gemseo / problems / uncertainty / ishigami

ishigami_space module¶

The uncertain space used in the Ishigami use case.

class gemseo.problems.uncertainty.ishigami.ishigami_space.IshigamiSpace(uniform_distribution_name=UniformDistribution.SCIPY)[source]¶

Bases: ParameterSpace

The uncertain space used in the Ishigami use case.

\(X_1,X_2,X_3\) are independent random variables uniformly distributed between \(-\pi\) and \(\pi\).

This uncertain space uses the class SPUniformDistribution.

See [IH90].

Parameters:

uniform_distribution_name (UniformDistribution) –

The name of the class implementing the uniform distribution.

By default it is set to “SPUniformDistribution”.

class DesignVariable(size=1, var_type=_DesignVariableType.FLOAT, l_b=None, u_b=None, value=None)¶

Bases: NamedTuple

A design variable.

Create new instance of DesignVariable(size, var_type, l_b, u_b, value)

Parameters:

size (int | None) –

By default it is set to 1.
var_type (NDArray[_DesignVariableType] | _DesignVariableType | None) –

By default it is set to “float”.
l_b (ndarray | None)
u_b (ndarray | None)
value (ndarray | None)

count(value, /)¶

Return number of occurrences of value.

index(value, start=0, stop=9223372036854775807, /)¶

Return first index of value.

Raises ValueError if the value is not present.

l_b: ndarray | None¶: Alias for field number 2

size: int | None¶: Alias for field number 0

u_b: ndarray | None¶: Alias for field number 3

value: ndarray | None¶: Alias for field number 4

var_type: NDArray[_DesignVariableType] | _DesignVariableType | None¶: Alias for field number 1

DesignVariableType¶: alias of _DesignVariableType

class UniformDistribution(value)[source]¶

Bases: StrEnum

The name of the class implementing the uniform distribution.

capitalize()¶

Return a capitalized version of the string.

More specifically, make the first character have upper case and the rest lower case.

casefold()¶

Return a version of the string suitable for caseless comparisons.

center(width, fillchar=' ', /)¶

Return a centered string of length width.

Padding is done using the specified fill character (default is a space).

count(sub[, start[, end]]) → int¶

Return the number of non-overlapping occurrences of substring sub in string S[start:end]. Optional arguments start and end are interpreted as in slice notation.

encode(encoding='utf-8', errors='strict')¶

Encode the string using the codec registered for encoding.

encoding: The encoding in which to encode the string.
errors: The error handling scheme to use for encoding errors. The default is ‘strict’ meaning that encoding errors raise a UnicodeEncodeError. Other possible values are ‘ignore’, ‘replace’ and ‘xmlcharrefreplace’ as well as any other name registered with codecs.register_error that can handle UnicodeEncodeErrors.

endswith(suffix[, start[, end]]) → bool¶

Return True if S ends with the specified suffix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. suffix can also be a tuple of strings to try.

expandtabs(tabsize=8)¶

Return a copy where all tab characters are expanded using spaces.

If tabsize is not given, a tab size of 8 characters is assumed.

find(sub[, start[, end]]) → int¶

Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.

Return -1 on failure.

format(*args, **kwargs) → str¶

Return a formatted version of S, using substitutions from args and kwargs. The substitutions are identified by braces (‘{’ and ‘}’).

format_map(mapping) → str¶

Return a formatted version of S, using substitutions from mapping. The substitutions are identified by braces (‘{’ and ‘}’).

index(sub[, start[, end]]) → int¶

Return the lowest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.

Raises ValueError when the substring is not found.

isalnum()¶

Return True if the string is an alpha-numeric string, False otherwise.

A string is alpha-numeric if all characters in the string are alpha-numeric and there is at least one character in the string.

isalpha()¶

Return True if the string is an alphabetic string, False otherwise.

A string is alphabetic if all characters in the string are alphabetic and there is at least one character in the string.

isascii()¶

Return True if all characters in the string are ASCII, False otherwise.

ASCII characters have code points in the range U+0000-U+007F. Empty string is ASCII too.

isdecimal()¶

Return True if the string is a decimal string, False otherwise.

A string is a decimal string if all characters in the string are decimal and there is at least one character in the string.

isdigit()¶

Return True if the string is a digit string, False otherwise.

