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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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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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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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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 ...
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Comparison of Kernel Home-Range Utilisation Distribution between samples with unequal numbers of locations

I have a dataset containing GPS coordinates for a sample of animals in the same location at the same time (think of an aerial snapshot of a cows in a field), at two different dates. I have estimated ...
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How to perform MCMC integration when no prior over the integrated function is available? [closed]

As far as I can tell, MCMC integration (e.g. VEGAS) is performed by sampling from a distribution proportional to $f(x)$ using MCMC, then building a density estimator $g(x)$ using these samples (for ...
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Mairhuber-Curtis Theorem, Adaptive Bases and Neural Networks

I am reading the paper Positive Definite Kernels: Past, Present and Future by GE Fasshauer in which he describes computing with PD kernels. In section 3.2 he mentions the Mairhuber-Curtis Theorem, ...
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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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Kernel density estimation (KDE) for data points with different variance

Consider the following situation: An experiment was done in 15 different conditions, and a value of the parameter 'A' was measured in each experiment (A can have any integer value between 0 and ...
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Data Normalization for Time Series Forecasting using Nonparametric Model

What are the specialities of applying locally weighted learning for time series forecasting? I am trying to apply a nonparametric model ($K$-NN Regression) to forecast daily load curve (entire time ...
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How to decide the weight of the locally logistic regression?

I have a problem of how to decide the weight of the logistic regression. My model is as below. For the general logistic regression, we have the likelihood of the model: $L(\theta)=\prod_{i=1}^{m}P ( ...
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How to compute Integrated Squared Error for kernel density estimation in R

I am working on R for kernel density estimation. I am testing different kernels and I need to evaluate them. I use next code for density estimation: ...
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downsampling a kde / combining kde and histogram

I'm calculating a KDE of one parameter (y, particle density) in bins of another parameter (x, distance from the origin). At ...
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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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Hypothesis testing in non-parametric regression

Say I have two processes/time series, $X = (X_{t_{1}},X_{t_{2}},\dots , X_{t_{n}})$ and $Y = (Y_{t_{1}},Y_{t_{2}},\dots , Y_{t_{n}})$ observed at times $t_i$ for $i=1,2,\dots, n$ where $0 < t_1 <...
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Equivalent Kernel - Bishop Chapter 3

I've been struggling to understand the Equivalent Kernel in Bishop's Pattern Recognition and Machine Learning book. Can somebody explain the following Figure (3.10) from chapter 3.3.3? (image taken ...
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Bilinear Process

how to generate a functional bilinear process such as: $ X_{n+1}= \int\psi(t,s)X_{n}(s)ds + \iint\phi(t,s,u)X_{n}(s)\varepsilon_{n}(u)dsdu + \varepsilon_{n+1}(t) $ ? About the first integral there ...
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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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What did Silverman (1981) mean by 'critical bandwidth'?

In the selection of a bandwidth for a Kernel Density Estimator, critical bandwidth according to my understanding is: "For every integer k, where 1<k<n, we ...
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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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Derive the estimator for the integrated squared bias $\int \left(\operatorname{E}\hat{f} - f\right)^2 $

This problem is found in p. 77 of Wand & Jones' (1995) book. If you are familiar with nonparametric estimation you may skip this introduction. Suppose we want to minimize the integrated squared ...
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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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Gaussian random fields: matrix and convolution sampling

I should be able to generate a stationary GRF from white noise in two different ways: multiplying the white noise vector by the square root of a covariance matrix with appropriate kernel; taking the ...
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157 views

Point process - intensity function vs probability density function

Suppose we have a point process in $\mathbb{R}$ with intensity $\lambda(x)$. Then, for a given compact set ${ S}$ we have $$\Lambda({ S})=\int_{\rm S} \lambda(x) \, dx,$$ where $\Lambda({ S})$ is ...
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How can I derive the function curve from a histogram of observed data

I'm analysing some datasets that produce heavy tailed data when plotted as a histogram. My initial goal was to attempt to fit a known distribution to my dataset. Thereafter I use to the properties of ...
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Is a kernel density estimate meaningful if > 25% of my data are duplicates?

The title pretty much says it all. I have data that consists of 80 samples but there are always at least four samples that have exactly the same value. I want to assess, whether the data is unimodal. ...
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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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Uniform kernel estimator

For practice, I'm trying to provide an estimation for a nonparametric model on dataset BMACS from library (npmlda). I'm having trouble to set up a kernel estimator with Uniform(-1/2,1/2) kernel and ...
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why not chosing always a spline?

I'm having a quite simple question: Why is a spline fit not the best choice everytime? In other words: How do I separate a spline fit from a kernel smoother or a polynomial in a meaningful way? I'm ...
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How to compare a new measurement to an existing multivariate distribution?

I have a dataset that describes the position and rotation of an object at different points in time using four dimensions. I want to use this sample of observations to get a sense of what positions and ...
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RKHS norm and Fourier transform link

In the notes here, it is stated that norms of some reproducing kernel Hilbert spaces can be written in terms of Fourier transforms, and this is often used to argue that a higher RKHS norm implies a ...
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Gradient Descent in Metric Learning for Kernel Regression (MLKR)

I am currently studying the Metric Learning for Kernel Regression (MLKR) algorithm (http://proceedings.mlr.press/v2/weinberger07a/weinberger07a.pdf). Let $\{(x_{1}, y_{1}), ..., (x_{N}, y_{N})\}$ ...
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Cluster/partition by time

I have a dataset of different events in time. I want to group/cluster/partition the data by datetime. A small example of the time of the data might be: [19-09-2018 12:00, 19-09-2018 12:01, 19-09-2018 ...
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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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Bandwidth Selection Methods

Are there more methods for calculating the bandwidth in a kernel regression? So far I found Cross Validation Methods and a Bayesian approach (by Zhang, Brooks and King).
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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
85 views

Python KDE plot for a value and not a count

I'm using a KDE plot to analyze the distribution of a sample population in terms of count by division. However, if I want to see how that distribution looks by some value (for example, dollar amount),...
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Are kernel Density estimation and gaussian blur related?

I have a set of points in a 2d space representing location of animals. I am interested in a probability heatmap which give lower values for cells far from these locations. I have seen many ...
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Kernel density estimation: kernel MISE (vs Epanechnikov)

In most places where I've looked, it generally says that the Epanechnikov kernel is optimal for kernel density estimation (KDE), in the sense that it minimizes the mean integrated squared error (MISE)....
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Is it possible to compute a bivariate (gaussian) kernel density estimation with a GAM model?

I regularly compute bivariate density estimation with Gaussian kernels, using functions specifically made for this (e.g., in R, the kde2d function). Recently reading literature on generalized ...
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47 views

Continously display probability for Bernoulli trials (kde?)

Is there a way to display probability for Bernoulli trials (outcome only 0 or 1). Data would look like this: timestamp,outcome t1,0 t2,1 t3,0 t4,0 I have ~1000 ...
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How to find the maximum probability of an hourly based variable ?

Let's say my dataset is composed by the time (hour) of when a user uses his TV, over a month. I may have something like this : ...
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63 views

Kernel Density Plot

I want to know what the underlying distribution of my data, I have used kernel density plot to find the distribution of my data. Code: ...
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375 views

Normalization of convolution kernel

I am trying to smooth a noisy one-dimensional physical signal, y, while retaining correspondence between the signal's amplitude and its units. I'm applying a ...
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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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Inconsistency of kernel intensity estimator

I have read that the kernel intensity estimator of a point process is not consistent, because the variance at any point is "of order 1". What does he mean by "of order 1"? I would like to understand ...