# LostTech.TensorFlow : API Documentation

Type SinhArcsinh

Namespace tensorflow.contrib.distributions

Parent TransformedDistribution

Interfaces ISinhArcsinh

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

### Public properties

#### objectloc get;

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

#### objectloc_dyn get;

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

#### objectscale get;

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

#### objectscale_dyn get;

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

#### objectskewness get;

Controls the skewness. `Skewness > 0` means right skew.

#### objectskewness_dyn get;

Controls the skewness. `Skewness > 0` means right skew.

#### objecttailweight get;

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

#### objecttailweight_dyn get;

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