A string is a digit string if all characters in the string are digits and there is at least one character in the string.

isidentifier()¶

Return True if the string is a valid Python identifier, False otherwise.

Call keyword.iskeyword(s) to test whether string s is a reserved identifier, such as “def” or “class”.

islower()¶

Return True if the string is a lowercase string, False otherwise.

A string is lowercase if all cased characters in the string are lowercase and there is at least one cased character in the string.

isnumeric()¶

Return True if the string is a numeric string, False otherwise.

A string is numeric if all characters in the string are numeric and there is at least one character in the string.

isprintable()¶

Return True if the string is printable, False otherwise.

A string is printable if all of its characters are considered printable in repr() or if it is empty.

isspace()¶

Return True if the string is a whitespace string, False otherwise.

A string is whitespace if all characters in the string are whitespace and there is at least one character in the string.

istitle()¶

Return True if the string is a title-cased string, False otherwise.

In a title-cased string, upper- and title-case characters may only follow uncased characters and lowercase characters only cased ones.

isupper()¶

Return True if the string is an uppercase string, False otherwise.

A string is uppercase if all cased characters in the string are uppercase and there is at least one cased character in the string.

join(iterable, /)¶

Concatenate any number of strings.

The string whose method is called is inserted in between each given string. The result is returned as a new string.

Example: ‘.’.join([‘ab’, ‘pq’, ‘rs’]) -> ‘ab.pq.rs’

ljust(width, fillchar=' ', /)¶

Return a left-justified string of length width.

Padding is done using the specified fill character (default is a space).

lower()¶

Return a copy of the string converted to lowercase.

lstrip(chars=None, /)¶

Return a copy of the string with leading whitespace removed.

If chars is given and not None, remove characters in chars instead.

static maketrans()¶

Return a translation table usable for str.translate().

If there is only one argument, it must be a dictionary mapping Unicode ordinals (integers) or characters to Unicode ordinals, strings or None. Character keys will be then converted to ordinals. If there are two arguments, they must be strings of equal length, and in the resulting dictionary, each character in x will be mapped to the character at the same position in y. If there is a third argument, it must be a string, whose characters will be mapped to None in the result.

partition(sep, /)¶

Partition the string into three parts using the given separator.

This will search for the separator in the string. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.

If the separator is not found, returns a 3-tuple containing the original string and two empty strings.

removeprefix(prefix, /)¶

Return a str with the given prefix string removed if present.

If the string starts with the prefix string, return string[len(prefix):]. Otherwise, return a copy of the original string.

removesuffix(suffix, /)¶

Return a str with the given suffix string removed if present.

If the string ends with the suffix string and that suffix is not empty, return string[:-len(suffix)]. Otherwise, return a copy of the original string.

replace(old, new, count=-1, /)¶

Return a copy with all occurrences of substring old replaced by new.

count
Maximum number of occurrences to replace. -1 (the default value) means replace all occurrences.

If the optional argument count is given, only the first count occurrences are replaced.

rfind(sub[, start[, end]]) → int¶

Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.

Return -1 on failure.

rindex(sub[, start[, end]]) → int¶

Return the highest index in S where substring sub is found, such that sub is contained within S[start:end]. Optional arguments start and end are interpreted as in slice notation.

Raises ValueError when the substring is not found.

rjust(width, fillchar=' ', /)¶

Return a right-justified string of length width.

Padding is done using the specified fill character (default is a space).

rpartition(sep, /)¶

Partition the string into three parts using the given separator.

This will search for the separator in the string, starting at the end. If the separator is found, returns a 3-tuple containing the part before the separator, the separator itself, and the part after it.

If the separator is not found, returns a 3-tuple containing two empty strings and the original string.

rsplit(sep=None, maxsplit=-1)¶

Return a list of the words in the string, using sep as the delimiter string.

sep
The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.

maxsplit
Maximum number of splits to do. -1 (the default value) means no limit.

Splits are done starting at the end of the string and working to the front.

rstrip(chars=None, /)¶

Return a copy of the string with trailing whitespace removed.

If chars is given and not None, remove characters in chars instead.

split(sep=None, maxsplit=-1)¶

Return a list of the words in the string, using sep as the delimiter string.

sep: The delimiter according which to split the string. None (the default value) means split according to any whitespace, and discard empty strings from the result.
maxsplit: Maximum number of splits to do. -1 (the default value) means no limit.

splitlines(keepends=False)¶

Return a list of the lines in the string, breaking at line boundaries.

