Function¶
- class Function(*args)¶
Function base class.
Notes
A function acts on points to produce points: .
A function enables to evaluate its gradient and its hessian when mathematically defined.
Examples
Create a Function from a list of analytical formulas and descriptions of the input vector and the output vector :
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x0', 'x1'], ... ['x0 + x1', 'x0 - x1']) >>> print(f([1, 2])) [3,-1]
Create a Function from strings:
>>> import openturns as ot >>> f = ot.SymbolicFunction('x', '2.0*sqrt(x)') >>> print(f(([16],[4]))) [ y0 ] 0 : [ 8 ] 1 : [ 4 ]
Create a Function from a Python function:
>>> def a_function(X): ... return [X[0] + X[1]] >>> f = ot.PythonFunction(2, 1, a_function) >>> print(f(((10, 5),(6, 7)))) [ y0 ] 0 : [ 15 ] 1 : [ 13 ]
See
PythonFunction
for further details.Create a Function from a Python class:
>>> class FUNC(OpenTURNSPythonFunction): ... def __init__(self): ... super(FUNC, self).__init__(2, 1) ... self.setInputDescription(['R', 'S']) ... self.setOutputDescription(['T']) ... def _exec(self, X): ... Y = [X[0] + X[1]] ... return Y >>> F = FUNC() >>> myFunc = Function(F) >>> print(myFunc((1.0, 2.0))) [3]
See
PythonFunction
for further details.Create a Function from another Function:
>>> f = ot.SymbolicFunction(ot.Description.BuildDefault(4, 'x'), ... ['x0', 'x0 + x1', 'x0 + x2 + x3'])
Then create another function by setting x1=4 and x3=10:
>>> g = ot.ParametricFunction(f, [3, 1], [10.0, 4.0], True) >>> print(g.getInputDescription()) [x0,x2] >>> print(g((1, 2))) [1,5,13]
Or by setting x0=6 and x2=5:
>>> g = ot.ParametricFunction(f, [3, 1], [6.0, 5.0], False) >>> print(g.getInputDescription()) [x3,x1] >>> print(g((1, 2))) [6,8,12]
Create a Function from another Function and by using a comparison operator:
>>> analytical = ot.SymbolicFunction(['x0','x1'], ['x0 + x1']) >>> indicator = ot.IndicatorFunction(ot.LevelSet(analytical, ot.Less(), 0.0)) >>> print(indicator([2, 3])) [0] >>> print(indicator([2, -3])) [1]
Create a Function from a collection of functions:
>>> functions = list() >>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'], ... ['x1^2 + x2', 'x1 + x2 + x3'])) >>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'], ... ['x1 + 2 * x2 + x3', 'x1 + x2 - x3'])) >>> myFunction = ot.AggregatedFunction(functions) >>> print(myFunction([1.0, 2.0, 3.0])) [3,6,8,0]
Create a Function which is the linear combination linComb of the functions defined in functionCollection with scalar weights defined in scalarCoefficientColl:
where and then the linear combination is:
>>> myFunction2 = ot.LinearCombinationFunction(functions, [2.0, 4.0]) >>> print(myFunction2([1.0, 2.0, 3.0])) [38,12]
Create a Function which is the linear combination vectLinComb of the scalar functions defined in scalarFunctionCollection with vectorial weights defined in vectorCoefficientColl:
If where and where
>>> functions=list() >>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'], ... ['x1 + 2 * x2 + x3'])) >>> functions.append(ot.SymbolicFunction(['x1', 'x2', 'x3'], ... ['x1^2 + x2'])) >>> myFunction2 = ot.DualLinearCombinationFunction(functions, [[2., 4.], [3., 1.]]) >>> print(myFunction2([1, 2, 3])) [25,35]
Create a Function from values of the inputs and the outputs:
>>> inputSample = [[1.0, 1.0], [2.0, 2.0]] >>> outputSample = [[4.0], [5.0]] >>> database = ot.DatabaseFunction(inputSample, outputSample) >>> x = [1.8]*database.getInputDimension() >>> print(database(x)) [5]
Create a Function which is the composition function :
>>> g = ot.SymbolicFunction(['x1', 'x2'], ... ['x1 + x2','3 * x1 * x2']) >>> f = ot.SymbolicFunction(['x1', 'x2'], ['2 * x1 - x2']) >>> composed = ot.ComposedFunction(f, g) >>> print(composed([3, 4])) [-22]
Methods
__call__
(*args)Call self as a function.
draw
(*args)Draw the output of function as a
Graph
.Accessor to the number of times the function has been called.
Accessor to the object's name.
Accessor to the description of the inputs and outputs.
Accessor to the evaluation function.
Accessor to the number of times the function has been called.
Accessor to the gradient function.
