plenoptic.process.kurtosis

plenoptic.process.kurtosis(x, mean=None, var=None, dim=None, keepdim=False)

Calculate sample estimate of tailedness (presence of outliers).

To help with interpretation:

• Kurtosis of univariate normal is 3.

• Smaller than 3: platykurtic (e.g. uniform distribution).

• Greater than 3: leptokurtic (e.g. Laplace distribution).

Parameters:

• x (Tensor) – The input tensor.

• mean (float | Tensor | None (default: None)) – Reuse a precomputed mean.

• var (float | Tensor | None (default: None)) – Reuse a precomputed variance.

• dim (int | list[int] | None (default: None)) – The dimension or dimensions to reduce.

• keepdim (bool (default: False)) – Whether to retain the reduced dimensions (as singletons) or not.

Return type:

Tensor

Returns:

out – The kurtosis tensor.

See also

variance

Calculate sample variance.

skew

Calculate sample skewness.

Examples

>>> import plenoptic as po
>>> import matplotlib.pyplot as plt
>>> import torch
>>> po.set_seed(42)
>>> x = torch.randn(10000)
>>> k = po.process.kurtosis(x)
>>> x_platy = torch.rand(10000) * 10 - 5
>>> k_platy = po.process.kurtosis(x_platy)
>>> x_lepto = torch.distributions.Laplace(loc=0.0, scale=1.0).sample((10000,))
>>> k_lepto = po.process.kurtosis(x_lepto)
>>> fig, (ax_platy, ax, ax_lepto) = plt.subplots(
...     1, 3, sharex=True, figsize=(12, 4)
... )
>>> _ = ax_platy.hist(x_platy.numpy(), bins=50)
>>> _ = ax_platy.set(
...     title=f"Platykurtic (Uniform)\nKurtosis: {k_platy:.4f}",
...     ylabel="Frequency",
...     xlim=(-5, 5),
... )
>>> _ = ax.hist(x.numpy(), bins=50)
>>> _ = ax.set(title=f"Standard Gaussian\nKurtosis: {k:.4f}")
>>> _ = ax_lepto.hist(x_lepto.numpy(), bins=50)
>>> _ = ax_lepto.set(title=f"Leptokurtic (Laplace)\nKurtosis: {k_lepto:.4f}")

(png, hires.png, pdf)

If you have precomputed the mean and/or variance, you can pass them and avoid recomputing:

>>> precomputed_mean = torch.mean(x)
>>> precomputed_var = variance(x)
>>> k = po.process.kurtosis(x, mean=precomputed_mean, var=precomputed_var)
>>> k
tensor(2.9354)

If you want to compute along a specific dimension, you can specify it:

>>> x = torch.randn(10000, 2)
>>> k = po.process.kurtosis(x, dim=0)
>>> k
tensor([3.0057, 2.9506])