Interval¶

class Interval(*args)¶

Numerical interval.

Available constructors:

Interval(dim)

Interval(lowerBound, upperBound, finiteLowerBound, finiteUpperBound)

Parameters:

dimint, $dim \geq 0$: Dimension of the interval. If only dim is mentioned, it leads to create the finite interval $[0, 1]^{dim}$ . By default, an empty interval is created.
lowerBound, upperBoundfloat or sequence of float of dimension dim: Define the interval $[lowerBound_0, upperBound_0]\times \dots \times [lowerBound_{dim-1}, upperBound_{dim-1}]$ . It is allowed to have $lowerBound_i \geq upperBound_i$ for some $i$ : it simply defines an empty interval. The lowerBound and the upperBound must be of the same type. If finiteLowerBound and finiteUpperBound are mentioned, they must be sequences.
finiteLowerBoundsequence of bool of dimension dim: Flags telling for each component of the lower bound whether it is finite or not.
finiteUpperBoundsequence of bool of dimension dim: Flags telling for each component of the upper bound whether it is finite or not.

Methods

`computeDistance`(*args)	Compute the Euclidean distance of a given point to the domain.
`contains`(*args)	Check if the given point is inside of the domain.
`getClassName`()	Accessor to the object's name.
`getDimension`()	Get the dimension of the domain.
`getFiniteLowerBound`()	Tell for each component of the lower bound whether it is finite or not.
`getFiniteUpperBound`()	Tell for each component of the upper bound whether it is finite or not.
`getLowerBound`()	Get the lower bound.
`getMarginal`(*args)	Marginal accessor.
`getName`()	Accessor to the object's name.
`getUpperBound`()	Get the upper bound.
`getVolume`()	Get the volume of the interval.
`hasName`()	Test if the object is named.
`intersect`(other)	Get the intersection with another interval.
`isEmpty`()	Check if the interval is empty.
`isNumericallyEmpty`()	Check if the interval is numerically empty.
`join`(other)	Get the smallest interval containing both the current interval and another one.
`numericallyContains`(point)	Check if the given point is inside of the discretization of the interval.
`setFiniteLowerBound`(finiteLowerBound)	Tell for each component of the lower bound whether it is finite or not.
`setFiniteUpperBound`(finiteUpperBound)	Tell for each component of the upper bound whether it is finite or not.
`setLowerBound`(lowerBound)	Set the lower bound.
`setName`(name)	Accessor to the object's name.
`setUpperBound`(upperBound)	Set the upper bound.

Notes

The meaning of a flag is: if flag $i$ is True, the corresponding component of the given bound is finite and its value is given by bound $i$ . If not, the corresponding component is infinite and its value is either $-\infty$ if bound $i < 0$ or $+\infty$ if bound $i \geq 0$ .

It is possible to add or subtract two intervals and multiply an interval by a scalar according to the following rules:

$I_1 + I_2 & = \{ \vect{x} + \vect{y} \, |\, \vect{x} \in I_1, \vect{y} \in I_2\} \\ I_1 - I_2 & = \{ \vect{x} - \vect{y} \, |\, \vect{x} \in I_1, \vect{y} \in I_2\} \\ \lambda I_1 & = \{ \lambda \vect{x} \, |\, \vect{x} \in I_1 \} \\$

Examples

>>> import openturns as ot
>>> # The finite interval [2,4] * [3,5]
>>> print(ot.Interval([2.0, 3.0], [4.0, 5.0]))
[2, 4]
[3, 5]
>>> # The right-unbounded interval [2, +inf[
>>> a = 2.0
>>> print(ot.Interval([a], [1], [True], [False]))
[2, (1) +inf[
>>> print(ot.Interval([1], [a], [False], [True]))
]-inf (1), 2]
>>> # Operations with intervals:
>>> interval1 = ot.Interval([2.0, 3.0], [5.0, 8.0])
>>> interval2 = ot.Interval([1.0, 4.0], [6.0, 13.0])
>>> # Addition
>>> print(interval1 + interval2)
[3, 11]
[7, 21]
>>> # Subtraction
>>> print(interval1 - interval2)
[-4, 4]
[-10, 4]
>>> # Multiplication
>>> print(interval1 * 3)
[6, 15]
[9, 24]

__init__(*args)¶

computeDistance(*args)¶

Compute the Euclidean distance of a given point to the domain.

