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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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Non-parametric estimation

This is some sort of an initial question may be I'm asking which may not have a fixed, straightforward answer. But this is very unclear to me. I have come up with some parameter estimation methods for ...
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Conditional Density Estimate Loss. Why the double integral?

I read RFCDE: Random Forests for Conditional Density Estimation. Just like it sounds, these folks trained random forests for making conditional density estimates. At inference time, density estimates ...
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bandwidth setting for density comparison

I intend to compare a list of density distributions. Following one published paper (with similar type of data and same objective), I learned that I have to use Gaussian kernel density estimator, and a ...
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1answer
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Properties of Kernel Density Estimators

Given Let $X \in \mathbb{R}$ be a real-valued random variable with theoretical probability density function (pdf) $f(x)$ and corresponding cumulative distribution function (cdf) $F(x)$. Let $X_1, X_2,...
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Integration of Kernel and density product

Im considering kernels of the form $$K_s(u) = A(s)k_s(u)I[\lvert u \lvert \leq 1]$$ and $$k_s(u) = (1-u^2)^s$$ with $r$'th derivative $$K_s^r(u) = A(s)\frac{d^r k_s(u)}{du^r}I[\lvert u \lvert \leq ...
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Expectation of derivative of kernel density estimator

I am trying to calculate the expectation of the $s$'th derivative of a kernel density estimator. This problem arises naturally when trying to estimate the derivative of a density, because one approach ...
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28 views

Is There an Optimal Bandwidth for Kernel Density Estimation at One Point

I am trying to estimate the density of a 2D distribution using KDE, but I only care about the accuracy right at the origin. I have read about methods to estimate the optimal bandwidth matrix when you ...
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1answer
19 views

Function dependent to nth order on a sample

From this paper: ... every multivariate density estimator that is in any reasonable sense nonparametric may be written in the form: $$ \hat{f}(y) = \frac{1}{n}\sum_{i=1}^n K_n(x_i, y) $$ ...
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1answer
297 views

Difference between function and distribution?

I know this is dumb question, but i am confused to understand it. I have constant function as $$y = exp(-x^2)$$ This also represent gaussian distribution with mean zero and variance of 0.5 Now if ...
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44 views

Are vanishing bias and variance enough for pointwise consistency for KDE-based estimation?

Question: Is the condition that asymptotic bias and asymptotic variance goes to zero for infinite samples sufficient to guarantee the pointwise consistency of an estimator based on plug-in kernel ...
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34 views

Can kernel density estimation be used to estimate a radial probability density?

I am trying to estimate the PDF of the radius of points distributed in a 2D plane. The points are distributed like this: I can produce a histogram of the radius data which looks like this: I want ...
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1answer
119 views

Credibility evaluation - how to model conditional continuous density from multiple variables of various types?

I recently got dataset for 37000 households with declared income and a few dozens of other variables of various types: continuous, discrete, binary. The task is to automatically (unsupervised) ...
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Plot the exact density of a transformation of a distribution

I would like to compute (and plot) the exact density of the following distribution: $ X_i \sim exp(-Exponential(\lambda)) - 0.5 $ I already have the estimated density for this distribution, but I ...
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Weighted kernel density

I would like to produce a 3-d plot based on density of 2-d data. This can be achieved for example in R using the kde2d function from the ...
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2answers
78 views

GLMM species count data with transects

I am trying to create a GLMM model which explains differences in abundance/count of three species of scorpion around a field reserve in different forest types. -I have 7 trails in different forest ...
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1answer
82 views

How to get a density from a forecast with prediction interval

Some reproducible code to have in your environment a time series and a possible forecast: ...
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Choosing Gaussian PDF basis bandwidth depending on number of bases and range of data

Summary (details below!) I have a basis expansion of $m$ (univariate) Gaussian PDFs to model the density of a sample $X$. The means of these PDFs are spaced equidistantly through the domain of $X$ ...
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Principled estimate of joint PDF given marginals and first and second-order statistics

(Trying to solve the same underlying problem as this.) Let's say we know the marginal probability density functions $p_i(x)$ of a set of zero-mean random variables $\{X_i\}_{i=1}^N$ as well as the ...
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3answers
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How long is a distribution considered normal?

I have a dataset of metric distances ($n=5800$) and plotted those as a histogram. My initial thought was that this distribution looks normal. But after performing a Shapiro Wilk test and plotting the ...
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What are the pitfals of using kernel density estimates to infer about the shape of an underlying distribution

I know that we cannot simply infer from the shape of a histogram to the shape of the underlying distribution, as the shape of the histogram is influenced by the choice of the intervals (Assessing ...
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1answer
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Is the inverted pi a standard symbol for d-separation?

I've seen the inverted pi symbol earlier but found it hard to find an explanation for it. Found it again in the context of d-separation between Xu and Xv here: https://en.m.wikipedia.org/wiki/...
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1answer
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Multiplying two event with probability density function, is it possible?

in my exercise, $X$ is the size of a tree trunk, and $X$ follows a normal distribution $\mathcal{N}(9,0.4)$, we want to know $P(8.8\le X\le11.2)$ So I though that I could do this: $P(8.8\le X \le11.2)...
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1answer
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How to determine Density Value of a Z-Score?

