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KerMor
0.9
Model order reduction for nonlinear dynamical systems and nonlinear approximation
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KerMor
0.9
Model order reduction for nonlinear dynamical systems and nonlinear approximation
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Radial Basis Function Kernel. More...
Radial Basis Function Kernel.
Uses the notation
\[\K(x,y) = e^{\frac{||x-y||^2}{\gamma^2}},\]
so be careful with the \(\gamma\) constant.
evaluate. The tests are run iter times and the mean value is plotted to the output.Definition at line 18 of file GaussKernel.m.
Public Member Functions | |
| GaussKernel (double Gamma) | |
| Creates a new GaussKernel. More... | |
| function K = | evaluate (matrix< double > x,matrix< double > y) |
| Evaluates the gaussian. More... | |
| function Nablax = | getNabla (colvec< double > x,matrix< double > y) |
| Method for first derivative evaluation. More... | |
| function dx = | evaluateD1 (r) |
| Method for first derivative evaluation. More... | |
| function ddx = | evaluateD2 (r) |
| Method for second derivative evaluation. More... | |
| function phi = | evaluateScalar (r) |
| Implements the required method from the IRotationInvariant interface. More... | |
| function g = | setGammaForDistance (dist, ep) |
| Computes the \(\gamma\) value for which the Gaussian is smaller than \(\epsilon\) in a distance of dist, i.e.
\[e^{-\frac{d^2}{\gamma}} < \epsilon\] Returns the computed value AND sets the kernel's Gamma property to this value. More... | |
| function copy = | clone () |
 Public Member Functions inherited from kernels.BellFunction | |
| BellFunction () | |
| function c = | getGlobalLipschitz () |
| Computes the absolute value of the first derivative at x0 Implements the template method from BaseKernel. More... | |
| function copy = | clone (copy) |
| The interface method with returns a copy of the current class instance. More... | |
| function rtmp = | ModifiedNewton (rstart, s) |
| virtual function dr = | evaluateD1 () |
| Method for first derivative evaluation. More... | |
| virtual function ddr = | evaluateD2 () |
| Method for second derivative evaluation. More... | |
| Public Member Functions inherited from kernels.ARBFKernel | |
| ARBFKernel () | |
| function K = | evaluate (matrix< double > x,matrix< double > y) |
| Evaluates the rotation and translation invariant kernel. More... | |
| function bool = | eq (B) |
| function r = | getSqDiffNorm (matrix< double > x,matrix< double > y) |
| Returns the weighted squared norm \(r\) of the difference \(\noG{x-y}^2/\gamma^2\). More... | |
| virtual function phir = | evaluateScalar (matrix< double > r) |
| Allows the evaluation of the function \(\phi(r)\) for scalar \(r\) directly. More... | |
| Public Member Functions inherited from kernels.BaseKernel | |
| BaseKernel () | |
| function fcn = | getLipschitzFunction () |
| Method that allows error estimators to obtain a lipschitz constant estimation function from this kernel. More... | |
| function bool = | eq (B) |
| Checks if a kernel equals another kernel. More... | |
| virtual function Nabla = | getNabla (x, y) |
| Computes the partial derivatives with respect to each component of the first argument. More... | |
| Public Member Functions inherited from KerMorObject | |
| KerMorObject () | |
| Constructs a new KerMor object. More... | |
| function | display () |
| disp(object2str(this)); More... | |
| function bool = | eq (B) |
| Checks equality of two KerMor objects. More... | |
| function bool = | ne (B) |
| Checks if two KerMorObjects are different. More... | |
| function cn = | getClassName () |
| Returns the simple class name of this object without packages. More... | |
| Public Member Functions inherited from DPCMObject | |
| DPCMObject () | |
| Creates a new DPCM object. More... | |
| DPCMObject () | |
Static Public Member Functions | |
| static function res = | test_InterpolGamma () |
| static function [ res , pm ] = | test_GaussKernel (pm) |
Static Protected Member Functions | |
| static function obj = | loadobj (obj) |
| Static Protected Member Functions inherited from kernels.BellFunction | |
| static function this = | loadobj (this, from) |
| As the constant properties are transient, they have to be re-computed upon loading. More... | |
| Static Protected Member Functions inherited from DPCMObject | |
| static function obj = | loadobj (obj, from) |
| Re-register any registered change listeners! More... | |
| static function obj = | loadobj (obj, from) |
Additional Inherited Members | |
| Public Attributes inherited from kernels.BellFunction | |
| r0 | |
| Point of maximum first derivative on scalar evaluation. More... | |
| NewtonTolerance = 1e-7 | |
| Error tolerance for modified newton iteration. More... | |
| MaxNewtonIterations = 5000 | |
| Hard break for iteration count of modified newton algorithm. More... | |
| rm | |
| The maximum ("right") value for any \(r_s\). More... | |
| Public Attributes inherited from kernels.ARBFKernel | |
| double | Gamma = 1 |
| Univariate scaling. More... | |
| Public Attributes inherited from kernels.BaseKernel | |
| matrix< double > | G = 1 |
| The matrix \(\vG\) that induces the state space scalar product \(\spG{x}{y}\) and norm \(\noG{x-y}\) to use. More... | |
| matrix< double > | P = "[]" |
| Projection/selection matrix \(\vP\) for argument components. More... | |
| logical | IsRBF = false |
| Flag that determines if the current kernel is a radial basis function, i.e. its evaluation is of the form \(\K(x,y) = \phi(\noG{x-y})\) for some scalar function \(\phi\). More... | |
| logical | IsScProd = false |
| Flag that determines if the current kernel bases on scalar product evaluations, i.e. are of the form \(\K(x,y) = \phi(\spG{x}{y})\) for some scalar function \(\phi\). More... | |