Line breaks are not included in the resulting list unless keepends is given and true.

startswith(prefix[, start[, end]]) → bool¶

Return True if S starts with the specified prefix, False otherwise. With optional start, test S beginning at that position. With optional end, stop comparing S at that position. prefix can also be a tuple of strings to try.

strip(chars=None, /)¶

Return a copy of the string with leading and trailing whitespace removed.

If chars is given and not None, remove characters in chars instead.

swapcase()¶

Convert uppercase characters to lowercase and lowercase characters to uppercase.

title()¶

Return a version of the string where each word is titlecased.

More specifically, words start with uppercased characters and all remaining cased characters have lower case.

translate(table, /)¶

Replace each character in the string using the given translation table.

table
Translation table, which must be a mapping of Unicode ordinals to Unicode ordinals, strings, or None.

The table must implement lookup/indexing via __getitem__, for instance a dictionary or list. If this operation raises LookupError, the character is left untouched. Characters mapped to None are deleted.

upper()¶

Return a copy of the string converted to uppercase.

zfill(width, /)¶

Pad a numeric string with zeros on the left, to fill a field of the given width.

The string is never truncated.

OPENTURNS = 'OTUniformDistribution'¶

SCIPY = 'SPUniformDistribution'¶

add_random_variable(name, distribution, size=1, interfaced_distribution='', interfaced_distribution_parameters=(), **parameters)¶

Add a random variable from a probability distribution.

Parameters:

name (str) – The name of the random variable.
distribution (str) – The name of a class implementing a probability distribution, e.g. "OTUniformDistribution" or "SPUniformDistribution", or an interface to a library of probability distributions, e.g. "OTDistribution" or "SPDistribution".
size (int) –
The dimension of the random variable. The parameters of the distribution are shared by all the components of the random variable.

By default it is set to 1.
interfaced_distribution (str) –
The name of the distribution in the library of probability distributions when distribution is the name of a class implementing an interface to this library.

By default it is set to “”.
interfaced_distribution_parameters (tuple[Any] | StrKeyMapping) –
The parameters of the distribution in the library of probability distributions when distribution is the name of a class implementing an interface to this library; if empty, use the default ones.

By default it is set to ().
**parameters (Any) – The parameters of the distribution; otherwise, use the default ones.

Return type:

None

Warning

The probability distributions must have the same DISTRIBUTION_FAMILY_ID. For instance, one cannot mix a random variable distributed as an OTUniformDistribution with identifier "OT" and a random variable distributed as a SPNormalDistribution with identifier "SP".

add_random_vector(name, distribution, size=0, interfaced_distribution='', interfaced_distribution_parameters=(), **parameters)¶

Add a d-length random vector from a probability distribution.

Warning

The probability distributions must have the same DISTRIBUTION_FAMILY_ID. For instance, one cannot mix a random vector using a OTUniformDistribution with identifier "OT" and a random vector using a SPNormalDistribution with identifier "SP".

Parameters:

name (str) – The name of the random vector.
distribution (str) – The name of a class implementing a probability distribution, e.g. "OTUniformDistribution" or "SPUniformDistribution", or an interface to a library of probability distributions, e.g. "OTDistribution" or "SPDistribution".
size (int) –
The length d of the random vector. If 0, deduce it from the parameters.

By default it is set to 0.
interfaced_distribution (str) –
The name of the distribution in the library of probability distributions when distribution is the name of a class implementing an interface to this library.

By default it is set to “”.
interfaced_distribution_parameters (tuple[list[Any]] | Mapping[str, list[Any]]) –
The parameters of the distribution in the library of probability distributions when distribution is the name of a class implementing an interface to this library. The values of the data structure (mapping or tuple) must be set either as [p_1,...,p_d] (one value per component of the random vector) or as [p] (one value for all the components) If empty, use the default ones.

By default it is set to ().
**parameters (list[Any]) – The parameters of the distribution, either as [p_1,...,p_d] (one value per component of the random vector) or as [p] (one value for all the components); otherwise, use the default ones.

Raises:

ValueError – When mixing probability distributions from different families, e.g. an OTDistribution and a SPDistribution or when the lengths of the distribution parameter collections are not consistent.