Accessor to the number of times the gradient of the function has been called.
Accessor to the hessian function.
Accessor to the number of times the hessian of the function has been called.
getId
()Accessor to the object's id.
Accessor to the underlying implementation.
Accessor to the description of the input vector.
Accessor to the dimension of the input vector.
getMarginal
(*args)Accessor to marginal.
getName
()Accessor to the object's name.
Accessor to the description of the output vector.
Accessor to the number of the outputs.
Accessor to the parameter values.
Accessor to the parameter description.
Accessor to the dimension of the parameter.
gradient
(inP)Return the Jacobian transposed matrix of the function at a point.
hessian
(inP)Return the hessian of the function at a point.
isLinear
()Accessor to the linearity of the function.
isLinearlyDependent
(index)Accessor to the linearity of the function with regard to a specific variable.
parameterGradient
(inP)Accessor to the gradient against the parameter.
setDescription
(description)Accessor to the description of the inputs and outputs.
setEvaluation
(evaluation)Accessor to the evaluation function.
setGradient
(gradient)Accessor to the gradient function.
setHessian
(hessian)Accessor to the hessian function.
setInputDescription
(inputDescription)Accessor to the description of the input vector.
setName
(name)Accessor to the object's name.
setOutputDescription
(inputDescription)Accessor to the description of the output vector.
setParameter
(parameter)Accessor to the parameter values.
setParameterDescription
(description)Accessor to the parameter description.
- __init__(*args)¶
- draw(*args)¶
Draw the output of function as a
Graph
.- Available usages:
draw(inputMarg, outputMarg, centralPoint, xiMin, xiMax, ptNb)
draw(firstInputMarg, secondInputMarg, outputMarg, centralPoint, xiMin_xjMin, xiMax_xjMax, ptNbs)
draw(xiMin, xiMax, ptNb)
draw(xiMin_xjMin, xiMax_xjMax, ptNbs)
- Parameters:
- outputMarg, inputMargint,
outputMarg is the index of the marginal to draw as a function of the marginal with index inputMarg.
- firstInputMarg, secondInputMargint,
In the 2D case, the marginal outputMarg is drawn as a function of the two marginals with indexes firstInputMarg and secondInputMarg.
- centralPointsequence of float
Central point with dimension equal to the input dimension of the function.
- xiMin, xiMaxfloat
Define the interval where the curve is plotted.
- xiMin_xjMin, xiMax_xjMaxsequence of float of dimension 2.
In the 2D case, define the intervals where the curves are plotted.
- ptNbint or list of ints of dimension 2
The number of points to draw the curves.
Notes
We note where and , with and .
In the first usage:
Draws graph of the given 1D outputMarg marginal as a function of the given 1D inputMarg marginal with respect to the variation of in the interval , when all the other components of are fixed to the corresponding components of the centralPoint . Then OpenTURNS draws the graph:
for any where is defined by the equation:
In the second usage:
Draws the iso-curves of the given outputMarg marginal as a function of the given 2D firstInputMarg and secondInputMarg marginals with respect to the variation of in the interval , when all the other components of are fixed to the corresponding components of the centralPoint . Then OpenTURNS draws the graph:
for any where is defined by the equation:
In the third usage:
The same as the first usage but only for function .
In the fourth usage:
The same as the second usage but only for function .
Examples
>>> import openturns as ot >>> from openturns.viewer import View >>> f = ot.SymbolicFunction('x', 'sin(2*pi_*x)*exp(-x^2/2)') >>> graph = f.draw(-1.2, 1.2, 100) >>> View(graph).show()
- getCallsNumber()¶
Accessor to the number of times the function has been called.
- Returns:
- calls_numberint
Integer that counts the number of times the function has been called since its creation.
- getClassName()¶
Accessor to the object’s name.
- Returns:
- class_namestr
The object class name (object.__class__.__name__).
- getDescription()¶
Accessor to the description of the inputs and outputs.
- Returns:
- description
Description
Description of the inputs and the outputs.
- description
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getDescription()) [x1,x2,y0]
- getEvaluation()¶
Accessor to the evaluation function.
- Returns:
- function
EvaluationImplementation
The evaluation function.
- function
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getEvaluation()) [x1,x2]->[2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6]
- getEvaluationCallsNumber()¶
Accessor to the number of times the function has been called.
- Returns:
- evaluation_calls_numberint
Integer that counts the number of times the function has been called since its creation.
- getGradient()¶
Accessor to the gradient function.
- Returns:
- gradient
GradientImplementation
The gradient function.
- gradient
- getGradientCallsNumber()¶
Accessor to the number of times the gradient of the function has been called.
