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|
Maps unconstrained reals to Cholesky-space correlation matrices.
Inherits From: Bijector
tfp.substrates.numpy.bijectors.CorrelationCholesky(
validate_args=False, name='correlation_cholesky'
)
Mathematical Details
This bijector provides a change of variables from unconstrained reals to a parameterization of the CholeskyLKJ distribution. The CholeskyLKJ distribution [1] is a distribution on the set of Cholesky factors of positive definite correlation matrices. The CholeskyLKJ probability density function is obtained from the LKJ density on n x n matrices as follows:
1 = int p(A | eta) dA = int Z(eta) * det(A) ** (eta - 1) dA = int Z(eta) L_ii ** {(n - i - 1) + 2 * (eta - 1)} ^dL_ij (0 <= i < j < n)
where Z(eta) is the normalizer; the matrix L is the Cholesky factor of the correlation matrix A; and ^dL_ij denotes the wedge product (or differential) of the strictly lower triangular entries of L. The entries L_ij are constrained such that each entry lies in [-1, 1] and the norm of each row is
- The norm includes the diagonal; which is not included in the wedge product. To preserve uniqueness, we further specify that the diagonal entries are positive.
The image of unconstrained reals under the CorrelationCholesky bijector is
the set of correlation matrices which are positive definite. A correlation
matrix
can be characterized as a symmetric positive semidefinite matrix with 1s on
the main diagonal.
For a lower triangular matrix L to be a valid Cholesky-factor of a positive
definite correlation matrix, it is necessary and sufficient that each row of
L have unit Euclidean norm [1]. To see this, observe that if L_i is the
ith row of the Cholesky factor corresponding to the correlation matrix R,
then the ith diagonal entry of R satisfies:
1 = R_i,i = L_i . L_i = ||L_i||^2
where '.' is the dot product of vectors and ||...|| denotes the Euclidean
norm.
Furthermore, observe that R_i,j lies in the interval [-1, 1]. By the
Cauchy-Schwarz inequality:
|R_i,j| = |L_i . L_j| <= ||L_i|| ||L_j|| = 1
This is a consequence of the fact that R is symmetric positive definite with
1s on the main diagonal.
We choose the mapping from x in R^{m} to R^{n^2} where m is the
(n - 1)th triangular number; i.e. m = 1 + 2 + ... + (n - 1).
L_ij = x_i,j / s_i (for i < j) L_ii = 1 / s_i
where s_i = sqrt(1 + x_i,0^2 + xi,1^2 + ... + x(i,i-1)^2). We can check that the required constraints on the image are satisfied.
Examples
bijector.CorrelationCholesky().forward([2., 2., 1.])
# Result: [[ 1. , 0. , 0. ],
[ 0.70710678, 0.70710678, 0. ],
[ 0.66666667, 0.66666667, 0.33333333]]
bijector.CorrelationCholesky().inverse(
[[ 1. , 0. , 0. ],
[ 0.70710678, 0.70710678, 0. ],
[ 0.66666667, 0.66666667, 0.33333333]])
# Result: [2., 2., 1.]
References
[1] Stan Manual. Section 24.2. Cholesky LKJ Correlation Distribution. https://mc-stan.org/docs/2_18/functions-reference/cholesky-lkj-correlation-distribution.html [2] Daniel Lewandowski, Dorota Kurowicka, and Harry Joe, "Generating random correlation matrices based on vines and extended onion method," Journal of Multivariate Analysis 100 (2009), pp 1989-2001.
Args | |
|---|---|
graph_parents
|
Python list of graph prerequisites of this Bijector.
|
is_constant_jacobian
|
Python bool indicating that the Jacobian matrix is
not a function of the input.
|
validate_args
|
Python bool, default False. Whether to validate input
with asserts. If validate_args is False, and the inputs are invalid,
correct behavior is not guaranteed.
|
dtype
|
tf.dtype supported by this Bijector. None means dtype is not
enforced. For multipart bijectors, this value is expected to be the
same for all elements of the input and output structures.
|
forward_min_event_ndims
|
Python integer (structure) indicating the
minimum number of dimensions on which forward operates.
|
inverse_min_event_ndims
|
Python integer (structure) indicating the
minimum number of dimensions on which inverse operates. Will be set to
forward_min_event_ndims by default, if no value is provided.
|
parameters
|
Python dict of parameters used to instantiate this
Bijector. Bijector instances with identical types, names, and
parameters share an input/output cache. parameters dicts are
keyed by strings and are identical if their keys are identical and if
corresponding values have identical hashes (or object ids, for
unhashable objects).
|
name
|
The name to give Ops created by the initializer. |
Raises | |
|---|---|
ValueError
|
If neither forward_min_event_ndims and
inverse_min_event_ndims are specified, or if either of them is
negative.
|
ValueError
|
If a member of graph_parents is not a Tensor.
|
Attributes | |
|---|---|
dtype
|
|
forward_min_event_ndims
|
Returns the minimal number of dimensions bijector.forward operates on.
