Kernel density estimation

Kernel density estimation. (C) Pseudotime values, binned data and kernel density estimate. The Tensor Euler deconvolution applies additional constraints to the Euler solution using all gravity vectors and the full gravity gradient tensor. Essentially this just means placing a smooth Details. Robust Kernel Density Estimation Let X1,,Xn ∈Rd be a random sample from a distribution F with a density f. We see that the kernel density estimate is smoother than the histogram, and A classical approach of density estimation is the histogram. Formally, the Kernel Density Estimation (KDE) problem is: preprocess a dataset P of npoints The kernel estimator The nearest neighbour method The variable kernel method Orthogonal series estimators Maximum penalized likelihood estimators General weight function estimators Bounded domains and directional data Discussion and bibliography 1. It automatically computes the optimal bandwidth parameter. The bottom left panel used the same data as in the upper left panel, only the adaptive kernel density estimator was used. Given a set of observations \((x_i)_{1\leq i \leq n}\). 0. Here we will talk about another approach{the kernel density estimator (KDE; sometimes called kernel density estimation). 1. Many plots are shown, all created using Python and the KDEpy library (https:// Aug 27, 2024 · Kernel Density Estimation¶. Feb 2, 2024 · Kernel Density Estimation (KDE) is a technique used to estimate the probability density function (PDF) of a continuous random variable. K(x) is called the kernel function, and is normalized to one (and so ĝ(x) is also normalized to 1). It is useful to recall (2. Process. Kernel density estimation is shown without a barrier (1) and with a barrier on both sides of the roads (2). The proposed estimator builds on existing ideas for adaptive smoothing by incorporating information from a pilot density estimate. Kernel density estimates are closely related to histograms but can be endowed with properties such as smoothness or continuity by using a suitable kernel. Density estimation is the reconstruction of the density function from a set of observed data. 7 : Triweight density estimates of running example, (Table 2); bandwidth h = 0_2,0_8,1. This example uses the KernelDensity class to demonstrate the principles of Kernel Density Estimation in one dimension. Revisit Intuition. Kernel density estimation (KDE) is a different approach to density estimation that address some of the issues of histogram density estimation. Read more in the User Guide. See the formula, the bandwidth parameter, the Gaussian and bisquare kernels, and the Mathematica code and plots. stats. The first step toward KDE is to focus on just one data point. Kernel density estimation is the process of estimating an unknown probability density function using a kernel function \(K(u)\). The first plot shows one of the problems with using histograms to visualize the density of points in 1D. References. The estimation attempts to infer characteristics of a population, based on a finite data set. Using a smoother kernel function K, such as a Gaussian density, leads to a smoother estimate fˆ K. While a histogram counts the number of data points in somewhat arbitrary regions, a kernel density estimate is a function defined as the sum of a kernel function on every data point. Density estimation based on histograms is also implemented in the pack-ages delt and ash. Nov 2, 2016 · In this tutorial, we’ll carry on the problem of probability density function inference but using another method: Kernel density estimation (KDE). 5 %âãÏÓ 1 0 obj > endobj 2 0 obj > endobj 3 0 obj > endobj 4 0 obj > endobj 5 0 obj >/Metadata 1273 0 R/Outlines 1235 0 R/Pages 10 0 R/StructTreeRoot 975 0 R A kernel distribution is defined by a smoothing function and a bandwidth value, which control the smoothness of the resulting density curve. Learn how to estimate the probability density function of a random variable using kernel smoothing and bandwidth selection. In this, the objective is to calculate the unknown density f h (x) using the equation given below: where, K -> kernel (non-negative function) h -> bandwidth (smoothing parameter, h > 0) K h -> scaled kernel f h(x) -> density (to calculate) n -> no. 1 for an alternative. ksdensity estimates the density at 100 points for univariate data, or 900 points for bivariate data. We assume the observations are a random sampling of a probability distribution \(f\). The kernel’s bandwidth determines the estimate’s smoothness: if the bandwidth is too small, the estimate may include spurious bumps and wiggles; too large, and the estimate reveals little about the underlying distribution. The Kernel Density Estimation is a mathematic process of finding an estimate probability density function of a random variable. The probability density function is a fundamental concept