Return type:

None

add_variable(name, size=1, var_type=_DesignVariableType.FLOAT, l_b=None, u_b=None, value=None)¶

Add a variable to the design space.

Parameters:

name (str) – The name of the variable.
size (int) –
The size of the variable.

By default it is set to 1.
var_type (DesignVariableType | Sequence[DesignVariableType]) –
Either the type of the variable or the types of its components.

By default it is set to “float”.
l_b (Number | Iterable[Number] | None) – The lower bound of the variable. If None, use \(-\infty\).
u_b (Number | Iterable[Number] | None) – The upper bound of the variable. If None, use \(+\infty\).
value (Number | Iterable[Number] | None) – The default value of the variable. If None, do not use a default value.

Raises:

ValueError – Either if the variable already exists or if the size is not a positive integer.

Return type:

None

array_to_dict(x_array)¶

Convert a design array into a dictionary indexed by the variables names.

Parameters:: x_array (ndarray) – A design value expressed as a NumPy array.
Returns:: The design value expressed as a dictionary of NumPy arrays.
Return type:: dict[str, ndarray]

build_joint_distribution(copula=None)¶

Build the joint probability distribution.

Parameters:: copula (Any | None) – A copula distribution defining the dependency structure between random variables; if None, consider an independent copula.
Return type:: None

check()¶

Check the state of the design space.

Raises:: ValueError – If the design space is empty.
Return type:: None

check_membership(x_vect, variable_names=None)¶

Check whether the variables satisfy the design space requirements.

Parameters:

x_vect (Mapping[str, ndarray] | ndarray) – The values of the variables.
variable_names (Sequence[str] | None) – The names of the variables. If None, use the names of the variables of the design space.

Raises:

ValueError – Either if the dimension of the values vector is wrong, if the values are not specified as an array or a dictionary, if the values are outside the bounds of the variables or if the component of an integer variable is not an integer.

Return type:

None

clear() → None. Remove all items from D.¶

compute_samples(n_samples=1, as_dict=False)¶

Sample the random variables and return the realizations.

Parameters:

n_samples (int) –
A number of samples.

By default it is set to 1.
as_dict (bool) –
The type of the returned object. If True, return a dictionary. Otherwise, return an array.

By default it is set to False.

Returns:

The realizations of the random variables, either stored in an array or in a dictionary whose values are the names of the random variables and the values are the evaluations.

Return type:

dict[str, ndarray] | ndarray

dict_to_array(design_values, variable_names=None)¶

Convert a mapping of design values into a NumPy array.

Parameters:

design_values (Mapping[str, ndarray]) – The mapping of design values.
variable_names (Iterable[str] | None) – The design variables to be considered. If None, consider all the design variables.

Returns:

The design values as a NumPy array.

Return type:

ndarray

Notes

The data type of the returned NumPy array is the most general data type of the values of the mapping design_values corresponding to the keys iterable from variables_names.

evaluate_cdf(value, inverse=False)¶

Evaluate the cumulative density function (or its inverse) of each marginal.

Parameters:

value (dict[str, ndarray]) – The values of the uncertain variables passed as a dictionary whose keys are the names of the variables.
inverse (bool) –
The type of function to evaluate. If True, compute the cumulative density function. Otherwise, compute the inverse cumulative density function.

By default it is set to False.

Returns:

A dictionary where the keys are the names of the random variables and the values are the evaluations.

Return type:

dict[str, ndarray]

extend(other)¶

Extend the design space with another design space.

Parameters:: other (DesignSpace) – The design space to be appended to the current one.
Return type:: None

extract_deterministic_space()¶

Define a new DesignSpace from the deterministic variables only.

Returns:: A DesignSpace defined by the deterministic variables only.
Return type:: DesignSpace

extract_uncertain_space(as_design_space=False)¶

Define a new DesignSpace from the uncertain variables only.

Parameters:

as_design_space (bool) –

If False, return a ParameterSpace containing the original uncertain variables as is; otherwise, return a DesignSpace where the original uncertain variables are made deterministic. In that case, the bounds of a deterministic variable correspond to the limits of the support of the original probability distribution and the current value correspond to its mean.

By default it is set to False.

Returns:

A ParameterSpace defined by the uncertain variables only.

Return type:

DesignSpace | ParameterSpace

filter(keep_variables, copy=False)¶

Filter the design space to keep a subset of variables.

