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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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Estimating Probability Density for Sample

I have a dataset of over 20,000+ samples. The objective here is to define a distribution for the sample so that I can plot all possible outcomes. However, I am unable to find an appropriate ...
Ahmed Jyad's user avatar
1 vote
1 answer
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Intensity outliers/anomalies in 2D plot

I wonder what kind of method better to use to see outliers on z value of 2D plot. For example, I have measurements of x and y values both in range of 1 to 16 with step of 1. Next I calculate how many ...
Zoomman'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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Help in simulation for bivariate residual entropy

I am reading a research paper research paper. Let $X=(X_1,X_2)$ be a bivariate random vector with survival function $\bar{F}(x_1,x_2)$. The the condition residual entropy for the condition ...
Unknown's user avatar
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0 answers
65 views

Linear regression with smoothed time-series as independent and dependent variables

I'm pretty sure I'm misunderstanding something quite obvious here but I'm rather confused. I have multiple time-series that have been smoothed with a gaussian kernel. My goal is to regress the time-...
Thomas's user avatar
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7 votes
2 answers
883 views

Why does re-scaling my density plot using counts change the y-axis so much?

When I make a histogram I get the actual distribution of my samples, with the appropriate counts, but when I try making a density plot the scales go up to 800, and when I try using ...
maglorismyspiritanimal's user avatar
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Is there a way to accommodate multiple nominal datasets in one-class classification with KDE?

I have 50 sets of time series data, which are collected from 50 'good' runs of the fabrication process, and I would like to utilize all of these nominal datasets to train my model. From what I`ve ...
wruskrappy's user avatar
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33 views

Implementing Convolution Function for Gaussian Kernel in Python for PDF Estimation

I am currently working on estimating a probability density function (PDF) nonparametrically using a Gaussian kernel. My goal is to determine the optimal bandwidth $h$ that minimizes the cross-...
Tim's user avatar
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34 views

Image Blur - Disc Kernel [closed]

I'm trying to use the blur() function from the spatstat package in R to blur an image. One ...
BurlyPotatoMan's user avatar
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62 views

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
2 votes
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25 views

Exchanging integrals with inner products with kernel mean embeddings

I am doing some reading on kernel mean embeddings. In particular I am reading the survey paper by Muandet et al. On page 27 (Section 3.1) the authors begin a gentle introduction to kernel mean ...
Nick Bishop's user avatar
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63 views

The meaning of probability density functions' product followed by an integration

Scipy's KDE object allows integration of a function multiplied by another KDE object. I assume that this is meant to be used for the estimation of distance between two distributions. As far as I ...
Gideon Kogan's user avatar
1 vote
1 answer
28 views

Estimating the Distribution of the data

I wanted to know the distribution of some discrete points and they don't follow any particular distribution according to the graphical and other methods. So, I thought of applying non-parametric ...
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2 votes
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111 views

Textbook Recommendation other than ESL [duplicate]

My current background is as follows: (core subjects only) Math : Linear Algebra, Analysis, (half of) Measure TheoryStats : Mathematical Statistics, Regression Analysis, Multivariate Analysis "...
jason 1's user avatar
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3 votes
2 answers
106 views

Positive Semidefinite Kernel in RKHS

The following shows part of the page 170 of The Element of Statistical Learning that I want to make clear. The solution can be characterized in two equivalent ways $$\min_{c_j}\sum_{i=1}^N(y_i - \...
jason 1's user avatar
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2 votes
1 answer
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Regularization Problem and Reproducing Kernel Hilbert Space

The following shows part of the page 169 of The Element of Statistical Learning that I want to make clear. We have $$\min_{f \in \mathcal H_K}[\sum_{i = 1}^NL(y_i, f(x_i)) + \lambda\Vert f\Vert_{\...
jason 1's user avatar
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Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?

The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar polynomials) can be used to interpolate a continuous function $f$ at by its value at ...
Ma Joad's user avatar
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1 answer
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Addition of asymmetrical uncertainties for use in KDE?