Parameters:

point or samplesequence of float or 2-d sequence of float: Point or Sample with the same dimension as the current domain’s dimension.

Returns:

distancefloat or Sample: Euclidean distance of the point to the domain.

contains(*args)¶

Check if the given point is inside of the domain.

Parameters:

point or samplesequence of float or 2-d sequence of float: Point or Sample with the same dimension as the current domain’s dimension.

Returns:

isInsidebool or sequence of bool: Flag telling whether the given point is inside of the domain.

getClassName()¶

Accessor to the object’s name.

Returns:

class_namestr: The object class name (object.__class__.__name__).

getDimension()¶

Get the dimension of the domain.

Returns:

dimint: Dimension of the domain.

getFiniteLowerBound()¶

Tell for each component of the lower bound whether it is finite or not.

Returns:

flagsBoolCollection: If the $i^{th}$ element is False, the corresponding component of the lower bound is infinite. Otherwise, it is finite.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([2.0, 3.0], [4.0, 5.0], [True, False], [True, True])
>>> print(interval.getFiniteLowerBound())
[1,0]

getFiniteUpperBound()¶

Tell for each component of the upper bound whether it is finite or not.

Returns:

flagsBoolCollection: If the $i^{th}$ element is False, the corresponding component of the upper bound is infinite. Otherwise, it is finite.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([2.0, 3.0], [4.0, 5.0], [True, False], [True, True])
>>> print(interval.getFiniteUpperBound())
[1,1]

getLowerBound()¶

Get the lower bound.

Returns:

lowerBoundPoint: Value of the lower bound.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([2.0, 3.0], [4.0, 5.0], [True, False], [True, True])
>>> print(interval.getLowerBound())
[2,3]

getMarginal(*args)¶

Marginal accessor.

Parameters:

indexint or sequence of int: Index or indices of the selected components.

Returns:

intervalInterval: The marginal interval.

getName()¶

Accessor to the object’s name.

Returns:

namestr: The name of the object.

getUpperBound()¶

Get the upper bound.

Returns:

upperBoundPoint: Value of the upper bound.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([2.0, 3.0], [4.0, 5.0], [True, False], [True, True])
>>> print(interval.getUpperBound())
[4,5]

getVolume()¶

Get the volume of the interval.

Returns:

volumefloat: Volume contained within interval bounds.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([2.0, 3.0], [4.0, 5.0], [True, False], [True, True])
>>> print(interval.getVolume())
4.0

hasName()¶

Test if the object is named.

Returns:

hasNamebool: True if the name is not empty.

intersect(other)¶

Get the intersection with another interval.

Parameters:

otherIntervalInterval: Interval of the same dimension.

Returns:

intervalInterval: An interval corresponding to the intersection of the current interval with otherInterval.

Examples

>>> import openturns as ot
>>> interval1 = ot.Interval([2.0, 3.0], [5.0, 8.0])
>>> interval2 = ot.Interval([1.0, 4.0], [6.0, 13.0])
>>> print(interval1.intersect(interval2))
[2, 5]
[4, 8]

isEmpty()¶

Check if the interval is empty.

Returns:

isEmptybool: True if the interior of the interval is empty.

Examples

>>> import openturns as ot
>>> interval = ot.Interval([1.0, 2.0], [1.0, 2.0])
>>> interval.setFiniteLowerBound([True, False])
>>> print(interval.isEmpty())
False

isNumericallyEmpty()¶

Check if the interval is numerically empty.

Returns:

isEmptybool: Flag telling whether the interval is numerically empty, i.e. if its numerical volume is inferior or equal to $\epsilon$ (defined in the ResourceMap: $\epsilon$ = Domain-SmallVolume).

Examples

>>> import openturns as ot
>>> interval = ot.Interval([1.0, 2.0], [1.0, 2.0])
>>> print(interval.isNumericallyEmpty())
True

join(other)¶

Get the smallest interval containing both the current interval and another one.

Parameters:

otherIntervalInterval: Interval of the same dimension.

Returns:

intervalInterval: Smallest interval containing both the current interval and otherInterval.

Examples

>>> import openturns as ot
>>> interval1 = ot.Interval([2.0, 3.0], [5.0, 8.0])
>>> interval2 = ot.Interval([1.0, 4.0], [6.0, 13.0])
>>> print(interval1.join(interval2))
[1, 6]
[3, 13]

numericallyContains(point)¶

Check if the given point is inside of the discretization of the interval.

