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b/vignettes/ScoringForecast.md new file mode 100644 index 00000000..febd5994 --- /dev/null +++ b/vignettes/ScoringForecast.md @@ -0,0 +1,244 @@ +--- +title: "Untitled" +output: github_document +--- + + +This vignette illustrates several examples of how to score a seasonal forecast using different functions in s2dverification. + + +### 1- Load dependencies + +This example requires the following system libraries: + +- libssl-dev +- libnecdf-dev +- cdo + + +The **s2dverification R package** should be loaded by running the following lines in R, onces it is integrated into CRAN mirror. + +```r +library(s2dverification) +library(devtools) +``` + + + +### 2- Define the problem and loading the data with the corresponding parameters + +In this vignette we will quantify how skilled a seasonal forecast is. The model will be the EUROSIP multi-model seasonal forecasting system and our real truth will be observations represented by the ERAinterim reanalysis dataset. + +For more information about both datasets see the next documentation: +- [**ERAinterim**](https://www.ecmwf.int/en/forecasts/datasets/archive-datasets/reanalysis-datasets/era-interim). +- [**EUROSIP system**](https://www.ecmwf.int/en/forecasts/documentation-and-support/long-range/seasonal-forecast-documentation/eurosip-user-guide/multi-model). + +Both datasets can be downloaded from the corresponding server. However in this example we will use the BSC esarchive database. Files path parameters for the Load() function are defined as follows: + +```r +path_exp_EUROSIP <- list(name = 'system4_m1', + path = file.path('/esarchive/exp/EUROSIP/ecmwf', + '$EXP_NAME$/monthly_mean', + '$VAR_NAME$_f6h/$VAR_NAME$_$START_DATE$.nc')) + +path_obs_eraI <- list(name = 'erainterim', + path = file.path('/esarchive/recon/ecmwf', + '$OBS_NAME$/monthly_mean', + '$VAR_NAME$_f6h/$VAR_NAME$_$YEAR$$MONTH$.nc')) +``` + +Our case of study will be predicting the North Atlantic Oscillation index (**NAO**(https://en.wikipedia.org/wiki/North_Atlantic_oscillation)), usually defined as the difference in the sea level pressure anomaly between the permanent High pressure system over the Azores Island and the Low pressure system over Iceland. Changes in the NAO index have a direct impact on European season weather since it controls the intensity and pathways of the storms in the continent; Positive NAO values are related with stronger storms and thus wet winters in Central and Northern Europe and its Atlantic facade. Negative NAO values correspond to reduced storm activity and rainfall in Northern Europe but increased in the South and Mediterranean Sea. Especially during the months of November to April, the NAO is responsible for much of the variability of weather in the North Atlantic region, affecting wind speed and wind direction changes, changes in temperature and moisture distribution and the intensity, number and track of storms. + +For this vignette we will use the whole North Atlantic as geographical domain and we will select all forecasts starting in November for the 1980 - 2010 time period. The EUROSIP system provides 7-months long window forecast inluding 51 different members. + +```r +#North Atlantic domain definition +lonNA <- c(-90, 40) +latNA <- c(20, 80) + +#Selected time periods: +startDates <- paste(1990:2000, '1101', sep = '') + +Loading sea level pressure maps for the whole North Atlantic and for the specific period +data <- Load(var = 'psl', + exp = list(path_exp_EUROSIP), + obs = list(path_obs_eraI), + sdates = startDates, + output = 'lonlat', + lonmin = lonNA[1], lonmax = lonNA[2], + latmin = latNA[1], latmax = latNA[2]) + + +#Retrieving observational and forecast data +obs <- data$obs +exp <- data$mod + +``` + + +### 3- NAO index, Forecast vs Observations + +As a first step we will compare the predicted intensity of the NAO index for observations and for the forecast system. + +In s2dverification the NAO index is computed using a common alternative definition; The NAO is defined as the first mode of variability for the sea level pressure in the North Atlantic region [20N-80N, 80W-40E]. The NAO() function requires as input anomalies that are previously computed using Ano_CrossValid() function. + +```r +#Computing anomalies along sdates dimension +ano <- Ano_CrossValid(exp, obs) + +#Computing NAO index as the first mode of slp variability in the North Atlantic +nao <- NAO(ano$ano_exp, ano$ano_obs, data$lon, data$lat) + +#For clarification, we normalize the NAO index for each member dividing by the corresponding standard deviation +nao_exp_sd <- apply(nao$NAO_exp, MARGIN = 1, sd) +nao_obs_sd <- sd(nao$NAO_obs) +nao_exp_n <- nao$NAO_exp / nao_exp_sd +nao_obs_n <- nao$NAO_obs / nao_obs_sd + + +# Finally plot the NAO index using Box plots for all decembers. +PlotBoxWhisker(nao_exp_n, nao_obs_n, toptitle = "NAO index, DJF", ytitle = "NAO index (PC1) psl", + monini = 12, yearini = 1980, freq = 1, expname = "EUROSIP", obsname = "EraInterim", + fileout = "NAO_BoxWhisker_Original.png") + +``` + + + +The figure above does not represent a good agreement between observations (blue line) and forecast (whisker boxes) due to the large dispersion through the 51 model members. The NAO signal is too noisy within the model ensemble thus almost disappearing (close to 0). + + +###4 Quantifying the skillfulness of the prediction. The RMSSS + +Let's quantify how good or bad it is this prediction looking also for a better understanding of what is happening. To do so, we will use the Root Mean Square Error (RMSE), that it is just the squared and then rooted difference between the predicted and the true value. + +s2dverification includes the RMS() function to directly calculate the RMSE, however, in order to get a better understanding of the output values we will use the RMSSS() function (Root Mean Square Error Skill Score). The RMSSS equals the RMSE but it is normalized by a reference RMSE, usually asociated with the intrinsic variability of the system (for example the standard deviation of the climatology). In s2dverification the score is computed such the best RMSSS score would be 1 (RMSE equals 0), while if RMSSS equals 0, the RMSE equals the variability reference of the system (i.e. the standard deviation). RMSSS can also be negative, meaning RMSE is greater than the variability reference. + +```r +#Calculating a RMSSS for the ensemble of members average from the mean NAO timeseries of all members +rmsss_m = RMSSS(var_exp = array(apply(nao_exp_n, MARGIN = 2, FUN = mean), dim = c(1,11)), + var_obs = nao_obs_n, pval = F) + +#Computing the RMSSS for each member +rmsss =RMSSS(var_exp = nao_exp_n, var_obs = nao_obs_n, pval = F) + +#Plotting..... +layout(matrix(c(1, 2), 1 , 2, byrow = TRUE)) + +#Plotting RMSSS for all members +plot(rmsss, type = 'h', lwd = 5, col = 'grey', xlab = 'Members', mgp = c(3,1,0), + main = sprintf('RMSSS for each member (RMSSS average %1.3f )', rmsss_m), + ylim = c(-1,1)) +grid(col ='grey') +lines(rep(0, Lmember), lwd = 1, col = 'black') + +#Plotting boxplot for good members +igood = which(rmsss > 0.2) +rmsss_m_goods = RMSSS(var_exp = array(apply(nao_exp_n[igood,], MARGIN = 2, FUN = mean), dim = c(1,11)), + var_obs = nao_obs_n, pval = F) + +PlotBoxWhisker(nao_exp_n[igood,], nao_obs_n, + toptitle = sprintf("NAO index, good members only (RMSSS=%1.2f)", rmsss_m_goods), + ytitle = "NAO index (PC1) psl", + monini = 12, yearini = 1980, freq = 1, + expname = "EUROSIP", obsname = "EraInterim", + fileout = NULL) +``` + + + + +The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. The general RMSSS for the whole ensemble is 0.015, what means a not very useful ensemble prediction. + +However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 45 and 47 are used. Now most marked NAO events are correctly predicted giving a RMSSS of 0.66 for this 2-member ensemble. + + +###4 Quantifying the skillfulness of the prediction. The Brier Score + +Another option to quantify the goodness of this prediction is using the BrierScore() function. The BS measures the accuracy of a probabilistic categorical prediction. BS scores perfect prediction when BS equals 0 and worst score corresponds to 1 value. + +Moreover, the BS can be descomposed in 3 terms: reliability, resolution (negative contribution) and uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). + +In the case of the NAO index, we can define a NAO event always that the index is greater than 0. Therefore, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored 2-member ensemble of previous section. + + +```r + +#Defining NAO events +Lmember <- length(exp[1, , 1, 1, 1, 1]) +Lsdates <- length(startDates) + +#For observations +nao_event_obs <- rep(0, Lsdates) +nao_event_obs[which(nao_obs_n > 0)] <- 1 + +#For all members +nao_event_exp <- array(0, dim = c(Lmember, Lsdates)) +nao_event_exp[which(nao_exp_n > 0)] = 1 +nao_event_exp_prob <- apply(nao_event_exp, MARGIN = 2, mean) + +BS <- BrierScore(nao_event_obs, nao_event_exp_prob) + +BSscore <- BS$bs +BSrel <- BS$rel +BSres <- BS$res +BSunc <- BS$unc + +#For good members nly +nao_event_exp_good <- array(0, dim = c(length(igood), Lsdates)) +nao_event_exp_good[which(nao_exp_n[igood,] > 0)] = 1 +nao_event_exp_good_prob <- apply(nao_event_exp_good, MARGIN = 2, mean) + +BS_good <- BrierScore(nao_event_obs, nao_event_exp_good_prob) + +BSscore_good <- BS_good$bs +BSrel_good <- BS_good$rel +BSres_good <- BS_good$res +BSunc_good <- BS_good$unc + + +#Plotting NAO events and prediction probabilities +layout(matrix(c(1, 2), 1, 2, byrow = TRUE)) + +#For all ensemble members +plot(BS$obs - 0.5, type='p', lwd = 5, col = 'red', + xlab = 'NAO events', yaxt = 'n', ylab = 'NAO probabilities', mgp = c(3,1,0)) +grid(col = 'grey') +lines(BS$obs - 0.5, type = 'h', lwd = 5, col = 'red', yaxt = 'n') +lines(BS$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') +lines(BS$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') +axis(2, at = seq(-0.5, 0.5, 0.25), labels = seq(0, 1, 0.25), mgp = c(3,1,0)) +lines(rep(0, Lmember), lwd=2, col = 'black') +title('Predictions for all ensemble members') + +#For good ensemble members only +plot(BS_good$obs - 0.5, type = 'p', lwd = 5, col = 'red', + xlab = 'NAO events', yaxt = 'n', ylab = 'NAO probabilities', mgp = c(3,1,0)) +grid(col = 'grey') +lines(BS_good$obs - 0.5, type = 'h', lwd = 5, col = 'red', yaxt = 'n') +lines(BS_good$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') +lines(BS_good$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') +axis(2, at = seq(-0.5, 0.5, 0.25), labels = seq(0, 1, 0.25), mgp = c(3, 1, 0)) +lines(rep(0, Lmember), lwd = 2, col = 'black') +title('Predictions for good ensemble members') + +``` + + + +Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the best members. + +For the whole ensemble, the results are: +Total BS = 0.239 +Reliability = 0.021 +Resolution = 0.012 +Uncertainty = 0.248 + +For the ensemble of best members, the results are: +Total BS = 0.091 +Reliability = 0.0 +Resolution = 0.157 +Uncertainty = 0.248 + +As expected, the ensemble of best members presents a clearly better Brier Score (lower value). While the uncertainty is obviously the same for the two cases, since the observations to compare with are the same, the reliability and resolution scores are better for the second case. Specially the resolution scores much better in the 2-member ensemble due to the agreement of all the members in correctly detecting positive and negative NAO events. + -- GitLab From 02c2938fcb1f2376605d5ca3774cb45248256e1d Mon Sep 17 00:00:00 2001 From: jpena Date: Wed, 13 Mar 2019 16:34:58 +0100 Subject: [PATCH 02/18] Minor edit in the vignette --- vignettes/ScoringForecast.md | 28 ++++++++++++++-------------- 1 file changed, 14 insertions(+), 14 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index febd5994..e3284753 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -7,7 +7,7 @@ output: github_document This vignette illustrates several examples of how to score a seasonal forecast using different functions in s2dverification. -### 1- Load dependencies +### 1-Load dependencies This example requires the following system libraries: @@ -25,7 +25,7 @@ library(devtools) -### 2- Define the problem and loading the data with the corresponding parameters +### 2-Define the problem and loading the data with the corresponding parameters In this vignette we will quantify how skilled a seasonal forecast is. The model will be the EUROSIP multi-model seasonal forecasting system and our real truth will be observations represented by the ERAinterim reanalysis dataset. @@ -47,7 +47,7 @@ path_obs_eraI <- list(name = 'erainterim', '$VAR_NAME$_f6h/$VAR_NAME$_$YEAR$$MONTH$.nc')) ``` -Our case of study will be predicting the North Atlantic Oscillation index (**NAO**(https://en.wikipedia.org/wiki/North_Atlantic_oscillation)), usually defined as the difference in the sea level pressure anomaly between the permanent High pressure system over the Azores Island and the Low pressure system over Iceland. Changes in the NAO index have a direct impact on European season weather since it controls the intensity and pathways of the storms in the continent; Positive NAO values are related with stronger storms and thus wet winters in Central and Northern Europe and its Atlantic facade. Negative NAO values correspond to reduced storm activity and rainfall in Northern Europe but increased in the South and Mediterranean Sea. Especially during the months of November to April, the NAO is responsible for much of the variability of weather in the North Atlantic region, affecting wind speed and wind direction changes, changes in temperature and moisture distribution and the intensity, number and track of storms. +Our case of study will be predicting the North Atlantic Oscillation index ([**NAO**](https://en.wikipedia.org/wiki/North_Atlantic_oscillation)), usually defined as the difference in the sea level pressure anomaly between the permanent High pressure system over the Azores Island and the Low pressure system over Iceland. Changes in the NAO index have a direct impact on European season weather since it controls the intensity and pathways of the storms in the continent; Positive NAO values are related with stronger storms and thus wet winters in Central and Northern Europe and its Atlantic facade. Negative NAO