MeanStandardDeviationTradeoffMeasure distribution¶

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../../_images/otrobopt-MeanStandardDeviationTradeoffMeasure-1.png

class otrobopt.MeanStandardDeviationTradeoffMeasure(*args)¶

Mean/variance tradeoff measure function.

It is built from a parametric function $f$ and a distribution $\cD$ of density function $p$ .

In case of a continuous distribution:

$\mu_k = \int_{Supp(\cD)} f_k(x, \theta) p(\theta) d \theta Var_k = \int_{Supp(\cD)} f_k(x, \theta)^2 p(\theta) d \theta - \mu^2 M_{f_k, \cD, \alpha_k}(x) = (1-\alpha_k) \mu_k + \alpha_k \sqrt{Var_k}$

In case of a discrete distribution:

$\mu_k = \sum_{\theta_i \in Supp(\cD)} f_k(x, \theta_i) p_i Var_k = \sum_{\theta_i \in Supp(\cD)} f_k(x, \theta_i)^2 p_i - {\mu_k}^2 M_{f_k, \cD, \alpha_k}(x) = (1-\alpha_k) \mu_k + \alpha_k \sqrt{Var_k}$

Parameters:

functionopenturns.Function: Parametric function
distributionopenturns.Distribution: Distribution of parameters
alphasequence of float, $\alpha_k \in [0,1]$: Tradeoff coefficient

Examples

>>> import openturns as ot
>>> import otrobopt
>>> thetaDist = ot.Normal(2.0, 0.1)
>>> f_base = ot.SymbolicFunction(['x', 'theta'], ['x*theta'])
>>> f = ot.ParametricFunction(f_base, [1], [1.0])
>>> measure = otrobopt.MeanStandardDeviationTradeoffMeasure(f, thetaDist, [0.4])
>>> x = [1.0]
>>> print(measure(x))
[1.24]

Methods

`draw`(*args)	Draw the output of function as a `Graph`.
`getAlpha`()	Tradeoff coefficient accessor.
`getCallsNumber`()	Accessor to the number of times the function has been called.
`getCheckOutput`()	Accessor to the output verification flag.
`getClassName`()	Accessor to the object's name.
`getDescription`()	Accessor to the description of the inputs and outputs.
`getDistribution`()	Distribution accessor.
`getFunction`()	Function accessor.
`getInputDescription`()	Accessor to the description of the inputs.
`getInputDimension`()	Accessor to the number of the inputs.
`getIntegrationAlgorithm`()	Integration algorithm accessor.
`getMarginal`(*args)	Accessor to marginal.
`getName`()	Accessor to the object's name.
`getOutputDescription`()	Accessor to the description of the outputs.
`getOutputDimension`()	Accessor to the number of the outputs.
`getPDFThreshold`()	PDF threshold accessor.
`getParameter`()	Accessor to the parameter values.
`getParameterDescription`()	Accessor to the parameter description.
`getParameterDimension`()	Accessor to the dimension of the parameter.
`hasName`()	Test if the object is named.
`isActualImplementation`()	Accessor to the validity flag.
`isLinear`()	Accessor to the linearity of the evaluation.
`isLinearlyDependent`(index)	Accessor to the linearity of the evaluation with regard to a specific variable.
`parameterGradient`(inP)	Gradient against the parameters.
`setAlpha`(alpha)	Tradeoff coefficient accessor.
`setCheckOutput`(checkOutput)	Accessor to the output verification flag.
`setDescription`(description)	Accessor to the description of the inputs and outputs.
`setDistribution`(distribution)	Distribution accessor.
`setFunction`(function)	Function accessor.
`setInputDescription`(inputDescription)	Accessor to the description of the inputs.
`setIntegrationAlgorithm`(algorithm)	Integration algorithm accessor.
`setName`(name)	Accessor to the object's name.
`setOutputDescription`(outputDescription)	Accessor to the description of the outputs.
`setPDFThreshold`(threshold)	PDF threshold accessor.
`setParameter`(parameters)	Accessor to the parameter values.
`setParameterDescription`(description)	Accessor to the parameter description.
`setStopCallback`(callBack[, state])	Set up a stop callback.

