Probability distribution constructed from predicted quantiles

Wraps a set of predicted quantiles into a proper distribution with:

  • Monotonicity enforcement (fixes quantile crossing)

  • Tail extrapolation (exponential or GPD) with data-inferred parameters

  • Analytical statistics (CDF, PDF, CRPS, mean, variance)

Setup linear spline segments

Setup tail parameters

Compute quantile function Q(α) = F^(-1)(α)

Expand batch parameter to match alpha shape

Left tail quantile

Right tail quantile

GPD left tail

GPD right tail

Piecewise linear quantile interpolation

Compute CDF F(z) = P(Z ≤ z)

Expand batch parameter to match z shape

CDF in left tail region

CDF in right tail region

CDF in spline region

Compute dQ/dα derivative

Left tail derivative

Right tail derivative

Spline derivative

Compute log PDF

Compute PDF

Compute mean E[Z]

Analytical mean for exponential tails

Analytical mean for GPD tails

Compute variance

Analytical variance for exponential tails

Analytical variance for GPD tails

Compute standard deviation

Compute analytical CRPS (Continuous Ranked Probability Score)

CRPS contribution from left tail

Exponential left tail CRPS

GPD left tail CRPS

CRPS contribution from right tail

Exponential right tail CRPS

GPD right tail CRPS

CRPS contribution from spline region

Numerical CRPS via pinball loss (for validation)

Draw samples from the distribution

QuantileDistribution(
  quantiles,
  alpha_levels = NULL,
  tail_type = "exp",
  fix_crossing = TRUE,
  crossing_method = "sort"
)

Arguments

param

Tensor. Parameter estimates of the tail function.

alpha

Tensor. Probability levels.

alpha_z

Tensor. Probability levels.

z

Tensor. Observation values. Shape: (*batch_shape,).

num_quantiles

Integer. Number of quantile levels for integration.

sample_shape

torch.Size. Shape of the sample.

Value

Tensor. Quantile values Q(α).

Tensor. CDF values in [0, 1].

Tensor. Log probability density values.

Tensor. PDF values (non-negative).

Tensor. Expected value. Shape: (*batch_shape,).

Tensor. Variance (non-negative). Shape: (*batch_shape,).

Tensor. Standard deviation. Shape: (*batch_shape,).

Tensor. CRPS values (non-negative, lower is better). Shape: same as z.

Tensor. Approximate CRPS values. Shape: (*batch_shape,).

Tensor. Samples from the distribution.