values correspond to reduced storm activity and rainfall in Northern Europe but increased in the South and Mediterranean Sea. Especially during the months of November to April, the NAO is responsible for much of the variability of weather in the North Atlantic region, affecting wind speed and wind direction changes, changes in temperature and moisture distribution and the intensity, number and track of storms. For this vignette we will use the whole North Atlantic as geographical domain and we will select all forecasts starting in November for the 1980 - 2010 time period. The EUROSIP system provides 7-months long window forecast inluding 51 different members. @@ -76,7 +76,7 @@ exp <- data$mod ``` -### 3- NAO index, Forecast vs Observations +### 3-NAO index, Forecast vs Observations As a first step we will compare the predicted intensity of the NAO index for observations and for the forecast system. @@ -108,7 +108,7 @@ PlotBoxWhisker(nao_exp_n, nao_obs_n, toptitle = "NAO index, DJF", ytitle = "NAO The figure above does not represent a good agreement between observations (blue line) and forecast (whisker boxes) due to the large dispersion through the 51 model members. The NAO signal is too noisy within the model ensemble thus almost disappearing (close to 0). -###4 Quantifying the skillfulness of the prediction. The RMSSS +### 4-Quantifying the skillfulness of the prediction. The RMSSS Let's quantify how good or bad it is this prediction looking also for a better understanding of what is happening. To do so, we will use the Root Mean Square Error (RMSE), that it is just the squared and then rooted difference between the predicted and the true value. @@ -153,7 +153,7 @@ The above figure shows very different RMSSS for different members (left plot). M However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 45 and 47 are used. Now most marked NAO events are correctly predicted giving a RMSSS of 0.66 for this 2-member ensemble. -###4 Quantifying the skillfulness of the prediction. The Brier Score +### 5-Quantifying the skillfulness of the prediction. The Brier Score Another option to quantify the goodness of this prediction is using the BrierScore() function. The BS measures the accuracy of a probabilistic categorical prediction. BS scores perfect prediction when BS equals 0 and worst score corresponds to 1 value. @@ -229,16 +229,16 @@ title('Predictions for good ensemble members') Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the best members. For the whole ensemble, the results are: -Total BS = 0.239 -Reliability = 0.021 -Resolution = 0.012 -Uncertainty = 0.248 +Total BS = 0.239, +Reliability = 0.021, +Resolution = 0.012, +Uncertainty = 0.248. For the ensemble of best members, the results are: -Total BS = 0.091 -Reliability = 0.0 -Resolution = 0.157 -Uncertainty = 0.248 +Total BS = 0.091, +Reliability = 0.0, +Resolution = 0.157, +Uncertainty = 0.248. As expected, the ensemble of best members presents a clearly better Brier Score (lower value). While the uncertainty is obviously the same for the two cases, since the observations to compare with are the same, the reliability and resolution scores are better for the second case. Specially the resolution scores much better in the 2-member ensemble due to the agreement of all the members in correctly detecting positive and negative NAO events. -- GitLab From fa8d60f8975eb38290888b67c4122a1d012565ab Mon Sep 17 00:00:00 2001 From: jpena Date: Wed, 13 Mar 2019 16:53:51 +0100 Subject: [PATCH 03/18] Minor edits 2 --- vignettes/ScoringForecast.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index e3284753..53e1d5a7 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -57,7 +57,7 @@ lonNA <- c(-90, 40) latNA <- c(20, 80) #Selected time periods: -startDates <- paste(1990:2000, '1101', sep = '') +startDates <- paste(1980:1990, '1101', sep = '') Loading sea level pressure maps for the whole North Atlantic and for the specific period data <- Load(var = 'psl', -- GitLab From 0a9a167c53bfcf0fc47245d8dda07a1974fb8b18 Mon Sep 17 00:00:00 2001 From: jpena Date: Wed, 13 Mar 2019 17:29:00 +0100 Subject: [PATCH 04/18] Minor edits 3 --- vignettes/ScoringForecast.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 53e1d5a7..d8fae202 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -80,7 +80,7 @@ exp <- data$mod As a first step we will compare the predicted intensity of the NAO index for observations and for the forecast system. -In s2dverification the NAO index is computed using a common alternative definition; The NAO is defined as the first mode of variability for the sea level pressure in the North Atlantic region [20N-80N, 80W-40E]. The NAO() function requires as input anomalies that are previously computed using Ano_CrossValid() function. +In s2dverification the NAO index is computed using a common alternative definition; The NAO is defined as the first mode of variability for the sea level pressure in the North Atlantic region [20N-80N, 80W-40E]. The `NAO()` function requires as input anomalies that are previously computed using `Ano_CrossValid()` function. ```r #Computing anomalies along sdates dimension @@ -112,7 +112,7 @@ The figure above does not represent a good agreement between observations (blue Let's quantify how good or bad it is this prediction looking also for a better understanding of what is happening. To do so, we will use the Root Mean Square Error (RMSE), that it is just the squared and then rooted difference between the predicted and the true value. -s2dverification includes the RMS() function to directly calculate the RMSE, however, in order to get a better understanding of the output values we will use the RMSSS() function (Root Mean Square Error Skill Score). The RMSSS equals the RMSE but it is normalized by a reference RMSE, usually asociated with the intrinsic variability of the system (for example the standard deviation of the climatology). In s2dverification the score is computed such the best RMSSS score would be 1 (RMSE equals 0), while if RMSSS equals 0, the RMSE equals the variability reference of the system (i.e. the standard deviation). RMSSS can also be negative, meaning RMSE is greater than the variability reference. +s2dverification includes the `RMS()` function to directly calculate the RMSE, however, in order to get a better understanding of the output values we will use the `RMSSS()` function (Root Mean Square Error Skill Score). The RMSSS equals the RMSE but it is normalized by a reference RMSE, usually asociated with the intrinsic variability of the system (for example the standard deviation of the climatology). In s2dverification the score is computed such the best RMSSS score would be 1 (RMSE equals 0), while if RMSSS equals 0, the RMSE equals the variability reference of the system (i.e. the standard deviation). RMSSS can also be negative, meaning RMSE is greater than the variability reference. ```r #Calculating a RMSSS for the ensemble of members average from the mean NAO timeseries of all members @@ -155,7 +155,7 @@ However, we can select the members that present a better RMSSS, i.e. those close ### 5-Quantifying the skillfulness of the prediction. The Brier