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Topic: Basis function

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	POV-Ray Hall of Fame
		The basis function for the "terrain" in this image is a "sombrero" function that has been displaced vertically by a noise function.
		The grooves are made by subtracting the function for the "tubes" from the function for the "hills" with the help of a "blobbing" function; the hyperbolic tangent function.
		The fourth function, which is given as the minor radius to the torus function, also controls the colour of the pigment.
	hof.povray.org (1122 words)

	An In-Depth Look at 'Level-Two' of the Feeling Function
		The feeling function picks up on the over-all sense of the situation as a whole, and emotion is thus conceived as 'the positive or negative after-effect' of the general attitude that one has toward a situation (or event), apprehended as a whole.
		Propensities toward attraction and avoidance form the basis of the subject's 'evaluation' of any given object; they are closely connected to 'pain' and 'pleasure' as subjective reactions to the presence of various types of object.
		the well-known 'emergency' function of emotion in fulfilling a 'homeostatic' requirement as part of the 'wisdom of the body' is widely presented and discussed in detail in the literature.
	tap3x.net /ENSEMBLE/mpage3e.html (3832 words)

	Space-Frequency Localized Basis Function Networks for Nonlinear System Estimation and Control (Site not responding. Last check: )
		Approximation models employing basis functions that are localized in both space and spatial frequency admit a measure of the approximated function's spatial frequency content that is not directly dependent on reconstruction error.
		An asymptotic bound on the error in the network's reconstruction is derived and shown to be dependent solely on the minimum approximation error associated with the steady state node configuration.
		In addition, prior bounds on the temporal bandwidth of the system to be identified or controlled are used to develop a criterion for on-line selection of radial and ridge wavelet basis functions, thus reducing the rate of increase in network's size with the dimension of the state vector.
	web.mit.edu /nsl/www/abstracts/neurocomp95-abst.html (313 words)

	Radial Basis Function Networks
		The output of the neuron is then formed by applying the basis function to this distance.
		(2.10) the basis function is chosen to be the Gaussian bell function.
		Although this function is the most commonly used basis function, other basis functions may be chosen.
	documents.wolfram.com /applications/neuralnetworks/NeuralNetworkTheory/2.5.2.html (369 words)

	No Title
		Basis functions are not usually atomic orbitals, and in many cases, they do not even resemble orbitals of isolated atoms.
		It comes as a consequence of the fact that basis functions on a given center are usually not orthogonal to each other and "core" basis functions on different centers overlap to a great extent - situation not likely to occur with true atomic orbitals.
		DZ sets have two basis functions per orbital, etc. Since valence orbitals of atoms are more affected by forming a bond than the inner (core) orbitals, more basis functions are assigned frequently to describe valence orbitals.
	www.ccl.net /cca/documents/basis-sets/basis.html (5465 words)

	Variation Method for the Particle in a Box
		Basis functions are only useful if the required overlap and hamiltonian matrix elements (integrals) can be readily computed.
		For the polynomial basis functions of the example, this is straightforward though tedious.
		For the example, with M=3 (four basis functions) the lowest root is 1.00001471 (an error of only ~15 ppm from the exact ground state energy).
	www.sci.wsu.edu /idea/quantum/var_PIB.htm (1150 words)

	[No title]
		A logistic activation function is used for the output, with scaling factors determined so that the range encountered in the training set is mapped to a restricted part of the logistic functions (0,1) range (e.g.
		The use of the sum-squared network error function means that points lying far from the others have a disproportionate influence on the position of the hyperplanes used in regression.
		Thus, the reliability function (commonly denoted as R(t)) is the complement to the cumulative distribution function (i.e., R(t)=1-F(t)); the reliability function is also sometimes referred to as the survivorship or survival function (since it describes the probability of not failing or surviving until a certain time t; e.g., see Lee, 1992).
	www.statsoft.com /textbook/glosr.html (4576 words)

	Gaussian-Type Functions
		for each basis function on the atom are optimized and hence the size of the orbital that is now expressed as a linear combination of two basis functions
		For example, the 3-21G basis set has one contracted Gaussian function that is a linear combination of three primitive Gaussian functions for each inner-shell atomic orbital and two basis functions, one contracted Gaussian function that is a linear combination of two primitive Gaussians and one primitive Gaussian function, for each valence orbital.
		The 6-31G* basis set represents each inner-shell orbital with one contracted Gaussian function that is a linear combination of six primitive Gaussian functions and each valence orbital with two basis functions, one contracted Gaussian function that is a linear combination of three primitive Gaussians and one primitive Gaussian function.
	www.chm.davidson.edu /ronutt/che401/Gaussian/Gaussian.htm (599 words)

	B-spline Basis Functions: Important Properties
		The above shows the basis functions of degree 5 where the left end and right end knots have multiplicity 6, while all middle knots are simple (left on top row).
		At each internal knot of multiplicity k, the number of non-zero basis functions is at most p - k + 1, where p is the degree of the basis functions.
		, the number of non-zero basis functions at a knot of multiplicity k is at most (p + 1) - k = p - k + 1.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/spline/bspline-property.html (583 words)

