A constant for the real line.

The positive real numbers.

The negative real numbers.

abstract type AbstractInterval

Abstract supertype for all univariate intervals. It is not specified whether they are open or closed.

Affine(α, β)

Mapping $ℝ → ℝ$ using $x ↦ α⋅x + β$.

$α > 0$ is enforced, see Negation.

ArrayTransformation(transformation, dims)
ArrayTransformation(transformation, dims...)

Apply transformation to a vector, returning an array of the given dimensions.

domain, image, and isincreasing return the corresponding values for the underlying transformation.

abstract type ContinuousTransformation <: Function

Continuous bijection $D ⊂ ℝ^n→ I ⊂ ℝ^n$ or $D ⊂ ℝ → I ⊂ ℝ$.

CorrelationCholeskyFactor(n)

Cholesky factor of a correlation matrix of size n.

Exp()

Mapping $ℝ → ℝ⁺$ using $x ↦ \exp(x)$.

abstract type GroupedTransformation <: ContinuousTransformation

Abstract type for grouped transformations.

A grouped transformation takes a vector, and transforms contiguous blocks of elements to some output type, determined by the specific transformation type.

All subtypes support

• length: return the length of the vector that can be used as an argument

• callable object for the transformation

• logjac, and inverse,

• domain and image, which may have specific interpretation for their result types depending on the concrete subtype.

InvRealCircle()

Mapping $(-1,1) → ℝ$ using $x ↦ x/√(1-x^2)$.

Log()

Mapping $ℝ → ℝ⁺$ using $x ↦ \exp(x)$.

Logistic()

Mapping $ℝ → (0,1)$ using $x ↦ 1/(1+\exp(-x))$.

Logit()

Mapping $(0,1) → ℝ$ using $x ↦ \log(x/(1-x))$.

Negation()

Mapping $ℝ → ℝ$ using $x ↦ -x$.

NegativeRay(right)

The real numbers below right. See ℝ⁻.

PositiveRay(left)

The real numbers above left. See ℝ⁺.

RealCircle()

Mapping $ℝ → (-1,1)$ using $x ↦ x/√(1+x^2)$.

RealLine()

The real line. Use the constant ℝ.

Segment(left, right)

The real numbers between left and right, with $-∞ < \text{left} < \text{right} < ∞$ enforced.

TransformDistribution(distribution, transformation)

Given a transformation and a distribution, create a transformed distribution object that has the distribution of transformation(x) with x ∼ distribution.

The transformation object is callable with the same syntax as transformation. It also supports methods rand, length.

See also logpdf_in_domain and logpdf_in_image.

TransformLogLikelihood(ℓ, transformation::Union{Tuple, GroupedTransformation})

TransformLogLikelihood(ℓ, transformations...)

Return a callable that

1. transforms its vector argument using a grouped transformation to a set of values,

2. calls ℓ (which should return a scalar) with this tuple.

3. returns the result above corrected by the log Jacobians.

Useful when ℓ is a log-likelihood function with a restricted domain, and transformations is used to trasform to this domain from $ℝ^n$.

See also get_transformation, get_distribution, Distributions.logpdf, and logpdf_in_domain.

TransformationTuple(transformations::Tuple)
TransformationTuple(transformations...)

A tuple of ContinuousTransformations. Given a vector of matching length, each takes as many reals as needed, and returns the result as a tuple.

abstract type TransformationWrapper <: Function

Wrap a transformation to achieve some specialized functionality.

Supports length, get_transformation, and other methods depending on the subtype.

UnitVector(n)

Transform n-1 real numbers to a unit vector of length n, under the Euclidean norm.

abstract type UnivariateTransformation <: ContinuousTransformation

Univariate monotone transformation, either increasing or decreasing on the whole domain (thus, a bijection).

Exp()

Mapping $ℝ → ℝ⁺$ using $x ↦ \exp(x)$.

Identity (as an affine transformation).

InvRealCircle()

Mapping $(-1,1) → ℝ$ using $x ↦ x/√(1-x^2)$.

Log()

Mapping $ℝ → ℝ⁺$ using $x ↦ \exp(x)$.

Logistic()

Mapping $ℝ → (0,1)$ using $x ↦ 1/(1+\exp(-x))$.

Logit()

Mapping $(0,1) → ℝ$ using $x ↦ \log(x/(1-x))$.

Negation()

Mapping $ℝ → ℝ$ using $x ↦ -x$.

RealCircle()

Mapping $ℝ → (-1,1)$ using $x ↦ x/√(1+x^2)$.

affine_bridge(interval1, interval1)

Return an affine transformation between two intervals of the same type.

all_finite(x)

Test if a numerical argument is made up of finite real numbers.

bridge(dom, img, [transformation])

Return a transformation that maps dom to img.