Score -Another option to quantify the goodness of this prediction is using the BrierScore() function. The BS measures the accuracy of a probabilistic categorical prediction. BS scores perfect prediction when BS equals 0 and worst score corresponds to 1 value. +Another option to quantify the goodness of this prediction is using the `BrierScore()` function. The BS measures the accuracy of a probabilistic categorical prediction. BS scores perfect prediction when BS equals 0 and worst score corresponds to 1 value. Moreover, the BS can be descomposed in 3 terms: reliability, resolution (negative contribution) and uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). -- GitLab From 4e50ac719ddd7c5b5720ea924616cf98dfcb742e Mon Sep 17 00:00:00 2001 From: jpena Date: Thu, 14 Mar 2019 16:08:07 +0100 Subject: [PATCH 05/18] Format edits in figures --- vignettes/NAOindex_81to91.png | Bin 0 -> 50688 bytes vignettes/NAOpredictions.png | Bin 33207 -> 38181 bytes vignettes/RMSSSforNAOprediction.png | Bin 0 -> 84060 bytes vignettes/ScoringForecast.md | 104 ++++++++++++++++------------ 4 files changed, 58 insertions(+), 46 deletions(-) create mode 100644 vignettes/NAOindex_81to91.png create mode 100644 vignettes/RMSSSforNAOprediction.png diff --git a/vignettes/NAOindex_81to91.png b/vignettes/NAOindex_81to91.png new file mode 100644 index 0000000000000000000000000000000000000000..7ee1220690eab76395e8d4dcde3f11a1c4d76553 GIT binary patch literal 50688 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Changes in the NAO index have a direct impact on European season weather since it controls the intensity and pathways of the storms in the continent; Positive NAO values are related with stronger storms and thus wet winters in Central and Northern Europe and its Atlantic facade. Negative NAO values correspond to reduced storm activity and rainfall in Northern Europe but increased in the South and Mediterranean Sea. Especially during the months of November to April, the NAO is responsible for much of the variability of weather in the North Atlantic region, affecting wind speed and wind direction changes, changes in temperature and moisture distribution and the intensity, number and track of storms. -For this vignette we will use the whole North Atlantic as geographical domain and we will select all forecasts starting in November for the 1980 - 2010 time period. The EUROSIP system provides 7-months long window forecast inluding 51 different members. +For this vignette we will use the whole North Atlantic as geographical domain and we will select all forecasts starting in November for the 1981 - 1991 time period. The EUROSIP system provides 7-months long window forecast inluding 51 different members. ```r #North Atlantic domain definition @@ -57,7 +57,7 @@ lonNA <- c(-90, 40) latNA <- c(20, 80) #Selected time periods: -startDates <- paste(1980:1990, '1101', sep = '') +startDates <- paste(1981:1991, '1101', sep = '') Loading sea level pressure maps for the whole North Atlantic and for the specific period data <- Load(var = 'psl', @@ -90,20 +90,20 @@ ano <- Ano_CrossValid(exp, obs) nao <- NAO(ano$ano_exp, ano$ano_obs, data$lon, data$lat) #For clarification, we normalize the NAO index for each member dividing by the corresponding standard deviation -nao_exp_sd <- apply(nao$NAO_exp, MARGIN = 1, sd) +nao_exp_sd <- apply(nao$NAO_exp, MARGIN = 1, sd, na.rm = T) nao_obs_sd <- sd(nao$NAO_obs) nao_exp_n <- nao$NAO_exp / nao_exp_sd nao_obs_n <- nao$NAO_obs / nao_obs_sd # Finally plot the NAO index using Box plots for all decembers. -PlotBoxWhisker(nao_exp_n, nao_obs_n, toptitle = "NAO index, DJF", ytitle = "NAO index (PC1) psl", - monini = 12, yearini = 1980, freq = 1, expname = "EUROSIP", obsname = "EraInterim", - fileout = "NAO_BoxWhisker_Original.png") - +PlotBoxWhisker(nao_exp_n, nao_obs_n, toptitle = "NAO index, DJF", + monini = 12, yearini = 1981, freq = 1, + drawleg = F, fileout = NULL) +legend(x = 3.8, y = 2.6, c('EUROSIP', 'EraInterim'), col = c(2, 4), pch = 15) ``` - + The figure above does not represent a good agreement between observations (blue line) and forecast (whisker boxes) due to the large dispersion through the 51 model members. The NAO signal is too noisy within the model ensemble thus almost disappearing (close to 0). @@ -112,9 +112,12 @@ The figure above does not represent a good agreement between observations (blue Let's quantify how good or bad it is this prediction looking also for a better understanding of what is happening. To do so, we will use the Root Mean Square Error (RMSE), that it is just the squared and then rooted difference between the predicted and the true value. -s2dverification includes the `RMS()` function to directly calculate the RMSE, however, in order to get a better understanding of the output values we will use the `RMSSS()` function (Root Mean Square Error Skill Score). The RMSSS equals the RMSE but it is normalized by a reference RMSE, usually asociated with the intrinsic variability of the system (for example the standard deviation of the climatology). In s2dverification the score is computed such the best RMSSS score would be 1 (RMSE equals 0), while if RMSSS equals 0, the RMSE equals the variability reference of the system (i.e. the standard deviation). RMSSS can also be negative, meaning RMSE is greater than the variability reference. +s2dverification includes the `RMS()` function to directly calculate the RMSE, however, in order to get a better understanding of the output values we will use the `RMSSS()` function (Root Mean Square Error Skill Score). The RMSSS equals the RMSE but it is normalized by a reference RMSE, usually asociated with the intrinsic variability of the system (for example the standard deviation of the climatology). In s2dverification the score is computed such that the best RMSSS score would be 1 (RMSE equals 0), while if RMSSS equals 0, the RMSE equals the variability reference of the system (i.e. the standard deviation). RMSSS can also be negative, meaning RMSE is greater than the variability reference. ```r +#Defining NAO events +Lmember <- length(exp[1, , 1, 1, 1, 1]) + #Calculating a RMSSS for the ensemble of members average from the mean NAO timeseries of all members rmsss_m = RMSSS(var_exp = array(apply(nao_exp_n, MARGIN = 2, FUN = mean), dim = c(1,11)), var_obs = nao_obs_n, pval = F) @@ -128,29 +131,29 @@ layout(matrix(c(1, 2), 1 , 2, byrow = TRUE)) #Plotting RMSSS for all members plot(rmsss, type = 'h', lwd = 5, col = 'grey', xlab = 'Members', mgp = c(3,1,0), main = sprintf('RMSSS for each member (RMSSS average %1.3f )', rmsss_m), - ylim = c(-1,1)) + ylim = c(-1,1), cex.lab=2, cex.axis=2, cex.main = 1.9) grid(col ='grey') lines(rep(0, Lmember), lwd = 1, col = 'black') #Plotting boxplot for good members -igood = which(rmsss > 0.2) +igood = which(rmsss > 0.1) rmsss_m_goods = RMSSS(var_exp = array(apply(nao_exp_n[igood,], MARGIN = 2, FUN = mean), dim = c(1,11)), var_obs = nao_obs_n, pval = F) PlotBoxWhisker(nao_exp_n[igood,], nao_obs_n, - toptitle = sprintf("NAO index, good members only (RMSSS=%1.2f)", rmsss_m_goods), - ytitle = "NAO index (PC1) psl", - monini = 12, yearini = 1980, freq = 1, - expname = "EUROSIP", obsname = "EraInterim", - fileout = NULL) + toptitle = sprintf('NAO index, selected-members (RMSSS=%1.2f)', rmsss_m_goods), + monini = 12, yearini = 1981, freq = 1, + drawleg = F, fileout = NULL) +legend(x = 4.95, y = 2.4, c('EUROSIP', 'EraInterim'), + col = c(2, 4), pch = 15, cex = 0.9, lty = 0) ``` - + The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. The general RMSSS for the whole ensemble is 0.015, what means a not very useful ensemble prediction. -However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 45 and 47 are used. Now most marked NAO events are correctly predicted giving a RMSSS of 0.66 for this 2-member ensemble. +However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8, 23, 40 and 45 are used. Now most marked NAO events are correctly predicted giving a RMSSS of 0.66 for this selected-members ensemble. ### 5-Quantifying the skillfulness of the prediction. The Brier Score @@ -159,13 +162,12 @@ Another option to quantify the goodness of this prediction is using the `BrierSc Moreover, the BS can be descomposed in 3 terms: reliability, resolution (negative contribution) and uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). -In the case of the NAO index, we can define a NAO event always that the index is greater than 0. Therefore, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored 2-member ensemble of previous section. +In the case of the NAO index, we can define a NAO event always that the index is greater than 0. Therefore, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. ```r -#Defining NAO events -Lmember <- length(exp[1, , 1, 1, 1, 1]) +#Defining number of sdates Lsdates <- length(startDates) #For observations @@ -184,17 +186,17 @@ BSrel <- BS$rel BSres <- BS$res BSunc <- BS$unc -#For good members nly -nao_event_exp_good <- array(0, dim = c(length(igood), Lsdates)) -nao_event_exp_good[which(nao_exp_n[igood,] > 0)] = 1 -nao_event_exp_good_prob <- apply(nao_event_exp_good, MARGIN = 2, mean) +#For selected members only +nao_event_exp_sel <- array(0, dim = c(length(isel), Lsdates)) +nao_event_exp_sel[which(nao_exp_n[isel, ] > 0)] = 1 +nao_event_exp_sel_prob <- apply(nao_event_exp_sel, MARGIN = 2, mean) -BS_good <- BrierScore(nao_event_obs, nao_event_exp_good_prob) +BS_sel <- BrierScore(nao_event_obs, nao_event_exp_sel_prob) -BSscore_good <- BS_good$bs -BSrel_good <- BS_good$rel -BSres_good <- BS_good$res -BSunc_good <- BS_good$unc +BSscore_sel <- BS_sel$bs +BSrel_sel <- BS_sel$rel +BSres_sel <- BS_sel$res +BSunc_sel <- BS_sel$unc #Plotting NAO events and prediction probabilities @@ -209,36 +211,46 @@ lines(BS$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') lines(BS$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') axis(2, at = seq(-0.5, 0.5, 0.25), labels = seq(0, 1, 0.25), mgp = c(3,1,0)) lines(rep(0, Lmember), lwd=2, col = 'black') -title('Predictions for all ensemble members') +title('Predictions for all members ensemble') -#For good ensemble members only -plot(BS_good$obs - 0.5, type = 'p', lwd = 5, col = 'red', +#For selected-member ensemble only +plot(BS_sel$obs - 0.5, type = 'p', lwd = 5, col = 'red', xlab = 'NAO events', yaxt = 'n', ylab = 'NAO probabilities', mgp = c(3,1,0)) grid(col = 'grey') -lines(BS_good$obs - 0.5, type = 'h', lwd = 5, col = 'red', yaxt = 'n') -lines(BS_good$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') -lines(BS_good$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') +lines(BS_sel$obs - 0.5, type = 'h', lwd = 5, col = 'red', yaxt = 'n') +lines(BS_sel$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') +lines(BS_sel$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') axis(2, at = seq(-0.5, 0.5, 0.25), labels = seq(0, 1, 0.25), mgp = c(3, 1, 0)) lines(rep(0, Lmember), lwd = 2, col = 'black') -title('Predictions for good ensemble members') +title('Predictions for selected-members ensemble') ``` -Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the best members. +Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the selected members. For the whole ensemble, the results are: -Total BS = 0.239, -Reliability = 0.021, -Resolution = 0.012, +Total BS = 0.213, +Reliability = 0.138, +Resolution = 0.172, Uncertainty = 0.248. -For the ensemble of best members, the results are: -Total BS = 0.091, -Reliability = 0.0, -Resolution = 0.157, +For the ensemble with the selected members, the results are: +Total BS = 0.136, +Reliability = 0.015, +Resolution = 0.126, Uncertainty = 0.248. -As expected, the ensemble of best members presents a clearly better Brier Score (lower value). While the uncertainty is obviously the same for the two cases, since the observations to compare with are the same, the reliability and resolution scores are better for the second case. Specially the resolution scores much better in the 2-member ensemble due to the agreement of all the members in correctly detecting positive and negative NAO events. +As expected, the slected-members ensemble presents a clearly better Brier Score (lower value). While the uncertainty is obviously the same for the two cases, since the observations to compare with are the same, the reliability and resolution scores are better for the second case. Specially the resolution scores much better in the selected-member ensemble due to the agreement of all the members in correctly detecting positive and negative NAO events. + + +### 6-Appendix: Forecast scores ranges + +SCORE RMSSS | BS (total) | Reliability | Resolution | Uncertainty + +Best 0 0 0 1 related with obs. +---- +Worst -Inf 1 1 0 related with obs. + -- GitLab From 61c401a2580af4587faabff3d1fcec6d4497a3a1 Mon Sep 17 00:00:00 2001 From: jpena Date: Thu, 14 Mar 2019 16:18:57 +0100 Subject: [PATCH 06/18] Adding formatted table --- vignettes/ScoringForecast.md | 6 +++++- 1 file changed, 5 insertions(+), 1 deletion(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 6ace894e..d7c7eac2 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -253,4 +253,8 @@ Best 0 0 0 1 rela ---- Worst -Inf 1 1 0 related with obs. - +| ds | dsaas | sa | df | sd | +|----|-------|-----|----|-----| +| ad | dad | add | ad | daa | +| | | | | | +| | | | | | -- GitLab From e1759aea945db21cb02d8fdfa3f638004bb2149c Mon Sep 17 00:00:00 2001 From: jpena Date: Thu, 14 Mar 2019 16:23:24 +0100 Subject: [PATCH 07/18] Adding formatted table 2 --- vignettes/ScoringForecast.md | 15 ++++----------- 1 file changed, 4 insertions(+), 11 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index d7c7eac2..72482a1c 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -247,14 +247,7 @@ As expected, the slected-members ensemble presents a clearly better Brier Score ### 6-Appendix: Forecast scores ranges -SCORE RMSSS | BS (total) | Reliability | Resolution | Uncertainty - -Best 0 0 0 1 related with obs. ----- -Worst -Inf 1 1 0 related with obs. - -| ds | dsaas | sa | df | sd | -|----|-------|-----|----|-----| -| ad | dad | add | ad | daa | -| | | | | | -| | | | | | +| SCORE | RMSSS | BS (total) | Reliability | Resolution | Uncertainty | 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which(rmsss > 0.1) +rmsss_m_sel = RMSSS(var_exp = array(apply(nao_exp_n[isel, ], MARGIN = 2, FUN = mean), dim = c(1,11)), var_obs = nao_obs_n, pval = F) -PlotBoxWhisker(nao_exp_n[igood,], nao_obs_n, - toptitle = sprintf('NAO index, selected-members (RMSSS=%1.2f)', rmsss_m_goods), +PlotBoxWhisker(nao_exp_n[isel, ], nao_obs_n, + toptitle = sprintf('NAO index, selected-members (RMSSS=%1.2f)', rmsss_m_sel), monini = 12, yearini = 1981, freq = 1, drawleg = F, fileout = NULL) legend(x = 4.95, y = 2.4, c('EUROSIP', 'EraInterim'), @@ -239,10 +239,10 @@ Uncertainty = 0.248. For the ensemble with the selected members, the results are: Total BS = 0.136, Reliability = 0.015, -Resolution = 0.126, +Resolution = 0.127, Uncertainty = 0.248. -As expected, the slected-members ensemble presents a clearly better Brier Score (lower value). While the uncertainty is obviously the same for the two cases, since the observations to compare with are the same, the reliability and resolution scores are better for the second case. Specially the resolution scores much better in the selected-member ensemble due to the agreement of all the members in correctly detecting positive and negative NAO events. +As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. As expected from the RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact than although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence or not of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). ### 6-Appendix: Forecast scores ranges -- GitLab From 48053ea7226a466ba6456e6f6057f4d7f4a356ef Mon Sep 17 00:00:00 2001 From: jpena Date: Thu, 14 Mar 2019 16:58:54 +0100 Subject: [PATCH 09/18] More edits 2 --- vignettes/RMSSSforNAOprediction.png | Bin 84060 -> 0 bytes 1 file changed, 0 insertions(+), 0 deletions(-) delete mode 100644 vignettes/RMSSSforNAOprediction.png diff --git a/vignettes/RMSSSforNAOprediction.png b/vignettes/RMSSSforNAOprediction.png deleted file mode 100644 index e472047cdb29889b874d4b25a5882c04b6682772..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 84060 zcmdSBhdbB*`!=p3t58<5lZdqJ9hI4pq!5y1uLzliQb;l@Sw+a66(MB{S!EN-dPzq1 zxX;)7{r&vz&wU*C?+>_N$I+4fdOjbI>v3J@b)M(-e5kFdyqju26$uH+ZdH{tIwT}J zkC2dTx1rdMudF{gGLHW!9aZ$5Nl0iu6aQ_ibx6BTLc&F&dPYIl{RN@l?FvKfmi!;d 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abcc3576..750d2828 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -242,7 +242,7 @@ Reliability = 0.015, Resolution = 0.127, Uncertainty = 0.248. -As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. As expected from the RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact than although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence or not of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). +As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. As expected from the RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence or not of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). ### 6-Appendix: Forecast scores ranges -- GitLab From 43e35340a97efbb41516538852db52e46f5b7ac7 Mon Sep 17 00:00:00 2001 From: jpena Date: Thu, 14 Mar 2019 17:10:29 +0100 Subject: [PATCH 12/18] More edits 5 --- vignettes/ScoringForecast.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 750d2828..ddb7723e 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -151,9 +151,9 @@ legend(x = 4.95, y = 2.4, c('EUROSIP', 'EraInterim'), -The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. The general RMSSS for the whole ensemble is 0.015, what means a not very useful ensemble prediction. +The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. **The general RMSSS for the whole ensemble is 0.015**, what means a not very useful ensemble prediction. -However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8, 23, 40 and 45 are used. Now most marked NAO events are correctly predicted giving a RMSSS of 0.66 for this selected-members ensemble. +However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8, 23, 40 and 45 are used. Now most marked NAO events are correctly predicted giving a **RMSSS of 0.66 for this selected-members ensemble**. ### 5-Quantifying the skillfulness of the prediction. The Brier Score @@ -231,13 +231,13 @@ title('Predictions for selected-members ensemble') Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the selected members. For the whole ensemble, the results are: -Total BS = 0.213, +**Total BS = 0.213,** Reliability = 0.138, Resolution = 0.172, Uncertainty = 0.248. For the ensemble with the selected members, the results are: -Total BS = 0.136, +**Total BS = 0.136,** Reliability = 0.015, Resolution = 0.127, Uncertainty = 0.248. @@ -249,5 +249,5 @@ As expected, the selected-members ensemble presents a better Brier Score (lower | SCORE | RMSSS | BS (total) | Reliability | Resolution | Uncertainty | |:-----:|:-----:|:----------:|:-----------:|:----------:|:--------------:| -| Best | 0 | 0 | 0 | 1 | Obs. dependant | +| Best | 1 | 0 | 0 | 1 | Obs. dependant | | Worst | -Inf | 1 | 1 | 0 | Obs. dependant | -- GitLab From 44a79e45a6a783d08c7eaaef1ef0affa2208c414 Mon Sep 17 00:00:00 2001 From: jpena Date: Fri, 15 Mar 2019 10:24:40 +0100 Subject: [PATCH 13/18] More edits 6 --- vignettes/NAOpredictions.png | Bin 38181 -> 33781 bytes vignettes/ScoringForecast.md | 17 ++++++++--------- 2 files changed, 8 insertions(+), 9 deletions(-) diff --git a/vignettes/NAOpredictions.png b/vignettes/NAOpredictions.png index 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zzw(8vWFayVkr&eSu3^Zeb>V&ik~=6$Dsc($O;lGWoZo?;B8~Nme|7ga)&(!DFDHWW zo*C4^mmZiO+Oha+Evf0e=0>)Iqt_N^4OLVs5V2vTz8~yN^GI1TQe3UVbm(=E)uO~g zzfI^pnt*d~7~`r=zj_cLGpp9G_{xcQI*6{H4RBP4AVc+e6mVoXxlQBA+TCXpi;eq`gyeid3=NeNqrNPN(@lyWGDD=N1&~PX(YLT; zA`%JA@%-6#H1!(!@#7T&2>LEzPe9X`pLp>nl@d4D{2uGV!XRc9DCv9WXW<|Z|JiFZ zeQ|;OaG31#-z8^JIXON)P#yCTv4^mNZ3)ORk37i;*2Og>6Fv&(h$Z3t-2=!fPB)g! z&#`yV60hu%iXJo7P8uhtX(UurSmGw1;8#S9G8)*{Y=mi2JdVEo_;GT^3pOB(T*}VltBHO8 z#i}Rh-ownyOn^t8s?YDCM=_)x@V|kuRwd$)IKgLrcSnC;FvtR~Zf>9lw{5s4i22Wl z`R_DxMzEY}XF}lJxd&57u4>zvf4j_2erf&s6|3)))CWiPC)j?Qh_((pBCk$0f!L1u z3c=uSylvDrm@+S>>PiEu20s_a$vd8&3+;>ZkwdgEM1h!qod|O#kt<4SYE@7#D6xwp zp0vE}@s37pl{$DZ8(u_fZV^(3jZzc-_<6)SWG9%Um+xHTMjKCP?yeW~K46K=UOp%? n(A)m!PQ3r+o9_QC4VYYfs{BG}rv6AF@rh&dD&!>Dvp4?