	Functional Data Analysis - The Basics of Basis Functions - Page1
		Moreover, however it is that we design our mathematical function, we will want to do any computation that is needed to fit the data quickly and with a minimum of programming.
		We need, therefore, a set of basic functional building blocks that can be stacked on top of one another so as to have the features that we need.
		A system developed in early nineteenth century France is the Fourier basis, consisting of 1 and a series of pairs of sines and cosines of increasing frequency; that is, 1, sin(ωt), cos(ωt), sin(2ωt), cos(2ωt), sin(3ωt), cos(3ωt),....
	www.psych.mcgill.ca /misc/fda/ex-basis-a1.html (558 words)

	Basis Function Help Page (Site not responding. Last check: )
		The B-Spline Basis Functions are used to determine how much affect each control point has on the parametric B-Spline curve at any given point in time.
		Every basis function is defined over the entire life "time" of the curve.
		Notice that the sum of all of the basis functions that affect the curve is always equal to 1.0.
	www.cs.utah.edu /~dav/curve_ed/basis_func_help.html (251 words)

	Radial basis function networks according to Moody and Darken
		Radial basis function networks according to Moody and Darken
		The THEN-parts of the fuzzy rules or, alternatively, the output weights of the RBFN, are set by pseudo-inverse computation such that the summed square error (4) for a given training data set is minimized.
		This multi-phase approach is straight-forward and is often reported to be much faster than, e.g., the backpropagation training of multi-layer perceptrons for the same data.
	www.ki.inf.tu-dresden.de /~fritzke/FuzzyPaper/node5.html (604 words)

	Basis Set Study
		6-31G* is a polarized type basis set where the "6" stands for the number of basis functions required for determining the s orbital, and the "31" stands forthe two types of basis functions required to determine the p orbital.
		The number of basis functions of a molecule are the number of equations it takes to determine a Single Point Energy task for that molecule.
		We were not able to find a linear equation relating the number of basis functions to the amount of CPU time because the equation is quadratic i.e., the data follows a parabola pattern because one of the values in the equation is squared.
	www.shodor.org /succeed/projects/compchem/cchem97/basis (1046 words)

	Estimating the hemodynamic response function (“Deconvolution”)
		Having fewer basis functions means there are fewer parameters (weights) to estimate in the GLM, which improves the reliability of the estimates.
		Regardless of which type of basis functions are chosen to model the HR for a given condition, a decision still needs to be made as to how many basis functions are required.
		In summary, it is probably best to limit the number of basis functions to the minimum number required to reasonably accurately model the range of expected HR shapes in your experiment.
	brainimaging.waisman.wisc.edu /~tjohnstone/Deconvolution.htm (1710 words)

	B-spline Basis Functions: Definition
		There are two interesting properties that are not part of the Bézier basis functions, namely: (1) the domain is subdivided by knots, and (2) basis functions are not non-zero on the entire interval.
		In fact, each B-spline basis function is non-zero on a few adjacent subintervals and, as a result, B-spline basis functions are quite "local".
		= 3, knot spans 0, 1 and 2 are [0,1), [1,2), [2,3) and the basis functions of degree 0 are N
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/spline/B-spline/bspline-basis.html (957 words)

	An Introduction to Wavelets: What are Basis Functions?
		It is simpler to explain a basis function if we move out of the realm of analog (functions) and into the realm of digital (vectors) (*).
		The best basis vectors have the valuable extra property that the vectors are perpendicular, or orthogonal to each other.
		The sines and cosines are the basis functions in this example, and the elements of Fourier synthesis.
	www.amara.com /IEEEwave/IW_basis.html (359 words)

	Advanced ab initio - Glossary
		In ab initio MO theory, the basis functions can take many forms and are not necessarily related to the atomic orbitals.
		Basis set - The complete set of basis functions which is used to represent the molecular orbitals.
		Because of the different r values at which they have their maxima, these pairs of basis functions are often called 'inner' and 'outer' functions.
	www.chem.swin.edu.au /modules/mod8/define.html (1253 words)

	ARANZ: Radial Basis Function Theory
		Radial basis functions are presented as a practical solution to the problem of interpolating incomplete surfaces derived from three-dimensional (3D) medical graphics.
		Radial basis functions impose few restrictions on the geometry of the interpolation centers and are suited to problems where the interpolation centers do not form a regular grid.
		In this paper radial basis functions are fitted to depth-maps of the skull's surface, obtained from X-ray CT data using ray-tracing techniques.
	aranz.com /research/modelling/theory (760 words)

