KernelExpansion: Base class for state-space kernel expansions. More...

Detailed Description

KernelExpansion: Base class for state-space kernel expansions.

The KernelExpansion class represents a function \(f\) in the space induced by a given kernel \(\K\) as

\[f(x) = \sum\limits_{i=1}^N c_i\K(x,x_i)\]

This class has been introduced in order to allow standalone use of kernel expansions in approximation contexts outside of the intended model reduction scheme.

Author: Daniel Wirtz

Date: 2011-07-07

Change in 0.6:: (Daniel Wirtz, 2012-04-25) Removed the RotationInvariant property from the kernel expansions and introduced kernels.BaseKernel.IsRBF and kernels.BaseKernel.IsScProd to indicate the properties on a per-kernel basis.

Change in 0.5:: (Daniel Wirtz, 2011-07-28) Fixed the evaluate method, it had an argument Centers.xi too much.

New in 0.5:: (Daniel Wirtz, 2011-07-07) Added this class.

Changes from former class CompwiseKernelCoreFun

Change in 0.4:

(Daniel Wirtz, 2011-05-03)

Removed the off property as proper RKHS functions dont have an offset
No more setter for Ma property - too slow even during offline phase

New in 0.3:: (Daniel Wirtz, 2011-04-21) Integrated this class to the property default value changed supervision system Property classes and levels. This class now inherits from KerMorObject and has an extended constructor registering any user-relevant properties using KerMorObject.registerProps.

Change in 0.3:: (Syed Ammar, 2011-04-16) Implemented Setter for the property off

This class is part of the framework

KerMor - Model Order Reduction using Kernels

Homepage http://www.morepas.org/software/index.html
Documentation http://www.morepas.org/software/kermor/index.html
License KerMor license conditions

Definition at line 18 of file KernelExpansion.m.

Public Member Functions
	KernelExpansion ()
	Default constructor. More...

function fx =	evaluate (matrix x, varargin)
	Evaluates the kernel expansion. More...

function phi =	getKernelVector (matrix x, varargin)
	Evaluates the kernel expansion. More...

function row =	getKernelMatrixColumn (integer idx,matrix x, varargin)
	Evaluates the kernel expansion. More...

function J =	getStateJacobian (x, varargin)
	Evaluates the jacobian matrix of this function at the given point. More...

function K =	getKernelMatrix ()
	Computes the kernel matrix for the currently set center data. More...

function v =	scalarProductWith (f)

function	normalize ()

function	setCentersFromATD (data.ApproxTrainData atd,rowvec< integer > idx)
	Sets the centers according to the indices `idx` of the data.ApproxTrainData. More...

function kernels.KernelExpansion kexp =	toTranslateBase ()
	Returns the same kernel expansion with the canonical translate base used. More...

function subexp =	getSubExpansion (arg)

function c =	getGlobalLipschitz (double t,colvec< double > mu)
	For some error estimators, a global Lipschitz constant estimation is performed. This function has to yield the global lipschitz constant for the spatial/state part of a CoreFunction \(f(x,t,mu)\). More...

function target =	project (V, W)

function copy =	clone (copy)
	The interface method with returns a copy of the current class instance. More...

function	clear ()
	Removes all centers and coefficients from the expansion and leaves the associated kernels untouched. More...

function sum =	plus (B)
	Adds two kernel expansions and will clone the frist argument's class as result type. More...

function diff =	minus (B)
	Adds two kernel expansions and will clone the frist argument's class as result type. More...

function neg =	uminus ()

function	exportToDirectory (dir)

function json =	toJSON (filename)

function res =	test_getStateJacobianInstance ()

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 ()

Public Member Functions inherited from general.AProjectable
function handle target =	project (matrix< double > V,matrix< double > W,handle target)
	Returns a NEW INSTANCE of the projected object that does not rely on data of the old one via references (everything must be copied to ensure separability of reduced(=projected) versions and full versions, unless. More...

