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Element of a basis for a function space
In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as
Basis_function
Type of mathematical function
In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input
Radial_basis_function
Machine learning kernel function
In machine learning, the radial basis function kernel, or RBF kernel, is a popular kernel function used in various kernelized learning algorithms. In particular
Radial_basis_function_kernel
Method in approximation theory
Radial basis function (RBF) interpolation is an advanced method in approximation theory for constructing high-order accurate interpolants of unstructured
Radial basis function interpolation
Radial_basis_function_interpolation
Type of artificial neural network
modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the
Radial_basis_function_network
Set of functions used to represent the electronic wave function
computational chemistry, a basis set is a set of functions (called basis functions) that is used to represent the electronic wave function in the Hartree–Fock
Basis_set_(chemistry)
First known wavelet basis
"square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over
Haar_wavelet
Set of vectors used to define coordinates
square-integrable functions cannot be represented as finite linear combinations of these basis functions, which therefore do not comprise a Hamel basis. Every Hamel
Basis_(linear_algebra)
In machine learning, a Hyper basis function network, or HyperBF network, is a generalization of radial basis function (RBF) networks concept, where the
Hyper_basis_function_network
Method of representing curves and surfaces in computer graphics
the basis function of degree one is a triangle function. It rises from zero to one, then falls to zero again. While it rises, the basis function of the
Non-uniform_rational_B-spline
Numerical method for solving physical or engineering problems
chooses basis functions. We used piecewise linear basis functions in our discussion, but it is common to use piecewise polynomial basis functions. Separate
Finite_element_method
automatic basis function construction, also known as basis discovery, is a technique that finds a set of general-purpose (task-independent) basis functions to
Automatic basis function construction
Automatic_basis_function_construction
Mathematical description of quantum state
a basis wave function. If spin is taken into account, there are two basis functions for each triple. The function space thus has a countable basis. The
Wave_function
Numerical method in computational electromagnetics
combination of pre-defined basis functions; generally, the coefficients of these basis functions are the sought unknowns. Green's functions and Galerkin method
Method of moments (electromagnetics)
Method_of_moments_(electromagnetics)
Method of data interpolation and smoothing
radial basis functions. Given a set of control points { c i , i = 1 , 2 , … , K } {\displaystyle \{c_{i},i=1,2,\ldots ,K\}} , a radial basis function defines
Thin_plate_spline
Topics referred to by the same term
and a put option Basis function Basis (linear algebra) Dual basis Orthonormal basis Schauder basis Basis (universal algebra) Basis of a matroid Generating
Basis
Vector basis functions
Raviart–Thomas basis functions are vector basis functions used in finite element and boundary element methods. They are regularly used as basis functions when working
Raviart–Thomas basis functions
Raviart–Thomas_basis_functions
Mathematical function
quantum chemistry to form basis sets. Gaussian functions arise by composing the exponential function with a concave quadratic function: f ( x ) = exp ( α
Gaussian_function
Mathematical function of a linear operator
the function space has an orthonormal basis given by the set of functions {u1(t), u2(t), …, un(t)}, where n may be infinite. For the orthonormal basis, ⟨
Eigenfunction
In functional analysis, a Hilbert space
space of functions in which point evaluation is a continuous linear functional. Specifically, a Hilbert space H {\displaystyle H} of functions from a set
Reproducing kernel Hilbert space
Reproducing_kernel_Hilbert_space
Function used in signal processing
processing and statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside
Window_function
Mathematical technique used in data compression and analysis
function ψ ∈ L 2 ( R ) {\displaystyle \psi \,\in \,L^{2}(\mathbb {R} )} is called an orthonormal wavelet if it can be used to define a Hilbert basis,
Wavelet_transform
Method for estimating new data within known data points