| Public Attributes inherited from DPCMObject | |
| WorkspaceVariableName = "" | |
| The workspace variable name of this class. Optional. More... | |
| ID = "[]" | |
| An ID that allows to uniquely identify this DPCMObject (at least within the current MatLab session/context). More... | |
| PropertiesChanged = "[]" | |
| The Dictionary containing all the property settings as key/value pairs. More... | |
| Public Attributes inherited from handle | |
| addlistener | |
| Creates a listener for the specified event and assigns a callback function to execute when the event occurs. More... | |
| notify | |
| Broadcast a notice that a specific event is occurring on a specified handle object or array of handle objects. More... | |
| delete | |
| Handle object destructor method that is called when the object's lifecycle ends. More... | |
| disp | |
| Handle object disp method which is called by the display method. See the MATLAB disp function. More... | |
| display | |
| Handle object display method called when MATLAB software interprets an expression returning a handle object that is not terminated by a semicolon. See the MATLAB display function. More... | |
| findobj | |
| Finds objects matching the specified conditions from the input array of handle objects. More... | |
| findprop | |
| Returns a meta.property objects associated with the specified property name. More... | |
| fields | |
| Returns a cell array of string containing the names of public properties. More... | |
| fieldnames | |
| Returns a cell array of string containing the names of public properties. See the MATLAB fieldnames function. More... | |
| isvalid | |
| Returns a logical array in which elements are true if the corresponding elements in the input array are valid handles. This method is Sealed so you cannot override it in a handle subclass. More... | |
| eq | |
| Relational functions example. See details for more information. More... | |
| transpose | |
| Transposes the elements of the handle object array. More... | |
| permute | |
| Rearranges the dimensions of the handle object array. See the MATLAB permute function. More... | |
| reshape | |
| hanges the dimensions of the handle object array to the specified dimensions. See the MATLAB reshape function. More... | |
| sort | |
| ort the handle objects in any array in ascending or descending order. More... | |
| Protected Member Functions inherited from kernels.BellFunction | |
| function [
g , dg , pi , pl , pr ] = | optFun (r, s, n, dn) |
| Protected Member Functions inherited from KerMorObject | |
| function | checkType (obj, type) |
| Object typechecker. More... | |
| Protected Member Functions inherited from DPCMObject | |
| function | registerProps (varargin) |
| Call this method at any class that defines DPCM observed properties. More... | |
| function | registerProps (varargin) |
| Protected Attributes inherited from kernels.BaseKernel | |
| fG = 1 | |
| fP = "[]" | |
| kernels.GaussKernel.GaussKernel | ( | double | Gamma | ) |
Creates a new GaussKernel.
| Gamma | The Gamma property to use. Default: 1 |
Definition at line 50 of file GaussKernel.m.
References handle.addlistener, kernels.ARBFKernel.Gamma, and DPCMObject.registerProps().
| function copy = kernels.GaussKernel.clone | ( | ) |
Definition at line 177 of file GaussKernel.m.
References kernels.ARBFKernel.Gamma.
Evaluates the gaussian.
If \(y_j\) is set, the dimensions of \(x_i\) and \(y_j\) must be equal for all \(i,j\).
| x | First set \(x_i \in \R^d\) of \(n\) vectors |
| y | Second set \(y_j \in \R^d\) of \(m\) vectors. If y is empty \(y_i = x_i\) and \(n=m\) is assumed. |
| K | An evaluation matrix \(K \in \R^{n\times m}\) of the evaluated gaussians with entries \(K_{i,j} = e^{-\norm{x_i-y_j}{G}^2/\gamma^2}\). |
Implements kernels.BaseKernel.
Definition at line 69 of file GaussKernel.m.
References kernels.ARBFKernel.Gamma, and kernels.ARBFKernel.getSqDiffNorm().
Referenced by getNabla().
| function dx = kernels.GaussKernel.evaluateD1 | ( | r | ) |
Method for first derivative evaluation.
Definition at line 115 of file GaussKernel.m.
References kernels.ARBFKernel.Gamma.
| function ddx = kernels.GaussKernel.evaluateD2 | ( | r | ) |
Method for second derivative evaluation.
Definition at line 126 of file GaussKernel.m.
References kernels.ARBFKernel.Gamma.
| function phi = kernels.GaussKernel.evaluateScalar | ( | r | ) |
Implements the required method from the IRotationInvariant interface.
Definition at line 137 of file GaussKernel.m.
Method for first derivative evaluation.
| x | The state variable vector \(\vx(t)\) |
| y | The output variable vector \(\vy(t) = \vC(t,\vmu)\vx(t),~t=0\ldots T\) (general notation) |
Definition at line 88 of file GaussKernel.m.
References evaluate(), kernels.BaseKernel.fP, and kernels.ARBFKernel.Gamma.
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staticprotected |
Definition at line 200 of file GaussKernel.m.
| function g = kernels.GaussKernel.setGammaForDistance | ( | dist, | |
| ep | |||
| ) |
Computes the \(\gamma\) value for which the Gaussian is smaller than \(\epsilon\) in a distance of dist, i.e.
\[e^{-\frac{d^2}{\gamma}} < \epsilon\]
Returns the computed value AND sets the kernel's Gamma property to this value.
| dist | The target distance at which the gaussian is smaller than ep |
| ep | The \(\epsilon\) value. If not given, \(\epsilon\)=eps (machine precision) is assumed. |
| g | The computed gamma |
Definition at line 149 of file GaussKernel.m.
References KerMor.App(), and kernels.ARBFKernel.Gamma.
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static |
Definition at line 255 of file GaussKernel.m.
References k, handle.reshape, X, and Y.
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static |
Definition at line 215 of file GaussKernel.m.
References handle.disp, and k.
1.8.6