Evaluación de la autonomía del BC

Contrary to BMA, being based on mixture distribution, the ensemble model output statis- tics (EMOS) approach, which is also known as non-homogeneous regression, fits a single parametric distribution g employing summary statistics from the ensemble. Again, for a fixed location and prediction horizon, let y be a weather quantity of interest and x1, . . . , xM the corresponding ensemble member forecasts. The EMOS predictive distribution then has the general form

y|x1, . . . , xM ∼ g(y|x1, . . . , xM), (3.2) where the parameters of the parametric family of probability distributions on the right-hand side of (3.2) depend on all ensemble members simultaneously.

EMOS implementations are available for temperature, pressure, u- and v-wind (Gneiting et al., 2005), precipitation (Wilks, 2009; Scheuerer, 2014), wind speed (Thorarinsdottir and Gneiting, 2010; Möller, 2013; Lerch and Thorarinsdottir, 2013; Baran and Lerch, 2014) and wind gusts (Thorarinsdottir and Johnson, 2012), respectively. An overview is given in Table 3.2.

Concerning the choice of the training period and the composition of the training data, respectively, the same comments hold as for BMA.

Table 3.2: EMOS settings for univariate weather quantities y. In the case of precipitation amount, we refer to y1/2∈ R₊ for the truncated logistic distribution approach of Wilks (2009), as it applies to root-transformed precipitation accumulations. This table follows the similar summary in Table 2 in Schefzik et al. (2013).

Weather quantity Range Distribution (g)

Temperature y∈ R Normal

Pressure y∈ R Normal

u- and v-Wind y∈ R Normal

Precipitation amount y1/2∈ R₊ Truncated logistic

y∈ R+ Generalized extreme value left-censored at zero Wind speed y∈ R₊ Truncated normal with cut-off at zero

y∈ R₊ Gamma

y∈ R₊ Generalized extreme value

y∈ R₊ Log-normal

use the regression model

y = a + b1x1+ . . . + bMxM+ ε with ε∼ N (0, c + ds2), (3.3) where a, b1, . . . , bM ∈ R and c, d ∈ R0+ are parameters that need to be estimated, and

s2 _:= 1

M M



m=1

(x_m− ¯x)2 _{denotes the empirical ensemble variance, with the empirical ensemble}

mean ¯x := _M1 M

m=1

x_m. The corresponding Gaussian predictive EMOS distribution is then given by

y|x1, . . . , xM ∼ N (a + b1x1+ . . . + bMxM, c + ds2), (3.4) having mean a + b1x1+ . . . + bMxM and variance c + ds2. While a is a bias correction term, the regression coefficients b1, . . . , bM reflect the performance of the ensemble members over the training period relative to the other members, as well as the correlations between the ensemble members. If the ensemble members can be considered exchangeable, it needs to be assumed that the regression coefficients are equal, that is, b := b1 = · · · = bM. In this case,

y|x1, . . . , xM ∼ N (a + b(x1+· · · + xM), c + ds2)

= N (a + b M ¯x, c + ds2). (3.5)

The parameters a, b1, . . . , bM, c and d are estimated by minimizing the training CRPS, which is based on Formula (2.4) and expressed as an analytic function of the parameters. This estimation technique comes within an optimum score frame, which generalizes the classical maximum likelihood notion (Gneiting et al., 2005).

An EMOS predictive PDF in the case of temperature has been presented in Figure 1.4 in the introductory chapter, and another example for pressure is shown in Figure 3.3.

For precipitation, Scheuerer (2014) provides an EMOS variant using the generalized ex- treme value (GEV) distribution with CDF

G(y) := ⎧ ⎪ ⎨ ⎪ ⎩

exp−1 + ξy−μ_σ −1/ξ for ξ = 0, exp  − exp−(y−μ) σ   for ξ = 0,

Figure 3.3: 24 hour ahead EMOS predictive normal PDF for pressure at Frankfurt, valid 2:00 am on 20 June 2010. The 50-member ECMWF ensemble forecast is shown in red, and the verifying observation in blue, while the black lines indicate the 10th, 50th and 90th percentiles of the EMOS predictive distribution.