(statistics) Lagrange interpolation Missing data Newton–Cotes formulas Radial basis function interpolation Simple rational approximation Smoothing Sheppard, William
Interpolation
Non-parametric regression technique
line in the formula for ozone above is one basis function multiplied by its coefficient. Each basis function B i ( x ) {\displaystyle B_{i}(x)} takes one
Multivariate adaptive regression spline
Multivariate_adaptive_regression_spline
Spatial statistical signal analysis
empirical orthogonal function (EOF) analysis is a decomposition of a signal or data set in terms of orthogonal basis functions which are determined from
Empirical orthogonal functions
Empirical_orthogonal_functions
Classification of Artificial Neural Networks (ANNs)
radial basis function (RBF, also called a kernel function) is applied to the distance to compute the weight (influence) for each neuron. The radial basis function
Types of artificial neural networks
Types_of_artificial_neural_networks
Spline function
In numerical analysis, a B-spline (short for basis spline) is a type of spline function designed to have minimal support (overlap) for a given degree,
B-spline
Overview of and topical guide to machine learning
Algorithm for Rule Set Production Growing self-organizing map Hyper basis function network IDistance k-means clustering k-nearest neighbors algorithm Kernel
Outline_of_machine_learning
Real function on a Euclidean space whose value depends only on distance from the origin
as the radial portion of the Fourier transform. Radial basis function "Radial Basis Function - Machine Learning Concepts". Machine Learning Concepts
Radial_function
Coordinate change in linear algebra
multivariate function whose variables are the coordinates on some basis of the vector on which the function is applied. When the basis is changed, the
Change_of_basis
One hundredth of one percentage point
A basis point (often abbreviated as bp, often pronounced as "bip" or "beep") is one hundredth of 1 percentage point. Changes of interest rates are often
Basis_point
Concept in mathematics
}r_{j}(x)^{k_{j}}} Walsh functions and trigonometric functions are both systems that form a complete, orthonormal set of functions, an orthonormal basis in the Hilbert
Walsh_function
Type of vector space in math
different basis functions for a space such as L2([0, 1]). In many circumstances, it is desirable not to decompose a function into trigonometric functions, but
Hilbert_space
Functions of an angle
mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of
Trigonometric_functions
Classification algorithm
of high-dimensional whitening representations using basis function systems (B-splines, Fourier basis, etc.). Decorrelation Principal component analysis
Whitening_transformation
of functions related to Fourier analysis. Such transformations map a function to a set of coefficients of basis functions, where the basis functions are
List of Fourier-related transforms
List_of_Fourier-related_transforms
Mathematical function
Slepian functions are a class of spatio-spectrally concentrated functions that form an orthogonal basis for bandlimited or spacelimited spaces. That is
Slepian_function
Decompositions of inner product spaces into orthonormal bases
function into a sum of square integrable orthogonal basis functions. The standard Fourier series uses an orthonormal basis of trigonometric functions
Generalized_Fourier_series
proto-value functions (PVFs) are automatically learned basis functions that are useful in approximating task-specific value functions, providing a compact
Proto-value_function
Type of artificial neural network
activation functions have been proposed, including the rectifier and softplus functions. More specialized activation functions include radial basis functions (used
Feedforward_neural_network
Type of noise in computer graphics
ISBN 978-1-4398-9376-0. Worley, Steven (August 1996). "A cellular texture basis function". Proceedings of the 23rd annual conference on Computer graphics and
Worley_noise
Optical filter
with the i-th current basis function (i.e., dotting from the left and integrating over the domain of the i-th current basis function, thereby completing
Frequency_selective_surface
Computer-aided design approach
i^{th}} knot, n {\displaystyle n} is the number of functions, p {\displaystyle p} refers to the basis functions order. A knot divides the knot span into elements
Isogeometric_analysis
Method for solving continuous operator problems (such as differential equations)
problem by applying linear constraints determined by finite sets of basis functions. They are named after the Soviet mathematician Boris Galerkin. Often
