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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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Finding probability bucket which result in 90% correct classification

I have a dataframe of two columns, one of which contains probabilities of event X happening and the second column is whether or not X did occur as indicated by a 0,1. I would like to find the buckets ...
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Density Estimation Efficieny

My Question Let's say a set training samples like D from a discrete distribution like p(x) over a discrete variable vector like x is available. We don't have any prior knowledge about the form of p(x)...
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50 views

Estimate value with binomial distribution [closed]

We have some compound A diluted in a solution. In 200 trials, we find that when we mix $1$ $\mathrm{mm}^3$ our solution of A with some amount of some compound B, we get a reaction 185 times. How can I ...
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47 views

Density Estimation and Data Normalization

Is there any problem to first normalize data (for example, min-max one) then use kernel density estimation to get pdf of each sample? Thanks.
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23 views

Every point has the same probability?

I am reading "Pattern recognition and machine learning" by Cristopher Bishop. In Chapter 2.5.1 "Kernel density estimator", there is written that: Let us suppose that observations are being drawn ...
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1answer
28 views

Bayesian approach: ignoring the denominator leads to the conditional density equaling the joint density? [duplicate]

I know there are a lot of questions here about ignoring the denominator in a Bayesian approach, but I don't think mine is a duplicate of any of them. I am reading the book "Pattern recognition and ...
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1answer
34 views

What is the difference between probabilistic forecasting and quantile forecasting?

A probabilistic time series forecast outputs the entire distribution of the forecasted values for a given time point, instead of just a mean or a point forecast. A quantile forecast is a forecast ...
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6 views

Piecewise Pareto Distribution

What are the best practices for Piece-wise Pareto Distribution or maybe Pareto Mixture Model(?). Example: $x\in [0, 1) \Rightarrow \alpha=0.1$ $x\in [1, 10) \Rightarrow \alpha=0.5$ $x\in [10, 100) ...
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To what distribution it's similar? Looks like an exponential but it's not

To what distribution it's similar? Looks like an exponential but it's not. It's seems to have a property, that if I zoom it (xlim) then each time it has the same ...
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2answers
55 views

Can Kernel Density Estimation estimate an Exponential Distribution?

Can Kernel Density Estimation estimate an Exponential Distribution? I tried to performed to make experiments with various kernels like: "gaussian" and "exponential", but performance seems to be very ...
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Comparing Density Plot height (MachineLearning Classification)

I am working on a binary Machine Learning classification problem. My classifiers are really performing poorly because distribution of the 1 class is very similar to 0 class (dataset is imbalanced, 1 ...
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28 views

Substitution for unknown true density in 'Density Estimation Trees'

I'm having a hard time understanding parts of the derivation of the objective function for Density Estimation Trees (reference below) regarding the loss function. Taken from the article (Sec. 3.1): ...
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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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23 views

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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23 views

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
55 views

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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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
299 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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48 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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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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48 views

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
122 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
118 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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30 views

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
91 views

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
22 views

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
31 views

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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91 views

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
168 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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60 views

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
715 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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37 views

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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53 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
533 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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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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102 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
87 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
83 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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284 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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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
40 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
140 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 $a_i$ ...