/usr/include/openturns/swig/ExponentialModel_doc.i is in libopenturns-dev 1.7-3.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 | %feature("docstring") OT::ExponentialModel
"Multivariate stationary exponential covariance function.
Available constructors:
ExponentialModel(*spatialDim, amplitude, scale*)
ExponentialModel(*spatialDim, amplitude, scale, spatialCorrelation*)
ExponentialModel(*spatialDim, scale, spatialCovariance*)
ExponentialModel(*scale, amplitude*)
ExponentialModel(*scale, amplitude, spatialCorrelation*)
ExponentialModel(*scale, amplitude, spatialCovariance*)
Parameters
----------
spatialDim : int
Dimension of the domain :math:`\\\\cD`.
amplitude : sequence of float
Vector :math:`\\\\vect{a}` of dimension :math:`d`.
scale : sequence of float
Vector :math:`\\\\vect{\\\\lambda}` of dimension :math:`spatialDim`.
spatialCorrelation : :class:`~openturns.CorrelationMatrix`
Correlation matrix :math:`\\\\mat{R} \\\\in \\\\mathcal{S}^{+}_{d}(\\\\Rset)`, with values in [-1,1]
spatialCovariance : :class:`~openturns.CovarianceMatrix`
Covariance matrix :math:`C^{stat} \\\\in \\\\mathcal{M}_{+}^{*}(\\\\Rset^d)`.
Notes
-----
The Exponential model defines the stationary covariance function
:math:`C^{stat}(\\\\vect{\\\\tau}) = C(\\\\vect{s}, \\\\vect{s}+\\\\vect{\\\\tau}) \\\\forall (\\\\vect{s},\\\\vect{\\\\tau}) \\\\in \\\\cD`
such that :
.. math::
\\\\forall \\\\vect{\\\\tau} \\\\in \\\\cD,\\\\quad
C^{stat}( \\\\vect{\\\\tau} )=\\\\left[\\\\mat{A}\\\\mat{\\\\Delta}( \\\\vect{\\\\tau} ) \\\\right] \\\\,\\\\mat{R}\\\\, \\\\left[ \\\\mat{\\\\Delta}( \\\\vect{\\\\tau} )\\\\mat{A}\\\\right]
where :math:`\\\\mat{R} \\\\in \\\\mathcal{M}_{d \\\\times d}([-1, 1])` is a correlation
matrix, :math:`\\\\mat{\\\\Delta}( \\\\vect{\\\\tau} ) \\\\in \\\\mathcal{M}_{d \\\\times d}(\\\\Rset)`
is defined by:
.. math::
\\\\mat{\\\\Delta}( \\\\vect{\\\\tau} )= \\\\mbox{Diag}(e^{-\\\\lambda_1|\\\\tau|/2}, \\\\dots, e^{-\\\\lambda_d|\\\\tau|/2})
and :math:`\\\\mat{A}\\\\in \\\\mathcal{M}_{d \\\\times d}(\\\\Rset)` is defined by:
.. math::
\\\\mat{A}= \\\\mbox{Diag}(a_1, \\\\dots, a_d)
with :math:`\\\\lambda_i>0` and :math:`a_i>0` for any :math:`i`.
We call :math:`\\\\vect{a}` the amplitude vector and :math:`\\\\vect{\\\\lambda}` the
scale vector.
We define the spatial covariance matrix :math:`\\\\mat{C}^{spat}` by:
.. math::
\\\\forall \\\\vect{t} \\\\in \\\\cD,\\\\quad
\\\\mat{C}^{spat} = \\\\Expect{X_{\\\\vect{t}}\\\\Tr{X}_{\\\\vect{t}}}
= \\\\mat{A}\\\\,\\\\mat{R}\\\\, \\\\mat{A}
- In the first usage, we fix the dimension spatial dimension :math:`n`, the
scale :math:`\\\\vect{a}` and the amplitude :math:`\\\\vect{\\\\lambda}`. By default,
:math:`\\\\mat{R}=\\\\mat{Id}(d)`. The dimension :math:`d` is deduced.
- In the second usage, we fix the dimension spatial dimension :math:`n`, the
scale :math:`\\\\vect{a}`, the amplitude :math:`\\\\vect{\\\\lambda}` and the spatial
correlation matrix :math:`\\\\mat{R}`. The dimension :math:`d` is deduced.
- In the third usage, the dimension spatial dimension :math:`n`, the scale
:math:`\\\\vect{a}`, the amplitude :math:`\\\\vect{\\\\lambda}` and the spatial
covariance matrix :math:`C^{stat}`. The dimension :math:`d` is deduced.
Examples
--------
Create an *ExponentialModel* from the amplitude and scale:
>>> import openturns as ot
>>> amplitude = [1., 2.]
>>> scale = [4., 5.]
>>> spatialDimension = 2
>>> myCovarianceModel = ot.ExponentialModel(spatialDimension, amplitude, scale)
Create an *ExponentialModel* from the amplitude, scale and correlation matrix:
>>> amplitude = [1., 2.]
>>> scale = [4., 5.]
>>> spatialCorrelation = ot.CorrelationMatrix(2)
>>> spatialCorrelation[0,1] = 0.8
>>> spatialDimension = 2
>>> myCovarianceModel = ot.ExponentialModel(spatialDimension, amplitude, scale)
Create an *ExponentialModel* from the amplitude, scale and covariance matrix:
>>> amplitude = [1., 2.]
>>> scale = [4., 5.]
>>> spatialCovariance = ot.CovarianceMatrix(2)
>>> spatialCovariance[0,0] = 4.0
>>> spatialCovariance[1,1] = 5.0
>>> spatialCovariance[0,1] = 1.2
>>> spatialDimension = 2
>>> myCovarianceModel = ot.ExponentialModel(spatialDimension, scale, spatialCovariance)"
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