Static Public Member Functions
static function res =	test_getStateJacobian ()

Public Attributes
kernels.BaseKernel	Kernel
	The Kernel to use for system variables. More...

rowvec	Ma_norms
	The norms of the coefficient matrix of the kernel expansion. More...

double	NativeNorm
	Returns the norm \(\|\|f\|\|^2_{\H^q}\) of \(f\) in the RKHS \(\H^q,\quad q\in\N\). More...

colvec< double >	ComponentNorms
	Returns the native space norms \(\|\|f_j\|\|^2_\H\) for each component function \(j \equiv\)`size(Ma,1)` More...

	MBnd
	Returns the M upper bound for this KernelExpansion, which is \(M := \max\limits_{j=1}^d\sum\limits_{i=1}^m \|c^d_i\|\). More...

logical	HasCustomBase
	Flag that indicates if a base other than the direct translate base is used for this kernel expansion. More...

	Size
	The size of the kernel expansion. More...

struct	Centers = struct("'xi',[]")
	The kernel centers used in the approximation. More...

matrix< double >	Ma = "[]"
	The coefficient data for each dimension. More...

matrix< double >	Base = 1
	This property lets the user store a custom base, for which the coefficients are set. The custom base tells which linear combinations of center elements belong to which base. 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...

Public Attributes inherited from general.AProjectable
	V
	The \(V\) matrix of the biorthogonal pair \(V,W\). More...

	W
	The \(W\) matrix of the biorthogonal pair \(V,W\). More...

Static Protected Member Functions
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)

Static Protected Member Functions inherited from general.AProjectable
static function obj =	loadobj (obj, from)

Protected Attributes
kernels.BaseKernel	fSK
	The inner (state) kernel object. More...

Additional Inherited Members
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)

Constructor & Destructor Documentation

kernels.KernelExpansion.KernelExpansion ( )

Default constructor.

Initializes default kernels.

Definition at line 249 of file KernelExpansion.m.

References fSK, KerMorObject.KerMorObject(), and DPCMObject.registerProps().

Referenced by testing.Speed.KernelExpCustomBaseEval().

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Member Function Documentation

function kernels.KernelExpansion.clear ( )

Removes all centers and coefficients from the expansion and leaves the associated kernels untouched.

Definition at line 487 of file KernelExpansion.m.

References Centers, and Ma.

Referenced by approx.algorithms.AAdaptiveBase.initExpansion().

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function copy = kernels.KernelExpansion.clone ( target )

virtual

The interface method with returns a copy of the current class instance.

Parameters

target If clone was called for a subclass of this class, target must contain the new instance of the subclass that is to be the cloned result.

Reimplemented from general.AProjectable.

Reimplemented in kernels.ParamTimeKernelExpansion.

Definition at line 475 of file KernelExpansion.m.

References Base, Centers, kernels.BaseKernel.clone(), fSK, and Ma.

Referenced by dscomponents.ParamTimeKernelCoreFun.clone(), approx.KernelEI.clone(), getSubExpansion(), project(), toTranslateBase(), and uminus().

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function fx = kernels.KernelExpansion.evaluate	(	matrix	x,
			varargin
	)

Evaluates the kernel expansion.

Parameters

x	The state space vector(s) to evaluate at
varargin	Dummy variable to also allow calls to this class with \(t_i,\mu_i\) parameters as in ParamTimeKernelExpansion.

Return values

fxi	The evaluation \(f(x) = \sumi c_i \K(x,x_i)\)

Definition at line 271 of file KernelExpansion.m.

References Ma, and t.

Referenced by dscomponents.ParamTimeKernelCoreFun.evaluate(), approx.KernelEI.evaluateCoreFun(), dscomponents.ParamTimeKernelCoreFun.evaluateMulti(), approx.algorithms.ABase.getError(), and test_getStateJacobianInstance().

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function kernels.KernelExpansion.exportToDirectory ( dir )

Definition at line 528 of file KernelExpansion.m.