- Returns:
- gradient_calls_numberint
Integer that counts the number of times the gradient of the Function has been called since its creation. Note that if the gradient is implemented by a finite difference method, the gradient calls number is equal to 0 and the different calls are counted in the evaluation calls number.
- getHessian()¶
Accessor to the hessian function.
- Returns:
- hessian
HessianImplementation
The hessian function.
- hessian
- getHessianCallsNumber()¶
Accessor to the number of times the hessian of the function has been called.
- Returns:
- hessian_calls_numberint
Integer that counts the number of times the hessian of the Function has been called since its creation. Note that if the hessian is implemented by a finite difference method, the hessian calls number is equal to 0 and the different calls are counted in the evaluation calls number.
- getId()¶
Accessor to the object’s id.
- Returns:
- idint
Internal unique identifier.
- getImplementation()¶
Accessor to the underlying implementation.
- Returns:
- implImplementation
A copy of the underlying implementation object.
- getInputDescription()¶
Accessor to the description of the input vector.
- Returns:
- description
Description
Description of the input vector.
- description
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getInputDescription()) [x1,x2]
- getInputDimension()¶
Accessor to the dimension of the input vector.
- Returns:
- inputDimint
Dimension of the input vector .
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getInputDimension()) 2
- getMarginal(*args)¶
Accessor to marginal.
- Parameters:
- indicesint or list of ints
Set of indices for which the marginal is extracted.
- Returns:
- marginal
Function
Function corresponding to either or , with and .
- marginal
- getName()¶
Accessor to the object’s name.
- Returns:
- namestr
The name of the object.
- getOutputDescription()¶
Accessor to the description of the output vector.
- Returns:
- description
Description
Description of the output vector.
- description
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getOutputDescription()) [y0]
- getOutputDimension()¶
Accessor to the number of the outputs.
- Returns:
- number_outputsint
Dimension of the output vector .
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getOutputDimension()) 1
- getParameterDescription()¶
Accessor to the parameter description.
- Returns:
- parameter
Description
The parameter description.
- parameter
- getParameterDimension()¶
Accessor to the dimension of the parameter.
- Returns:
- parameterDimensionint
Dimension of the parameter.
- gradient(inP)¶
Return the Jacobian transposed matrix of the function at a point.
- Parameters:
- pointsequence of float
Point where the Jacobian transposed matrix is calculated.
- Returns:
- gradient
Matrix
The Jacobian transposed matrix of the function at point.
- gradient
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2']) >>> print(f.gradient([3.14, 4])) [[ 13.5345 1 ] [ 4.00001 1 ]]
- hessian(inP)¶
Return the hessian of the function at a point.
- Parameters:
- pointsequence of float
Point where the hessian of the function is calculated.
- Returns:
- hessian
SymmetricTensor
Hessian of the function at point.
- hessian
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6','x1 + x2']) >>> print(f.hessian([3.14, 4])) sheet #0 [[ 20 -0.00637061 ] [ -0.00637061 0 ]] sheet #1 [[ 0 0 ] [ 0 0 ]]
- isLinear()¶
Accessor to the linearity of the function.
- Returns:
- linearbool
True if the function is linear, False otherwise.
- isLinearlyDependent(index)¶
Accessor to the linearity of the function with regard to a specific variable.
- Parameters:
- indexint
The index of the variable with regard to which linearity is evaluated.
- Returns:
- linearbool
True if the function is linearly dependent on the specified variable, False otherwise.
- parameterGradient(inP)¶
Accessor to the gradient against the parameter.
- Returns:
- gradient
Matrix
The gradient.
- gradient
- setDescription(description)¶
Accessor to the description of the inputs and outputs.
- Parameters:
- descriptionsequence of str
Description of the inputs and the outputs.
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> print(f.getDescription()) [x1,x2,y0] >>> f.setDescription(['a','b','y']) >>> print(f.getDescription()) [a,b,y]
- setEvaluation(evaluation)¶
Accessor to the evaluation function.
- Parameters:
- function
EvaluationImplementation
The evaluation function.
- function
- setGradient(gradient)¶
Accessor to the gradient function.
- Parameters:
- gradient_function
GradientImplementation
The gradient function.
- gradient_function
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> f.setGradient(ot.CenteredFiniteDifferenceGradient( ... ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceGradient-DefaultEpsilon'), ... f.getEvaluation()))
- setHessian(hessian)¶
Accessor to the hessian function.
- Parameters:
- hessian_function
HessianImplementation
The hessian function.
- hessian_function
Examples
>>> import openturns as ot >>> f = ot.SymbolicFunction(['x1', 'x2'], ... ['2 * x1^2 + x1 + 8 * x2 + 4 * cos(x1) * x2 + 6']) >>> f.setHessian(ot.CenteredFiniteDifferenceHessian( ... ot.ResourceMap.GetAsScalar('CenteredFiniteDifferenceHessian-DefaultEpsilon'), ... f.getEvaluation()))
- setInputDescription(inputDescription)¶
Accessor to the description of the input vector.