Parameters:

keep_variables (str | Iterable[str]) – The names of the variables to be kept.
copy (bool) –
If True, then a copy of the design space is filtered, otherwise the design space itself is filtered.

By default it is set to False.

Returns:

Either the filtered original design space or a copy.

Raises:

ValueError – If the variable is not in the design space.

Return type:

DesignSpace

filter_dimensions(name, dimensions)¶

Filter the design space to keep a subset of dimensions for a variable.

Parameters:

name (str) – The name of the variable.
dimensions (Iterable[int]) – The dimensions of the variable to be kept, between \(0\) and \(d-1\) where \(d\) is the number of dimensions of the variable.

Returns:

The filtered design space.

Raises:

ValueError – If a dimension does not exist.

Return type:

DesignSpace

classmethod from_csv(file_path, header=())¶

Create a design space from a CSV file.

Parameters:

file_path (str | Path) – The path to the CSV file.
header (Iterable[str]) –
The names of the fields saved in the file. If empty, read them in the file.

By default it is set to ().

Returns:

The design space defined in the file.

Raises:

ValueError – If the file does not contain the minimal variables in its header.

Return type:

DesignSpace

classmethod from_file(file_path, hdf_node_path='', **options)¶

Create a design space from a file.

Parameters:

file_path (str | Path) – The path to the file. If the extension starts with “hdf”, the file will be considered as an HDF file.
hdf_node_path (str) –
The path of the HDF node from which the database should be imported. If empty, the root node is considered.

By default it is set to “”.
**options (Any) – The keyword reading options.

Returns:

The design space defined in the file.

Return type:

DesignSpace

classmethod from_hdf(file_path, hdf_node_path='')¶

Create a design space from an HDF file.

Parameters:

file_path (str | Path) – The path to the HDF file.
hdf_node_path (str) –
The path of the HDF node from which the database should be imported. If empty, the root node is considered.

By default it is set to “”.

Returns:

The design space defined in the file.

Return type:

DesignSpace

get(k[, d]) → D[k] if k in D, else d. d defaults to None.¶

get_active_bounds(x_vec=None, tol=1e-08)¶

Determine which bound constraints of a design value are active.

Parameters:

x_vec (ndarray | None) – The design value at which to check the bounds. If None, use the current design value.
tol (float) –
The tolerance of comparison of a scalar with a bound.

By default it is set to 1e-08.

Returns:

Whether the components of the lower and upper bound constraints are active, the first returned value representing the lower bounds and the second one the upper bounds, e.g.

(
    {
        "x": array(are_x_lower_bounds_active),
        "y": array(are_y_lower_bounds_active),
    },
    {
        "x": array(are_x_upper_bounds_active),
        "y": array(are_y_upper_bounds_active),
    },
)

where:

are_x_lower_bounds_active = [True, False]
are_x_upper_bounds_active = [False, False]
are_y_lower_bounds_active = [False]
are_y_upper_bounds_active = [True]

Return type:

tuple[dict[str, ndarray], dict[str, ndarray]]

get_current_value(variable_names=None, complex_to_real=False, as_dict=False, normalize=False)¶

Return the current design value.

If the names of the variables are empty then an empty data is returned.

Parameters:

variable_names (Sequence[str] | None) – The names of the design variables. If None, use all the design variables.
complex_to_real (bool) –
Whether to cast complex numbers to real ones.

By default it is set to False.
as_dict (bool) –
Whether to return the current design value as a dictionary of the form {variable_name: variable_value}.

By default it is set to False.
normalize (bool) –
Whether to normalize the design values in \([0,1]\) with the bounds of the variables.

By default it is set to False.

Returns:

The current design value.

Raises:

ValueError – If names in variable_names are not in the design space.

Return type:

ndarray | dict[str, ndarray]

Warning

For performance purposes, get_current_value() does not return a copy of the current value. This means that modifying the returned object will make the DesignSpace inconsistent (the current design value stored as a NumPy array and the current design value stored as a dictionary of NumPy arrays will be different). To modify the returned object without impacting the DesignSpace, you shall copy this object and modify the copy.

Examples using IshigamiSpace¶

Comparing sensitivity indices

Correlation analysis

Morris analysis

Sobol’ analysis

Sobol' analysis