In this paper the authors are demonstrating the conversion of Ordinal data to Interval data. In the Step No. 7 on Page 360, they are talking about Density values of corresponding Z-values. They are ...
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Finding probability of a point using bivariate copula density

I have a data in the form $\textbf{N} \times 2$. I am using bivariate copula to model the joint density of this distribution. Firstly, I fit 2 marginal distributions independently on each column of ...
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1answer
136 views

MSE of Kernel Density Estimator

Erlang Kernel is used for density estimation. By using this estimates are pretty close to the real density on graph on the other side MSE is very large. But Author of Erlang Kernel stated that it will ...
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How to aggregate histograms for density estimation

Within a very large sensor network, each node does take measurements derived from a fixed number of samples taken at a high frequency from an instrument. The number of measurements send to an ...
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1answer
610 views

A Gaussian Mixture Model Is a Universal Approximator of Densities

When discussing the concept of mixtures of distributions in my machine learning textbook, the authors state the following: A Gaussian mixture model is a universal approximator of densities, in the ...
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Truncated Non-Parametric Density Estimation

I have cross-sectional data for individuals, $i$, and years, $t$. I need to estimate the density of a random variable, $X_{it}$, which is equal to $X_{it} = A_{it}/B_t$. My data is truncated in the ...
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47 views

Density curve in R - AUC bigger than 1

correct me if I'm wrong but I was expecting the area under the curve should be 1 for a probability density function. Can anybody explain why it's not always the case when using the ...
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1answer
444 views

Bandwidth parameters in multivariate KDE using scipy.stats.gaussian_kde

I am working on a project which involves implementing in Python two different density estimation functions over multivariate data; one using N-d histograms and the other using kernel density ...
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0answers
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How to estimate density given similar instances?

I'm trying to estimate the probability distribution of a random variable $X$ given values of $N$ parameters/qualities $Q_1, Q_2, ..., Q_N$, and historical data. The two extremes in estimation would ...
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89 views

Predicting probability distribution of value in time series of real numbers like Dow Jones?

While we are usually interested in predicting values of time series, it is often also valuable to predict probability distribution of the next value basing on its context - for example for risk ...
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1answer
67 views

Why isn't mean integrated square error (MISE) probability-weighted?

We often try to minimize the MISE of a KDE: $\text{E}_{\mathbb{P}^n}[\int (\hat{p}(x) - p(x))^2 dx]$. Why don't we instead try to minimize $\text{E}_{\mathbb{P}^n}[\int (\hat{p}(x) - p(x))^2 p(x) dx]$,...
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Any evidence that fitting distributions that are more concentrated is easier?

Suppose we have the budget to sample $n$ instances from an unknown distribution. If the distribution is more flat, the samples are spread and hence it feels harder to learn, compared to distributions ...
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1answer
79 views

Assessing the quality of a density estimate without knowing the true density

Assume that for a univariate sample $X$ you have a density estimate by a model you developed (in my case it's a density forecasting model) and you want to assess the quality of your models estimate. ...
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263 views

Does a density forecast add value beyond a point forecast when the loss function is given?

Density forecasts are more universal than point forecasts; they provide information on the whole predicted distribution of a random variable rather than on a concrete function thereof (such as ...
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2answers
605 views

Sample multivariate PDF from KDE with different norm [closed]

I am using KDE with a modified metric for the distance. The PDF is as expected (see below: color is the probability and the dot is the point used to fit the KDE). But due to the new metric, I cannot ...
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1answer
38 views

Mean Integrated Square Error

I'm evaluating several methods to estimate the density of an unknown distribution $f$ from observed data, among which kernel density estimation with distinct kernel functions, a mixture density ...
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1answer
128 views

Density estimation using (different) order statistics

I need to estimate a univariate distribution $F$ as flexibly as possible. However, I do not observe draws from $F$ directly. Each observation $x_i$ is the minimum of $a_i$ draws from $F$, where $a_i$ ...
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Estimating conditional pdf using kernel estimator

In a paper I'm reading they consider the logarithmic return $Y$ and the number of trades $T$ over a time periof of fixed size. They then want to show that $Y$ conditioned on $T$ is approximately ...
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1answer
66 views

convolution and deconvolution of random variables of different dimensions

Preliminary: Let's say we have $Y=X+Z$ ($Y$ is data, $X$ is latent variable and $Z$ is noise), where the random variables are all in $\mathbb{R}$. Then an inverse Fourier transform leads to \begin{...
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111 views

Density of features from density of principal components

I have used principal component analysis for dimensionality reduction on large number of features. But after some stack of unsupervised learning, I have calculated kernel densities for all these ...
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28 views

binomial distribution density asymptote

mathematically, why does this function have an asymptote around 0.63? is there an way to formally describe this asymptote? here's the pattern: two coin flips, what's the probability that at least ...
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1answer
57 views

Semi-supervised parametric density estimation

I am trying to learn a (neural) density estimator for a set of data p(x), however I know that the true distribution is a mixture of two other distributions, q(x) and z(x), with fixed mixture weight. ...
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2answers
179 views

ABC with non-uniform prior

I had asked some similar questions in the past, but I never got either the answers or the discussion I was hopping for. So I will rephrase the problem to see if I can understand it myself. I'm trying ...
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2answers
312 views

Quantile of kernel density estimator

I consider the kernel density estimation, and I want to find the quantile the quantile at some level (0.95) of the KDE. ...
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1answer
170 views

Kernel density estimation as a density

I have some troble understanding the Kernel density estimation. If a consider the next example: ...
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0answers
43 views

In the choice of bandwidth for kernel density estimator. Why usually minimize MISE instead of minimizing ISE?

Before presenting my question (which I already formulate in the title of this post) is important to establish the context of my problem: Let $\xi$ be a random variable with density function $f$ ...
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1answer
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Mean integrated total variation KDE

Background: Kernel Density Estimators (KDE), given $n$ i.i.d. samples of a random variable $X$, and when chosen with the appropriate window size $h^*$, have an asymptotic mean square integrated error (...
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How to estimate the observation level data generating when there is unequal sampling of data for each record

I am looking at bid amounts for cars sold at a sealed auction (only the seller has all the information). I am trying to predict the number and magnitude of offers that a car will receive. Some of the ...