tf.contrib.distributions.SinhArcsinh

Class SinhArcsinh

Inherits From: TransformedDistribution

Defined in tensorflow/contrib/distributions/python/ops/sinh_arcsinh.py.

The SinhArcsinh transformation of a distribution on (-inf, inf).

This distribution models a random variable, making use of a SinhArcsinh transformation (which has adjustable tailweight and skew), a rescaling, and a shift.

The SinhArcsinh transformation of the Normal is described in great depth in Sinh-arcsinh distributions. Here we use a slightly different parameterization, in terms of tailweight and skewness. Additionally we allow for distributions other than Normal, and control over scale as well as a "shift" parameter loc.

Mathematical Details

Given random variable Z, we define the SinhArcsinh transformation of Z, Y, parameterized by (loc, scale, skewness, tailweight), via the relation:

Y := loc + scale * F(Z) * (2 / F_0(2))
F(Z) := Sinh( (Arcsinh(Z) + skewness) * tailweight )
F_0(Z) := Sinh( Arcsinh(Z) * tailweight )

This distribution is similar to the location-scale transformation L(Z) := loc + scale * Z in the following ways:

  • If skewness = 0 and tailweight = 1 (the defaults), F(Z) = Z, and then Y = L(Z) exactly.
  • loc is used in both to shift the result by a constant factor.
  • The multiplication of scale by 2 / F_0(2) ensures that if skewness = 0 P[Y - loc <= 2 * scale] = P[L(Z) - loc <= 2 * scale]. Thus it can be said that the weights in the tails of Y and L(Z) beyond loc + 2 * scale are the same.

This distribution is different than loc + scale * Z due to the reshaping done by F:

  • Positive (negative) skewness leads to positive (negative) skew.
  • positive skew means, the mode of F(Z) is "tilted" to the right.
  • positive skew means positive values of F(Z) become more likely, and negative values become less likely.
  • Larger (smaller) tailweight leads to fatter (thinner) tails.
  • Fatter tails mean larger values of |F(Z)| become more likely.
  • tailweight < 1 leads to a distribution that is "flat" around Y = loc, and a very steep drop-off in the tails.
  • tailweight > 1 leads to a distribution more peaked at the mode with heavier tails.

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

To see the argument regarding multiplying scale by 2 / F_0(2),

P[(Y - loc) / scale <= 2] = P[F(Z) * (2 / F_0(2)) <= 2]
                          = P[F(Z) <= F_0(2)]
                          = P[Z <= 2]  (if F = F_0).

Properties

allow_nan_stats

Python bool describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

Returns:

  • allow_nan_stats: Python bool.

batch_shape

Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Returns:

  • batch_shape: TensorShape, possibly unknown.

bijector

Function transforming x => y.

distribution

Base distribution, p(x).

dtype

The DType of Tensors handled by this Distribution.

event_shape

Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

Returns:

  • event_shape: TensorShape, possibly unknown.

loc

The loc in `Y := loc + scale @ F(Z) * (2 / F(2)).

name

Name prepended to all ops created by this Distribution.

parameters

Dictionary of parameters used to instantiate this Distribution.

reparameterization_type

Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances distributions.FULLY_REPARAMETERIZED or distributions.NOT_REPARAMETERIZED.

Returns:

An instance of ReparameterizationType.

scale

The LinearOperator scale in `Y := loc + scale @ F(Z) * (2 / F(2)).

skewness

Controls the skewness. Skewness > 0 means right skew.

tailweight

Controls the tail decay. tailweight > 1 means faster than Normal.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

__init__

__init__(
    loc,
    scale,
    skewness=None,
    tailweight=None,
    distribution=None,
    validate_args=False,
    allow_nan_stats=True,
    name='SinhArcsinh'
)

Construct SinhArcsinh distribution on (-inf, inf).

Arguments (loc, scale, skewness, tailweight) must have broadcastable shape (indexing batch dimensions). They must all have the same dtype.

Args:

  • loc: Floating-point Tensor.
  • scale: Tensor of same dtype as loc.
  • skewness: Skewness parameter. Default is 0.0 (no skew).
  • tailweight: Tailweight parameter. Default is 1.0 (unchanged tailweight)
  • distribution: tf.Distribution-like instance. Distribution that is transformed to produce this distribution. Default is tf.distributions.Normal(0., 1.). Must be a scalar-batch, scalar-event distribution. Typically distribution.reparameterization_type = FULLY_REPARAMETERIZED or it is a function of non-trainable parameters. WARNING: If you backprop through a SinhArcsinh sample and distribution is not FULLY_REPARAMETERIZED yet is a function of trainable variables, then the gradient will be incorrect!
  • validate_args: Python bool, default False. When True distribution parameters are checked for validity despite possibly degrading runtime performance. When False invalid inputs may silently render incorrect outputs.
  • allow_nan_stats: Python bool, default True. When True, statistics (e.g., mean, mode, variance) use the value "NaN" to indicate the result is undefined. When False, an exception is raised if one or more of the statistic's batch members are undefined.
  • name: Python str name prefixed to Ops created by this class.

batch_shape_tensor

batch_shape_tensor(name='batch_shape_tensor')

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args:

  • name: name to give to the op

Returns:

  • batch_shape: Tensor.

cdf

cdf(
    value,
    name='cdf'
)

Cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

cdf(x) := P[X <= x]

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • cdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

copy

copy(**override_parameters_kwargs)

Creates a deep copy of the distribution.

Args:

  • **override_parameters_kwargs: String/value dictionary of initialization arguments to override with new values.

