# tf.contrib.distributions.bijectors.SinhArcsinh

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| [![](https://www.tensorflow.org/images/GitHub-Mark-32px.png) View source on GitHub](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/contrib/distributions/python/ops/bijectors/sinh_arcsinh.py#L67-L193) |

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

Inherits From: [`Bijector`](https://www.tensorflow.org/versions/r1.15/api_docs/python/tf/contrib/distributions/bijectors/Bijector)

    tf.contrib.distributions.bijectors.SinhArcsinh(
        skewness=None, tailweight=None, validate_args=False, name='SinhArcsinh'
    )

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](https://www.jstor.org/stable/27798865)
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.

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)`.

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| ## Args ||
|---|---|
| `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. |

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| ## Attributes ||
|---|---|
| `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. <br /> > [!NOTE] > **Note:** Jacobian matrix is either constant for both forward and inverse or neither. |
| `name` | Returns the string name of this `Bijector`. |
| `skewness` | The `skewness` in: `Y = Sinh((Arcsinh(X) + skewness) * tailweight)`. |
| `tailweight` | The `tailweight` in: `Y = Sinh((Arcsinh(X) + skewness) * tailweight)`. |
| `validate_args` | Returns True if Tensor arguments will be validated. |

## Methods

### `forward`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L753-L768)

    forward(
        x, name='forward'
    )

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

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| Args ||
|---|---|
| `x` | `Tensor`. The input to the "forward" evaluation. |
| `name` | The name to give this op. |

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| Returns ||
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| `Tensor`. ||

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

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### `forward_event_shape`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L677-L690)

    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.

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| Args ||
|---|---|
| `input_shape` | `TensorShape` indicating event-portion shape passed into `forward` function. |

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

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### `forward_event_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L653-L670)

    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`.

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

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

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### `forward_log_det_jacobian`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L973-L997)

    forward_log_det_jacobian(
        x, event_ndims, name='forward_log_det_jacobian'
    )

Returns both the forward_log_det_jacobian.

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| 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 `x.shape.ndims - event_ndims` dimensions. |
| `name` | The name to give this op. |

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

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| 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. |

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### `inverse`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L787-L804)

    inverse(
        y, name='inverse'
    )

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

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| Args ||
|---|---|
| `y` | `Tensor`. The input to the "inverse" evaluation. |
| `name` | The name to give this op. |

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| Returns ||
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| `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`. ||

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| Raises ||
|---|---|
| `TypeError` | if `self.dtype` is specified and `y.dtype` is not `self.dtype`. |
| `NotImplementedError` | if `_inverse` is not implemented. |

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### `inverse_event_shape`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L721-L734)

    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.

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| Args ||
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| `output_shape` | `TensorShape` indicating event-portion shape passed into `inverse` function. |

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

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### `inverse_event_shape_tensor`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L697-L714)

    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`.

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

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

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### `inverse_log_det_jacobian`

[View source](https://github.com/tensorflow/tensorflow/blob/v1.15.0/tensorflow/python/ops/distributions/bijector_impl.py#L871-L900)

    inverse_log_det_jacobian(
        y, event_ndims, name='inverse_log_det_jacobian'
    )

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)`.

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| 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 `y.shape.ndims - event_ndims` dimensions. |
| `name` | The name to give this op. |

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| Returns ||
|---|---|
| `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`. ||

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| Raises ||
|---|---|
| `TypeError` | if `self.dtype` is specified and `y.dtype` is not `self.dtype`. |
| `NotImplementedError` | if `_inverse_log_det_jacobian` is not implemented. |

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