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Questions tagged [kernel-smoothing]

Kernel smoothing techniques, such as kernel density estimation (KDE) and Nadaraya-Watson kernel regression, estimate functions by local interpolation from data points. Not to be confused with [kernel-trick], for the kernels used e.g. in SVMs.

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15 views

Behavior of kernel density estimation

Consider the random variable $X=YZ$, where $Y\sim\text{Normal}(0,1)$ and $Z\sim\text{log-Normal}(0,1)$ are independent. I wanted to assess the accuracy of kernel density estimates for the density ...
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Density estimation as an optimization problem

Density estimation is the estimation of a probability density function from observed data. Can some of the common approaches to density estimation, such as kernel density estimation, be formulated as ...
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Sampling from transformed KDE representation

Let's say that I have a random variable $X$ distributed according to some prior distribution $p(x)$, for simplicity assume that it's a Log Normal distribution. I then sample $N$ elements from this ...
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“Simple” boundary correction method in kernel density estimation

I'm new to kernel density estimation and have a rough idea on boundary bias. When correcting for boundaries, I tried to use boundary correction method as "simple" which is available in R. Once I ...
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How does one choose the bandwidth for the confidence intervals of a kernel density estimate?

My lecturer estimated a kernel density, then wanted to give confidence intervals for it. So, he used a bandwidth $$h_c = h_g \times \frac 1{n^{0.05}}$$ where $h_g$ was the bandwidth of the original ...
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Demmler-Reinsch basis for smoothing splines

I have seen some papers about using the so-called Demmler-Reinsch basis for smoothing spline because it is a basis for natural spline space and also Sobolev space. For example, these papers: A ...
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How to generate random samples from a 2D dataset?

Suppose I'm given a data set consisting of many pairs of $(x,y)$ values which are correlated in some arbitrary complex way. How would I go about 'generating' more pairs of $(x,y)$ coordinates which ...
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Fitting KDE with scikitlearn and pandas to plot. However, distributions lie outside the range of data

I am fitting a distribution of scores ranging from 1-13 for a set of data using scikitlearns KDE functions and Pandas plot.kde. I have set the bandwidth with a gridsearchCV method. However, when the ...
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Gaussian Kernel and Feature Space [duplicate]

I have been reading this paper for a few days. There is one section (Section 3.3) that confuses me. We start by gathering local features from training images of a particular class into a single ...
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MSE for the kernel-based HAC long-run covariance matrix itself (in the Frobenius norm sense)

Consider the stationary multivariate time series $X_1, \ldots, X_T$ and the HAC-consistent long-run covariance matrix estimator $$\hat{\Gamma} = \hat{\Gamma}_0 + \sum_{l=1}^{T-1} K\left(\frac{l}{h_T}\...
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How does this algorithm converge?

From https://en.wikipedia.org/wiki/Kernel_density_estimation, we have this formula for the optimal bandwidth for a kernel density estimator. Note that we need $R(f'')$ which is unknown. An ...
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A comparison of the global optimal binwidth and local optimal binwidth of the histogram estimator

Suppose we have $X_1, \dots, X_n$ to be an i.i.d sample with unknown pdf $f(x)$ and cdf $F(x)$, and define $\hat{f} (x)$ to be the histogram estimator. We also define its Mean Integrated Square Error ...
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Gaussian Kernel density estimation by hand

I'm trying to understand the logic behind kernel density estimation. I found the explanation in wikipedia very useful, but I'm not capable yet, of having a full understanding of this method so I want ...
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Minimizing MISE to find consistent estimator

Consider kernel regression estimation of the mean function $m$ of the process $$y_t = m(x_t) + \epsilon_t,$$ where $\epsilon_t$' s are correlated with covariance function $R(s,t) = \exp \{-\lambda|s-...
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Does local linear regression include a weighting Kernel?

3 minutes ago 1 Hey, I am applying a Regression Discontinuity Design (RDD) to estimate the effect of a policy change. In RDD I can apply the parametric approach (polynomial regression) and the non-...
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state of the art in density estimation

I have seen density estimation methods which are pretty old. Specifically, I am referring to Parzen Window method. When I read the original Parzen's paper, I was amazed by it's beauty and I know that ...
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Kernel density, why does my subset appear to have a larger spread than the original series?

I have a series that is 1500 observations long called alt_intercept. From it, I created a subset that contains values only if another series (called pvalue) is less ...
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why does y axis sometimes change from normal histogram to kernel density?

Consider the distributions I have plotted below. They are of the same variables, one in normal histogram form and another in kernel density (Epachanov). As far as I know, the auc of the kernel ...
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Kernel Density Estimation - Physical Interpretation?

I just read this article about the motivation for KDE. From what I understand, you are using Gaussian probability density distributions for each datapoint and then, depending on the selected kernel ...
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How can I estimate bivariate probability density for support restricted data?

I have a bivariate sample with the following kernel density estimation The issue is that there is actually a cutoff for log(Age) at about 2.5, so value greater than 2.5 has probability 0. The fitted ...
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Transformation of a random variable and KDEs — when is reweighting needed?