References Centers, Utils.ensureDir(), k, Kernel, Ma, Util.saveRealMatrix(), and Util.saveRealVector().

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function c = kernels.KernelExpansion.getGlobalLipschitz	(	double	t,
		colvec< double >	mu
	)

virtual

For some error estimators, a global Lipschitz constant estimation is performed. This function has to yield the global lipschitz constant for the spatial/state part of a CoreFunction \(f(x,t,mu)\).

Todo:: : what about cases where no lipschitz constants are available? somehow need to tell the error estimator etc.

Parameters

t	The current time(s) \(t \in [0, T]\)
mu	The currently used parameter \(\vmu\). Set to \([]\) if not used.

Implements dscomponents.IGlobalLipschitz.

Reimplemented in kernels.ParamTimeKernelExpansion.

Definition at line 450 of file KernelExpansion.m.

References kernels.BaseKernel.getGlobalLipschitz(), Kernel, and Ma_norms.

Referenced by dscomponents.ParamTimeKernelCoreFun.getGlobalLipschitz().

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function K = kernels.KernelExpansion.getKernelMatrix ( )

Computes the kernel matrix for the currently set center data.

Convenience method, equal to call evaluateAtCenters with passing the center data as arguments. However, this method is (possibly, depends on kernel implementation) faster as it calls the kernels kernels.BaseKernel.evaluate -method with one argument.

See Also: evaluateAtCenters

Definition at line 341 of file KernelExpansion.m.

References Centers, kernels.BaseKernel.evaluate(), and fSK.

Referenced by general.regression.KernelLS.init(), and general.regression.BaseScalarSVR.init().

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function row = kernels.KernelExpansion.getKernelMatrixColumn	(	integer	idx,
		matrix	x,
			varargin
	)

Evaluates the kernel expansion.

Parameters

idx	The column index
x	The state space vector(s) to evaluate at Default: this.Centers.xi
varargin	Dummy variable to also allow calls to this class with \(t_i,\mu_i\) parameters as in ParamTimeKernelExpansion.

Return values

phi	The kernel vector \(\varphi(x) =\left(\K(x,x_i)\right)_{i}\).

Definition at line 301 of file KernelExpansion.m.

References Centers, kernels.BaseKernel.evaluate(), and fSK.

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function phi = kernels.KernelExpansion.getKernelVector	(	matrix	x,
			varargin
	)

Evaluates the kernel expansion.

Parameters

x	The state space vector(s) to evaluate at
varargin	Dummy variable to also allow calls to this class with \(t_i,\mu_i\) parameters as in ParamTimeKernelExpansion.

Return values

phi	The kernel vector \(\varphi(x) =\left(\K(x,x_i)\right)_{i}\).

Definition at line 286 of file KernelExpansion.m.

References Centers, kernels.BaseKernel.evaluate(), and fSK.

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function J = kernels.KernelExpansion.getStateJacobian	(	x,
		varargin
	)

Evaluates the jacobian matrix of this function at the given point.

As this is a component-wise kernel expansion, the jacobian is easily computed using the coefficient vectors and the state variable nablas.

See Also: kernels.BaseKernel

Parameters

x	The state vector at which the jacobian is required.
varargin	Dummy parameter to satisfy the interface calls for kernels.ParamTimeKernelExpansion classes, which also give a \(t\) and \(\mu\) parameter.

Definition at line 321 of file KernelExpansion.m.

References Centers, Ma, and t.

Referenced by dscomponents.ParamTimeKernelCoreFun.getStateJacobian(), and test_getStateJacobianInstance().

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function subexp = kernels.KernelExpansion.getSubExpansion ( arg )

Definition at line 429 of file KernelExpansion.m.

References Centers, and clone().

Referenced by demos.ResponseSurfaceApprox.iterationPlots().

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function this = kernels.KernelExpansion.loadobj	(	this,
		from
	)

staticprotected

As the constant properties are transient, they have to be re-computed upon loading.