Parameters:

pointsequence of float: Point with the same dimension as the current domain’s dimension.

Returns:

isInsidebool: Flag telling whether the point is inside the interval bounds, not taking into account whether bounds are finite or not.

setFiniteLowerBound(finiteLowerBound)¶

Tell for each component of the lower bound whether it is finite or not.

Parameters:

flagssequence of bool: If the $i^{th}$ element is False, the corresponding component of the lower bound is infinite. Otherwise, it is finite.

Examples

>>> import openturns as ot
>>> interval = ot.Interval(2)
>>> interval.setFiniteLowerBound([True, False])
>>> print(interval)
[0, 1]
]-inf (0), 1]

setFiniteUpperBound(finiteUpperBound)¶

Tell for each component of the upper bound whether it is finite or not.

Parameters:

flagssequence of bool: If the $i^{th}$ element is False, the corresponding component of the upper bound is infinite. Otherwise, it is finite.

Examples

>>> import openturns as ot
>>> interval = ot.Interval(2)
>>> interval.setFiniteUpperBound([True, False])
>>> print(interval)
[0, 1]
[0, (1) +inf[

setLowerBound(lowerBound)¶

Set the lower bound.

Parameters:

lowerBoundsequence of float: Value of the lower bound.

Examples

>>> import openturns as ot
>>> interval = ot.Interval(2)
>>> interval.setLowerBound([-4, -5])
>>> print(interval)
[-4, 1]
[-5, 1]

setName(name)¶

Accessor to the object’s name.

Parameters:

namestr: The name of the object.

setUpperBound(upperBound)¶

Set the upper bound.

Parameters:

upperBoundsequence of float: Value of the upper bound.

Examples

>>> import openturns as ot
>>> interval = ot.Interval(2)
>>> interval.setUpperBound([4, 5])
>>> print(interval)
[0, 4]
[0, 5]

Examples using the class¶

Compute squared SRC indices confidence intervals

Fit a distribution by maximum likelihood

Model a singular multivariate distribution

Estimate a GEV on the Venice sea-levels data

Bandwidth sensitivity in kernel smoothing

Estimate a GPD on the Dow Jones Index data

Fitting a distribution with customized maximum likelihood

Estimate a GEV on the Port Pirie sea-levels data

Estimate a GPD on the daily rainfall data

Estimate a GEV on race times data

Estimate a GEV on the Fremantle sea-levels data

Truncate a distribution

Overview of univariate distribution management

Distribution manipulation

Create a customized distribution or copula

Draw minimum volume level sets

Export a field to VTK

Aggregate processes

Use the Box-Cox transformation

Create a custom covariance model

Draw a field

Create a process from random vectors and processes

Trend computation

Create a mesh

Create a linear model

Compute Sobol’ indices confidence intervals

Compute Sobol' indices confidence intervals

Conditional expectation of a polynomial chaos expansion

Polynomial chaos expansion cross-validation

Kriging : cantilever beam model

Gaussian Process Regression : cantilever beam model

Example of multi output Kriging on the fire satellite model

Kriging: choose a polynomial trend on the beam model

Kriging with an isotropic covariance function

Kriging: configure the optimization solver

Advanced Kriging

Kriging: metamodel with continuous and categorical variables

Validation of a Karhunen-Loeve decomposition

Metamodel of a field function

Use the FORM algorithm in case of several design points

Create a domain event

Cross Entropy Importance Sampling

Create an event based on a process

Estimate a process-based event probability

The HSIC sensitivity indices: the Ishigami model

The PlotDesign method

Plot Smolyak multi-indices

Various design of experiments

Plot the Smolyak quadrature

Use the Smolyak quadrature

Calibration of the Chaboche mechanical model

Sampling from an unnormalized probability density

Posterior sampling using a PythonDistribution

Customize your Metropolis-Hastings algorithm

Linear Regression with interval-censored observations

Bayesian calibration of hierarchical fission gas release models

Integrate a function with Gauss-Kronrod algorithm

Estimate a multivariate integral with IteratedQuadrature

Optimization with constraints

Optimization using NLopt

Mix/max search using optimization

Optimization using bonmin

Multi-objective optimization using Pagmo

Quick start guide to optimization

Optimization of the Rastrigin test function

Optimization using dlib

EfficientGlobalOptimization examples

OpenTURNS

An Open source initiative for the Treatment of Uncertainties, Risks'N Statistics

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Interval¶

Examples using the class¶