__init__(*args)¶

draw(*args)¶

Draw the output of function as a Graph.

Available usages:

draw(inputMarg, outputMarg, CP, xiMin, xiMax, ptNb, scale)

draw(firstInputMarg, secondInputMarg, outputMarg, CP, xiMin_xjMin, xiMax_xjMax, ptNbs, scale, isFilled)

draw(xiMin, xiMax, ptNb, scale)

draw(xiMin_xjMin, xiMax_xjMax, ptNbs, scale)

Parameters:

outputMarg, inputMargint, $outputMarg, inputMarg \geq 0$

outputMarg is the index of the marginal to draw as a function of the marginal with index inputMarg.

firstInputMarg, secondInputMargint, $firstInputMarg, secondInputMarg \geq 0$

In the 2D case, the marginal outputMarg is drawn as a function of the two marginals with indexes firstInputMarg and secondInputMarg.

CPsequence of float

Central point.

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 $ptNb > 0$

The number of points to draw the curves.

ptNbslist of int of dimension 2 $ptNbs_k > 0, k=1,2$

The number of points to draw the contour in the 2D case.

scalebool

scale indicates whether the logarithmic scale is used either for one or both axes:

ot.GraphImplementation.NONE or 0: no log scale is used,
ot.GraphImplementation.LOGX or 1: log scale is used only for horizontal data,
ot.GraphImplementation.LOGY or 2: log scale is used only for vertical data,
ot.GraphImplementation.LOGXY or 3: log scale is used for both data.

isFilledbool

isFilled indicates whether the contour graph is filled or not

Notes

We note $f: \Rset^n \rightarrow \Rset^p$ where $\vect{x} = (x_1, \dots, x_n)$ and $f(\vect{x}) = (f_1(\vect{x}), \dots,f_p(\vect{x}))$ , with $n\geq 1$ and $p\geq 1$ .

In the first usage:

Draws graph of the given 1D outputMarg marginal $f_k: \Rset^n \rightarrow \Rset$ as a function of the given 1D inputMarg marginal with respect to the variation of $x_i$ in the interval $[x_i^{min}, x_i^{max}]$ , when all the other components of $\vect{x}$ are fixed to the corresponding ones of the central point CP. Then it draws the graph: $t\in [x_i^{min}, x_i^{max}] \mapsto f_k(CP_1, \dots, CP_{i-1}, t, CP_{i+1} \dots, CP_n)$ .

In the second usage:

Draws the iso-curves of the given outputMarg marginal $f_k$ as a function of the given 2D firstInputMarg and secondInputMarg marginals with respect to the variation of $(x_i, x_j)$ in the interval $[x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}]$ , when all the other components of $\vect{x}$ are fixed to the corresponding ones of the central point CP. Then it draws the graph: $(t,u) \in [x_i^{min}, x_i^{max}] \times [x_j^{min}, x_j^{max}] \mapsto f_k(CP_1, \dots, CP_{i-1}, t, CP_{i+1}, \dots, CP_{j-1}, u, CP_{j+1} \dots, CP_n)$ .

In the third usage:

The same as the first usage but only for function $f: \Rset \rightarrow \Rset$ .

In the fourth usage:

The same as the second usage but only for function $f: \Rset^2 \rightarrow \Rset$ .

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()

getAlpha()¶

Tradeoff coefficient accessor.

Returns:

alphasequence of float in $(0,1)$: Tradeoff coefficient

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.

getCheckOutput()¶

Accessor to the output verification flag.

Returns:

check_outputbool: Whether to check return values for nan or inf.

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:

descriptionDescription: 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]

getDistribution()¶

Distribution accessor.