Multipart bijectors return structured |
graph_parents
|
Returns this Bijector's graph_parents as a Python list.
|
has_static_min_event_ndims
|
Returns True if the bijector has statically-known min_event_ndims.
|
inverse_min_event_ndims
|
Returns the minimal number of dimensions bijector.inverse operates on.
Multipart bijectors return structured |
is_constant_jacobian
|
Returns true iff the Jacobian matrix is not a function of x. |
name
|
Returns the string name of this Bijector.
|
parameters
|
Dictionary of parameters used to instantiate this Bijector.
|
trainable_variables
|
|
validate_args
|
Returns True if Tensor arguments will be validated. |
variables
|
|
Methods
forward
forward(
x, name='forward', **kwargs
)
Returns the forward Bijector evaluation, i.e., X = g(Y).
| Args | |
|---|---|
x
|
Tensor (structure). The input to the 'forward' evaluation.
|
name
|
The name to give this op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
| Returns | |
|---|---|
Tensor (structure).
|
| Raises | |
|---|---|
TypeError
|
if self.dtype is specified and x.dtype is not
self.dtype.
|
NotImplementedError
|
if _forward is not implemented.
|
forward_dtype
forward_dtype(
dtype=UNSPECIFIED, name='forward_dtype', **kwargs
)
Returns the dtype returned by forward for the provided input.
forward_event_ndims
forward_event_ndims(
event_ndims, **kwargs
)
Returns the number of event dimensions produced by forward.
forward_event_shape
forward_event_shape(
input_shape
)
Shape of a single sample from a single batch as a TensorShape.
Same meaning as forward_event_shape_tensor. May be only partially defined.
| Args | |
|---|---|
input_shape
|
TensorShape (structure) indicating event-portion shape
passed into forward function.
|
| Returns | |
|---|---|
forward_event_shape_tensor
|
TensorShape (structure) indicating
event-portion shape after applying forward. Possibly unknown.
|
forward_event_shape_tensor
forward_event_shape_tensor(
input_shape, name='forward_event_shape_tensor'
)
Shape of a single sample from a single batch as an int32 1D Tensor.
| Args | |
|---|---|
input_shape
|
Tensor, int32 vector (structure) indicating event-portion
shape passed into forward function.
|
name
|
name to give to the op |
| Returns | |
|---|---|
forward_event_shape_tensor
|
Tensor, int32 vector (structure)
indicating event-portion shape after applying forward.
|
forward_log_det_jacobian
forward_log_det_jacobian(
x, event_ndims, name='forward_log_det_jacobian', **kwargs
)
Returns both the forward_log_det_jacobian.
| Args | |
|---|---|
x
|
Tensor (structure). The input to the 'forward' Jacobian determinant
evaluation.
|
event_ndims
|
Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.forward_min_event_ndims. The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event, i.e.
it has shape rank(x) - event_ndims dimensions.
Multipart bijectors require structured event_ndims, such that
rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of
the structured input. Furthermore, the first event_ndims[i] of each
x[i].shape must be the same for all i (broadcasting is not allowed).
|
name
|
The name to give this op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
| Returns | |
|---|---|
Tensor (structure), if this bijector is injective.
If not injective this is not implemented.
|
| Raises | |
|---|---|
TypeError
|
if y's dtype is incompatible with the expected output dtype.
|
NotImplementedError
|
if neither _forward_log_det_jacobian
nor {_inverse, _inverse_log_det_jacobian} are implemented, or
this is a non-injective bijector.
|
inverse
inverse(
y, name='inverse', **kwargs
)
Returns the inverse Bijector evaluation, i.e., X = g^{-1}(Y).
| Args | |
|---|---|
y
|
Tensor (structure). The input to the 'inverse' evaluation.
|
name
|
The name to give this op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
| Returns | |
|---|---|
Tensor (structure), if this bijector is injective.