Galerkin_method
Artificial neural network node function
special class of activation functions known as radial basis functions (RBFs) are used in RBF networks. These activation functions can take many forms, but
Activation_function
Method in computational chemistry
is named. If the coefficients of the basis functions in the molecular orbital are Cμi for the μ'th basis function in the i'th molecular orbital, the density
Mulliken_population_analysis
Instrument for detecting variations in the Earth's magnetic field
Technology, vol. 32, no. 4, 2021, doi:10.1088/1361-6501/abc123. "Adaptive Basis Function Method for the Detection of an Undersurface Magnetic Anomaly Target
Magnetic_anomaly_detector
Type of data encoding
base band. Hence, the signal space can be represented by the single basis function ϕ ( t ) = 2 T b cos ( 2 π f t ) {\displaystyle \phi (t)={\sqrt {\frac
Phase-shift_keying
Tool in multivariate statistical analysis
the Matérn covariance, also called the Matérn kernel, is a covariance function used in spatial statistics, geostatistics, machine learning, image analysis
Matérn_covariance_function
Set of methods for supervised statistical learning
_{j})=(\mathbf {x} _{i}\cdot \mathbf {x} _{j}+r)^{d}} . Gaussian radial basis function: k ( x i , x j ) = exp ( − γ ‖ x i − x j ‖ 2 ) {\displaystyle k(\mathbf
Support_vector_machine
Algebraic structure in linear algebra
suitable basis functions as sufficient for approximation purposes, but also together with the Gram–Schmidt process, it enables one to construct a basis of orthogonal
Vector_space
Topics referred to by the same term
Rangierbahnhof, a railway marshalling yard in Germany Radial basis function in mathematics Radial basis function kernel or RBF kernel Random Block File, an OS-9 file
RBF
Matrix used in finite element analysis
determined so that the error in the approximation is orthogonal to each basis function φi: ∫ Ω φ i ⋅ f d x = − ∫ Ω φ i ∇ 2 u h d x = − ∑ j ( ∫ Ω φ i ∇ 2 φ
Stiffness_matrix
Point of reference in Euclidean space
space, a topological space with a distinguished point Radial basis function, a function depending only on the distance from the origin Madsen, David A
Origin_(mathematics)
Materials simulation software
code utilizing numeric atom-centered basis functions for its electronic structure calculations. The localized basis set enables the accurate treatment of
FHI-aims
Special mathematical functions defined on the surface of a sphere
spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, certain functions defined on the surface of a sphere can be written
Spherical_harmonics
Approximating an arbitrary function with a well-behaved one
theory Fitness approximation Kriging Least squares (function approximation) Radial basis function network Lakemeyer, Gerhard; Sklar, Elizabeth; Sorrenti
Function_approximation
Method of function approximation and data interpolation
polyharmonic spline is a linear combination of polyharmonic radial basis functions (RBFs) denoted by φ {\displaystyle \varphi } plus a polynomial term:
Polyharmonic_spline
Interpolation on functions of more than one variable
weighting Natural-neighbor interpolation Spline interpolation Radial basis function interpolation Barnes interpolation Bilinear interpolation Bicubic interpolation
Multivariate_interpolation
List of concepts in artificial intelligence
basis function network In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions
Glossary of artificial intelligence
Glossary_of_artificial_intelligence
Type of polynomial used in Numerical Analysis
as Daubechies wavelet) Lorentz 1953 Mathar, R.J. (2018). "Orthogonal basis function over the unit circle with the minimax property". Appendix B. arXiv:1802
Bernstein_polynomial
Vector quantization algorithm minimizing the sum of squared deviations
distance through a Gaussian RBF, obtains the hidden layer of a radial basis function network. This use of k-means has been successfully combined with simple
K-means_clustering
Basis sets used in quantum chemistry
systems within Hartree-Fock theory or density functional theory. The basis functions are linear combinations of n {\displaystyle n} primitive Gaussian-type
STO-nG_basis_sets
Topics referred to by the same term
as a polynomial function of these basis elements Orthonormal basis of a Hilbert space Hilbert basis (linear programming) Hilbert's basis theorem This disambiguation
Hilbert_basis
Linear function of explanatory variables used to predict a dependent variable