The transformation argument may be used to specify a particular transformation family, otherwise default_transformation is used.

default_transformation(dom, img)

Return a transformation from dom that can be mapped to img using affine_bridge.

domain(transformation)

Return the domain of the transformation.

get_distribution(t)

Return the wrapped distribution.

get_loglikelihood(t)

Return the log likelihood function.

get_transformation(d)

Return the transformation from a wrapper object.

image(transformation)

Return the image of the transformation.

inverse(t, x)

Return $t⁻¹(x)$.

inverse(t)

Return the transformation $t⁻¹$.

isincreasing(transformation)

Return true (false), when the transformation is monotonically increasing (decreasing).

lkj_correlation_cholesky_logpdf(L, η)

Log PDF of the LKJ distribution (Lewandowski et al 2009) for correlation matrices.

A correlation matrix $Ω=LL'$ has the density $|Ω|^(η-1)$. However, it is usually not necessary to construct $Ω$, so this function is parametrized in terms of a Cholesky decomposition L.

Note that this function does not check if L yields a valid correlation matrix. Use CorrelationCholeskyFactor for generating valid arguments.

Valid values are $η > 0$. When $η > 1$, the distribution is unimodal at the identity, while $0 < η < 1$ has a trough. $η = 2$ is recommended as a vague prior.

When $η = 1$, the density is uniform. Note however that the function should still be invoked, because of the Jacobian correction of the transformation.

logjac(t, x)

The log of the determinant of the Jacobian of t at x. ```

logpdf_in_domain(t, x)

The log pdf for a transformed distribution at t(x) in image, calculated in the domain without performing the transformation.

The log pdf is adjusted with the log determinant of the Jacobian, ie the following holds:

julia logpdf_in_image(t, t(x)) == logpdf_in_domain(t, x)

See logpdf_in_image.

logpdf_in_image(t, y)

The log pdf for a transformed distribution at y in image.

See also logpdf_in_domain.

transform(t, x)

Return $t(x)$. NOTE: this equivalent to the callable syntax t(x), which is also available.

y, logjac =

transform_and_logjac(t, x)

Return the transformed value and the determinant of the log Jacobian.

Equivalent to t(x), logjac(t, x), but may be faster because of reused components.

ungrouping_map(T, f, A)

Map the vector A elementwise using f, and collect the results in arrays or vectors organized as a single result from f. For example, if f returns tuples, this function returns tuples of vectors.

The shape of the result is determined by the first argument, which can be Vector or Array. The difference is that Array stacks arrays into arrays, while Vector does not.

Array may be more useful for summaries along dimensions, eg mean and similar.

width(s)

Width of a finite interval.

Return transformation(x) while incrementing logjac with the log Jacobian determinant.

Return transformation(x) while incrementing logjac with the log Jacobian determinant.

ComposedTransformation(f, g)

Compose two univariate transformations, resulting in the mapping $f∘g$, or `x ↦ f(g(x)).

Use the ∘ operator for construction.

struct NotRRStable <: ContinuousTransformations.RRStability

Trait that indicates that a univariate transformation is notRRStable.

abstract type RRStability

Trait that is useful for domain and image calculations. See RRStable.

struct RRStable <: ContinuousTransformations.RRStability

Trait that indicates that a univariate transformation

1. maps $ℝ$ to $ℝ$,

2. supports mapping intervals, and

3. maps subtypes of AbstractInterval to the same type.

RR_stability(?)

Return either the trait RRStable and NotRRStable.

Hyperbolic tangent transformation.

An affine stretch of LOGISTIC to (-1, 1).

_fma(x, y, z)

Placeholder for Base.fma until https://github.com/JuliaDiff/ReverseDiff.jl/issues/86 is fixed.

_maybe_segment(a, b)

Helper function for forming a segment when possible. Internal, not exported.

composed_domain(f_RR_stability, g_RR_stability, f, g)

composed_image(f_RR_stability, g_RR_stability, f, g)

make_ungrouped_container(?, representative_elt, len)

Make a container for ungrouped results.

The first argument, Vector or Array, determines the result of ungrouping.

representative_elt is a representative element, used to determine element type and stucture, and len is the length of the result.

rhs_string(transformation, term)

Return the formula representing the hand side of the transformation, with term as the argument.

triangle_length(n)

Number of elements in a triangle of an $n×n$ square matrix, including the diagonal.

ungroup_elt!(T, container, elt, i)

Save elt in container at index i. T determines the layout of container, conformably with make_ungrouped_container.

Define transform and logjac using transform_and_logjac.

Define an isapprox method, comparing the given fields in type T.

Define a singleton type with the given name and supertype (specified as name <: supertype), and a constant which defaults to the name in uppercase.