Questions tagged [radial-basis-functions]

Radial Basis Functions (RBFs) are commonly used for interpolating scattered data, in numerical meshfree simulation methods, and in artificial neural networks

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Prove that modified RBF function satisfies Mercer conditions.

Suppose that I have a modified RBF kernel function. $k(\mathbf{x},\mathbf{y}) = \exp{(-||\mathbf{x}-P\mathbf{y}||^2 })$ where $\mathbf{x},\mathbf{y}$ represent $d$ dimensional inputs and $P$ is the ...
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Ensuring Symmetry in Mixed Derivatives Using RBF-FD Method

Hello Mathematics Stack Exchange Community, I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a ...
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Proving Divergence Free Behavior of Matrix Valued Radial Basis Function

I'm am using radial basis functions to interpolate magnetic fields, which are divergence free. I have found several research papers that state that the following takes a scalar valued Radial Basis ...
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How to estimate shape parameter of spherical radial basis function

In this paper about positive definite kernels they introduce the extension of the radial basis function (RBF or Gaussian Kernel) $$ K_{rbf}(x,y) = e^{-\epsilon^2||x-y||^2} $$ for the unit sphere $$ K_{...
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Thin Plate Spline RBF Interpolation understanding

I have been looking at radial basis function interpolation: $f(x) = \sum w_i \phi_i(||x-x_i||)$ and examining the different kernels e.g. $\phi(r) = e^{-(\epsilon r) ^2}$ which are generally maximised ...
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How does the Radial Basis Functions transform multimodal distributions to normal distributions?

In the book Hands-On Machine Learning by Aurélien Géron, in chapter two, the author states: Another approach to transforming multimodal distributions is to add a feature for each of the modes (at ...
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What kernel to use for Gaussian Process Regression on data with large flat region?

The problem I'm working on has a region that's largely accurately modeled by a GP model using a squared exponential kernel. However, there is a large region in the "truth" model that is ...
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represent polynomials as linear combinations of a kind of radial basis functions

Recently, I'm reading a paper Riesz representation theorem, Borel measures and subsystems of second-order arithmetic. In page 2, the author said that polynomials are linear combinations of basic ...
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Why is multiquadratic radial basis function with larger shape parameter smoother?

I am not into mathematics so a request to answer in simpler terms. I am using radial basis function (RBF) for interpolating some data points using multiquadratic basis function. Considering $n$ data ...
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How to compute weights in thin plate spline radial basis function?

Consider the problem of approximating a function $f:R^n\rightarrow R$ using radial basis function(RBF). We are given with $p$ points $\{x^1,\ldots,x^p\}$ and their function values $\{y_1,\ldots,y_p\}$....
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Radial basis functions for spectral solution of PDE in spherical coordinates

I want to solve the following PDE defined in 3D-space and time: $(z \partial_t-F(t)\partial_z)f(t,\vec{x})+C[f]=S(t,z,r),$ where $r=\sqrt{x^2+y^2+z^2}$ and $C[f]$ is a linear integral operator. The ...
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Which values of the exponent yield only real values of this function on the given domain

Given the following function: $$ y = \frac{x^p}{x^p + (1 - x)^p}, $$ where $x,p\in\mathbb{R}$, I want to know which values of $p$ yields $y\in\mathbb{R}$ for all $x$. Can anyone help me with this? ...
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Approximating the inverse of a stochastic function

Assume that I have a stochastic function $f(x)=g(x)+\epsilon$, where $f:\mathbb{R}^D\rightarrow \mathbb{R}^D$ is composed of a deterministic function $g:\mathbb{R}^D\rightarrow \mathbb{R^D}$ and some ...
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finding best fit surface passing through points using RBF or bspline methods

I have some points in 3D (x, y and z) and want to fit a surface through them. I prefer to use bsplines for finding the best surface among my points. My final goal is the equation of the surface. I ...
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Does a positive definite and radial function imply its Fourier transform is nonnegative?

I am thinking about this question: Does a positive definite and radial function imply its Fourier transform is nonnegative? I know that the converse is correct. That is, we can apply the inverse ...
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Univariate Kernel in Gaussian Process

I cant seem to get my head around the behaviour of different kernels in an univariate case when using a gaussian process regressor. For example, this is the RBF kernel: $$K(x, x')= exp( - \frac{ ||x-x'...
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Showing that a function is constant on a disc D

Recall that a function $f(z)$ is called radial if it is constant along the circles of center $0$. Let $f$ be a radial holomorphic function defined on the unit disc $D$. Show that $f$ is constant. (...
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Radial Basis Function interpolation: why the multi-quadric basis function increase with distance?

I'm trying to understand the underlying logic in the Radial Basis Function interpolation. I understood that we estimate the value of the underlying function in any unknown point as $ y(\vec{x}) = \...
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Radial Basis Functions in $\mathbb R^3$ - How to interpolate?

I am fairly new to radial basis functions. I get the concept, but I need a little explanation or coaching when it comes to the actual application. Let's assume that I have $n$ points in $\mathbb R^3$ ...
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Weighted blend of multiple quaternions (Slime and Sasquash)

I am doing a R&d project for my uni and I am struggling with understandin the algorithm prsented in this work at page 147-148: http://alumni.media.mit.edu/~aries/papers/johnson_phd.pdf I have a ...
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How interpolate a huge cloud of scattered 3D points?