If not injective, returns the k-tuple containing the unique
k points (x1, ..., xk) such that g(xi) = y.
|
| Raises | |
|---|---|
TypeError
|
if y's structured dtype is incompatible with the expected
output dtype.
|
NotImplementedError
|
if _inverse is not implemented.
|
inverse_dtype
inverse_dtype(
dtype=UNSPECIFIED, name='inverse_dtype', **kwargs
)
Returns the dtype returned by inverse for the provided input.
inverse_event_ndims
inverse_event_ndims(
event_ndims, **kwargs
)
Returns the number of event dimensions produced by inverse.
inverse_event_shape
inverse_event_shape(
output_shape
)
Shape of a single sample from a single batch as a TensorShape.
Same meaning as inverse_event_shape_tensor. May be only partially defined.
| Args | |
|---|---|
output_shape
|
TensorShape (structure) indicating event-portion shape
passed into inverse function.
|
| Returns | |
|---|---|
inverse_event_shape_tensor
|
TensorShape (structure) indicating
event-portion shape after applying inverse. Possibly unknown.
|
inverse_event_shape_tensor
inverse_event_shape_tensor(
output_shape, name='inverse_event_shape_tensor'
)
Shape of a single sample from a single batch as an int32 1D Tensor.
| Args | |
|---|---|
output_shape
|
Tensor, int32 vector (structure) indicating
event-portion shape passed into inverse function.
|
name
|
name to give to the op |
| Returns | |
|---|---|
inverse_event_shape_tensor
|
Tensor, int32 vector (structure)
indicating event-portion shape after applying inverse.
|
inverse_log_det_jacobian
inverse_log_det_jacobian(
y, event_ndims, name='inverse_log_det_jacobian', **kwargs
)
Returns the (log o det o Jacobian o inverse)(y).
Mathematically, returns: log(det(dX/dY))(Y). (Recall that: X=g^{-1}(Y).)
Note that forward_log_det_jacobian is the negative of this function,
evaluated at g^{-1}(y).
| Args | |
|---|---|
y
|
Tensor (structure). The input to the 'inverse' Jacobian determinant
evaluation.
|
event_ndims
|
Number of dimensions in the probabilistic events being
transformed. Must be greater than or equal to
self.inverse_min_event_ndims. The result is summed over the final
dimensions to produce a scalar Jacobian determinant for each event, i.e.
it has shape rank(y) - event_ndims dimensions.
Multipart bijectors require structured event_ndims, such that
rank(y[i]) - rank(event_ndims[i]) is the same for all elements i of
the structured input. Furthermore, the first event_ndims[i] of each
x[i].shape must be the same for all i (broadcasting is not allowed).
|
name
|
The name to give this op. |
**kwargs
|
Named arguments forwarded to subclass implementation. |
| Returns | |
|---|---|
ildj
|
Tensor, if this bijector is injective.
If not injective, returns the tuple of local log det
Jacobians, log(det(Dg_i^{-1}(y))), where g_i is the restriction
of g to the ith partition Di.
|
| Raises | |
|---|---|
TypeError
|
if x's dtype is incompatible with the expected inverse-dtype.
|
NotImplementedError
|
if _inverse_log_det_jacobian is not implemented.
|
__call__
__call__(
value, name=None, **kwargs
)
Applies or composes the Bijector, depending on input type.
This is a convenience function which applies the Bijector instance in
three different ways, depending on the input:
- If the input is a
tfd.Distributioninstance, returntfd.TransformedDistribution(distribution=input, bijector=self). - If the input is a
tfb.Bijectorinstance, returntfb.Chain([self, input]). - Otherwise, return
self.forward(input)
| Args | |
|---|---|
value
|
A tfd.Distribution, tfb.Bijector, or a (structure of) Tensor.
|
name
|
Python str name given to ops created by this function.
|
**kwargs
|
Additional keyword arguments passed into the created
tfd.TransformedDistribution, tfb.Bijector, or self.forward.
|
| Returns | |
|---|---|
composition
|
A tfd.TransformedDistribution if the input was a
tfd.Distribution, a tfb.Chain if the input was a tfb.Bijector, or
a (structure of) Tensor computed by self.forward.
|
Examples
sigmoid = tfb.Reciprocal()(
tfb.Shift(shift=1.)(
tfb.Exp()(
tfb.Scale(scale=-1.))))
# ==> `tfb.Chain([
# tfb.Reciprocal(),
# tfb.Shift(shift=1.),
# tfb.Exp(),
# tfb.Scale(scale=-1.),
# ])` # ie, `tfb.Sigmoid()`
log_normal = tfb.Exp()(tfd.Normal(0, 1))
# ==> `tfd.TransformedDistribution(tfd.Normal(0, 1), tfb.Exp())`
tfb.Exp()([-1., 0., 1.])
# ==> tf.exp([-1., 0., 1.])
__eq__
__eq__(
other
)
Return self==value.