My data have asymmetrical confidence intervals (e.g. 100 with a 95% lower bound of 50 and 95% upper bound of 300). I want to perform a KDE on all my data where the bandwidth is determined by the ...
lmm's user avatar
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1 vote
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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
634 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
0 answers
31 views

Strong consistency of kernel density estimator

I am studying the book Nonparametric and Semiparametric Models written by Wolfgang Hardle and have difficulty with the following exercise: $\textbf{Exercise 3.13}$ Show that $\hat{f_h}^{(n)}(x) \...
graham's user avatar
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0 answers
30 views

KDE-like technique to learn a continuous distribution from samples subject to specific noise

There's a continuous-valued random variable $X$ with distribution $f_X$. Normally, we're given a bunch of i.i.d. samples $X_1, \ldots, X_n$, and we try to give an estimate $\hat{f}_X$ of the ...
chausies's user avatar
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0 answers
29 views

Kernel Density Estimation, Bandwidth Tuning, Independence, and Comparison

like many of us here, I turn to kernel density estimation when I need a nonparametric estimate of a numerical feature's distribution, and in an attempt to assume as little as possible, I usually use ...
user3163829's user avatar
1 vote
0 answers
41 views

Sampling subsets of a given PDF with controllable sum and frequency

I am working with a dataset with $n$ samples and $d$ features for each sample. I would like to be able to sample "nice" subsets of this dataset with specific properties. Assume that this ...
person17381's user avatar
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265 views

How do you chose the standard deviation for a Gaussian kernel in KDE? [duplicate]

The routine explanation of a KDE plot is: You chose a kernel (let's say a Gaussian) You center a kernel at each data point You average all kernels I get that (2) means that a Gaussian curve is "...
Nick's user avatar
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1 answer
22 views

How should I parse the definition of a regular kernel?

I was reading a paper on kernel regression, and the paper defines a non-negative kernel as regular if there exist $b > 0$ and $r > 0$, such that: (1) $K(x) \ge b I\{x \in S_{0,r}\}$ (2) $\int \...
kirafa's user avatar
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1 vote
1 answer
318 views

Kernel Smoothing for Time Series data [closed]

I have generated a time series data set of measurements that are a bit noisy and I want to apply kernel smoothing to the data. My time series data is not regular however, meaning that the time ...
Jade131621's user avatar
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0 answers
22 views

Comparing models with transformation from discrete to continuous

I have two models to fit a set of categorical features. One uses an encoding followed by a Kernel Density Estimation (with cross-validated bandwidth search) to make a continuous distribution. I am ...
Daniel Kagan's user avatar
4 votes
3 answers
1k views

How to interpret peaks in probability density function?

If a probability density function (created using kernel density estimation) exhibits peaks (not necessarily the mode), can we infer the presence of clusters or subgroups in the data?
Amit S's user avatar
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2 votes
1 answer
56 views

Gronwall's inequality

I am reading the article. I am getting stuck with the first proof proposition 4 on page 32. To be more specific, they understood the reason why they obtained $F(x) \le \frac{2K}{1-\frac{2R\epsilon}{\...
Pipnap's user avatar
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1 vote
1 answer
145 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
1 vote
0 answers
108 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
55 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
  • 389
0 votes
1 answer
68 views

Implementing (R-)ALoKDE algorithm for data streams density estimation

I'm trying to implement the (R-)ALoKDE algorithm for the density estimation of the data streams. The algorithm has been published and presented in [1, 2]. Although the algorithm seems simple, I'm ...
Tomasz Rybotycki's user avatar
1 vote
0 answers
161 views

PSM kernel matching and bandwith: what observations are used

I'm using PSM with an Epanechnikov Kernel and a bandwith of 0.06. I'm confused about which observations are matched. I thought it was (broadly) like a wheighted radius matching, where every control ...
Anis's user avatar
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2 votes
1 answer
56 views

How are these simulated sample means created/plotted in R?

