# tfp.experimental.substrates.numpy.bijectors.SinhArcsinh

`Y = g(X) = Sinh( (Arcsinh(X) + skewness) * tailweight ) * multiplier`.

Inherits From: `Bijector`

For `skewness in (-inf, inf)` and `tailweight in (0, inf)`, this transformation is a diffeomorphism of the real line `(-inf, inf)`. The inverse transform is `X = g^{-1}(Y) = Sinh( ArcSinh(Y) / tailweight - skewness )`.

The `SinhArcsinh` transformation of the Normal is described in Sinh-arcsinh distributions This Bijector allows a similar transformation of any distribution supported on `(-inf, inf)`.

#### Meaning of the parameters

• If `skewness = 0` and `tailweight = 1`, this transform is the identity.
• Positive (negative) `skewness` leads to positive (negative) skew.
• positive skew means, for unimodal `X` centered at zero, the mode of `Y` is "tilted" to the right.
• positive skew means positive values of `Y` become more likely, and negative values become less likely.
• Larger (smaller) `tailweight` leads to fatter (thinner) tails.
• Fatter tails mean larger values of `|Y|` become more likely.
• If `X` is a unit Normal, `tailweight < 1` leads to a distribution that is "flat" around `Y = 0`, and a very steep drop-off in the tails.
• If `X` is a unit Normal, `tailweight > 1` leads to a distribution more peaked at the mode with heavier tails.
• The `multiplier` term is equal to 2 / F_0(2) where F_0 is the bijector with `skewness = 0`. This is important for CDF values of distributions.SinhArcsinh.

To see the argument about the tails, note that for `|X| >> 1` and `|X| >> (|skewness| * tailweight)**tailweight`, we have `Y approx 0.5 X**tailweight e**(sign(X) skewness * tailweight)`.

`skewness` Skewness parameter. Float-type `Tensor`. Default is `0` of type `float32`.
`tailweight` Tailweight parameter. Positive `Tensor` of same `dtype` as `skewness` and broadcastable `shape`. Default is `1` of type `float32`.
`validate_args` Python `bool` indicating whether arguments should be checked for correctness.
`name` Python `str` name given to ops managed by this object.

`dtype` dtype of `Tensor`s transformable by this distribution.
`forward_min_event_ndims` Returns the minimal number of dimensions bijector.forward operates on.
`graph_parents` Returns this `Bijector`'s graph_parents as a Python list.
`inverse_min_event_ndims` Returns the minimal number of dimensions bijector.inverse operates on.
`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`.
`skewness` The `skewness` in: `Y = Sinh((Arcsinh(X) + skewness) * tailweight)`.
`tailweight` The `tailweight` in: `Y = Sinh((Arcsinh(X) + skewness) * tailweight)`.
`trainable_variables`

`validate_args` Returns True if Tensor arguments will be validated.
`variables`

## Methods

### `forward`

View source

Returns the forward `Bijector` evaluation, i.e., X = g(Y).

Args
`x` `Tensor`. The input to the 'forward' evaluation.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`Tensor`.

Raises
`TypeError` if `self.dtype` is specified and `x.dtype` is not `self.dtype`.
`NotImplementedError` if `_forward` is not implemented.

### `forward_dtype`

View source

Returns the dtype of the output of the forward transformation.

Args
`dtype` `tf.dtype`, or nested structure of `tf.dtype`s, of the input to `forward`.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`tf.dtype` or nested structure of `tf.dtype`s of the output of `forward`.

### `forward_event_shape`

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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` indicating event-portion shape passed into `forward` function.

Returns
`forward_event_shape_tensor` `TensorShape` indicating event-portion shape after applying `forward`. Possibly unknown.

### `forward_event_shape_tensor`

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Shape of a single sample from a single batch as an `int32` 1D `Tensor`.

Args
`input_shape` `Tensor`, `int32` vector indicating event-portion shape passed into `forward` function.
`name` name to give to the op

Returns
`forward_event_shape_tensor` `Tensor`, `int32` vector indicating event-portion shape after applying `forward`.

### `forward_log_det_jacobian`

View source

Returns both the forward_log_det_jacobian.

Args
`x` `Tensor`. 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.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`Tensor`, if this bijector is injective. If not injective this is not implemented.

Raises
`TypeError` if `self.dtype` is specified and `y.dtype` is not `self.dtype`.
`NotImplementedError` if neither `_forward_log_det_jacobian` nor {`_inverse`, `_inverse_log_det_jacobian`} are implemented, or this is a non-injective bijector.

### `inverse`

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Returns the inverse `Bijector` evaluation, i.e., X = g^{-1}(Y).

Args
`y` `Tensor`. The input to the 'inverse' evaluation.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`Tensor`, 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 `self.dtype` is specified and `y.dtype` is not `self.dtype`.
`NotImplementedError` if `_inverse` is not implemented.

### `inverse_dtype`

View source

Returns the dtype of the output of the inverse transformation.

Args
`dtype` `tf.dtype`, or nested structure of `tf.dtype`s, of the input to `inverse`.
`name` The name to give this op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`tf.dtype` or nested structure of `tf.dtype`s of the output of `inverse`.

### `inverse_event_shape`

View source

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` indicating event-portion shape passed into `inverse` function.

Returns
`inverse_event_shape_tensor` `TensorShape` indicating event-portion shape after applying `inverse`. Possibly unknown.

### `inverse_event_shape_tensor`

View source

Shape of a single sample from a single batch as an `int32` 1D `Tensor`.

Args
`output_shape` `Tensor`, `int32` vector indicating event-portion shape passed into `inverse` function.
`name` name to give to the op

Returns
`inverse_event_shape_tensor` `Tensor`, `int32` vector indicating event-portion shape after applying `inverse`.

### `inverse_log_det_jacobian`

View source

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`. 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.
`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 `self.dtype` is specified and `y.dtype` is not `self.dtype`.
`NotImplementedError` if `_inverse_log_det_jacobian` is not implemented.

### `__call__`

View source

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:

1. If the input is a `tfd.Distribution` instance, return `tfd.TransformedDistribution(distribution=input, bijector=self)`.
2. If the input is a `tfb.Bijector` instance, return `tfb.Chain([self, input])`.
3. Otherwise, return `self.forward(input)`

Args
`value` A `tfd.Distribution`, `tfb.Bijector`, or a `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 `Tensor` computed by `self.forward`.

#### Examples

``````sigmoid = tfb.Reciprocal()(
tfb.AffineScalar(shift=1.)(
tfb.Exp()(
tfb.AffineScalar(scale=-1.))))
# ==> `tfb.Chain([
#         tfb.Reciprocal(),
#         tfb.AffineScalar(shift=1.),
#         tfb.Exp(),
#         tfb.AffineScalar(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.])
``````