Returns:

  • distribution: A new instance of type(self) initialized from the union of self.parameters and override_parameters_kwargs, i.e., dict(self.parameters, **override_parameters_kwargs).

covariance

covariance(name='covariance')

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued distribution, it is calculated as,

Cov[i, j] = Covariance(X_i, X_j) = E[(X_i - E[X_i]) (X_j - E[X_j])]

where Cov is a (batch of) k x k matrix, 0 <= (i, j) < k, and E denotes expectation.

Alternatively, for non-vector, multivariate distributions (e.g., matrix-valued, Wishart), Covariance shall return a (batch of) matrices under some vectorization of the events, i.e.,

Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]

where Cov is a (batch of) k' x k' matrices, 0 <= (i, j) < k' = reduce_prod(event_shape), and Vec is some function mapping indices of this distribution's event dimensions to indices of a length-k' vector.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • covariance: Floating-point Tensor with shape [B1, ..., Bn, k', k'] where the first n dimensions are batch coordinates and k' = reduce_prod(self.event_shape).

cross_entropy

cross_entropy(
    other,
    name='cross_entropy'
)

Computes the (Shannon) cross entropy.

Denote this distribution (self) by P and the other distribution by Q. Assuming P, Q are absolutely continuous with respect to one another and permit densities p(x) dr(x) and q(x) dr(x), (Shanon) cross entropy is defined as:

H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)

where F denotes the support of the random variable X ~ P.

Args:

Returns:

  • cross_entropy: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of (Shanon) cross entropy.

entropy

entropy(name='entropy')

Shannon entropy in nats.

event_shape_tensor

event_shape_tensor(name='event_shape_tensor')

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

Args:

  • name: name to give to the op

Returns:

  • event_shape: Tensor.

is_scalar_batch

is_scalar_batch(name='is_scalar_batch')

Indicates that batch_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_batch: bool scalar Tensor.

is_scalar_event

is_scalar_event(name='is_scalar_event')

Indicates that event_shape == [].

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • is_scalar_event: bool scalar Tensor.

kl_divergence

kl_divergence(
    other,
    name='kl_divergence'
)

Computes the Kullback--Leibler divergence.

Denote this distribution (self) by p and the other distribution by q. Assuming p, q are absolutely continuous with respect to reference measure r, the KL divergence is defined as:

KL[p, q] = E_p[log(p(X)/q(X))]
         = -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
         = H[p, q] - H[p]

where F denotes the support of the random variable X ~ p, H[., .] denotes (Shanon) cross entropy, and H[.] denotes (Shanon) entropy.

Args:

Returns:

  • kl_divergence: self.dtype Tensor with shape [B1, ..., Bn] representing n different calculations of the Kullback-Leibler divergence.

log_cdf

log_cdf(
    value,
    name='log_cdf'
)

Log cumulative distribution function.

Given random variable X, the cumulative distribution function cdf is:

log_cdf(x) := Log[ P[X <= x] ]

Often, a numerical approximation can be used for log_cdf(x) that yields a more accurate answer than simply taking the logarithm of the cdf when x << -1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • logcdf: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_prob

log_prob(
    value,
    name='log_prob'
)

Log probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • log_prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

log_survival_function

log_survival_function(
    value,
    name='log_survival_function'
)

Log survival function.

Given random variable X, the survival function is defined:

log_survival_function(x) = Log[ P[X > x] ]
                         = Log[ 1 - P[X <= x] ]
                         = Log[ 1 - cdf(x) ]

Typically, different numerical approximations can be used for the log survival function, which are more accurate than 1 - cdf(x) when x >> 1.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

mean

mean(name='mean')

Mean.

mode

mode(name='mode')

Mode.

param_shapes

param_shapes(
    cls,
    sample_shape,
    name='DistributionParamShapes'
)

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

Args:

  • sample_shape: Tensor or python list/tuple. Desired shape of a call to sample().
  • name: name to prepend ops with.

Returns:

dict of parameter name to Tensor shapes.

param_static_shapes

param_static_shapes(
    cls,
    sample_shape
)

param_shapes with static (i.e. TensorShape) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample(). Assumes that the sample's shape is known statically.

Subclasses should override class method _param_shapes to return constant-valued tensors when constant values are fed.

Args:

  • sample_shape: TensorShape or python list/tuple. Desired shape of a call to sample().

Returns:

dict of parameter name to TensorShape.

Raises:

  • ValueError: if sample_shape is a TensorShape and is not fully defined.

prob

prob(
    value,
    name='prob'
)

Probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • prob: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

quantile

quantile(
    value,
    name='quantile'
)

Quantile function. Aka "inverse cdf" or "percent point function".

Given random variable X and p in [0, 1], the quantile is:

quantile(p) := x such that P[X <= x] == p

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

  • quantile: a Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

sample

sample(
    sample_shape=(),
    seed=None,
    name='sample'
)

Generate samples of the specified shape.

Note that a call to sample() without arguments will generate a single sample.

Args:

  • sample_shape: 0D or 1D int32 Tensor. Shape of the generated samples.
  • seed: Python integer seed for RNG
  • name: name to give to the op.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

stddev

stddev(name='stddev')

Standard deviation.

Standard deviation is defined as,

stddev = E[(X - E[X])**2]**0.5

where X is the random variable associated with this distribution, E denotes expectation, and stddev.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • stddev: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

survival_function(
    value,
    name='survival_function'
)

Survival function.

Given random variable X, the survival function is defined:

survival_function(x) = P[X > x]
                     = 1 - P[X <= x]
                     = 1 - cdf(x).

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

Tensor of shape sample_shape(x) + self.batch_shape with values of type self.dtype.

variance

variance(name='variance')

Variance.

Variance is defined as,

Var = E[(X - E[X])**2]

where X is the random variable associated with this distribution, E denotes expectation, and Var.shape = batch_shape + event_shape.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

  • variance: Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().