I have a cloud consist of a million scattered 3d points. I want to get a uniform cloud of 3d points. I think to interpolate in blocks. However, as shown in the figure grid, there is a problem of ...
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RBF kernel mapping

I was reading that the Gaussian/RBF kernel maps its input onto the surface of normalized hypersphere. Our RBF kernel given by: $k(x,z) = exp(\frac{- ||x-z||^2}{2\sigma^2})$ Can anyone explain why ...
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How to find the particular solution for Augmented Thin Plate Splines in the context of the Dual Reciprocity Boundary Element Method

In the dual reciprocity boundary element method (DRBEM) the non-homogeneous terms are expanded in terms of radial basis functions. This expansion involves approximating the solution to the linear ...
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Matrix operations for RBF solver

GOAL: I got from someone the python code for an RBF solver. The solver stores 9 transformation matrix (each of which, once decomposed, have tx, ty and tz set at 0 and sx, sy and sz set to 1, so only ...
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How does one approximate a second derivative with ATPS interpolation

When using the Dual Reciprocity Boundary Element Method ( or any radial basis function method ) to solve a nonlinear differential equation it is necessary to approximate some derivatives of a ...
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Radial Basis Function

I have this question and I need help; What is the effect of using a more hidden layer on the performance of approximation of the RBF network?
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How can I use RBF interpolation on a highly stretched rectangular domain?

I performed a 2D parametric analysis where one variable is much larger than the other. Basically I sampled a function in many points: let's say 5 points for $x_1$ and 5 points for $x_2$, where the ...
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Confusion between Wendland RBF functions - missing Wendland functions

I need to compute Wendland functions for a project, and got confused between the formula to construct Wendland functions $\phi(d,k)$ where d is the dimension So in the original paper introducing ...
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How to understand min max constraints?

I am trying to understand how to minimize the given function. In the paper [1] t says, ${{f}_{i}}\left( x \right)$ should be minimized in subject to: $\left\| x-{{x}_{j}} \right\|\ge \beta \text{ }{{...
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Understanding the use of Radial Basis Function in Linear Regression

I am attempting to understand the use of Radial Basis Functions (RBFs) as used in linear regression. Building the problem: RBFs can be used as a means of separating data which is not linearly ...
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Integral calculation of Fourier transform of a function

In this paper, an oscillatory radial basis function is introduced and its properties are considered. The RBF is as follows: $$\phi_d=\frac{J_{d/2-1}(\epsilon r)}{(\epsilon r)^{d/2-1}},\quad d=2,3,4,......
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Absolute value of an RBF distance is less than the absolute value of an actual distance

I have a radial basis function with a linear kernel $f(r)=r$ in $3D.$ I constructed the surface based on this RBF and noticed that the absolute value of actual distance from any point to the ...
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Gradient descent in $n$-dimensional space in the context of an RBF network

I am trying to implement an algorithm to perform gradient descent in a $n$-dimensional space in the context of an RBF network. My network has 5 inputs and 1 output. It has the following Gaussian ...
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Positive weights in Radial Basis Functions

Let $\phi$ be a positive definite radial kernel in $\mathbb{R}^d$. I have a point cloud with positions $(\mathbf{x}_i)_{i\in I}$. RBF interpolation of real valued data $f = (f_i)_{i\in I}$ on this ...
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Radial Basis Function RBF Gaussian based Interpolation

Based on short description below (an image), how do I find the highlighted f function value? I understand that it is a value associated with the vertex, sorry I am not a good math student to ...
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Is normalized RBF always better than RBF

The question is as the title. Mathematically, I want to know does the following inequation always hold for any vector $\mathbf b$? $\mathbf b^T \mathbf B \mathbf B^+ \mathbf b \, \ge \, \mathbf b^T \...
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Interpolation with RBF

I have a function that is continuous and differentiable over $\mathbb{R}$ and its support is the whole real line. I want to approximate it through a linear combination of Gaussian functions. I know ...
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Radial Basis Functions Interpolation

$ \let\oldcdot\cdot \renewcommand{\cdot}{\!\oldcdot\!} \newcommand{\e}{\varepsilon} \renewcommand{\p}{\varphi} \renewcommand{\p}{\varphi} \renewcommand{\vp}{\vec{\boldsymbol\p}(x)} \newcommand{\P}{\...
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Symmetry Of Differentiation Matrix

I have a problem computing numerically the eigenvalues of Laplace-Beltrami operator. I use meshfree Radial Basis Functions (RBF) approach to construct differentiation matrix $D$. Testing my code on ...
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Showing that a thin-plate spline RBF approximation is real analytic

I am finishing my Ph.D. dissertation in engineering and I would like to show a simple proof. I am having troubles formalizing my ideas into a proof though. I think in a mathematics paper this concept ...
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The positive-definite-ness of RBF kernel

In Micchelli's paper Interpolation of Scattered Data: Distance Matrices and Conditionally Positive Definite Functions it mentioned that the RBF kernel $e^{-\alpha^2\|x^i-x^j\|^2/2}$ is positive ...
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Radial Basis Function and Neural Networks

I need a simple explanation about what is the radial basis function? And what is the relationship between the radial basis function and neural networks? And are there any simple examples to explain ...
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