Similarity-based semilocal estimation of post-processing models
Sebastian Lerch and S´andor Baran German Probability and Statistics Days, Freiburg, March 2018
Probabilistic weather forecasts
Weather forecasts rely on numerical weather prediction (NWP) models that represent the physics and chemistry of the atmosphere based on systems of partial differential equations. However, there are major sources of uncertainty, including uncertainty about initial conditions and physical models. Carefully designed ensembles of NWP model runs seek to quantify uncertainty. However, despite their undisputed success, ensemble forecasts are subject to biases and lack calibration.
Example: Wind speed forecasts at Frankfurt airport
8 6 4 2
Forecast
10
12
14
Ensemble forecasts Observation
2011−03−20
2011−03−30
2011−04−09 Date
2011−04−19
Ensemble forecasts Observation
0
5
Forecast
10
15
Example: Wind speed forecasts at Frankfurt airport
2010−10−14
2010−10−24
2010−11−03 Date
2010−11−13
Statistical post-processing of ensemble forecasts
Ensemble forecasts typically fail to to represent the full model uncertainty and require statistical post-processing. I
Basic idea: Exploit structure in past forecast-observation pairs to correct the systematic errors in the model output by fitting distributional regression models.
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Post-processing combines physical weather models and statistical distributional regression modeling.
Non-homogeneous regression (NR) models Fit a parametric predictive distribution, y |x1 , . . . , xm ∼ Fθ (y |x1 , . . . , xm ), with parameters θ ∈ Rd depending on the ensemble forecasts through link functions, θ = g (x1 , . . . , xm ).
Non-homogeneous regression (NR) models Fit a parametric predictive distribution, y |x1 , . . . , xm ∼ Fθ (y |x1 , . . . , xm ), with parameters θ ∈ Rd depending on the ensemble forecasts through link functions, θ = g (x1 , . . . , xm ). This requires choosing an appropriate I
parametric model Fθ , g (S´andor Baran’s talk)
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loss function for parameter estimation (Bernhard Klar’s talk)
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training set (this talk)
An NR model for wind speed
y |x1 , . . . , xm ∼ Fθ = N[0,∞) (µ, σ 2 ), ¯ , c + d S 2 ), θ = (µ, σ 2 ) = g (x1 , . . . , xm ) = (a + b X ¯ denotes the ensemble mean, S 2 the ensemble variance. where X Parameters a, b, c, d are estimated over a rolling training period minimizing the CRPS.
Thorarinsdottir, T.L. and Gneiting, T. (2010) Probabilistic forecasts of wind speed: Ensemble model output statistics by using heteroscedastic censored regression. Journal of the Royal Statistical Society Series A, 173, 371–388.
Parameter estimation methods
Thereby, decisions regarding the spatial composition of training sets have to be made. I
regional estimation: data from all available stations are composited to form a single training set for all stations
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local estimation: consider only forecast cases from the single observation station of interest
Local estimation accounts for spatial variability and results in better predictions, but requires long training periods. Depending on the data, both options can be undesirable.
GLAMEPS data
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Grand Limited Area Model Ensemble Prediction System
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short-range multi-model EPS
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complex model structure combining different physical models and initialization times
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18 h ahead forecasts of 10 m wind speed for 1738 observation stations in Europe and Northern Africa
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data for October–November 2013, and March–May 2014
Locations of observation stations ●
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Longitude
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Structure of GLAMEPS ensemble I
52 members, combination of subensembles from 4 numerical models, each contributing I I I
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1 control forecast 6 perturbed members 6 perturbed members (lagged by 6 h)
12 groups of members which should have distinct NR coefficients bi , i.e., µ=a+
12 X i=1
¯i bi X
in the TN model
Structure of GLAMEPS ensemble I
52 members, combination of subensembles from 4 numerical models, each contributing I I I
I
1 control forecast 6 perturbed members 6 perturbed members (lagged by 6 h)
12 groups of members which should have distinct NR coefficients bi , i.e., µ=a+
12 X
¯i bi X
in the TN model
i=1 I
we consider the following variants and simplifications I I I
full model: account for all 12 groups, 15 parameters lag-ignoring model: ignore time-lagging, 11 parameters simplified model: ignore existence of groups, 4 parameters
Challenges in post-processing GLAMEPS forecasts I
regional parameter estimation does not account for variability I
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local parameter estimation causes numerical issues I
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single set of coefficients undesirable for large heterogeneous ensemble domain lack of training data (c.f. Hemri et al. (2014): optimal training period lengths of 365–1816 days) large number of parameters leads to numerical problems only simplified model can be estimated successfully
ideal solution: compute re-forecasts of past cases with current model, however, usually impossible in practice due to large computational costs of generating ensemble forecasts
Hemri, S., Scheuerer, M., Pappenberger, F., Bogner, K. and Haiden, T. (2014) Trends in the predictive performance of raw ensemble weather forecasts. Geophysical Research Letters, 41, 9197–9205.
Similarity-based semi-local parameter estimation
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basic idea: augment training data for a given station with data from stations with similar characteristics
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thereby, combine advantages of regional and local estimation I I I
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locally adaptive, but parsimonious complex models can be estimated without numerical issues improved predictive performance
choice of similar stations is based on I I
distance functions clustering
Distance-based model estimation I
Two-step approach 1. compute pair-wise similarities (distances) d(i, j) between stations 2. add corresponding forecast cases from the L most similar stations to the training set for the station of interest
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distances are based on characteristics of stations, distribution of observations, and forecast errors of the ensemble
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follows basic idea of Hamill et al. (2008)
Hamill, T.M., Hagedorn, R. and Whitaker J.S. (2008) Probabilistic Forecast Calibration Using ECMWF and GFS Ensemble Reforecasts. Part II: Precipitation. Monthly Weather Review, 136, 2620–2632.
Example 1: Geographical locations (Hamill et al., 2008) d
(1)
q (i, j) = (Xi − Xj )2 + (Yi − Yj )2 .
Euclidean distance of locations ●
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70
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●
● ●
●
Latitude
●●●
60
40 ● ●
30
●
● ●
−20
●
●
●
● ●
● ●●●
60
40 ● ●
30
●
●
● ●
20
Longitude
40
● ●
● ● ●●
60
●
● ●
● ●●●● ● ●●● ●● ●●● ●● ● ● ●●● ●● ● ●● ●●●● ●● ● ● ● ●●● ●● ● ●● ● ● ●●● ● ● ●● ●● ●●● ●● ●● ● ● ●● ● ● ● ● ●●● ●● ●● ● ●● ● ● ●● ● ●●● ● ● ● ●● ●
50
●
0
●
●
●
● ●
● ● ● ● ●● ●● ● ●● ● ●● ● ●● ● ●●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ● ●
50
●●● ●● ● ●
● ● ● ● ● ●
●
●
Latitude
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●
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−20
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0
20
Longitude
40
60
How many similar stations should be added? 0.94
Distance 1: Geographical locations ● ● ● ●
0.92
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0.90
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0.88
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0.86
CRPS
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0.84
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0
20
40
60
Number of similar stations
CRPS of GLAMEPS: 1.058; best regional model: 0.955; best local model: 0.790
simpl., n = 25 simpl., n = 50 simpl., n = 80 lag−ign., n = 25 lag−ign., n = 50 lag−ign., n = 80 full, n = 25 full, n = 50 full, n = 80
Example 2: Station climatology + forecast errors d (4) (i, j) =
1 X ˆ 1 X ˆ e ˆje (x) , Gi (x) − G Fi (x) − Fˆj (x) + 0 |S| |S | x∈S x∈S 0 | {z } | {z } station climatology
forecast error of ensemble
where Fˆi = empirical CDF of observations at station i; S = {0, 0.5, . . . , 14.5, 15}; ˆie = empirical CDF of forecast errors of ensemble mean; S 0 = {−10, −9.5, . . . , 10} G ● ●
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50
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40
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60
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50
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30
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Longitude
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30 40
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20
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40
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0
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●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ●●● ● ● ●● ●● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ●● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●●●●● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ● ● ●● ● ●● ● ●●● ●●● ●● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ● ●●● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ●● ●● ● ● ● ● ●●● ●●●● ● ●●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ●●● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●●●●● ●● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ●●●●●● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ●●●● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●●●● ●●● ●●●●●● ● ●● ● ● ● ● ● ●● ●● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ●●● ● ● ●●● ●● ● ●●● ●●●● ● ● ● ● ● ●● ● ● ●● ●● ● ●●●●● ●●● ●● ● ● ● ● ●● ●● ● ●●● ● ● ●● ●● ●● ●●● ● ● ●●●●●●●●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ●●● ●● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●●● ● ● ● ●●● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ●●●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ●●● ● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●●●●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ●● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ●● ●● ● ●● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ●● ● ● ●●● ●●● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
●
● ●
●
●
60
● ●
70
●
Latitude
Latitude
● ●●●
● ● ●
●
● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● ● ● ●●● ● ● ●● ●● ● ● ●● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ●● ● ● ●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ●● ● ●● ●● ●● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●●● ●● ● ●● ● ● ● ● ● ● ● ● ● ●●●● ● ● ●● ● ● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ● ● ●● ● ●● ● ●● ● ● ● ●● ● ● ● ●●●●● ●● ●● ● ● ● ● ●●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ●● ● ● ●● ● ●● ● ●●● ●●● ●● ● ● ● ● ● ● ●● ● ● ●● ●● ● ●● ● ● ● ●●● ● ●● ● ● ● ●●● ● ● ● ●● ● ● ● ● ●● ●● ● ● ● ● ● ● ●● ● ●●● ● ● ●● ●● ● ● ● ● ●●● ●●●● ● ●●● ● ● ●●● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ●●● ● ● ● ●● ●●● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●●● ● ● ● ●● ●● ● ● ● ●● ● ● ● ● ● ●● ● ● ●●●●● ●● ●●● ●● ● ● ● ● ● ● ● ● ●● ● ●●●●●● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ●●●● ●● ●● ● ● ●● ●● ● ● ● ●● ● ● ●● ● ●●● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ●● ● ●●● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ●●● ● ● ●●●● ●●● ●●●●●● ● ●● ● ● ● ● ● ●● ●● ● ●●● ● ● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ●● ●●● ● ● ●●● ●● ● ●●● ●●●● ● ● ● ● ● ●● ● ● ●● ●● ● ●●●●● ●●● ●● ● ● ● ● ●● ●● ● ●●● ● ● ●● ●● ●● ●●● ● ● ●●●●●●●●● ●● ●●●● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ● ●● ●●● ●● ●●● ● ● ●● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ●● ● ●● ●●● ● ● ● ● ● ● ● ● ●●● ●● ● ● ● ● ●●● ● ● ● ●●● ● ●● ● ● ●● ●● ● ●● ● ● ● ● ● ●● ● ●● ●● ● ● ● ● ● ●●● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ●● ● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ●●●●● ●● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ●● ● ●● ● ● ● ●●● ● ● ● ● ●●● ● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ●●●● ● ● ● ● ● ●● ● ● ● ●●● ● ●● ●●●●●●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●● ●● ● ● ● ● ● ●●● ● ●● ● ● ● ●● ●● ●● ● ●● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ●● ● ●● ● ● ● ●● ● ● ● ● ● ●● ●●● ● ● ●● ● ● ● ●● ● ● ●●● ●●● ● ● ●●● ●● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ●● ● ● ●● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ●● ● ●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ●● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●●● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
● ● ● ● ● ●
●●
● ●
70
60
−20
●
●
0
20
Longitude
40
60
How many similar stations should be added? 0.795
Distance 4: Climatology + forecast errors
●
● ●
●
0.785
● ● ● ●
● ●
CRPS
● ●
● ●
● ● ● ●
●
0.775
●
● ●
●
● ● ●
● ●
●
●
● ● ●
0.765
●
0
simpl., n = 25 simpl., n = 50 simpl., n = 80
20
40
60
Number of similar stations
CRPS of GLAMEPS: 1.058; best regional model: 0.955; best local model: 0.790 (gray dashed line)
lag−ign., n = 25 lag−ign., n = 50 lag−ign., n = 80 full, n = 25 full, n = 50 full, n = 80
Clustering-based model estimation
I
application of k-means clustering 1. determine clusters based on various feature sets. 2. parameters are estimated separately for each cluster using only data from stations within the given cluster.
I
feature sets for clustering depend on I I I
I
distribution of the observations distribution of forecast errors of the ensemble combination thereof
computationally much more efficient than distance-based semi-local model estimation
Example: Station climatology + forecast errors Feature set given by equidistant quantiles of distribution of observations + equidistant quantiles of distribution of forecast errors of ensemble mean ●
●
●
●●
● ●
70
●
● ●
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●
●
● ●
● ●
●
Latitude
●●●
60
50
40 ● ●
30
●
● ●
−20
● ●
●
●
0
20
Longitude
40
60
How many clusters? 0.78 0.80 0.82 0.84 0.86 0.88 0.90
CRPS
Feature set 3: Climatology + forecast errors
● ● ● ●
● ● ●
● ● ●
● ● ● ●
● ● ●
0
● ● ● ●
20
● ● ●
● ●
40
● ● ●
● ●
60
●
● ●
●
●
●
● ●
80
● ●
100
Number of clusters
all models estimated with a fixed number of 24 features
simpl., n = 25 simpl., n = 50 simpl., n = 80 lag−ign., n = 25 lag−ign., n = 50 lag−ign., n = 80 full, n = 25 full, n = 50 full, n = 80
Computational aspects & summary
I
semi-local models outperform local and regional models: mean CRPS distance-based semi-local
I
clustering-based < local < regional semi-local
computational costs regional <
I
<
clustering-based < local < semi-local
distance-based semi-local
allow for estimation of complex models without numerical stability issues, straightforward to implement
Outlook I
extension towards multivariate model estimation utilizing spatio-temporal dependecies (Annette M¨ oller’s talk)
I
comparison and combination with methods based on historical analogs
I
investigate alternative, meteorologically meaningful similarity measures
Lerch, S. and Baran, S. (2017) Similarity-based semilocal estimation of post-processing models, Journal of the Royal Statistical Society Series C, 66, 29–51
Outlook I
extension towards multivariate model estimation utilizing spatio-temporal dependecies (Annette M¨ oller’s talk)
I
comparison and combination with methods based on historical analogs
I
investigate alternative, meteorologically meaningful similarity measures
Lerch, S. and Baran, S. (2017) Similarity-based semilocal estimation of post-processing models, Journal of the Royal Statistical Society Series C, 66, 29–51
Thank you for your attention.