in statistics. Aug 17, 2020 · Learn how to use kernel density estimation (KDE) to estimate the probability density function of a random variable. The KDE is one of the most famous method for density estimation. Kernel density estimation (KDE), is used to estimate the probability density of a data sample. Feb 14, 2018 · We review recent advances in modal regression studies using kernel density estimation. This is close enough to the Kernel estimation method. Silverman, B. Kernel Density Estimation# This example shows how kernel density estimation (KDE), a powerful non-parametric density estimation technique, can be used to learn a generative model for a dataset. . (7) The animation shows how bandwidth and kernel affect the density estimate, and how the kernels are rescaled densities with modes at the data points. May 1, 2024 · Kernel density estimation of critical region identification. The most well-known tool to do this is the histogram. For any real values of x, the kernel density estimator's formula is given by Feb 3, 2019 · Kernel Density Estimation (커널 밀도 추정) 03 Feb 2019 | kernel density estimation KDE 커널 밀도 추정 Kernel Density Estimation (커널 밀도 추정) CNN을 이용한 실험을 했는데 직관적으로는 결과가 좋아졌지만 왜 좋아졌는지에 대한 구체적 이유를 규명하기 위해 공부해 봤다. The goal is to estimate f(x). Apr 30, 2020 · Kernel Density Estimators. This example, based on work by John Firebaugh, shows times between eruptions of Old Faithful. To ensure that bfKDE(x)is a density 2. While kernel density estimation produces a probability distribution, the height of the curve at each point gives a density, not a probability. The kernel function is evaluated for each datapoint separately, and these partial results are summed to form the KDE. This video gives a brief, graphical introduction to kernel density estimation. We can also specify the bandwidth by setting the value of the bw_method parameter. In histogram density estimation, the density is proportional to: Kernel density estimation of 100 normally distributed random numbers using different smoothing bandwidths. Nov 7, 2018 · Kernel density estimation is a technique for estimation of probability density function that is a must-have enabling the user to better analyse the studied probability distribution than when using Kernel density estimation extrapolates data to an estimated population probability density function. See examples of KDE with different kernels, bandwidths, and distance metrics. This paper proposes a k-means-based supervised-learning clustering method termed supervised kernel-density-estimation k-means. All about Kernel Density Estimation (KDE) in data science. KDE surpasses traditional histograms by utilizing a smooth kernel function, thereby offering a continuous Apr 15, 2021 · However, these centroids have a structure usually represented by a list of quantized vectors, so that kernel density estimation models can better represent complex data distributions. A well-constructed density estimate can give valuable indication of such features as skewness and multimodality in the underlying density function. Kernel density estimation (KDE), also known as the Parzen’s window (Parzen,1962), is one of the most well-known approaches to estimate the underlying proba-bility density function of a dataset. The estimation works best for a unimodal distribution; bimodal or multi-modal distributions tend to be 在统计学中，核密度估计（kernel density estimation，KDE）是将核平滑应用于概率密度估计，即以核为权重来估计随机变量的概率密度函数的非参数方法。KDE 解决了一个基本的数据平滑问题，即根据有限的数据样本对总体进行推断。 The function scipy. A barrier can be used to alter the influence of a feature while calculating kernel density. The density functions obtained using the kernel density estimation described the arrival time and the energy consumed in commuting, defining the EVs cluster to be charged or discharged. any transformation has to give PDFs which integrate to 1 and don’t ever go negative • The answer… Kernel Density Estimation (KDE) nal kernel density estimate is the sum of the Gaussians, which makes the blue curve. We can directly calculate the density at a point x by summing the kernel response for each data point. 3 Kernel Density Estimation STAT/BIOSTAT 527, University of Washington Emily Fox April 18th, 2013 ©Emily Fox 2013 Module 2: Splines and Kernel Methods Motivating Kernel Methods ©Emily Fox 2013 2 ! Recall original goal from Lecture 1: " We don’t actually know the data-generating mechanism • Use data to get local point-wise density estimates which can be combined to get an overall density estimate • Smooth • At least smoother than a ‘jagged’ histogram • Preserves real probabilities, i. Kick-start your project with my new book Probability for Machine Learning , including step-by-step tutorials and the Python source code files for all examples. 