Returns:

distributionopenturns.Distribution: Parameters distribution.

getFunction()¶

Function accessor.

Returns:

functionopenturns.Function: Parametric function.

getInputDescription()¶

Accessor to the description of the inputs.

Returns:

descriptionDescription: Description of the inputs.

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 number of the inputs.

Returns:

number_inputsint: Number of inputs.

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

getIntegrationAlgorithm()¶

Integration algorithm accessor.

Returns:

algorithmopenturns.IntegrationAlgorithm: Algorithm used to compute the exact value of the measure.

getMarginal(*args)¶

Accessor to marginal.

Parameters:

indicesint or list of ints: Set of indices for which the marginal is extracted.

Returns:

marginalFunction: Function corresponding to either $f_i$ or $(f_i)_{i \in indices}$ , with $f:\Rset^n \rightarrow \Rset^p$ and $f=(f_0 , \dots, f_{p-1})$ .

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getOutputDescription()¶

Accessor to the description of the outputs.

Returns:

descriptionDescription: Description of 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.getOutputDescription())
[y0]

getOutputDimension()¶

Accessor to the number of the outputs.

Returns:

number_outputsint: Number of outputs.

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

getPDFThreshold()¶

PDF threshold accessor.

Returns:

thresholdfloat: PDF threshold under which the contribution is not computed.

getParameter()¶

Accessor to the parameter values.

Returns:

parameterPoint: The parameter values.

getParameterDescription()¶

Accessor to the parameter description.

Returns:

parameterDescription: The parameter description.

getParameterDimension()¶

Accessor to the dimension of the parameter.

Returns:

parameter_dimensionint: Dimension of the parameter.

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

isActualImplementation()¶

Accessor to the validity flag.

Returns:

is_implbool: Whether the implementation is valid.

isLinear()¶

Accessor to the linearity of the evaluation.

Returns:

linearbool: True if the evaluation is linear, False otherwise.

isLinearlyDependent(index)¶

Accessor to the linearity of the evaluation 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 evaluation is linearly dependent on the specified variable, False otherwise.

parameterGradient(inP)¶

Gradient against the parameters.

Parameters:

xsequence of float: Input point

Returns:

parameter_gradientMatrix: The parameters gradient computed at x.

setAlpha(alpha)¶

Tradeoff coefficient accessor.

Parameters:

alphasequence of float in $(0,1)$: Tradeoff coefficient

setCheckOutput(checkOutput)¶

Accessor to the output verification flag.

Parameters:

check_outputbool: Whether to check return values for nan or inf.

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]

setDistribution(distribution)¶

Distribution accessor.

Parameters:

distributionopenturns.Distribution: Parameters distribution.

setFunction(function)¶

Function accessor.

Parameters:

functionopenturns.Function: Parametric function.

setInputDescription(inputDescription)¶

Accessor to the description of the inputs.

Returns:

descriptionDescription: Description of the inputs.

setIntegrationAlgorithm(algorithm)¶

Integration algorithm accessor.

Parameters:

algorithmopenturns.IntegrationAlgorithm: Algorithm used to compute the exact value of the measure.

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setOutputDescription(outputDescription)¶

Accessor to the description of the outputs.

Returns:

descriptionDescription: Description of the outputs.

setPDFThreshold(threshold)¶

PDF threshold accessor.

Parameters:

thresholdfloat: PDF threshold under which the contribution is not computed.

setParameter(parameters)¶

Accessor to the parameter values.

Parameters:

parametersequence of float: The parameter values.

setParameterDescription(description)¶

Accessor to the parameter description.

Parameters:

parameterDescription: The parameter description.

setStopCallback(callBack, state=None)¶

Set up a stop callback.

Can be used to programmatically stop an evaluation.

Parameters:

callbackcallable: Returns a bool deciding whether to stop or continue.

MeanStandardDeviationTradeoffMeasure distribution¶

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