fixed set of nonlinear functions are used to transform the value(s) of a data point, these functions are known as basis functions. An example is polynomial
Linear_predictor_function
Technique used in signal processing and data compression
reflects their different underlying algebraic structures: the DFT basis functions are the irreducible representations of the cyclic group, making the
Discrete_cosine_transform
Mathematical framework
consistent, a function interpolating these constraints can be constructed by selecting n {\displaystyle n} linearly independent basis functions such as monomials
Theory of functional connections
Theory_of_functional_connections
Computational tool
earlier. For example, the Haar basis was given in 1909, and Georg Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called
Schauder_basis
Method in physics
positions τ α {\displaystyle {\boldsymbol {\tau }}_{\alpha }} , an LAPW basis function is characterized by a reciprocal lattice vector G {\displaystyle \mathbf
Linearized augmented-plane-wave method
Linearized_augmented-plane-wave_method
Statistics concept
polynomial basis-functions are used along with new basis functions, such as splines, radial basis functions, and wavelets. These families of basis functions offer
Polynomial_regression
Computational method for solving partial differential equations
first attempt to extend radial basis function (RBF), which was then quite popular in scattered data processing and function approximation, to the solution
Kansa_method
Family of solutions to related differential equations
function is expanded in the basis of the functions Jα(x uα,m) for fixed α and varying m. An analogous relationship for the spherical Bessel functions
Bessel_function
Method for solving differential equations
standard function inner product with respect to some weighting function r ( x ) {\displaystyle r(x)} which is determined usually by the basis function set
Method of mean weighted residuals
Method_of_mean_weighted_residuals
Class of analytical partial differential equation-solving processes
spectral methods, but complement the basis by an additional pseudo-spectral basis, which allows representation of functions on a quadrature grid[definition
Pseudo-spectral_method
Mathematical function defined piecewise by polynomials
In mathematics, a spline is a function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial
Spline_(mathematics)
Function for integral Fourier-like transform
representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set or frame of
Wavelet
Non-cryptographic hash function
(or FNV) is a non-cryptographic hash function created by Glenn Fowler, Landon Curt Noll, and Kiem-Phong Vo. The basis of the FNV hash algorithm was taken
Fowler–Noll–Vo_hash_function
Error in quantum-chemistry calculations due to overlapping basis sets
another, their basis functions overlap. Each monomer "borrows" functions from other nearby components, effectively increasing its basis set and improving
Basis_set_superposition_error
Type of artificial neural network architecture
functions instead of B-spline to create new KAN variants. The choice of functional basis strongly influences the performance of KANs. Common function
Kolmogorov–Arnold_Networks
Statistical method for investigating the dominant modes of variation of functional data
functional data. Using this method, a random function is represented in the eigenbasis, which is an orthonormal basis of the Hilbert space L2 that consists of
Functional principal component analysis
Functional_principal_component_analysis
Mathematical function
number of basis functions generally required in a Gaussian calculation. For reasons of convenience, many quantum chemistry programs work in a basis of Cartesian
Gaussian_orbital
Method of representing a random variable
(Y)<\infty } ) as a function of an M {\displaystyle M} -dimensional random vector X {\displaystyle \mathbf {X} } , using a polynomial basis that is orthogonal
Polynomial_chaos
Algorithm in computational number theory
Arageli as the function lll_reduction_int fpLLL as a stand-alone implementation FLINT as the function fmpz_lll GAP as the function LLLReducedBasis Macaulay2
Lenstra–Lenstra–Lovász lattice basis reduction algorithm
Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm
Engineering model
generalized Bayesian approaches; gradient-enhanced kriging (GEK); radial basis function; support vector machines; space mapping; artificial neural networks
Surrogate_model
Property of artificial neural networks
J.; Sandberg, I. W. (1991). "Universal Approximation Using Radial-Basis-Function Networks". Neural Computation. 3 (2): 246–257. doi:10.1162/neco.1991