- Parameters:
- description
Description
Description of the input vector.
- description
- setName(name)¶
Accessor to the object’s name.
- Parameters:
- namestr
The name of the object.
- setOutputDescription(inputDescription)¶
Accessor to the description of the output vector.
- Parameters:
- description
Description
Description of the output vector.
- description
- setParameter(parameter)¶
Accessor to the parameter values.
- Parameters:
- parametersequence of float
The parameter values.
- setParameterDescription(description)¶
Accessor to the parameter description.
- Parameters:
- parameter
Description
The parameter description.
- parameter
Examples using the class¶
Estimate Wilks and empirical quantile
Compute squared SRC indices confidence intervals
Model a singular multivariate distribution
Estimate a GEV on the Port Pirie sea-levels data
Estimate a GEV on the Fremantle sea-levels data
Estimate a GEV on race times data
Kolmogorov-Smirnov : understand the p-value
Kolmogorov-Smirnov : get the statistics distribution
Estimate a non stationary covariance function
Create a conditional distribution
Create your own distribution given its quantile function
Generate random variates by inverting the CDF
Overview of univariate distribution management
Create a functional basis process
Create a custom covariance model
Use the Box-Cox transformation
Create a process from random vectors and processes
Create a linear least squares model
Create a general linear model metamodel
Over-fitting and model selection
Apply a transform or inverse transform on your polynomial chaos
Fit a distribution from an input sample
Polynomial chaos over database
Compute grouped indices for the Ishigami function
Create a polynomial chaos metamodel by integration on the cantilever beam
Advanced polynomial chaos construction
Create a polynomial chaos metamodel
Create a polynomial chaos for the Ishigami function: a quick start guide to polynomial chaos
Polynomial chaos expansion cross-validation
Polynomial chaos is sensitive to the degree
Compute Sobol’ indices confidence intervals
Kriging: propagate uncertainties
Kriging : multiple input dimensions
Kriging : cantilever beam model
Kriging: choose an arbitrary trend
Kriging the cantilever beam model using HMAT
Example of multi output Kriging on the fire satellite model
Kriging : generate trajectories from a metamodel
Kriging: choose a polynomial trend on the beam model
Kriging with an isotropic covariance function
Kriging: metamodel of the Branin-Hoo function
Sequentially adding new points to a kriging
Kriging :configure the optimization solver
Kriging: choose a polynomial trend
Kriging: metamodel with continuous and categorical variables
Viscous free fall: metamodel of a field function
Estimate moments from Taylor expansions
Use the post-analytical importance sampling algorithm
Specify a simulation algorithm
Estimate a flooding probability
Exploitation of simulation algorithm results
Use the FORM algorithm in case of several design points
Non parametric Adaptive Importance Sampling (NAIS)
Use the FORM - SORM algorithms
Test the design point with the Strong Maximum Test
Time variant system reliability problem
Create unions or intersections of events
An illustrated example of a FORM probability estimate
Using the FORM - SORM algorithms on a nonlinear function
Estimate Sobol indices on a field to point function
Parallel coordinates graph as sensitivity tool
Estimate Sobol’ indices for a function with multivariate output
Sobol’ sensitivity indices from chaos
The HSIC sensitivity indices: the Ishigami model
Example of sensitivity analyses on the wing weight model
Create a linear combination of functions
Increase the output dimension of a function
Increase the input dimension of a function
Defining Python and symbolic functions: a quick start introduction to functions
Create a multivariate basis of functions from scalar multivariable functions
Define a connection function with a field output
Link to a computer code with coupling tools
Generate flooding model observations
Calibrate a parametric model: a quick-start guide to calibration
Generate observations of the Chaboche mechanical model
Calibration without observed inputs
Calibration of the logistic model
Calibration of the deflection of a tube
Calibration of the flooding model
Calibration of the Chaboche mechanical model
Gibbs sampling of the posterior distribution
Sampling from an unnormalized probability density
Posterior sampling using a PythonDistribution
Bayesian calibration of a computer code
Bayesian calibration of the flooding model
Customize your Metropolis-Hastings algorithm
Linear Regression with interval-censored observations
Compute leave-one-out error of a polynomial chaos expansion
Compute confidence intervals of a regression model from data
Compute confidence intervals of a univariate noisy function
Mix/max search and sensitivity from design
Mix/max search using optimization
Multi-objective optimization using Pagmo
Quick start guide to optimization
Optimization of the Rastrigin test function
EfficientGlobalOptimization examples
Plot the log-likelihood contours of a distribution