2. Parameters: bandwidth float or {“scott”, “silverman”}, default=1. For example, Sala-i-Martin (2006) used the KDE to estimate income distribution and showed reductions in global inequality during the 1980s and 1990s; Quah (1993) examined the distribution of output per work across countries and showed that it evolved from unimodal to bimodal Calculates a magnitude-per-unit area from point or polyline features using a kernel function to fit a smoothly tapered surface to each point or polyline. The upper right panel is the estimate using the Group 1 data. The estimate is based on a normal kernel function, and is evaluated at equally-spaced points, xi, that cover the range of the data in x. This task is called density estimation. Kernel density estimation (KDE), also known as the Parzen's window [1], is one of the most well-known approaches to estimate the underlying probability density function of a data-set. In this blog, we look into the foundation of KDE and demonstrate how to use it with a simple application. However, it has certain drawbacks: proper description of the density using narrow kernels needs large data samples, whereas if the kernel width is large, boundaries and narrow structures tend to be smeared. Kernel smoothing, or kernel density estimation methods (KDE methods) of the type described have a variety of applications: probability distribution estimation; exploratory data analysis; point data smoothing; creation of continuous surfaces from point data in order to combine or compare these with other datasets that are continuous Jun 28, 2022 · In this paper, by using weighted kernel density estimation (KDE) to approximate the continuous probability density function (PDF) of the portfolio loss, and to compute the corresponding approximated Conditional Value-at-Risk (CVaR), a KDE-based distributionally robust mean-CVaR portfolio optimization model is investigated. However, the demanded/supplied power depends on the market price signal. Learn how to use kernel density estimation (KDE) to model the distribution of data points in any dimension. 1 Univariate Density Estimation To best appreciate why one might consider using nonparametric meth­ ods to estimate a PDF, we begin with an illustrative example, the We present a new adaptive kernel density estimator based on linear diffusion processes. Density Estimation for Statistics and Data Analysis. See examples, definitions, formulas, and applications of kernel density estimation in statistics, signal processing, and thermodynamics. Application available here . Often shortened to KDE, it’s a technique that let’s you create a smooth curve given a set of data. Kernel density estimation is a convenient way to estimate the probability density of a distribution given the sample of data points. What is density estimation? The probability density function is a fundamental Jan 30, 2023 · Conventional Euler deconvolution is widely used for interpreting profile, grid, and ungridded potential field data. A kernel is a probability density function (pdf) f(x) which is symmetric around the y axis, i. of samples Apr 8, 2021 · We test the performance of Roundtrip in a series of experiments, including simulation studies and real data studies. Dec 22, 2017 · This chapter describes the kernel density estimation technique that can be considered a smoothed version of the Parzen windows presented in the Chapter 2. 2 Kernel Density Estimation in S Direct Algorithm In the histogram case, we have to define a binmesh which represents a discretization of the real line. We denote the kernel density estimate with bandwidth (smoothing parameter) h by fˆ h(x) = 1 nh Xn j=1 K x−X j h . Kernel density estimation can in principle be used in any number of dimensions. Dec 1, 2017 · 1. Oct 13, 2020 · Ex: Kernel Density Estimators, SVMs, Decision Trees. This goes hand in hand with the fact that this kind of estimator is now provided by many software packages. Details. Consider a kernel map k: Rd Rd![0;1]. If bandwidth is a float, it defines the bandwidth of the kernel. The upper left panel shows a kernel density estimate using a normal kernel based on the Group 2 data in Table 3. Different Kernels for Density Estimation. If we query the density at m measurement points, this Aug 1, 2023 · KDE estimates the probability distribution of a random variable. Kernel density estimation is a technique that estimates the probability density function of the data points randomly in a sample space. On the other hand, since about three decades the discussion on bandwidth selection has been going on. Kernel density estimation is a way to estimate the probability density function (PDF) of a random variable in a non-parametric way. There is a vast amount of literature on bandwidth selection. Let’s revisit the intuition that we developed for nonparametric density Kernal Density Estimation A kernel density estimator (in 1D) is defined by Here x i is the ith of N data points. However we choose the interval length, a histogram will always look wiggly, because it is a stack of rectangles (think bricks again). Lecture 6: Density Estimation: Histogram and Kernel Density Estimator Instructor: Yen-Chi Chen Reference: Section 6 of All of Nonparametric Statistics. 