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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

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

41 questions
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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 Fourier transform formula, and the definition of the…
user431550
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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 radial basis function on Matlab? Tracking too many…
thomson
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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 definte…
Ziyuan
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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 separable (see example scatter-plot by link containing…
MarkMark
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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}{\boldsymbol \varPhi} \newcommand{\O}{\boldsymbol…
Vlad
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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 when $r = 0$ i.e. so the weights assigned to the…
Governor
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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 functions with rational coefficients, while the basic…
ZyS
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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 points, my RBF model looks like:…
Maaz
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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 details of this equation are not very important.…
Benoit
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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}) = \sum \lambda_j \phi(r_j) $ where $r_j$ is the…
Luca
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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 these functions (Wendland, H. (1995). Piecewise…
Nguyen Thi Lan Huong
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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 simple manifolds (e.g. on a unit sphere) shows that…
Vlad
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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 is obvious, but I think a simple proof will go a…
Dr. Papa
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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 permutation of the elements of the vector…
flammmes
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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 bivariate function $f(x, y)$ at a set of points…
Rule
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