Universal approximation theorem
Universal_approximation_theorem
Mathematical function whose derivative exists
or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable
Differentiable_function
Physical spaces representing position and momentum, Fourier-transform duals
operator as a set of basis functions, the resulting wave function ϕ ( k ) {\displaystyle \phi (\mathbf {k} )} is said to be the wave function in momentum space
Position_and_momentum_spaces
Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way
The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function, gate function, unit pulse, or the normalized
Rectangular_function
Primarily Arizonan charter school operator
are made up of BASIS Charter Schools, BASIS Independent Schools, and BASIS International Schools. The first BASIS Curriculum School, BASIS Tucson, was founded
Basis_Schools
Quantization giving rise to photons
a particle in the corresponding basis function and an annihilation operator annihilates a particle in this function. In the case of EM fields the required
Quantization of the electromagnetic field
Quantization_of_the_electromagnetic_field
Type of feedforward neural network
activation functions have been proposed, including the rectifier and softplus functions. More specialized activation functions include radial basis functions (used
Multilayer_perceptron
Thought experiment in quantum mechanics
live-action television, cartoons, music, and webcomics. Physics portal Basis function Cat state Complementarity (physics) Double-slit experiment Elitzur–Vaidman
Schrödinger's_cat
Concept in probability theory and statistics
it provides the basis of an alternative route to analytical results compared with working directly with probability density functions or cumulative distribution
Moment_generating_function
coordinate basis vector eα and the partial derivative operator ∂/∂xα, under the interpretation of vectors as operators acting on functions. A local condition
Holonomic_basis
Technique in quantum chemistry
wavefunctions. In a mathematical sense, these wave functions are the basis set of functions, the basis functions, which describe the electrons of a given atom
Linear combination of atomic orbitals
Linear_combination_of_atomic_orbitals
Specific linear basis (mathematics)
same linear span as an orthonormal basis may not be a basis at all. For instance, any square-integrable function on the interval [ − 1 , 1 ] {\displaystyle
Orthonormal_basis
Statistics models class
method="REML") which uses a basis size of 100 for the smooth of t {\displaystyle t} . The specification of distribution and link function uses the `family' objects
Generalized_additive_model
Association of one output to each input
mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the
Function_(mathematics)
Set of learning techniques in machine learning
clusters as features, perhaps after transforming them through a radial basis function (a technique that has been used to train RBF networks). Coates and Ng
Feature_learning
Approximation method in quantum physics
sounder theoretical basis by applying the variational principle to an ansatz (trial wave function) as a product of single-particle functions. In 1930, Slater
Hartree–Fock_method
BASIS FUNCTION
BASIS FUNCTION
Boy/Male
Muslim
King, Basil the herb (1)
Surname or Lastname
English
English : probably a variant spelling of Bevis.
Boy/Male
Tamil
King, Basil the herb
Boy/Male
Turkish
Intelligent.
Male
English
 English form of French Basile, BASIL means "king." Also sometimes given as an herb name.
Boy/Male
Indian
Smiling, Happy
Boy/Male
Muslim
Smiling, Happy
Boy/Male
Muslim Arabic
Smiling.
Female
Hebrew
 Variant spelling of Hebrew Basya, BASIA means "daughter of God."
Boy/Male
Greek American English
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Surname or Lastname
English and French
English and French : from a medieval personal name, ultimately from Greek Basileios ‘royal’. The name was borne by a 4th-century bishop of Caesarea in Cappadocia, regarded as one of the four Fathers of the Eastern Church; he wrote important theological works and established a rule for religious orders of monks. Various other saints are also known under these and cognate names. The popularity of Vasili as a Russian personal name is largely due to the fact that this was the ecclesiastical name of St. Vladimir (956–1015), Prince of Kiev, who was chiefly responsible for the introduction of Christianity to Russia. As an American surname, this has also absorbed some Greek, Russian, and other derivatives of Greek Vasili.
