/usr/include/openturns/swig/LinearLeastSquares

%feature("docstring") OT::LinearLeastSquares
"First order polynomial response surface by least squares.

Available constructors:
    LinearLeastSquares(*dataIn, function*)

    LinearLeastSquares(*dataIn, dataOut*)

Parameters
----------
dataIn : 2-d sequence of float
    Input data.
function : :class:`~openturns.NumericalMathFunction`
    Function :math:`h` to be approximated.
dataOut : 2-d sequence of float
    Output data. If not specified, this sample is computed such as:
    :math:`dataOut = h(dataIn)`.

Notes
-----
Instead of replacing the model response :math:`h(\\\\vect{x})` for a *local*
approximation around a given set :math:`\\\\vect{x}_0` of input parameters as in
Taylor approximations, one may seek a *global* approximation of
:math:`h(\\\\vect{x})` 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 linear
function:

.. math::

    \\\\vect{y} \\\\, \\\\approx \\\\, \\\\widehat{h}(\\\\vect{x}) \\\\,
                      = \\\\, \\\\sum_{j=0}^{n_X} \\\\; a_j \\\\; \\\\psi_j(\\\\vect{x})

where :math:`(a_j  \\\\, , \\\\, j=0, \\\\cdots,n_X)` is a set of unknown coefficients
and the family :math:`(\\\\psi_j,j=0,\\\\cdots, n_X)` gathers the constant monomial
:math:`1` and the monomials of degree one :math:`x_i`. Using the vector
notation :math:`\\\\vect{a} \\\\, = \\\\, (a_{0} , \\\\cdots , a_{n_X} )^{\\\\textsf{T}}` and
:math:`\\\\vect{\\\\psi}(\\\\vect{x}) \\\\, = \\\\, (\\\\psi_0(\\\\vect{x}), \\\\cdots, \\\\psi_{n_X}(\\\\vect{x}) )^{\\\\textsf{T}}`,
this rewrites:

.. math::

    \\\\vect{y} \\\\, \\\\approx \\\\, \\\\widehat{h}(\\\\vect{x}) \\\\,
                      = \\\\, \\\\vect{a}^{\\\\textsf{T}} \\\\; \\\\vect{\\\\psi}(\\\\vect{x})

A *global* approximation of the model response over its whole definition domain
is sought. To this end, the coefficients :math:`a_j` may be computed using a
least squares regression approach. In this context, an experimental design
:math:`\\\\vect{\\\\cX} =(x^{(1)},\\\\cdots,x^{(N)})`, i.e. a set of realizations of
input parameters is required, as well as the corresponding model evaluations
:math:`\\\\vect{\\\\cY} =(y^{(1)},\\\\cdots,y^{(N)})`.

The following minimization problem has to be solved:

.. math::

    \\\\mbox{Find} \\\\quad \\\\widehat{\\\\vect{a}} \\\\quad \\\\mbox{that minimizes}
      \\\\quad \\\\cJ(\\\\vect{a}) \\\\, = \\\\, \\\\sum_{i=1}^N \\\\;
                                \\\\left(
                                y^{(i)} \\\\; - \\\\;
                                \\\\vect{a}^{\\\\textsf{T}} \\\\vect{\\\\psi}(\\\\vect{x}^{(i)})
                                \\\\right)^2

The solution is given by:

.. math::

    \\\\widehat{\\\\vect{a}} \\\\, = \\\\, \\\\left(
                               \\\\vect{\\\\vect{\\\\Psi}}^{\\\\textsf{T}} \\\\vect{\\\\vect{\\\\Psi}}
                               \\\\right)^{-1} \\\\;
                               \\\\vect{\\\\vect{\\\\Psi}}^{\\\\textsf{T}}  \\\\; \\\\vect{\\\\cY}

where:

.. math::

    \\\\vect{\\\\vect{\\\\Psi}} \\\\, = \\\\, (\\\\psi_{j}(\\\\vect{x}^{(i)}) \\\\; , \\\\; i=1,\\\\cdots,N \\\\; , \\\\; j = 0,\\\\cdots,n_X)

See also
--------
QuadraticLeastSquares, LinearTaylor, QuadraticTaylor

Examples
--------
>>> import openturns as ot
>>> formulas = ['cos(x1 + x2)', '(x2 + 1) * exp(x1 - 2 * x2)']
>>> myFunc = ot.NumericalMathFunction(['x1', 'x2'], ['y1', 'y2'], 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.LinearLeastSquares(data, myFunc)
>>> myLeastSquares.run()
>>> responseSurface = myLeastSquares.getResponseSurface()
>>> print(responseSurface([0.1,0.1]))
[0.854471,1.06031]"

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%feature("docstring") OT::LinearLeastSquares::getDataIn
"Get the input data.

Returns
-------
dataIn : :class:`~openturns.NumericalSample`
    Input data."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::getConstant
"Get the constant vector of the approximation.

Returns
-------
constantVector : :class:`~openturns.NumericalPoint`
    Constant vector of the approximation, equal to :math:`a_0`."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::getDataOut
"Get the output data.

Returns
-------
dataOut : :class:`~openturns.NumericalSample`
    Output data. If not specified in the constructor, the sample is computed
    such as: :math:`dataOut = h(dataIn)`."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::setDataOut
"Set the output data.

Parameters
----------
dataOut : 2-d sequence of float
    Output data."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::getInputFunction
"Get the function.

Returns
-------
function : :class:`~openturns.NumericalMathFunction`
    Function :math:`h` to be approximated."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::getLinear
"Get the linear matrix of the approximation.

Returns
-------
linearMatrix : :class:`~openturns.Matrix`
    Linear matrix of the approximation of the function :math:`h`."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::getResponseSurface
"Get an approximation of the function.

Returns
-------
approximation : :class:`~openturns.NumericalMathFunction`
    An approximation of the function :math:`h` by Linear Least Squares."

// ---------------------------------------------------------------------

%feature("docstring") OT::LinearLeastSquares::run
"Perform the least squares approximation."
libopenturns-dev 1.7-3 / usr / include / openturns / swig / LinearLeastSquares_doc.i
