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

Inherits From: TransformedDistribution, Distribution

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)
F(Z) := Sinh( (Arcsinh(Z) + skewness) * tailweight ) * (2 / F_0(2))
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).

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. Must have a batch shape to which the shapes of loc, scale, skewness, and tailweight all broadcast. Default is tfd.Normal(batch_shape, 1.), where batch_shape is the broadcasted shape of the parameters. 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.

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.

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.

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.

experimental_shard_axis_names The list or structure of lists of active shard axis names.
loc The loc in Y := loc + scale @ F(Z).
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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

scale The LinearOperator scale in Y := loc + scale @ F(Z).
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.



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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.

name name to give to the op

batch_shape Tensor.


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Cumulative distribution function.

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

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

value float or double Tensor.
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

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


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Creates a deep copy of the distribution.

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

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).


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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.

name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.