# 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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### Kernel Smoothing for Time Series data

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 ...
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### 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 ...
765 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?
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 $F(x) \le \frac{2K}{1-\frac{2R\epsilon}{\... 1 vote 1 answer 88 views ### kernel density estimation on 2D data with rotational symmetry My question is: what is the appropriate way to apply a kernel density estimator (KDE) to a 2D dataset that has a rotational symmetry? Specifically, I have the points ($x_i$,$y_i$) and want the ... 0 votes 0 answers 71 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 ... 1 vote 0 answers 35 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 ... 0 votes 1 answer 44 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 ... 0 votes 0 answers 37 views ### Adaptive (variable) bandwidth kernel density estimation for univariate data To overcome some problems with a global single bandwidth on my data with the R base function density(), I would like to try out adaptive (aka variable) bandwidths. ... 1 vote 0 answers 118 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 ... 1 vote 1 answer 39 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 ... 8 votes 1 answer 167 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 ... 4 votes 1 answer 125 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 ... 1 vote 0 answers 110 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$f''$... 1 vote 0 answers 59 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 ... 0 votes 1 answer 27 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 ... 1 vote 0 answers 45 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 ... 2 votes 1 answer 80 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 ... 1 vote 0 answers 98 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 ... 1 vote 1 answer 129 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 $$A_x(y):=\sum_{i=1}^k\varphi_{... 0 votes 0 answers 55 views ### Storing a probability distribution without saving single values I saw this question "Storing a probability distribution without saving single values" on stackexchange and thought it deserved a statistical answer. Example Scenario I could see this problem ... 1 vote 0 answers 62 views ### Gasser Müller estimator for estimating the derivative m'(x) of a nonparametric regression function I would like to compare the performance of the Gasser Müller estimator with other estimators for estimating the the derivative m'(x) of the regression function m(x). Let's say we have the ... 1 vote 0 answers 25 views ### Maximum bias for NW estimator when r(x) is Lipschitz (question 17, chapter 5 All of Non-Parametric Statistics) The general condition is that Y_i = r(X_i) + \epsilon_i, and we want to estimate r using Nadaraya–Watson kernel regression. We additionally assume r\colon [0,1] \to \mathbb{R} is lipschitz, so |... 1 vote 0 answers 147 views ### Bias of kernel density estimator of pdf f, where f has bounded first derivative f' Let's say the kernel density estimator is given by$$\hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K\left(\frac{X_i-x}{h_n}\right),$$where h_n \to 0, nh_n \to \infty, K a symmetric probability ... 0 votes 0 answers 37 views ### Kernel Density Estimator: Misunderstanding in Taylor Series and the bias of KDE [duplicate] Let's say the kernel density estimator is given by \hat f(x) = \frac{1}{nh_n} \sum_{i=1}^n K(\frac{X_i-x}{h_n}), where h_n \to 0, nh_n \to \infty, K a symmetric probability distribution ... 2 votes 1 answer 258 views ### my density/kde plots don't show what I expected I thought I understood kde/density plots until this problem. I have a dataset with two columns, Diff and Var with 5 million rows ... 1 vote 0 answers 101 views ### Plotting Kernel Density Estimates on Same Axes in R Here: https://www.python-graph-gallery.com/74-density-plot-of-several-variables Python's seaborn library is used to illustrate how to plot multiple kernel density ... 1 vote 0 answers 27 views ### Closeness of two estimators of median under non parametric setup in a large sample situation Median Regression under non-parametric set-up (Nadaraya Watson Estimate) Data: \{(Y_i,X_i):1\le i\le n\} Interested in estimating \phi(x)=\text{median}(Y|X=x). Possible estimates are Minimize the ... 3 votes 1 answer 282 views ### KDE bandwidth estimation in R and Python I am trying to estimate the bandwidth parameter of a multivariate KDE in R and then use the estimate to evaluate the KDE in Python. The reason for this somewhat convoluted approach is that my project ... 1 vote 1 answer 64 views ### Using KDE to approximate a Price vs Quantity curve I am trying to approximate Price-vs-Quantity (P-Q) curve of a dynamic product (think Hotels, Airlines etc). As you can imagine, if you take a hotel property, the price of rooms (assuming the same room ... 1 vote 1 answer 132 views ### How the kernel regression works? I am working on propensity score matching using the Kernel Nadaraya–Watson kernel regression. But I am looking to understand the logic of estimation; First, we estimate the Kernel density of each unit ... 1 vote 1 answer 109 views ### How to estimate the conditional probability p(y|x) if y and x are both continuous but y is discrete given x? For example, P(Y=f_1(x)|X=x)=g_1(x), P(Y=f_2(x)|X=x)=1-g_1(x). (The functions f1,f2 are unknown and need to be learned.) How can I estimate such a conditional probability? I guess that kernel ... 2 votes 0 answers 158 views ### Question regarding Kernel Density Estimation bandwidth selection (Scott's rule) I'm studying KDE and got trouble understanding Scott's rule or Silverman's rule for bandwidth selection. I saw that the optimal bandwidth is the value that minimizes Mean Integrated Squared Error (... 1 vote 0 answers 42 views ### Mathematical underpinnings of variations on Silverman's bandwidth When using a Gaussian kernel to estimate the distribution of a Gaussian-distributed x, the bandwidth that minimizes the mean integrated squared error is:$$h=\left(\frac{4 \hat{\sigma}^5}{3n}\right)^... 0 votes 1 answer 276 views ### How to convert units of KDE into physical units I have a sample of people's heights. If I histogram it, I would usually divide by the bin width, so that my result becomes independent of my bin width, and I would label the axis something like (... 0 votes 0 answers 54 views ### Compromise between ranked and kernel-based estimate of empirical distribution (for estimating likelihood of single value) Setup I have: About fifty values$x_1,…,x_n$sampled from the same but unknown distribution$X$. I have no useful theoretical insight on the nature of that distribution. Empirically, the distribution ... 3 votes 0 answers 405 views ### Pros and cons of Nadaraya–Watson estimator vs. RKHS method? Recently I've been reading some materials about nonparametric methods. Two methods related to the word "kernel" rasied my interest-- Nadaraya–Watson estimator and RKHS method. What's the ... 0 votes 1 answer 188 views ### Spatial analysis / kernel density in R, what kernel describes this distribution of points? So, I have a scenario where I want to model the probability distribution / density of random displacement using kernel density, but having trouble finding resources on how the math works. I'm working ... 0 votes 0 answers 39 views ### How to prove symmetry of a Uniform kernel? I am trying to prove this kernel is valid, $$K(x) = \frac{1}{2}I(-1 < x < 1)$$ So far I can integrate to 1, but how do I prove $$k(x) = k(-x)$$ Also, how do we satisfy that k(x) is$\ge$0 for ... 1 vote 0 answers 45 views ### Kernel Density: How do the terms 'global' and 'pilot' translate? I nearly most of the articles on kernel smoothing or concepts that use kernel density estimations, authors speak of 'pilot' and 'global'. https://link.springer.com/article/10.1023/A:1008925425102 &... 2 votes 1 answer 479 views ### Is Kernel-Regression parametric or non-parametric? As the title says, is kernel regression a parametric or non-parametric method, and how can this be motivated/explained? 1 vote 1 answer 71 views ### How to efficiently bias future samples? My goal is to hit all of my 'targets' through a random variable$Y : [0,1] \rightarrow \mathbb{R}$. An explicit form of$Y$is unknown but I am able to take a sample point as pass it to$Y$and get ... 1 vote 0 answers 148 views ### Intuition for bandwidth and degrees of freedom in kernel smoothers For Kernel smoothers such as local polynomial regression smoothers (the Nadaraya-Watson smoother), we consider$y = m(x) + \epsilon$, for$m(x)$some smooth function we are trying to estimate and$\...
I'm trying to understand how to perform a KDE on a bounded support, i.e. with lower and upper boundary, when using a kernel that is specifically designed to ensure consistency/$h^2$-bias at the ...