-yiB7D diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index ddb7723e..ea82b44e 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -80,7 +80,7 @@ exp <- data$mod As a first step we will compare the predicted intensity of the NAO index for observations and for the forecast system. -In s2dverification the NAO index is computed using a common alternative definition; The NAO is defined as the first mode of variability for the sea level pressure in the North Atlantic region [20N-80N, 80W-40E]. The `NAO()` function requires as input anomalies that are previously computed using `Ano_CrossValid()` function. +In s2dverification the NAO index is computed using a common alternative definition; The NAO is defined as the first mode of variability (computed using Single Value Decomposition method) for the sea level pressure in the North Atlantic region [20N-80N, 80W-40E]. The `NAO()` function requires as input anomalies that are previously computed using `Ano_CrossValid()` function. ```r #Computing anomalies along sdates dimension @@ -105,7 +105,7 @@ legend(x = 3.8, y = 2.6, c('EUROSIP', 'EraInterim'), col = c(2, 4), pch = 15) -The figure above does not represent a good agreement between observations (blue line) and forecast (whisker boxes) due to the large dispersion through the 51 model members. The NAO signal is too noisy within the model ensemble thus almost disappearing (close to 0). +The figure above does not represent a good agreement between observations (blue line) and forecast (whisker boxes) due to the large dispersion through the 51 model members. The NAO signal is too weak due to the large dispersion among ensemble members thus almost disappearing (close to 0). ### 4-Quantifying the skillfulness of the prediction. The RMSSS @@ -153,14 +153,14 @@ legend(x = 4.95, y = 2.4, c('EUROSIP', 'EraInterim'), The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. **The general RMSSS for the whole ensemble is 0.015**, what means a not very useful ensemble prediction. -However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8, 23, 40 and 45 are used. Now most marked NAO events are correctly predicted giving a **RMSSS of 0.66 for this selected-members ensemble**. +However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8th, 23rd, 40th and 45th are used. Now most marked NAO events are correctly predicted giving a **RMSSS of 0.66 for this selected-members ensemble**. ### 5-Quantifying the skillfulness of the prediction. The Brier Score -Another option to quantify the goodness of this prediction is using the `BrierScore()` function. The BS measures the accuracy of a probabilistic categorical prediction. BS scores perfect prediction when BS equals 0 and worst score corresponds to 1 value. +Another option to quantify the goodness of this prediction is using the `BrierScore()` function. The BS measures the accuracy of a probabilistic categorical prediction calculating the mean squared difference between the predicted probability and the actual observation. BS scores perfect prediction when BS equals 0 (no difference between prediction and observation) and worst score corresponds to value of 1. -Moreover, the BS can be descomposed in 3 terms: reliability, resolution (negative contribution) and uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). +Moreover, the BS can be descomposed in 3 terms: BS = Reliability - Resolution + Uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). In the case of the NAO index, we can define a NAO event always that the index is greater than 0. Therefore, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. @@ -211,7 +211,7 @@ lines(BS$pred - 0.5, type = 'h', lwd = 4, col = 'blue', yaxt = 'n') lines(BS$pred - 0.5, type = 'p', lwd = 3, col = 'blue', yaxt = 'n') axis(2, at = seq(-0.5, 0.5, 0.25), labels = seq(0, 1, 0.25), mgp = c(3,1,0)) lines(rep(0, Lmember), lwd=2, col = 'black') -title('Predictions for all members ensemble') +title('Predictions for all-members ensemble') #For selected-member ensemble only plot(BS_sel$obs - 0.5, type = 'p', lwd = 5, col = 'red', @@ -228,15 +228,14 @@ title('Predictions for selected-members ensemble') -Following the analysis of the RMSSS, we compute the BS for the whole ensemble and for the ensemble with the selected members. -For the whole ensemble, the results are: +For the all-members ensemble, the results are: **Total BS = 0.213,** Reliability = 0.138, Resolution = 0.172, Uncertainty = 0.248. -For the ensemble with the selected members, the results are: +For the selected-members ensemble, the results are: **Total BS = 0.136,** Reliability = 0.015, Resolution = 0.127, -- GitLab From 4f6a0218a9c0bdde00f91272d51efcf6c2d09f7d Mon Sep 17 00:00:00 2001 From: jpena Date: Fri, 15 Mar 2019 10:29:35 +0100 Subject: [PATCH 14/18] More edits 7 --- vignettes/ScoringForecast.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index ea82b44e..9a3ed3d1 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -241,7 +241,7 @@ Reliability = 0.015, Resolution = 0.127, Uncertainty = 0.248. -As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. As expected from the RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence or not of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). +As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. In agreement with the shown RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). ### 6-Appendix: Forecast scores ranges -- GitLab From a11e2e289cac36c58fb6ddd425b83cdac9dfe715 Mon Sep 17 00:00:00 2001 From: jpena Date: Fri, 15 Mar 2019 10:56:46 +0100 Subject: [PATCH 15/18] More edits 8 --- vignettes/ScoringForecast.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 9a3ed3d1..28eb5959 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -151,7 +151,7 @@ legend(x = 4.95, y = 2.4, c('EUROSIP', 'EraInterim'), -The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. **The general RMSSS for the whole ensemble is 0.015**, what means a not very useful ensemble prediction. +The above figure shows very different RMSSS for different members (left plot). Most of them have RMSSS close to 0, thus the prediction error is close to the system variability. **The RMSSS for the whole ensemble is 0.091**, what means a not very useful ensemble prediction. However, we can select the members that present a better RMSSS, i.e. those closer to 1, and recompute the ensemble RMSSS. This is shown in the right plot where only members 8th, 23rd, 40th and 45th are used. Now most marked NAO events are correctly predicted giving a **RMSSS of 0.66 for this selected-members ensemble**. -- GitLab From cf879cf84a0eabb9450435f7e5265a294a3344a1 Mon Sep 17 00:00:00 2001 From: jpena Date: Fri, 15 Mar 2019 12:32:03 +0100 Subject: [PATCH 16/18] More edits 9 --- vignettes/ScoringForecast.md | 8 +++++--- 1 file changed, 5 insertions(+), 3 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 28eb5959..c5a35062 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -135,7 +135,7 @@ plot(rmsss, type = 'h', lwd = 5, col = 'grey', xlab = 'Members', mgp = c(3,1,0), grid(col ='grey') lines(rep(0, Lmember), lwd = 1, col = 'black') -#Plotting boxplot for good members +#Plotting boxplot for selected members with higher rmsss isel = which(rmsss > 0.1) rmsss_m_sel = RMSSS(var_exp = array(apply(nao_exp_n[isel, ], MARGIN = 2, FUN = mean), dim = c(1,11)), var_obs = nao_obs_n, pval = F) @@ -162,7 +162,7 @@ Another option to quantify the goodness of this prediction is using the `BrierSc Moreover, the BS can be descomposed in 3 terms: BS = Reliability - Resolution + Uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). -In the case of the NAO index, we can define a NAO event always that the index is greater than 0. Therefore, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. +In the case of the NAO index, we can discretize it in to a categorical variable defining a NAO event always that the index is greater than 0 and asociating the event with the value 1. Negative NAO index values then correspond to the non-ocurrance of a NAO event and thus they are assigned the value 0. For the forecast, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. ```r @@ -241,7 +241,9 @@ Reliability = 0.015, Resolution = 0.127, Uncertainty = 0.248. -As expected, the selected-members ensemble presents a better Brier Score (lower value). The uncertainty is obviously the same for the two cases, since the observations to compare with are the same. In agreement with the shown RMSSS results, the reliability is much better (lower) for the second case since the prediction probabilities are much closer to the NAO ocurrence distribution than for the all-members ensemble prediction (see right plot in the figure above). However, the resolution scores are not that different. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence of the NAO event. The all-members ensemble instead, although with a higher RMSSS/reliability tends to predict better the occurrence of positive/negative NAO events (left plot). +To put the total Brier Score values in to context, note that the BS definition is such that in general a forecast that always gives a 50% probability prediction for all cases would have a BS of 0.25. + +Taking this into account, the all-member ensemble prediction with a BS of 0.213 seems only a bit better than flipping a coin in terms of Brier Scoring. The selected-members ensemble instead presents a clear improvement in the total Brier Score (lower value). This is due to the better scoring (lower value) achieved by the reliability term, in agreement with the previous RMSSS results and also shown in the right plot of the figure above; The prediction probabilities are much closer to the NAO ocurrence distribution. However, the resolution scores are not that different between the all-members and the selected-members ensemble. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence of the NAO event. The all-members ensemble instead, even with a higher RMSSS/reliability, tends to predict better the occurrence of positive/negative NAO events (left plot). Finally, the uncertainty is obviously the same for the two cases, since the observations to compare with are the same. ### 6-Appendix: Forecast scores ranges -- GitLab From f8379b7569a6e27d22511304115b3dfc80dc260a Mon Sep 17 00:00:00 2001 From: jpena Date: Fri, 15 Mar 2019 12:41:18 +0100 Subject: [PATCH 17/18] More edits 10 --- vignettes/ScoringForecast.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index c5a35062..069009ea 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -162,7 +162,7 @@ Another option to quantify the goodness of this prediction is using the `BrierSc Moreover, the BS can be descomposed in 3 terms: BS = Reliability - Resolution + Uncertainty. Reliability refers to how close the predicted probabilities are to the true probabilities (i.e. the lower the reliability, the lower the difference between prediction and observation and the better the score). Resolution relates with the difference between the predicted observation and the climatology (i.e. the higher the resolution, the greater the ability of the forecast to predict events different from the climatology). Uncertainty represents the inherent uncertainty of ocurring the event (i.e. the higher the uncertainty, the more difficult to correctly predict the event). -In the case of the NAO index, we can discretize it in to a categorical variable defining a NAO event always that the index is greater than 0 and asociating the event with the value 1. Negative NAO index values then correspond to the non-ocurrance of a NAO event and thus they are assigned the value 0. For the forecast, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. +In the case of the NAO index, we can discretize it in to a categorical variable defining a NAO event always that the index is greater than 0 and asociating the event with the value 1. Negative NAO index values then correspond to the non-ocurrence of a NAO event and thus they are assigned the value 0. For the forecast, we can convert the NAO index prediction of all ensemble members in to a NAO prediction probability computing the proportion of members with a positive NAO prediction. For comparison we do the same for the good scored selected-members ensemble of previous section. ```r @@ -243,7 +243,7 @@ Uncertainty = 0.248. To put the total Brier Score values in to context, note that the BS definition is such that in general a forecast that always gives a 50% probability prediction for all cases would have a BS of 0.25. -Taking this into account, the all-member ensemble prediction with a BS of 0.213 seems only a bit better than flipping a coin in terms of Brier Scoring. The selected-members ensemble instead presents a clear improvement in the total Brier Score (lower value). This is due to the better scoring (lower value) achieved by the reliability term, in agreement with the previous RMSSS results and also shown in the right plot of the figure above; The prediction probabilities are much closer to the NAO ocurrence distribution. However, the resolution scores are not that different between the all-members and the selected-members ensemble. This is due to the fact that although the selected-members ensemble has a lower RMSSS/reliability, it does not always correctly predict the ocurrence of the NAO event. The all-members ensemble instead, even with a higher RMSSS/reliability, tends to predict better the occurrence of positive/negative NAO events (left plot). Finally, the uncertainty is obviously the same for the two cases, since the observations to compare with are the same. +Taking this into account, the all-members ensemble prediction with a BS of 0.213 seems only a bit better than flipping a coin in terms of Brier Scoring. The selected-members ensemble instead presents a clear improvement in the total Brier Score (lower value). This is due to the better scoring (lower value) achieved by the reliability term in this ensemble, in agreement with the previous RMSSS results and also shown in the right plot of the figure above; The prediction probabilities are much closer to the NAO ocurrence distribution. However, the resolution scores are not that different between the all-members and the selected-members ensemble. This is due to the fact that even with a notable difference in RMSSS and reliability, the two ensembles mostly predict equally the correct occurrence of positive/negative NAO events (left plot). Finally, the uncertainty is obviously the same for the two cases, since the observations to compare with are the same. ### 6-Appendix: Forecast scores ranges -- GitLab From 027a13bb1cde296f1c891d9d962f33b062e4e1e1 Mon Sep 17 00:00:00 2001 From: jpena Date: Tue, 26 Mar 2019 17:18:33 +0100 Subject: [PATCH 18/18] Minor edits 11 --- vignettes/ScoringForecast.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/vignettes/ScoringForecast.md b/vignettes/ScoringForecast.md index 069009ea..2f3679f7 100644 --- a/vignettes/ScoringForecast.md +++ b/vignettes/ScoringForecast.md @@ -59,7 +59,7 @@ latNA <- c(20, 80) #Selected time periods: startDates <- paste(1981:1991, '1101', sep = '') -Loading sea level pressure maps for the whole North Atlantic and for the specific period +#Loading sea level pressure maps for the whole North Atlantic and for the specific period data <- Load(var = 'psl', exp = list(path_exp_EUROSIP), obs = list(path_obs_eraI), -- GitLab

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