5However, core-sets preserve all KDE values with high probability, while simple random sampling only preserves the KDE of any individual query with high probability. , a non-parametric method to estimate the probability density function of a random variable based on kernels as Sep 24, 2019 · In this note, I am going to use Gaussian kernel function to estimate kernel density and to optimize bandwidth using example data sets. default disperses the mass of the empirical distribution function over a regular grid of at least 512 points and then uses the fast Fourier transform to convolve this approximation with a discretized version of the kernel and then uses linear approximation to evaluate the density at the specified points. The Kernel Density Estimator (KDE) has been a commonly used tool for the analysis of economic data. e. We begin with a discussion of basic properties of KDE: the convergence rate under various metrics, density derivative estimation, and bandwidth selection. The blue curve is Basic Concepts. 我們再次回到身高的例子，假設x=160，我們原先就是逐一計算x=160與任一樣本xi的距離，最後除以全部 An example comparing four plots of data. This can be useful if you want to visualize just the “shape” of some data, as a kind of continuous replacement for the discrete histogram. A density estimate or density estimator is just a fancy word for a guess: We are trying to guess the density function f that describes well the randomness of the data. The KDE is often used for data fitting when the potential probability density cannot be obtained. These new samples reflect the underlying model of the data. See examples, plots and formulas for different kernels, smoothing parameters and bias-variance trade-offs. What is Kernel Density Estimation? Kernel density • The histogram is a very simple form of density estimation, but has several drawbacks –The density estimate depends on the starting position of the bins •For multivariate data, the density estimate is also affected by the orientation of the bins –The discontinuities of the estimate are not due to the underlying Nov 16, 2023 · Now, this density estimate seems to model the data very well. Learn more about how Kernel Density works. KDE is a n Apr 2, 2021 · Kernel Density Estimation 核密度估計 (KDE) 與直方圖的目的相同，我們所希望看到的是數據的”機率密度函數”，但KDE是較為平滑的方法。我們可以把數據視為多個kernel結合的結果，多個kernel可以擬合出較為平滑的分布曲線。 Kernel density estimation will always produce a smooth curve, which would be misleading in these situations. The kernel density estimate of f, also called the Parzen window estimate, is a nonparametric estimate given by bf KDE (x)= 1 n n ∑ i=1 kσ(x,Xi) where kσis a kernel function with bandwidth σ. 4, and 4. Kernel density estimation is a nonparametric technique for density estimation i. See also a Kernel Density Estimation is a method to estimate the frequency of a given value given a random sample. Kernel Density Estimator. 1. The kernel density estimate was obtained by solving our continuous PDE by our discrete numerical method with the “four corners matrix” in . Although a good part of the discussion is about nonparametric regression Kernel Density Estimation. The algorithm used in density. In addition, we propose a new plug-in bandwidth selection method that is free from the arbitrary normal reference rules used by existing methods. In addition, the package np includes routines for estimating multivariate conditional densities using kernel methods. With this generative model in place, new samples can be drawn. The kernel density estimator is the estimated pdf of a random variable. Related topics. First, the most popular kernel types are presented together with a number of basic definitions both for uni- and multivariate cases and then a review of performance criteria is provided, starting with the univariate case and then extended to What we've landed on in the last two plots is what's called kernel density estimation in one dimension: we have placed a "kernel"—a square or "tophat"-shaped kernel in the former, a Gaussian kernel in the latter—at the location of each point, and used their sum as an estimate of density. Namely, we observe X 1; ;X Dec 8, 2020 · In both cases (kernel regression and kernel density estimation), we have to select a kernel and a bandwidth. Usually a \(d\)-dimensional kernel \(K_d\) of the product form \[\begin{equation*} K_d(u) = \prod_{i=1}^d K_1(u_i) \end{equation*}\] is used. As the frequently used kernels have similar shapes (see Figure 7), the choice of the bandwidth is more crucial. The K-NN density estimation is, where d k (x) is the Euclidean distance from the sample to its k th nearest neighbor. , estimation of probability density functions, which is one of the fundamental questions in statistics. See Kernel Distribution for more information. bandwidth s, the univariate kernel density estimate is defined as: f(x)= 1 ns n å i=1 K x xi s (1) We focus on the case where K is the normal (Gaussian) density K(x)= p1 2p e x2=2. Fish Icon:https://www. D ensity Estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function[2 Aug 2, 2021 · KDEとは？ ”カーネル密度推定（カーネルみつどすいてい、英: kernel density estimation）は、統計学において、確率変数の確率密度関数を推定するノンパラメトリック手法のひとつ（Wikipedia）”とされており、機械学習などで様々に応用されています。 6. New York: Chapman and Hall, 1986. The kernel density estimate is then Jan 17, 2023 · In today's blog, we examine a very useful data analytics tool, kernel density estimation. The parameter h is called the “bandwidth”, and scales the width of the kernel. Apr 13, 2024 · In my previous post, we discussed a neat intuition for nonparametric density estimation and introduced a nonparametric method, histograms. The idea is to start out with a parametric density before you do your kernel density estimation, so that your actual kernel density estimation will be a correction to the original parametric Dec 12, 2012 · This comprehensive review summarizes the most important theoretical aspects of kernel density estimation and provides an extensive description of classical and modern data analytic methods to Kernel density estimation is a fundamental problem with numerous applications in machine learning, statistics and data analysis [FG96, SS01, JKPV11, SZK14, GPPV+14, ACMP15, GB17]. gaussian_kde works for both uni-variate and multi-variate data. scikit-learn allows kernel density estimation using different kernel functions: Mar 6, 2022 · 標準高斯函數 核密度估計(Kernel Density Estimation). In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i. Kernel density estimation is a really useful statistical tool with an intimidating name. The equation for Gaussian kernel is: Where xi is the observed PDF-1. Learn how to use kernel density estimation (KDE) to model the probability distribution of a dataset. gaussian_kde computes the kernel density function with the gaussian kernel. Modal regression is an alternative approach for investigating the relationship between a response variable and its covariates. Compare KDE with parametric and histogram methods, and explore the bias-variance tradeoff of bandwidth selection. Introduction. kde uses kernel density estimation to estimate the pdf. Mar 29, 2023 · The density of a sample depends on the value of k and the distance of the kth nearest neighbor from the sample. and bivariate density estimation via kernel smoothing, while the ks package allows for multivariate density estimation for up to 6 dimensions. As a nonparametric estimation method, KDE has become a popular development direction for modern statistics because it is not affected by prior model selection. Jun 1, 2020 · 1. It can be viewed as a generalisation of histogram density estimation with improved statistical properties. 6 Density Estimation in Higher Dimensions. The units on the density axis are a common source of confusion. com/search?format=search&icon_color=red&last_filter=icon_color&last_v 2. Kernel density estimation# A common task in statistics is to estimate the probability density function (PDF) of a random variable from a set of data samples. 2 Kernel Density Estimation in S 49 <"l 0 N 0 0 0 0 -5 0 5 -5 o 0 0 0 -5 o 5 Figure 2. Kernels are used in kernel density estimation to estimate random variables' density functions, or in kernel regression to estimate the conditional expectation of a random variable. Density estimation is the problem of reconstructing the probability density function using a set of given data points. INTROUCTION 1. While F(x) can be estimated by the EDF Fˆ(x),we cannot set fˆ(x)=d dx Fˆ(x) since Fˆ(x) is a step function. We present simulation examples Aug 5, 2024 · One of the most common nonparametric approach is known as Kernel Density Estimation. example [f,xf,bw] = kde Jan 1, 2022 · Kernel Density Estimation If we consider the norm of a dataset should fit certain kind of probability distribution, the anomaly are those that we should see them rarely, or in a very low probability. Its distributional uncertainty set (DUS) is defined indirectly by Kernel Density Estimation in Practice¶ The free parameters of kernel density estimation are the kernel, which specifies the shape of the distribution placed at each point, and the kernel bandwidth, which controls the size of the kernel at each point. Let’s again compute kernel density functions for the example above. This makes sense because the pdf is a derivative of the cdf and the derivative of F at x only depends on the behavior of F locally at the point x and this local bevahior of F at x is reflected by the Apr 12, 2017 · This tutorial provides a gentle introduction to kernel density estimation (KDE) and recent advances regarding confidence bands and geometric/topological