	Basis Sets
		All basis set equations in the form STO-NG (where N represents the number of GTOs combined to approximate the STO) are considered to be "minimal" basis sets.
		Previously with the minimal basis sets, we approximated all orbitals to be of the same shape.
		As atoms are brought close together, their charge distribution causes a polarization effect (the positive charge is drawn to one side while the negative charge is drawn to the other) which distorts the shape of the atomic orbitals.
	www.shodor.org /chemviz/basis/teachers/background.html (1466 words)

	Plotting of Stimulus Pulse Trains and Basis Functions
		These are functions that extend over a relatively short (event-related) or long (epoch-related) period of time, and are convolved with the stimulus pulse functions to arrive at the linear regressor(s) representing what the brain response should look like at each voxel.
		In SPM99, this is a pre-canned function equal to the sum of two beta functions and extending for 32 seconds, which is used by many investigators as a model of the Hemodynamic Response Function.
		The "hrf" basis function is fixed in shape, though you can add time and dispersion derivative functions to it to create a basis set that may be a more accurate model.
	www-psych.stanford.edu /~gablab/internal/SPM99/LabEnhancements/StimBasisPlotting.html (564 words)

	Basis and Shape Functions
		It is apparent that this basis function satisfies the requirements of being square-integrable over the domain, and that it has the required differentiability; i.e., that first derivatives exist.
		By the same reasoning we could assemble in a piece-wise manner basis functions for each of the nodes in the domain from precisely two of the shape functions from the two adjacent elements.
		First we notice that integrals of a single basis function, or of a basis function times itself, need only be integrated over the nonzero region of the basis function.
	www.umcs.maine.edu /~shamis/papers/theory/node7.html (628 words)

	Radial Basis Function Network (Site not responding. Last check: )
		Using the training samples, the weights multiplying each of the gaussian basis functions arecalculated using the pseudo-inverse (yielding the minimum least-squares solution).
		The resulting network is then used to approximate the function between the given "sample" points.
		Adjust the number of centers and the standard deviation (width of Gaussian) for the Gaussians used in the approximation.
	diwww.epfl.ch /mantra/tutorial/english/rbf/html/index.html (491 words)

	Basis Sets
		It is important to choose a basis set large enough to give a good description of the molecular wave function.
		Typically, the basis functions are centered on the atoms, and so sometimes they are called ``atomic orbitals.'' However, it is important to note that this does not imply that they are actually solutions to the electronic Schrödinger equation for the atom.
		The STO-3G basis is a very well-known minimal basis set which contracts 3 Gaussian functions to approximate the more accurate (but more difficult to compute) Slater type orbitals.
	vergil.chemistry.gatech.edu /courses/chem4681/background/node5.html (654 words)

	Basis Function and Activation Function (Site not responding. Last check: )
		The mapping is mathematically described by a basis function f to yield a new activation value
		For an analytical study, the connection networks are mathematically represented by a basis function u(w,x), where w stands for the weight matrix, and x for the input vector.
		The net value as expressed by the basis function, U(w,x), will be inmediatly transformed by a nonlinear activation function of the neuron.
	www.gc.ssr.upm.es /inves/neural/ann1/concepts/basis.htm (213 words)

	Fourier transform theory
		The Fourier transform is simply a method of expressing a function (which is a point in some infinite dimensional vector space of functions) in terms of the sum of its projections onto a set of basis functions.
		We can show that any two basis functions of different frequencies are orthogonal by calculating their inner product and showing that it is 0.
		This is the basis by which image restoration for blur due to focus or motion is done.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/OWENS/LECT4/node2.html (1260 words)

	Chapter 8. Curves and Surfaces
		The distribution of basis functions in parameter space is controlled by the knot sequence (also referred to as the knot vector, or the knots).
		This figure also illustrates another important NURBS relationship: at any point where the curve is defined, the sum of all basis functions is equal to 1.
		In Figure 8-7, for example, the basis function beginning at 0 and ending at 4 spans four segments (knot 0 to knot 1; knot 1 to knot 2; knot 2 to knot 3; and knot 3 to knot 4).
	www-evasion.imag.fr /Membres/Francois.Faure/doc/inventorMentor/sgi_html/ch08.html (3232 words)

	Data processing: vehicles, navigation, and relative location inventions new
		The at least one backpropagation network is coupled to the at least one radial basis function for the purpose of determining a vehicle-specific...
		Agent: Caterpillar C/o Liell & Mcneil Attorneys PC Brake function based on controlling according to acceleration: A brake controller function to optimally brake a wheel of a vehicle in motion, such as an aircraft.
		A functional module is provided by which a desired wheel torque course is determined for reaching the desired vehicle velocity, the course at least...
	www.freshpatents.com /Data-processing--vehicles-navigation-and-relative--dtnewntc701.php (7198 words)

	rbfh.com RBF Homepage
		RBF Homepage = Radial basis function, a family of functions used in radial basis function networks;
		A radial basis function (RBF) is a real-valued function whose value depends only on the distance from the origin.
		In artificial neural networks radial basis functions are utilized as activation functions.
	rbfh.com (489 words)

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