# 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 ...
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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 ...
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### 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 $\sigma_i^2 (X_i)$ ...
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### 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 $G$?
1 vote
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 ...
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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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1 vote
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### 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 ...
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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 ...
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### 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 ...
1 vote
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 ...
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### 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 ...
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1 vote
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### 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, ...
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1 vote
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### 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 ...
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### 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 ...
• 636
1 vote
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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### 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 ...
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1 vote
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### 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 ...
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 ...
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### 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 ...
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1 vote
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### 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 ...
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### 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 ...
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 ...
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1 vote
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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). (...
1 vote
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### 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 &...
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### 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_{\...
1 vote
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 ...
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