where the parameters μ, σ and ξ characterize location, scale and shape, respectively, of the GEV distribution. For ξ < 0, y > μ− (σ/ξ), one sets G(y) := 1, while for ξ > 0, y <

μ− (σ/ξ), one defines G(y) := 0. Scheuerer (2014) assumes ξ ∈ (−0.278, 1) because then,

the GEV has positive skew and its mean η exists, where

η =



μ + σΓ(1−ξ)−1_ξ for ξ= 0,

μ + σγ for ξ = 0,

with the Gamma function Γ and the Euler-Mascheroni constant γ ≈ 0.5772. In order to be appropriate for modeling precipitation amounts, Scheuerer (2014) considers the GEV distribution to be left-censored at zero, such that all mass below zero is assigned to exactly zero, with predictive CDF

˜

G(y) :=



G(y) for y≥ 0, 0 for y < 0.

According to Scheuerer (2014), this distribution is non-negative and exactly zero with pos- itive probability if either ξ≤ 0 or ξ > 0 and μ < σ/ξ. In his model, Scheuerer (2014) links the parameters η and σ from the GEV distribution left-censored at zero to the raw ensemble forecastx := (x1, . . . , xM) by setting

η := α₀+ α1x + α¯ 21{x=0}

and

σ := β0+ β1MD(x),

where ¯x is the ensemble mean,

1_{x=0}:= 1 M M  m=1 1_{xm=0}

the fraction of zero precipitation and

MD(x) := 1

M2

M

 _M

the ensemble mean difference. The parameters α0, α1, α2, β0, β1 and ξ are then estimated

by minimizing the training CRPS according to the formulas (2.5a) and (2.5b), respectively. Details about the implementation can be found in Scheuerer (2014).

Alternatively to the approach of Scheuerer (2014), truncated logistic distributions can be employed to model precipitation, motivated as follows. Predictions for the probability of the precipitation amount exceeding a certain threshold have been constructed using logistic regression (Wilks and Hamill, 2007; Hamill et al., 2008). However, if a full predictive distri- bution is desired, this approach fails because it is typically inconsistent across thresholds, violating the monotonicity restriction for CDFs. To address this shortcoming, Wilks (2009) introduced a technique, in which the postprocessed predictive CDF takes the form

G(y|x1, . . . , xM) =

exp(a + b1x1+ . . . + bMxM + h(y)) 1 + exp(a + b1x1+ . . . + bMxM + h(y))

, (3.6)

with parameters a, b1, . . . , bM, and with h increasing strictly monotonically and without bounds as a function of the precipitation accumulation y≥ 0, whereas G(y|x1, . . . , xM) = 0 for y < 0. If h is chosen to be linear, mixtures of a point mass at zero and a truncated logistic distribution are obtained. In view of the parametric family in (3.6), the approach of Wilks (2009) can be considered an EMOS method. In fact, Wilks (2009) applies the truncated logistic distributions to root-transformed precipitation amounts y1/2_{. More generally, the} use of transformations xα

1, . . . , xαM and yβ, respectively, with α, β ∈ R+, in model (3.6) is

feasible (Gneiting, 2014). The coefficients a, b1, . . . , bM and those included in the model for

h need to be fitted from appropriate training data. Generalizations allowing for interaction

terms are discussed by Ben Bouallègue (2013).

The EMOS approach for temperature was employed by Hagedorn et al. (2008) and Kann et al. (2009), among others, while Lerch and Thorarinsdottir (2013) provide a review and comparison of EMOS regression models for wind speed.

Locally adaptive methods for estimating EMOS parameters in the case of temperature are provided by Scheuerer and Büermann (2014) and Scheuerer and König (2014), respectively, who introduce further refinements of EMOS employing geostatistical methods, in which the postprocessing at individual stations varies in space. Similar locally adaptive EMOS models for wind speed are considered in Möller (2013).

In addition, Möller (2014) proposes a spatially adaptive extension of univariate EMOS for temperature called Markovian EMOS due to the Markovian dependence structure induced by the Gaussian Markov random field representations of the Gaussian fields used to model the parameters in the approach.