Benjamin Lang$\ast $ 
Institut für Geographie, Universität Augsburg 
Version from June 22, 2015
1 Introduction
Teleconnections like the North Atlantic Oscillation (NAO) denote recurring and persistent, largescale patterns of pressure and circulation anomalies that spans vast geographical areas. Since they have a strong inﬂuence on the weather activity of aﬀected regions, their predictability on interannual to decadal time scales is of great interest. The VADYtele plugin evaluates the predictability of indices of wellknown teleconnections, like the North Atlantic Oscillation (NAO), the Paciﬁc North American Pattern (PNA) or the Southern Oscillation Index (SOI).
In section 2, the methods concerning the calculation of the teleconnection indices and the applied skill scores are described. Sections 3 and 4 explain the input and the output, respectively, of the VADYtele plugin.
2 Methods
2.1 Teleconnections
Several teleconnection indices from both the Northern Hemisphere (e. g. NAO, PNA) and the Southern Hemisphere (e. g. SOI, SAM) can be analysed concerning their predictability on interannual to decadal time scales (cf. Table 1). They include diﬀerent calculation methods – zonal means, grid point values or (Rotated) Principal Component Analysis ((R)PCA) – but all of them are based on monthly / seasonally aggregated anomaly ﬁelds of sea level pressure (psl) or geopotential heights (zg).
Please note the indicated references for detailed information about the calculation methods of the particular indices.
Index (abbr.)  Index  Calculation  Reference 
EA  East Atlantic Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
EAWR  East Atlantic / Western Russia Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
EPNP  East Paciﬁc / North Paciﬁc Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
NAOCPC  North Atlantic Oscillation (after CPC)  RPCA  Barnston and Livezey [1987] 
NAOHUR  North Atlantic Oscillation (after Hurrell)  PCA  Hurrell [1995] 
PNA  Paciﬁc / North American Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
POL  Polar / Eurasia Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
SAM  Southern Annular Mode  Zonal means  Nan and Li [2003] 
SCAND  Scandinavia Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
SOI  Southern Oscillation Index  Grid point values  Trenberth [1984] 
TNH  Tropical / Northern Hemisphere Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
WP  West Paciﬁc Pattern (after CPC)  RPCA  Barnston and Livezey [1987] 
2.2 Skill scores
2.2.1 Correlation (CORR)
The Pearson productmoment correlation coeﬃcient is calculated between the observation and the forecast ($COR{R}_{fc,\tau}$) or the reference ($COR{R}_{ref,\tau}$) for a speciﬁc lead time $\tau $.
Figure 1 shows selected results for CORR.
2.2.2 Mean Squared Error Skill Score (MSESS)
The Mean Squared Error Skill Score (MSESS) is based on the Mean Squared Error (MSE), which is calculated for the diﬀerent lead years $\tau $ of the forecast ($MS{E}_{fc,\tau}$ ; shown in (1)) and the reference ($MS{E}_{ref,\tau}$):
$$MS{E}_{fc,\tau}=\frac{1}{I}\sum _{i=1}^{I}{\left({F}_{\tau ,i}{O}_{t\left(\tau ,i\right)}\right)}^{2}$$  (1) 
${F}_{\tau ,i}$ is the mean forecast derived from the ensemble for initialisation $i$ for a speciﬁc forecast lead time $\tau $ and ${O}_{t\left(\tau ,i\right)}$ the observation for time $t\left(\tau ,i\right)$, corresponding to the time of initialisation $i$ and forecast lead time $\tau $.
Finally, MSESS is calculated [Goddard et al., 2013] by relating forecast to reference:
$$MSES{S}_{\tau}=1\frac{MS{E}_{fc,\tau}}{MS{E}_{ref,\tau}}$$  (2) 
Figure 2 shows selected results for MSESS.
2.2.3 Ranked Probability Skill Score (RPSS)
The Ranked Probability Skill Score (RPSS) is based on the Ranked Probability Score (RPS), which is calculated for the diﬀerent lead years $\tau $ of the forecast ($RP{S}_{fc,\tau}$ ; shown in 3) and the reference ($RP{S}_{ref,\tau}$):
$$RP{S}_{fc,\tau}=\frac{1}{I}\sum _{i=1}^{I}\sum _{k=1}^{K}{\left({F}_{\tau ,i,k}{O}_{t\left(\tau ,i\right),k}\right)}^{2}$$  (3) 
${F}_{\tau ,i,k}$ is the cumulative probability derived from the ensemble for initialisation $i$ within class $k$ (with three equiprobable classes (below normal, normal, above normal), i.e. $K=3$) for a speciﬁc forecast lead time $\tau $, that is the fraction of ensemble members forecasting the occurrence of class $k$ or lower. ${O}_{t\left(\tau ,i\right),k}$ is the cumulative probability of class $k$ from observations for the time $t\left(\tau ,i\right)$, corresponding to the time of initialisation $i$ and forecast lead time $\tau $ and eﬀectively the Heaviside step function with ${O}_{t\left(\tau ,i\right),k}=0$ if a value within a class higher than $k$ is observed or else ${O}_{t\left(\tau ,i\right),k}=1$ [Kruschke et al., 2014].
Since it is biased for ﬁnite ensemble size $m$, the VADYtele plugin calculates the RPS with hypothetical ensemble size $M$ as estimation of $RP{S}_{fc,\tau ,m}$ [Ferro et al., 2008]:
$$RP{S}_{fc,\tau ,M}=RP{S}_{fc,\tau ,m}\frac{M{m}_{i}}{M\left({m}_{i}1\right)I}\sum _{i=1}^{I}\sum _{k=1}^{K}{F}_{\tau ,i,k}\left(1{F}_{\tau ,i,k}\right)$$  (4) 
Therefore, the comparison of forecasts and references with diﬀerent ensemble sizes is possible and RPSS can be calculated as:
$$RPS{S}_{\tau}=1\frac{RP{S}_{fc,\tau}}{RP{S}_{ref,\tau}}$$  (5) 
Figure 3 shows selected results for RPSS.
3 Input
This sections describes the various options for the plugin. For this purpose, table 2 lists and explains all possible options. You have to choose i.a. the FORECAST (“1”), REFERENCE (“2”) and OBSERVATION of your choice (Please note: REFERENCE and OBSERVATION are remapped to the resolution of the FORECAST).
Output  Output directory 
mandatory  default: /scratch/$user/evaluation_system/output/vadytele/ 
Index  Teleconnection index for evaluation 
mandatory 

