alfi::dist Namespace Reference

Enumerations
enum class	Type {   GENERAL ,   UNIFORM ,   QUADRATIC ,   CUBIC ,   CHEBYSHEV ,   CHEBYSHEV_STRETCHED ,   CHEBYSHEV_ELLIPSE ,   CHEBYSHEV_ELLIPSE_STRETCHED ,   CIRCLE_PROJECTION ,   ELLIPSE_PROJECTION ,   LOGISTIC ,   LOGISTIC_STRETCHED ,   ERF ,   ERF_STRETCHED }

Functions
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	uniform (SizeT n, Number a, Number b)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	quadratic (SizeT n, Number a, Number b)
	Generates a distribution of n points on the segment [a, b] using two quadratic polynomials.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	cubic (SizeT n, Number a, Number b)
	Generates a distribution of n points on the segment [a, b] using cubic polynomial.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	chebyshev (SizeT n, Number a, Number b)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	chebyshev_stretched (SizeT n, Number a, Number b)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	chebyshev_ellipse (SizeT n, Number a, Number b, Number ratio)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	chebyshev_ellipse_stretched (SizeT n, Number a, Number b, Number ratio)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	circle_proj (SizeT n, Number a, Number b)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	ellipse_proj (SizeT n, Number a, Number b, Number ratio)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	logistic (SizeT n, Number a, Number b, Number steepness)
	Generates a distribution of n points on the interval (a, b) using the logistic function.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	logistic_stretched (SizeT n, Number a, Number b, Number steepness)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	erf (SizeT n, Number a, Number b, Number steepness)
	Generates a distribution of n points on the interval (a, b) using the error function.
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	erf_stretched (SizeT n, Number a, Number b, Number steepness)
template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>
Container< Number >	of_type (Type type, SizeT n, Number a, Number b, Number parameter=NAN)

Enumeration Type Documentation

Type

enum class alfi::dist::Type

strong

Enumerator
GENERAL
UNIFORM
QUADRATIC
CUBIC
CHEBYSHEV
CHEBYSHEV_STRETCHED
CHEBYSHEV_ELLIPSE
CHEBYSHEV_ELLIPSE_STRETCHED
CIRCLE_PROJECTION
ELLIPSE_PROJECTION
LOGISTIC
LOGISTIC_STRETCHED
ERF
ERF_STRETCHED

Function Documentation

uniform()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::uniform	(	SizeT	n,
		Number	a,
		Number	b )

quadratic()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::quadratic	(	SizeT	n,
		Number	a,
		Number	b )

Generates a distribution of n points on the segment [a, b] using two quadratic polynomials.

The following transform function \(f\):

\[ f(x) = \begin{cases} (x+1)^2 - 1, & x \leq 0 \\ -(x-1)^2 + 1, & x > 0 \end{cases} \]

is applied to n points uniformly distributed on the segment [-1, 1], mapping them onto the same segment.
The resulting points are then linearly mapped to the target segment [a, b].

Parameters

n	number of points
a	left boundary of the segment
b	right boundary of the segment

Returns

a container with n points distributed on the segment [a, b] according to the transform function

cubic()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::cubic	(	SizeT	n,
		Number	a,
		Number	b )

Generates a distribution of n points on the segment [a, b] using cubic polynomial.

The following transform function \(f\):

\[ f(x) = -0.5x^3 + 1.5x \]

is applied to n points uniformly distributed on the segment [-1, 1], mapping them onto the same segment.
The resulting points are then linearly mapped to the target segment [a, b].

Parameters

n	number of points
a	left boundary of the segment
b	right boundary of the segment

Returns

a container with n points distributed on the segment [a, b] according to the transform function

chebyshev()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::chebyshev	(	SizeT	n,
		Number	a,
		Number	b )

chebyshev_stretched()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::chebyshev_stretched	(	SizeT	n,
		Number	a,
		Number	b )

chebyshev_ellipse()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::chebyshev_ellipse	(	SizeT	n,
		Number	a,
		Number	b,
		Number	ratio )

chebyshev_ellipse_stretched()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::chebyshev_ellipse_stretched	(	SizeT	n,
		Number	a,
		Number	b,
		Number	ratio )

circle_proj()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::circle_proj	(	SizeT	n,
		Number	a,
		Number	b )

ellipse_proj()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::ellipse_proj	(	SizeT	n,
		Number	a,
		Number	b,
		Number	ratio )

logistic()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::logistic	(	SizeT	n,
		Number	a,
		Number	b,
		Number	steepness )

Generates a distribution of n points on the interval (a, b) using the logistic function.

The following transform function \(f\):

\[ f(x) = \frac{1}{1 + e^{-steepness \cdot x}} \]

is applied to n points uniformly distributed on the interval (-1, 1), mapping them onto the (0, 1) interval.
The resulting points are then linearly mapped to the target interval (a, b).

The slope of the transform function \(f\) at x = 0 is determined by steepness and is given by:

\[ \left. \dv{f}{x} \right\vert_{x=0} = \frac14 \cdot steepness \]

Note

Extreme points don't lie on the interval (a, b) boundaries.

Parameters

n	number of points
a	left boundary of the interval
b	right boundary of the interval
steepness	determines the slope of the transform function at x = 0

Returns

a container with n points distributed on the interval (a, b) according to the transform function

logistic_stretched()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::logistic_stretched	(	SizeT	n,
		Number	a,
		Number	b,
		Number	steepness )

erf()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::erf	(	SizeT	n,
		Number	a,
		Number	b,
		Number	steepness )

Generates a distribution of n points on the interval (a, b) using the error function.

The following transform function \(f\):

\[ f(x) = \operatorname{erf}(steepness \cdot x) \]

is applied to n points uniformly distributed on the interval (-1, 1), mapping them onto the same interval.
The resulting points are then linearly mapped to the target interval (a, b).

The slope of the transform function \(f\) at x = 0 is determined by steepness and is given by:

\[ \left. \dv{f}{x} \right\vert_{x=0} = \frac2\pi \cdot steepness \]

Note

Extreme points don't lie on the interval (a, b) boundaries.

Parameters

n	number of points
a	left boundary of the interval
b	right boundary of the interval
steepness	determines the slope of the transform function at x = 0

Returns

a container with n points distributed on the interval (a, b) according to the transform function

erf_stretched()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::erf_stretched	(	SizeT	n,
		Number	a,
		Number	b,
		Number	steepness )

of_type()

template<typename Number = DefaultNumber, template< typename, typename... > class Container = DefaultContainer>

Container< Number > alfi::dist::of_type	(	Type	type,
		SizeT	n,
		Number	a,
		Number	b,
		Number	parameter = NAN )