I had a student point out this image in Learning Statistics with Jamovi that is also in Learning Statistics with R (Page 294 in the latter). I was going to send her a reply when I noticed something in ...
Shawn Hemelstrand's user avatar
8 votes
1 answer
215 views

Radial axis transformation in polar kernel density estimate

Consider a kernel density estimate of a continuous, non-negative random variable defined over the unit circle with no discontinuity between 360 and 0 degrees. Unlike in the most common KDE ...
Reinderien's user avatar
4 votes
1 answer
176 views

Proving that the bias of the derivative of Parzen-Rosenblatt (kernel density) estimator is of order $O(h^2) $ and $O(h)$ when $h$ approaches $0$

I came across this property that I don't get and I couldn't find the proof anywhere: Suppose we have a density $K$ of the standard normal distribution and $K'$ its derivative. Suppose that the density ...
wageeh's user avatar
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1 vote
0 answers
225 views

Bias of kernel density estimator

This question is related to a calculation on page 20 of Kernel Smoothing by Wand and Jones. i) Let $f:\mathbb{R}\to \mathbb{R}$ be a density for a real-valued random variable, and assume that $f''$ ...
Abm's user avatar
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1 vote
0 answers
70 views

Methods describe the temporal consistency of kernel density data

I am working on a spatial time series analysis project. The task is to study the spatial distribution of point features (e.g., crime events, traffic accidents) over time. I aim to find the places with ...
Bright Chang's user avatar
0 votes
1 answer
31 views

I can't understand how this function works (linear filter with a filter kernel)

on page 13 of the Dayan and Abbott book onTheoretical Neuroscience, there is this formula $$r_{approx}(t)= \int\limits_{-\infty}^\infty{d\tau w(\tau) \rho(t-\tau)}$$ Let's assume that $w(t)$ is a ...
Vaaal's user avatar
  • 587
1 vote
0 answers
55 views

Are low-rank kernel approximations implementing implicit regularization?

Consider a kernel estimation problem as follows. We have functions $f^* \sim GP(0, C^*)$ drawn from a Gaussian process. We want to construct a kernel $K$ that does well in regressing functions drawn ...
Tanishq Kumar's user avatar
2 votes
1 answer
115 views

Reason why kernel density graphs are so different in Python versus R

I plotted the same data in R using geom_density, but the blip for "Yes" is much, much smaller in Python using kdeplot ...
Flying Spaghetti's user avatar
2 votes
0 answers
131 views

Propensity score non parametric estimation

In several papers, in the 'double machine learning' literature, the propensity score (a nuisance parameter) is estimated non parametrically. It is a bit unclear how this estimation is performed, as ...
mich95's user avatar
  • 111
2 votes
1 answer
140 views

Generate a point cloud distributed according to a Boltzmann-Gibbs distribution with prescribed marginals

Let $p$ be a probability density on $\Omega\in\mathcal B(\mathbb R^d)$ for some $d\in\mathbb N$ (I'm primarily interested in $\Omega=[0,1)^d$). We can approximate $p$ by $$A_x(y):=\sum_{i=1}^k\varphi_{...
0xbadf00d's user avatar
  • 293
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0 answers
90 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
124 views

Gasser Müller estimator for estimating the derivative $m'(x)$ of a nonparametric regression function

I would like to compare the performance of the Gasser Müller estimator with other estimators for estimating the the derivative $m'(x)$ of the regression function $m(x)$. Let's say we have the ...
Mathieu Rousseau's user avatar
1 vote
0 answers
38 views

Maximum bias for NW estimator when $r(x)$ is Lipschitz (question 17, chapter 5 All of Non-Parametric Statistics)

The general condition is that $Y_i = r(X_i) + \epsilon_i$, and we want to estimate $r$ using Nadaraya–Watson kernel regression. We additionally assume $r\colon [0,1] \to \mathbb{R}$ is lipschitz, so $|...
Phil's user avatar
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