Any subclasser MUST call this superclasses loadobj function explicitly!

Required fields of from:

Required fields of this:

Generated fields of this:

Definition at line 761 of file KernelExpansion.m.

References Base, Centers, fSK, KerMorObject.KerMorObject(), Kernel, and Ma.

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function diff = kernels.KernelExpansion.minus ( B )

Adds two kernel expansions and will clone the frist argument's class as result type.

Definition at line 510 of file KernelExpansion.m.

function kernels.KernelExpansion.normalize ( )

Definition at line 381 of file KernelExpansion.m.

References Ma, and NativeNorm.

function sum = kernels.KernelExpansion.plus ( B )

Adds two kernel expansions and will clone the frist argument's class as result type.

Definition at line 498 of file KernelExpansion.m.

Referenced by test_getStateJacobianInstance().

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function target = kernels.KernelExpansion.project	(	V,
		W
	)

Definition at line 455 of file KernelExpansion.m.

References all(), Centers, clone(), k, Ma, t, and general.AProjectable.V.

Referenced by dscomponents.ParamTimeKernelCoreFun.project(), and approx.KernelEI.project().

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function v = kernels.KernelExpansion.scalarProductWith ( f )

Definition at line 367 of file KernelExpansion.m.

References Centers, and Ma.

function kernels.KernelExpansion.setCentersFromATD	(	data.ApproxTrainData	atd,
		rowvec< integer >	idx
	)

Sets the centers according to the indices idx of the data.ApproxTrainData.

Parameters

atd	The approximation training data instance
idx	The indices of the entries in the approx train data to use

Required fields of atd:

Definition at line 386 of file KernelExpansion.m.

References Centers, and data.ApproxTrainData.xi.

static function res = kernels.KernelExpansion.test_getStateJacobian ( )

static

Definition at line 729 of file KernelExpansion.m.

References k.

function res = kernels.KernelExpansion.test_getStateJacobianInstance ( )

Definition at line 564 of file KernelExpansion.m.

References Centers, evaluate(), getStateJacobian(), k, Kernel, plus(), and X.

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function json = kernels.KernelExpansion.toJSON ( filename )

Definition at line 547 of file KernelExpansion.m.

References Base, Centers, HasCustomBase, Ma, and Util.matrixToJSON().

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function kernels.KernelExpansion kexp = kernels.KernelExpansion.toTranslateBase ( )

Returns the same kernel expansion with the canonical translate base used.

If no custom base is used, a simple copy will be returned.

Using a non-standard base may be the result of a Newton-base type approximation algorithm. However, evaluating the kernel expansion in a different base is less effective. If the coefficients of the direct translates are not too bad (i.e. large values with opposite signs), using the direct translate base might give the same results at an improved evaluation speed.

Return values

kexp	A cloned instance with the direct translate base.

Generated fields of kexp:

Definition at line 402 of file KernelExpansion.m.

References clone().

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function neg = kernels.KernelExpansion.uminus ( )

Definition at line 522 of file KernelExpansion.m.

References clone(), and Ma.

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Member Data Documentation

kernels.KernelExpansion.Base = 1

This property lets the user store a custom base, for which the coefficients are set. The custom base tells which linear combinations of center elements belong to which base.

In KerMor, this is typically set by the interpolating algorithms like approx.algorithm.VKOGA or general.interpolation.KernelInterpol

Property class optional:: A custom base for which the coefficients are set.

Default: 1

Note: This property has the MATLAB attribute SetObservable set to true.; Matlab documentation of property attributes.

Definition at line 210 of file KernelExpansion.m.

Referenced by clone(), loadobj(), and toJSON().

kernels.KernelExpansion.Centers = struct("'xi',[]")

The kernel centers used in the approximation.

This is the union of all center data used within the kernel function as a struct

with the following fields

xi The state variable centers
ti The times at which the state xi is taken
mui The parameter used to obtain the state xi

The only required field is xi, others can be set to [] if not used.

