class QuadraticLeastSquares(*args)

Second order polynomial response surface by least squares.

Available constructors:

Parameters: dataIn : 2-d sequence of float Input data. function : Function Function to be approximated. dataOut : 2-d sequence of float Output data. If not specified, this sample is computed such as: .

Notes

Instead of replacing the model response for a local approximation around a given set of input parameters as in Taylor approximations, one may seek a global approximation of over its whole domain of definition. A common choice to this end is global polynomial approximation.

We consider here a global approximation of the model response using a quadratic function:

where denotes the number of terms, is a set of unknown coefficients and the family gathers the constant monomial , the monomials of degree one , the cross-terms as well as the monomials of degree two . Using the vector notation and , this rewrites:

A global approximation of the model response over its whole definition domain is sought. To this end, the coefficients may be computed using a least squares regression approach. In this context, an experimental design , i.e. a set of realizations of input parameters is required, as well as the corresponding model evaluations .

The following minimization problem has to be solved:

The solution is given by:

where:

Examples

>>> import openturns as ot
>>> formulas = ['x1 * sin(x2)', 'cos(x1 + x2)', '(x2 + 1) * exp(x1 - 2 * x2)']
>>> myFunc = ot.SymbolicFunction(['x1', 'x2'], formulas)
>>> data  = [[0.5,0.5], [-0.5,-0.5], [-0.5,0.5], [0.5,-0.5]]
>>> data += [[0.25,0.25], [-0.25,-0.25], [-0.25,0.25], [0.25,-0.25]]
>>> myLeastSquares = ot.QuadraticLeastSquares(data, myFunc)
>>> myLeastSquares.run()
>>> responseSurface = myLeastSquares.getResponseSurface()
>>> print(responseSurface([0.1,0.1]))
[0.00960661,0.976781,1.0138]

Attributes: thisown The membership flag

Methods

 getClassName() Accessor to the object’s name. getConstant() Get the constant vector of the approximation. getDataIn() Get the input data. getDataOut() Get the output data. getId() Accessor to the object’s id. getInputFunction() Get the function. getLinear() Get the linear matrix of the approximation. getName() Accessor to the object’s name. getQuadratic() Get the quadratic term of the approximation. getResponseSurface() Get an approximation of the function. getShadowedId() Accessor to the object’s shadowed id. getVisibility() Accessor to the object’s visibility state. hasName() Test if the object is named. hasVisibleName() Test if the object has a distinguishable name. run() Perform the least squares approximation. setDataOut(dataOut) Set the output data. setName(name) Accessor to the object’s name. setShadowedId(id) Accessor to the object’s shadowed id. setVisibility(visible) Accessor to the object’s visibility state.
__init__(*args)

Initialize self. See help(type(self)) for accurate signature.

getClassName()

Accessor to the object’s name.

Returns: class_name : str The object class name (object.__class__.__name__).
getConstant()

Get the constant vector of the approximation.

Returns: constantVector : Point Constant vector of the approximation, equal to .
getDataIn()

Get the input data.

Returns: dataIn : Sample Input data.
getDataOut()

Get the output data.

Returns: dataOut : Sample Output data. If not specified in the constructor, the sample is computed such as: .
getId()

Accessor to the object’s id.

Returns: id : int Internal unique identifier.
getInputFunction()

Get the function.

Returns: function : Function Function to be approximated.
getLinear()

Get the linear matrix of the approximation.

Returns: linearMatrix : Matrix Linear matrix of the approximation of the function .
getName()

Accessor to the object’s name.

Returns: name : str The name of the object.
getQuadratic()

Get the quadratic term of the approximation.

Returns: tensor : SymmetricTensor Quadratic term of the approximation of the function .
getResponseSurface()

Get an approximation of the function.

Returns: approximation : Function An approximation of the function by Quadratic Least Squares.
getShadowedId()

Accessor to the object’s shadowed id.

Returns: id : int Internal unique identifier.
getVisibility()

Accessor to the object’s visibility state.

Returns: visible : bool Visibility flag.
hasName()

Test if the object is named.

Returns: hasName : bool True if the name is not empty.
hasVisibleName()

Test if the object has a distinguishable name.

Returns: hasVisibleName : bool True if the name is not empty and not the default one.
run()

Perform the least squares approximation.

setDataOut(dataOut)

Set the output data.

Parameters: dataOut : 2-d sequence of float Output data.
setName(name)

Accessor to the object’s name.

Parameters: name : str The name of the object.
setShadowedId(id)

Accessor to the object’s shadowed id.

Parameters: id : int Internal unique identifier.
setVisibility(visible)

Accessor to the object’s visibility state.

Parameters: visible : bool Visibility flag.
thisown

The membership flag