Boy/Male
Indian
Vast, Spacious, One who stretches, Enlarges
Boy/Male
Muslim
Clear
Boy/Male
Indian
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Boy/Male
Hindu
King, Basil the herb
Boy/Male
Greek
Royal. Kingly. St Basil the Great was Bishop of Caesarea in the latter half of the 4th century....
Surname or Lastname
English
English : from Old French bas(se) ‘low’, ‘short’ (Latin bassus ‘thickset’; see Basso), either a descriptive nickname for a short person or a status name meaning ‘of humble origin’, not necessarily with derogatory connotations.English : in some instances, from Middle English bace ‘bass’ (the fish), hence a nickname for a person supposedly resembling this fish, or a metonymic occupational name for a fish seller or fisherman.Scottish : habitational name from a place in Aberdeenshire, of uncertain origin.Jewish (Ashkenazic) : metonymic occupational name for a maker or player of bass viols, from Polish, Ukrainian, and Yiddish bas ‘bass viol’.German : see Basse.
Boy/Male
Muslim
Vision, Propitious, Auspicious, Prudent, Bringer of glad tidings
Boy/Male
Muslim
Vast, Spacious, One who stretches, Enlarges
Surname or Lastname
English
English : variant of Bayliss.Hungarian and Croatian (Bališ) : from the personal name Bali, a pet form of Baltazar or Balint.Perhaps also Greek : occupational status name from Turkish balija ‘workman’, ‘low-ranking man’.
BASIS FUNCTION
BASIS FUNCTION
Girl/Female
Arabic, Australian, Muslim
Literary Woman; Authoress; Cultured; Polite; Soft; A Literary Person; Civilized
Surname or Lastname
English
English : variant of Kay.
Girl/Female
Arabic, Urdu
River of Knowledge
Female
Egyptian
, the daughter of Aahmes II.
Boy/Male
British, English, French, Hebrew
Has Shown Favour; Variant of John; Jehovah has been Gracious; God is Gracious
Boy/Male
Muslim/Islamic
Blessed
Surname or Lastname
German
German : nickname for someone with a peculiarity of the back, Middle High German rucke.German : topographic name from a southern field name denoting a slight dome-shaped elevation.German : from the personal names Ruck, Rück, short forms of Rüdiger (see Rudiger).English : variant spelling of Rook.
Boy/Male
Gaelic
Scholar.
Boy/Male
British, English
Brave
Girl/Female
Latin American Italian
Lady.
BASIS FUNCTION
BASIS FUNCTION
BASIS FUNCTION
BASIS FUNCTION
BASIS FUNCTION
a.
One who sings, or the instrument which plays, bass.
v. & a.
Fixed foundation; established basis.
pl.
of Basis
n.
The striped bass. See Bass.
pl.
of Bass
n.
The quantity contained in a basin.
n.
The two American fresh-water species of black bass (genus Micropterus). See Black bass.
n.
The southern, red, or channel bass (Sciaena ocellata). See Redfish.
n.
An isolated or circumscribed formation, particularly where the strata dip inward, on all sides, toward a center; -- especially applied to the coal formations, called coal basins or coal fields.
pl.
of Oasis
a.
A bass, or deep, sound or tone.
n.
A basin.
n.
The striped bass. See Bass.
n.
The name given to several aromatic herbs of the Mint family, but chiefly to the common or sweet basil (Ocymum basilicum), and the bush basil, or lesser basil (O. minimum), the leaves of which are used in cookery. The name is also given to several kinds of mountain mint (Pycnanthemum).
n.
The deepest pedal stop, or the lowest tones of an organ; the fundamental or ground bass.
a.
Hence, basic; metallic; not acid; -- opposed to negative, and said of metals, bases, and basic radicals.
n.
Species of Serranus, the sea bass and rock bass. See Sea bass.