def probability_less_than(y_pdf, bins, y, bin_axis=1):
"""
Calculate the probability of a sample being less than a given
value for a tensor of predicted PDFs.
Args:
y_pdf: Tensor containing the predicted PDFs.
bins: The bin-boundaries corresponding to the predictions.
y: The sample value.
bin_axis: The index of the tensor axis which contains the predictions
for each bin.
Return:
Tensor with rank reduced by one compared to ``y_pdf`` and with
the values along ``bin_axis`` of ``y_pdf`` replaced with the
probability that a sample of the distribution is smaller than
the given value ``y``.
"""
if len(y_pdf.shape) == 1:
bin_axis = 0
n_y = y_pdf.shape[bin_axis]
n_b = len(bins)
_check_dimensions(n_y, n_b)
xp = get_array_module(y_pdf)
n = len(y_pdf.shape)
x = 0.5 * (bins[1:] + bins[:-1])
mask = x < y
shape = [1] * n
shape[bin_axis] = -1
mask = as_type(xp, reshape(xp, mask, shape), y_pdf)
return trapz(xp, mask * y_pdf, bins, bin_axis)
def posterior_mean(y_pdf, bins, bin_axis=1):
"""
Calculate posterior mean from predicted PDFs.
Args:
y_pdf: Tensor containing the predicted PDFs.
bins: The bin-boundaries corresponding to the predictions.
bin_axis: The index of the tensor axis which contains the predictions
for each bin.
Return:
Tensor with rank reduced by one compared to ``y_pdf`` and with
the values along ``bin_axis`` of ``y_pdf`` replaced with the
mean value of the PDF.
"""
if len(y_pdf.shape) == 1:
bin_axis = 0
n_y = y_pdf.shape[bin_axis]
n_b = len(bins)
_check_dimensions(n_y, n_b)
xp = get_array_module(y_pdf)
n = len(y_pdf.shape)
shape = [1] * n
shape[bin_axis] = -1
bins_r = reshape(xp, 0.5 * (bins[1:] + bins[:-1]), shape)
return trapz(xp, bins_r * y_pdf, bins, bin_axis)
def crps(y_pdf, y_true, bins, bin_axis=1):
r"""
Compute the Continuous Ranked Probability Score (CRPS) for a given
discrete probability density.
This function uses a piece-wise linear fit to the approximate posterior
CDF obtained from the predicted quantiles in :code:`y_pred` to
approximate the continuous ranked probability score (CRPS):
.. math::
CRPS(\mathbf{y}, x) = \int_{-\infty}^\infty (F_{x | \mathbf{y}}(x')
- \mathrm{1}_{x < x'})^2 \: dx'
Args:
y_pred: Tensor containing the predicted discrete posterior PDF
with the probabilities for different bins oriented along axis
``bin_axis`` in ``y_pred``.
y_true: Array containing the true point values.
bins: 1D array containing the bins corresponding to the probabilities
in ``y_pred``.
Returns:
Tensor of rank :math:`k - 1` containing the CRPS values for each of the
predictions in ``y_pred``.
"""
if len(y_pdf.shape) == 1:
bin_axis = 0
n_y = y_pdf.shape[bin_axis]
n_b = len(bins)
n_dims = len(y_pdf.shape)
_check_dimensions(n_y, n_b)
xp = get_array_module(y_pdf)
n = len(y_pdf.shape)
y_cdf = posterior_cdf(y_pdf, bins, bin_axis=bin_axis)
x = bins
shape = [1] * n_dims
shape[bin_axis] = -1
x = x.reshape(shape)
if len(y_true.shape) < len(y_pdf.shape):
y_true = y_true.unsqueeze(bin_axis)
i = as_type(xp, x > y_true, y_cdf)
crps = trapz(xp, (y_cdf - i) ** 2, x, bin_axis)
return crps
def posterior_mean(y_pred, bins, bin_axis=1):
if len(y_pred.shape) == 1:
bin_axis = 0
n_y = y_pred.shape[bin_axis]
n_b = len(bins)
_check_dimensions(n_y, n_b)
xp = get_array_module(y_pred)
n = len(y_pred.shape)
shape = [1] * n
shape[bin_axis] = -1
bins_r = reshape(xp, 0.5 * (bins[1:] + bins[:-1]), shape)
return trapz(xp, bins_r * y_pred, bins, bin_axis)
def probability_less_than(y_pred, bins, y, bin_axis=1):
if len(y_pred.shape) == 1:
bin_axis = 0
n_y = y_pred.shape[bin_axis]
n_b = len(bins)
_check_dimensions(n_y, n_b)
xp = get_array_module(y_pred)
n = len(y_pred.shape)
x = 0.5 * (bins[1:] + bins[:-1])
mask = x < y
shape = [1] * n
shape[bin_axis] = -1
mask = as_type(xp, reshape(xp, mask, shape), y_pred)
return trapz(xp, mask * y_pred, bins, bin_axis)
def posterior_mean(y_pred, quantiles, quantile_axis=1):
r"""
Computes the mean of the posterior distribution defined by an array
of predicted quantiles.
Args:
y_pred: A tensor of predicted quantiles with the quantiles located
along the axis given by ``quantile_axis``.
quantiles: The quantile fractions corresponding to the quantiles
located along the quantile axis.
quantile_axis: The axis along which the quantiles are located.
Returns:
Array containing the posterior means for the provided inputs.
"""
if len(y_pred.shape) == 1:
quantile_axis = 0
xp = get_array_module(y_pred)
x_cdf, y_cdf = cdf(y_pred, quantiles, quantile_axis=quantile_axis)
return trapz(xp, x_cdf, y_cdf, quantile_axis)
def test_trapz(backend):
array = arange(backend, 0, 10.1, 1)
result = trapz(backend, array, array, 0)
assert result == 50
