SpecsVerification (version 0.4-0)

FitAkdParameters: Fit parameters for affine kernel dressing

Description

The 5 parameters of affine kernel dressing are fitted by minimum CRPS estimation.

Usage

FitAkdParameters(ens, obs)

Arguments

ens

matrix, N rows, K columns. An archive of K-member ensemble forecasts for N time instances.

obs

vector of length N. The verifying observations corresponding to the N ensemble forecasts.

Value

• The function returns a list of parameters for affine kernel dressing.

Details

Affine Kernel Dressing transforms the discrete K-member forecast ensemble at time instance n, `ens[n, ]`, to a continuous distribution function for the target `y` by the following equation:

p(y|ens) = 1 / K * sum {dnorm(y, z.i, s)} where s = (4/3/K)^0.4 * (s1 + s2 * a^2 * var(ens)) and z.i = r1 + r2 * mean(ens) + a * ens

The parameters r1, r2, a, s1, s2 are fitted by minimizing the continuously ranked probability score (CRPS) of the ensemble forecasts `ens` with respect to the observations `obs`. The optimization is carried out using the R function `optim(...)`.

The dressing variance is a linear function of the ensemble variance, and can thus become negative for certain parameter settings. In order to ensure nonnegativity, a logarithmic barrier is introduced during the optimization. That is, the method does not minimize the mean CRPS, but the function:

mean(CRPS(ens, obs)) * (1 + 0.01 * barrier) where barrier = max(0, -log( minimal dressing variance ))

Since the evaluation of the CRPS is numerically expensive, the optimization can take a long time. Speed can be increased by optimizing the parameters only for a part of the forecast instances.

References

Broecker J. and Smith L. (2008). From ensemble forecasts to predictive distribution functions. Tellus (2008), 60A, 663--678. DOI 10.1111/j.1600-0870.2008.00333.x.

Examples

# Example:
  ens <- matrix(rnorm(500),100,5)
  obs <- rnorm(100)
  FitAkdParameters(ens, obs)