Property class data:: The expansion centers.

Default: struct(xi,[])

Note: This property has the MATLAB attribute SetObservable set to true.; Matlab documentation of property attributes.; This property has custom functionality when its value is changed.

Definition at line 170 of file KernelExpansion.m.

kernels.KernelExpansion.ComponentNorms

Returns the native space norms \(||f_j||^2_\H\) for each component function \(j \equiv\)size(Ma,1)

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 117 of file KernelExpansion.m.

kernels.KernelExpansion.fSK

protected

The inner (state) kernel object.

Default: kernels.GaussKernel;

Note: This property has non-standard access specifiers: SetAccess = Private, GetAccess = Protected; Matlab documentation of property attributes.

Definition at line 234 of file KernelExpansion.m.

Referenced by clone(), getKernelMatrix(), getKernelMatrixColumn(), kernels.ParamTimeKernelExpansion.getKernelVector(), getKernelVector(), kernels.ParamTimeKernelExpansion.getStateJacobian(), KernelExpansion(), and loadobj().

kernels.KernelExpansion.HasCustomBase

Flag that indicates if a base other than the direct translate base is used for this kernel expansion.

Default: false

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 143 of file KernelExpansion.m.

Referenced by toJSON().

kernels.KernelExpansion.Kernel

The Kernel to use for system variables.

Property class critical:: Correct choice of the system kernel greatly influences the function behaviour.

Default: kernels.GaussKernel

See Also: kernels

Note: This property has the MATLAB attribute Dependent set to true.; This property has the MATLAB attribute SetObservable set to true.; Matlab documentation of property attributes.

Definition at line 68 of file KernelExpansion.m.

Referenced by exportToDirectory(), kernels.ParamTimeKernelExpansion.getGlobalLipschitz(), getGlobalLipschitz(), loadobj(), and test_getStateJacobianInstance().

kernels.KernelExpansion.Ma = "[]"

The coefficient data for each dimension.

Property class data:: The expansion coefficients.

Default: []

Note: This property has the MATLAB attribute SetObservable set to true.; Matlab documentation of property attributes.

Definition at line 195 of file KernelExpansion.m.

Referenced by clear(), clone(), kernels.ParamTimeKernelExpansion.evaluate(), evaluate(), exportToDirectory(), kernels.ParamTimeKernelExpansion.getStateJacobian(), getStateJacobian(), loadobj(), normalize(), approx.KernelEI.project(), project(), scalarProductWith(), toJSON(), and uminus().

kernels.KernelExpansion.Ma_norms

The norms of the coefficient matrix of the kernel expansion.

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 89 of file KernelExpansion.m.

Referenced by kernels.ParamTimeKernelExpansion.getGlobalLipschitz(), and getGlobalLipschitz().

kernels.KernelExpansion.MBnd

Returns the M upper bound for this KernelExpansion, which is \(M := \max\limits_{j=1}^d\sum\limits_{i=1}^m |c^d_i|\).

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 131 of file KernelExpansion.m.

kernels.KernelExpansion.NativeNorm

Returns the norm \(||f||^2_{\H^q}\) of \(f\) in the RKHS \(\H^q,\quad q\in\N\).

Returns the native norm of the kernel expansion.

For \(q=1\) this equals the ComponentNorms output.

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 102 of file KernelExpansion.m.

Referenced by normalize().

kernels.KernelExpansion.Size

The size of the kernel expansion.

Note: This property has the MATLAB attribute Dependent set to true.; Matlab documentation of property attributes.; [readonly]

Definition at line 157 of file KernelExpansion.m.

The documentation for this class was generated from the following file:

core/+kernels/KernelExpansion.m

Detailed Description

Public Member Functions

Static Public Member Functions

Public Attributes

Static Protected Member Functions

Protected Attributes

Additional Inherited Members

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation