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# Score definitions and requirements

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The following scores are to be calculated for all parameters against both analysis (except mean sea-level pressure) and observation:

## Wind

### Mandatory:

• rms vector wind error
• mean error of wind speed

## Other parameters

### Mandatory

• Mean error
• Root mean square (rms) error
• Correlation coefficient between forecast and analysis anomalies (not required for obs)
• S1 score (only for MSLP and only against analysis)

• mean absolute error
• rms forecast and analysis anomalies (not required for observations)
• standard deviation of forecast and analysis fields (not required for observations)

# Definition

The following definitions should be used
Mean error

$M = \frac{1}{S_w} \sum_{i=1}^n w_i (x_f - x_v)_i$

where the sum of the weights

$S_w = \sum_{i=1}^n w_i$

Root mean square (rms) error

$rmse = \sqrt {\frac{1}{S_w} \sum_{i=1}^n w_i (x_f - x_v)_i^2 }$

Correlation coefficient between forecast and analysis anomalies

$r = \frac{\sum_{i=1}^n w_i (x_f-x_c-M_{f,c})_i (x_v-x_c-M_{v,c})_i}{\sqrt{\left(\sum_{i=1}^n w_i (x_f-x_c-M_{f,c})_i^2 \right) \left(\sum_{i=1}^n w_i (x_v-x_c-M_{v,c})_i^2 \right) }}$

rms vector wind error

$rmse = \sqrt {\frac{1}{S_w} \sum_{i=1}^n w_i (\vec{V}_f - \vec{V}_v)_i^2 }$

Mean absolute error

$MAE = \frac{1}{S_w} \sum_{i=1}^n w_i | x_f - x_v |_i$

rms anomaly

$rmsa = \sqrt {\frac{1}{S_w} \sum_{i=1}^n w_i (x - x_c)_i^2 }$

standard deviation of field

$sd = \sqrt {\frac{1}{S_w} \sum_{i=1}^n w_i (x - M_x)_i^2 }$

where

$M_x = \frac{1}{S_w} \sum_{i=1}^n w_i x_i$

S1 score

$S1 = 100 \frac{\sum_{i=1}^n w_i (e_g)_i}{\sum_{i=1}^n w_i (G_L)_i}$

Where:

$$x_f$$ = the forecast value of the parameter in question; $$x_v$$ = the corresponding verifying value; $$x_c$$ = the climatological value of the parameter; n = the number of grid points or observations in the verification area; $$M_{f,c}$$ = the mean value over the verification area of the forecast anomalies from climate; $$M_{v,c}$$ = the mean value over the verification area of the analysed anomalies from climate; $$\vec{V}_f$$ = the forecast wind vector; $$\vec{V}_v$$ = the corresponding verifying value;

The differentiation is approximated by differences computed on the verification grid:

$e_g = \left ( \left | \frac{\partial}{\partial x}(x_f-x_v)\right | + \left | \frac{\partial}{\partial y}(x_f-x_v)\right | \right )$ $G_L = \max \left ( \left | \frac{\partial x_f}{\partial x}\right | , \left | \frac{\partial x_v}{\partial x}\right | \right) + \max \left ( \left | \frac{\partial x_f}{\partial y}\right | , \left | \frac{\partial x_v}{\partial y}\right | \right)$

The weights w i applied at each grid point or observation location are defined as

• Verification against analyses: $$w_i = \cos \theta_i$$ , cosine of latitude at the the grid point i

• Verification against observations: $$w_i = 1/n$$ , all observations have equal weight

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