Project1  FORECAST project, e.g. ”baseline0”, ”baseline1”, ”prototype”. 
mandatory 

Product1  FORECAST product, e.g. ”output2”, ”output1”. 
mandatory 

Institute1  FORECAST institute of experiment, e.g. ”mpim”. 
mandatory 

Model1  FORECAST model of experiment, e.g. ”mpiesmlr”. 
mandatory 

Experiment1  Preﬁx for FORECAST experiment, e.g. ”decs4e” or ”decadal”. 
mandatory 

Ensemblemembers1  FORECAST ensemble members as list, e.g. ”r1i1p1,r2i1p1,r3i1p1” 
mandatory 

Project2  REFERENCE project, e.g. ”baseline0”, ”baseline1”, ”prototype” 
mandatory 

Product2  REFERENCE product, e.g. ”output2”, ”output1”. 
mandatory 

Institute2  REFERENCE institute of experiment, e.g. ”mpim”. 
mandatory 

Model2  REFERENCE model of experiment, e.g. ”mpiesmlr”. 
mandatory 

Experiment2  Preﬁx for REFERENCE experiment, e.g. ”decs4e” or ”decadal”. 
mandatory 

Ensemblemembers2  REFERENCE ensemble members as list, e.g. ”r1i1p1,r2i1p1,r3i1p1” 
mandatory 

Season  Season, e.g. ”MAM”, ”JJA”, or a comma separated list with the month(s) of interest, i.e. ”2,4,6,7” 
mandatory 

Observation  Observation, e.g. ”eraint” or ”merra” 
mandatory 

Aggregation  Level of aggregation: ”monthly” or ”seasonal” 
mandatory 

Integrative  Set TRUE for integrative calculation (with all lead years) of the selected index (default=False, i.e. index is calculated separately for all lead years. 
mandatory 

4 Output
The processed ﬁles can be found in the selected Output folder for every lead year $\tau $. It contains i. a. plots of the teleconnections patterns (ﬁgure 4), the teleconnection index (ﬁgure 5) and the rank histogram (ﬁgure 6).
The output also encompasses raw data ﬁles of the teleconnection index and the skill scores (CORR, MSESS, RPSS; given as netCDF and ASCII ﬁles, respectively), which can be used for further analysis (cf. ﬁgure 7).
References
A.G. Barnston and R.E. Livezey. Classiﬁcation, Seasonality and Persistence of LowFrequency Atmospheric Circulation Patterns. Mon. Weather Rev., 115: 1083–1126, 1987. doi: 10.1175/15200493(1987)115.
C.A.T. Ferro, D.S. Richardson, and A.P. Weigel. On the eﬀect of ensemble size on the discrete and continuous ranked probability scores. Meteorol. Appl., 15:19–24, 2008. doi: 10.1002/met.45.
L. Goddard, A. Kumar, A. Solomon, D. Smith, G. Boer, P. Gonzalez, V. Kharin, W. Merryﬁeld, C. Deser, S. J. Mason, B. P. Kirtman, R. Msadek, R. Sutton, E. Hawkins, T. Fricker, G. Hegerl, C. A. T. Ferro, D. B. Stephenson, G. A. Meehl, T. Stockdale, R. Burgman, A. M. Greene, Y. Kushnir, M. Newman, J. Carton, I. Fukumori, and T. Delworth. A veriﬁcation framework for interannualtodecadal predictions experiments. Clim. Dyn., 40:245–272, 2013. doi: 10.1007/s0038201214812.
J.W. Hurrell. Decadal Trends in the North Atlantic Oscillation: Regional Temperatures and Precipitation. Science, 269:676–679, 1995. doi: 10.1126/science.269.5224.676.
T. Kruschke, H.W. Rust, C. Kadow, G.C. Leckebusch, and U. Ulbrich. Evaluating decadal predictions of northern hemispheric cyclone frequencies. Tellus A, 66, 22830, 2014. doi: 10.3402/tellusa.v66.22830.
S. Nan and J. Li. The relationship between the summer precipitation in the Yangtze River valley and the boreal spring Southern Hemisphere annular mode. Geophys. Res. Lett., 30:2266–2269, 2003. doi: 10.1029/2003GL018381.
K.E. Trenberth. Signal Versus Noise in the Southern Oscillation. Mon. Weather Rev., 112:326–332, 1984. doi: 10.1175/15200493(1984)112.