features. W. Using NumXL, you can compute the Kernel density estimation with a parametric start was introduced by Hjort and Glad in Nonparametric Density Estimation with a Parametric Start (1995). The choice of kernel has relatively minor effects in practice, and we will in this chapter mostly use the Gaussian kernel, but see Exercise 2. Compare KDE with histograms and Gaussian mixture models (GMM) and see examples of KDE applications. For the density estimation task, we compared our method to the widely used Gaussian KDE as well as several neural density estimators, including MADE (), RealNVP (), and MAF (). An overview of the Density toolset; Understand density analysis; Kernel Density Dec 18, 2023 · Kernel density estimation (KDE) is a method for estimating the probability density function (PDF) of a random variable with an unknown distribution using a random sample drawn from that distribution. We first consider the kernel estimator: Jun 18, 2020 · An average cell follows the indicated path through the dataset. f(-x) = f(x). A kernel density estimation (KDE) is a non-parametric method for estimating the pdf of a random variable based on a random sample using some kernel K and some smoothing parameter (aka bandwidth) h > 0. The bandwidth of the kernel. It includes automatic bandwidth determination. May 1, 2024 · Kernel density estimation. The kernel density estimation problem can be formally stated as follows. The follow picture shows the KDE and the histogram of the faithful dataset in R. Aug 9, 2024 · Figure @ref{fig:psiDens} illustrates how the choice of bandwidth and kernel affects the resulting kernel density estimate. If bandwidth is a string, one of the estimation methods is implemented. The standard nonparametric method to estimate f(x) is based on smoothing using a kernel. 7) with the normal kernel. We now turn our focus to kernel density estimation, an effective nonparametric statistical method for estimating the underlying probability density function from a dataset [44]. Estimates that are linear combinations of such kernel functions centered at the data are called kernel density estimates. These algorithms use a series of different-sized moving windows to yield many solutions that can be employed to estimate the source treatments of the kernel estimation of a PDF discussed in this chapter are drawn from the two excellent monographs by Silverman (1986) and Scott (1992). KDE is a composite function made up of one kind of building block referred to as a kernel function. Kernel Density Estimation. Jun 2, 2013 · On the one hand, kernel density estimation has become a common tool for empirical studies in any research area. Kernel Density Estimation Let Xbe a random variable with continuous distribution F(x) and density f(x)=d dxF(x). Simple 1D Kernel Density Estimation#. KDE is a nonparametric density estimator requiring no assumption that the underlying density function is from a parametric family. In nonparametric statistics, a kernel is a weighting function used in non-parametric estimation techniques. Then, we introduce common approaches to the construction of confidence Dec 14, 2023 · Kernel Density Estimation¶. Learn the definition, properties and applications of kernel density estimation, a nonparametric method for estimating probability densities. freepik. It is a non-parametric method, meaning it does not assume Jul 24, 2020 · Nonparametric probability density estimation involves using a technique to fit a model to the arbitrary distribution of the data, like kernel density estimation. Introduction to kernel density estimation; Kernel Bandwidth Optimization フリーウェブアプリ データを入力すれば最適化なカーネルバンド幅を計算してカーネル密度推定値を出力します。 Free Online Software (Calculator) 任意のデータ列についてカーネル密度推定を行い描画する 核密度估计（英語： Kernel density estimation ，縮寫：KDE）是在概率论中用来估计未知的密度函数，属於非参数检验方法之一，由Rosenblatt (1955)和Emanuel Parzen(1962)提出，又名Parzen窗（Parzen window）。Ruppert和Cline基于数据集密度函数聚类算法提出修订的核密度估计方法。 Aug 17, 2020 · The kernel density estimate at x only depends on the observations that fall close to x (inside the bin \( (x-\frac{h}{2},x+\frac{h}{2}) \) ). Aug 15, 2023 · Each datapoint is given a brick, and KDE is the sum of all bricks. It’s called kernel density estimation because each data point is replaced with a kernel—a weighting function to estimate the pdf. Dec 8, 2020 · Learn how to use kernel density estimation (KDE) to smooth out the probability density function (PDF) of a univariate or multivariate data set. The first half of the plot is in agreement with the log-normal distribution and the second half of the plot models the normal distribution quite well. Abstract. Hence, it allows us to infer the probability density of a population, based on a finite dataset sampled from it. This post describes another (very popular) density estimation method called Kernel Density Estimation (KDE). hvjpiy zpm jqudl rpzja tmlv hiltdcj mzh iecvuo tzqlzph tjptaj