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from warnings import warn

import torch

from modulus.metrics.general import histogram

Tensor = torch.Tensor


[docs]def wasserstein_from_normal(
    mu0: Tensor, sigma0: Tensor, mu1: Tensor, sigma1: Tensor
) -> Tensor:
    """Compute the wasserstein distances between two (possibly multivariate) normal
    distributions.

    Parameters
    ----------
    mu0 : Tensor [B (optional), d1]
        The mean of distribution 0. Can optionally have a batched first dimension.
    sigma0 : Tensor [B (optional), d1, d2 (optional)]
        The variance or covariance of distribution 0. If mu0 has a batched dimension,
        then so must sigma0. If sigma0 is 2 dimension, it is assumed to be a covariance matrix
        and must be symmetric positive definite.
    mu1 : Tensor [B (optional), d1]
        The mean of distribution 1. Can optionally have a batched first dimension.
    sigma1 : Tensor [B (optional), d1, d2 (optional)]
        The variance or covariance of distribution 1. If mu1 has a batched dimension,
        then so must sigma1. If sigma1 is 2 dimension, it is assumed to be a covariance matrix
        and must be symmetric positive definite.

    Returns
    -------
    Tensor [B]
        The wasserstein distance between N(mu0, sigma0) and N(mu1, sigma1)
    """
    mu_ndim = mu0.ndim
    sigma_ndim = sigma0.ndim
    if sigma_ndim == mu_ndim:
        # Univariate normal distribution
        return (mu0 - mu1) ** 2 + (sigma0 + sigma1 - 2 * torch.sqrt(sigma0 * sigma1))

    else:
        # Multivariate normal distribution
        # Compute trace(sig0 + sig1 - 2*(sig0^1/2 * sig1 * sig0^1/2)^1/2) first

        # Compute sig0^1/2 first using eigen decomposition.
        vals0, vecs0 = torch.linalg.eigh(sigma0)
        if torch.any(vals0 < 0.0):
            warn(
                "Warning! Some eigenvalues are less than zero and matrix is not positive definite."
            )
            vals0 = torch.nn.functional.relu(vals0)
        sqrt_sig0 = torch.matmul(
            torch.matmul(vecs0, torch.diag_embed(torch.sqrt(vals0))),
            vecs0.transpose(-2, -1),
        )

        # Compute C = (sig0^1/2 * sig1 * sig0^1/2)
        C = torch.matmul(torch.matmul(sqrt_sig0, sigma1), sqrt_sig0)

        # Compute Csqrt = sqrt( C )
        vals0, vecs0 = torch.linalg.eigh(C)
        if torch.any(vals0 < 0.0):
            warn(
                "Warning! Some eigenvalues are less than zero and matrix is not positive definite."
            )
            vals0 = torch.nn.functional.relu(vals0)
        sqrtC = torch.matmul(
            torch.matmul(vecs0, torch.diag_embed(torch.sqrt(vals0))),
            vecs0.transpose(-2, -1),
        )

        # Compute T = tr(sig0 + sig1 - 2* sqrtC)
        if sigma_ndim > 2:
            T = torch.vmap(torch.trace)(sigma0 + sigma1 - 2 * sqrtC)
        else:
            T = torch.trace(sigma0 + sigma1 - 2 * sqrtC)

        return torch.norm((mu0 - mu1), p=2, dim=-1) ** 2 + T


[docs]def wasserstein_from_samples(x: Tensor, y: Tensor, bins: int = 10):
    """1-Wasserstein distances between two sets of samples, computed using
    the discrete CDF.

    Parameters
    ----------
    x : Tensor [S, ...]
        Tensor containing one set of samples. The wasserstein metric will be computed
        over the first dimension of the data.
    y : Tensor[S, ...]
        Tensor containing the second set of samples. The wasserstein metric will be computed
        over the first dimension of the data. The shapes of x and y must be compatible.
    bins : int, Optional.
        Optional number of bins to use in the empirical CDF. Defaults to 10.

    Returns
    -------
    Tensor
        The 1-Wasserstein distance between the samples x and y.
    """
    bin_edges, cdf_x = histogram.cdf(x, bins=bins)
    _, cdf_y = histogram.cdf(y, bins=bin_edges)
    return wasserstein_from_cdf(bin_edges, cdf_x, cdf_y)


[docs]def wasserstein_from_cdf(bin_edges: Tensor, cdf_x: Tensor, cdf_y: Tensor) -> Tensor:
    """1-Wasserstein distance between two discrete CDF functions

    This norm is typically used to compare two different forecast ensembles (for X and
    Y). Creates a map of distance and does not accumulate over lat/lon regions.
    Computes

    .. math::

        W(F_X, F_Y) = int[ |F_X(x) - F_Y(x)| ] dx

    where F_X is the empirical cdf of X and F_Y is the empirical cdf of Y.

    Parameters
    ----------
    bin_edges : Tensor
        Tensor containing bin edges. The leading dimension must represent the N+1 bin
        edges.
    cdf_x : Tensor
        Tensor containing a CDF one, defined over bins. The non-zeroth dimensions of
        bins and cdf must be compatible.
    cdf_y : Tensor
        Tensor containing a CDF two, defined over bins. Must be compatible with cdf_x in
        terms of bins and shape.

    Returns
    -------
    Tensor
        The 1-Wasserstein distance between cdf_x and cdf_y
    """
    return torch.sum(
        torch.abs(cdf_x - cdf_y) * (bin_edges[1, ...] - bin_edges[0, ...]), dim=0
    )