $$ p_{\theta, \Lambda}(y \mid \boldsymbol{x})=\frac{\exp \left(f_{\theta, \Lambda}(\boldsymbol{x})[y]\right)}{\sum_{k=1}^{n} \exp \left(f_{\...
Fengfan Zhou's user avatar
1 vote
0 answers
156 views

Kernel Density Estimation using a Two-Boundary-Kernel à la Jones

I'm trying to understand how to perform a KDE on a bounded support, i.e. with lower and upper boundary, when using a kernel that is specifically designed to ensure consistency/$h^2$-bias at the ...
trowraic's user avatar
14 votes
3 answers
1k views

Kernel Density Estimate for Cauchy

As far as I understand, kernel density estimation does not make any assumptions on the moments of the underlying density, and just requires smoothness. The Cauchy density function is quite smooth. ...
Greenparker's user avatar
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0 votes
0 answers
127 views

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 ...
knightontable's user avatar

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