Suppose I have been given some data $X$ that I wish to resample according to their empirical distribution. For whatever reason, I decided to transform these variables to some other space $Y = f(X)$ ...
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How to choose sample size from probability density for computing mutual information based on continuous variables

I need to compute mutual information gain based two continuous variables $X$ and $Y$ $I(X|Y) = \int_X\int_Y p_{x.y}(x,y) \log(\frac{p_{x.y}(x,y)}{p_{x}(x)p_{y}(y)})$. I have used Kernel Density ...
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ksmooth function in R

I'm trying to understand how ksmooth function in R works (I haven't really taken much statistics other than an introductory one at college, so sorry if this is a ...
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Find PDF(X,Y) from PDF(X) and PDF(Y)

given that X and Y are not mutually exclusive, is there anyway to calculate PDF(X,Y) from PDF(X) and PDF(Y)? Following are a few plots made from the dataset. In above image i have to find how PDF(11,...
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Reading kernel distribution plot vs typical histogram

Tasked with showing the distribution of a certain data set in a different way, I wanted to try to plot a kernel density. After seeing it however, my co-worker advised against it saying that because ...
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What does alpha in smoothing stand for

I want to apply Gaussian smoothing to a dataset and came across the smth.gaussian function in R. That besides the numerical input data requires two parameters: ...
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ROC curve with symmetrical kernel

I am trying to use kernels with ROC curve, and 'm succeed to plot them but now my query is about theoretical grounds, i.e. its bias, var, etc. I want to evaluate the theorems (1 & 2) in Pulit (...
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Kernel Density estimation with guassian kernel function why don't have indicator function like other kernels [closed]

Kernel Density estimation with guassian kernel function why don't have indicator function like other kernels.
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When to use non-parametric regression such as kernel, generalized additive model, spline, and polynomial?

I understand that kernel regression is a form of non-linear/non-parametric regression. However, I know you can also use generalized additive models for non-linear regression, as well as polynomials ...
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Python: “Normalizing” kde, so it always lines up with histogram

In Python, I am attempting to find a way to plot/rescale kde's so that they match up with the histograms of the data that they are fitted to: The above is a nice example of what I am going for, but ...
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What kind of kernel is used by statsmodels.nonparametric.kernel_regression.KernelReg?

I am doing multivariate nonparametric kernel regression using the Python function as mentioned in the title. The documentation can be found here: https://www.statsmodels.org/stable/generated/...
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How to calculate Kernel Density for Bootstrap Likelihood

I am attempting to write R code to generate bootstrap likelihood as described in section 3 of this paper https://arxiv.org/pdf/1510.07287.pdf. I am confident that I performed the bootstraps correct, ...
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Reproducing kernels: how do I numerically compute the decomposition?

Suppose I'm given a kernel, $$ K(x,y) : \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} $$ In order to describe/understand the (unique) associated RKHS, I seek its eigenfunctions, as per Mercer'...
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KDE that better-preserves percentile distributions

My understanding is that a Gaussian KDE, because the kernel is symmetric, preserves the statistical mean of a distribution. For my particular case, I'd really prefer a KDE that preserved the median, ...
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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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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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Kernel Density Estimation for non-parametric

I'm writing an R function to get the fitted values of the kernel density estimate. For that I use the computational formula of summation of ({n-1 h-1 K{(x - Xi)/h}}?) $$ \hat{f}(x) = \frac1{n h}\sum_{...
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449 views

How to choose the bandwidth of a KDE in python

Python's Sklearn module provides methods to perform Kernel Density Estimation. One of the challenges in Kernel Density Estimation is the correct choice of the kernel-bandwidth. I have come across ...
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Group comparison for bivariate distributions

For two groups A and B that consist of n and m individual samples. Each individual sample has a unique 2-dimensional joint probability density functions (PDFs)of two variables. These PDFs are ...
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102 views

Estimating the population median from a kernel density estimator

I have a 1-d kernel density estimate in the form of two vectors: x_grid is a vector of x-values at which the density function was sampled ...
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2answers
129 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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197 views

Isn't kernel ridge regression supposed to be the mean of gaussian processes?

I read a few times that the mean prediction of a GP should be equivalent to KRR. I tested this empirically and found (dataset is y=2x + gaussian noise): Two explanations for this come to mind: GP is ...
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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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Derivation of AMISE and Bandwidth

Given: Let $K(\cdot)$ be a bona fide kernel. Let $f$ be a pdf and $\widehat{f}_n$ is kernel density estimator with bandwidth $h$ based on a sample $X_1,X_2,\cdots,X_n$ of size $n$ draw iid from $f$. ...
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Contribution of a predictor in Nonparametric regression

Is there an equivalent to a beta weights in a nonparametric regression? I am using the NP package in R and running a local linear regression where my bandwidth estimates are produced using least ...
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63 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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GARCH model with t-innovations

I am modelling a time series with GARCH model with t-distributed error using RUGARCH package. My model is specified as: ...
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1answer
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What does bandwidth in kernel regression mean?

here https://stat.ethz.ch/R-manual/R-devel/library/stats/html/ksmooth.html is bandwidth explained as "the bandwidth. The kernels are scaled so that their quartiles (viewed as probability densities) ...
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Adaptive Parzen Estimator

I am trying to implement the Tree Parzen Estimator for hyperparameter optimization. I follow this paper https://papers.nips.cc/paper/4443-algorithms-for-hyper-parameter-optimization.pdf, in which ...
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Confidence Region of a multivariate KDE in Python?

I have an estimated bivariate kernel density based on a set of observations (𝑥11,𝑥12,...,𝑥1𝑛) and (𝑥21,𝑥22,...,𝑥2𝑛) and would like to draw confidence regions in the (𝑥1,𝑥2) space. This is in ...