Dirk Schlabing and András Bárdossy. Comparing Five Weather Generators in Terms of Entropy
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1 Dirk Schlabing and András Bárdossy Comparing Five Weather Generators in Terms of Entropy
2 Motivation 1
3 Motivation What properties of weather should be reproduced [...]? Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
4 Motivation What properties of weather should be reproduced [...]? Endless possibilities for choosing such properties Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
5 Motivation What properties of weather should be reproduced [...]? Endless possibilities for choosing such properties, so focus on application is necessary! Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
6 Motivation What properties of weather should be reproduced [...]? Endless possibilities for choosing such properties, so focus on application is necessary! Entropy might be relevant Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
7 Motivation What properties of weather should be reproduced [...]? Endless possibilities for choosing such properties, so focus on application is necessary! Entropy might be relevant Entropy is not only about (multivariate) distributions also applicable to spatial patterns and temporal sequences Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
8 Motivation What properties of weather should be reproduced [...]? Endless possibilities for choosing such properties, so focus on application is necessary! Entropy might be relevant Entropy is not only about (multivariate) distributions also applicable to spatial patterns and temporal sequences This talk is meant to be food for thought, rather than promoting/bashing specific WG s. Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 3
9 Setup & Weather Generators 2
10 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Daily data Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
11 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Daily data Diverse climates Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
12 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Daily data Diverse climates Stations selected for large difference between calibration and validation Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
13 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Daily data Diverse climates Stations selected for large difference between calibration and validation Calibration period: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
14 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Daily data Diverse climates Stations selected for large difference between calibration and validation Calibration period: Validation period: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
15 Setup: five European Climate Assessment & Dataset (ECA&D) weather stations Data: Variables: Daily data Diverse climates Stations selected for large difference between calibration and validation Calibration period: Validation period: Daily precipitation Daily maximum temperature (compromising on what all WG s can generate) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 5
16 Five weather generators Parametric Semi-Parametric Non-Parametric: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
17 Five weather generators Parametric Vector-Autoregressive Weather Generator (VG) Semi-Parametric Non-Parametric: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
18 Five weather generators Parametric Vector-Autoregressive Weather Generator (VG) Phase-Randomized Copula (WeatherCop) Semi-Parametric Non-Parametric: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
19 Five weather generators Parametric Vector-Autoregressive Weather Generator (VG) Phase-Randomized Copula (WeatherCop) Statistical Downscaling Model (SDSM) Semi-Parametric Non-Parametric: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
20 Five weather generators Parametric Vector-Autoregressive Weather Generator (VG) Phase-Randomized Copula (WeatherCop) Statistical Downscaling Model (SDSM) Semi-Parametric Long Ashton Research Station Weather Generator (LARS-WG) Non-Parametric: Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
21 Five weather generators Parametric Vector-Autoregressive Weather Generator (VG) Phase-Randomized Copula (WeatherCop) Statistical Downscaling Model (SDSM) Semi-Parametric Long Ashton Research Station Weather Generator (LARS-WG) Non-Parametric: K-Nearest-Neighbor (KNN) resampling Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 6
22 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
23 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X Transformation to Standard-Normal Φ 1 (F doy (X)) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
24 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
25 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
26 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit VAR-process A i,doy, COV doy (ϵ t ) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
27 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit VAR-process A i,doy, COV doy (ϵ t ) Simulate time series p ) y t = (A i,doy y t i + ˆϵ t i=1 Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
28 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit VAR-process A i,doy, COV doy (ϵ t ) Simulate time series p ) y t = (A i,doy y t i + ˆϵ t i=1 Re-transformation ˆX = F 1 (Φ(Y )) doy Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
29 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit VAR-process A i,doy, COV doy (ϵ t ) Simulate time series p ) y t = (A i,doy y t i + ˆϵ t i=1 Re-transformation ˆX = F 1 (Φ(Y )) doy Synthetic time series ˆX Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
30 Parametric 1: Vector-Autoregressive Weather Generator (VG) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit VAR-process A i,doy, COV doy (ϵ t ) Simulate time series p ) y t = (A i,doy y t i + ˆϵ t + m i=1 Scenario perturbation m, m t Re-transformation ˆX = F 1 (Φ(Y )) doy Synthetic time series ˆX Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 7
31 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
32 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit Copula per doy Obtain conditioned quantiles C doy, P i,t Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
33 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit Copula per doy Obtain conditioned quantiles C doy, P i,t Phase-randomize conditioned quantiles Re-correlate conditioned quantiles with fitted copula Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
34 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit Copula per doy Obtain conditioned quantiles C doy, P i,t Phase-randomize conditioned quantiles Re-correlate conditioned quantiles with fitted copula Re-transformation ˆX = F 1 (Φ(Y )) doy Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
35 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit Copula per doy Obtain conditioned quantiles C doy, P i,t Phase-randomize conditioned quantiles Re-correlate conditioned quantiles with fitted copula Re-transformation ˆX = F 1 (Φ(Y )) doy Synthetic time series ˆX Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
36 Parametric 2: Phase-Randomized Copula (WeatherCop) Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) In-fill negative rain Fit Copula per doy Obtain conditioned quantiles C doy, P i,t Phase-randomize conditioned quantiles Re-correlate changed conditioned quantiles with fitted copula Scenario perturbation Re-transformation ˆX = F 1 (Φ(Y )) doy Synthetic time series ˆX Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 8
37 Parametric 3: Statistical Downscaling Model (SDSM) Generation: multiple linear regression between large-scale predictors and local predictand + random error term Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 9
38 Parametric 3: Statistical Downscaling Model (SDSM) Generation: multiple linear regression between large-scale predictors and local predictand + random error term Temporal dependence due to predictor-series Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 9
39 Parametric 3: Statistical Downscaling Model (SDSM) Generation: multiple linear regression between large-scale predictors and local predictand + random error term Temporal dependence due to predictor-series Correlations between generated variables also only through correlations of predictors Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 9
40 Parametric 3: Statistical Downscaling Model (SDSM) Generation: multiple linear regression between large-scale predictors and local predictand + random error term Temporal dependence due to predictor-series Correlations between generated variables also only through correlations of predictors Using 4 NCEP predictors chosen by partial correlation Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 9
41 Semi-Parametric: Long Ashton Research Station Weather Generator (LARS-WG) Generates a series of dry and wet spells Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 10
42 Semi-Parametric: Long Ashton Research Station Weather Generator (LARS-WG) Generates a series of dry and wet spells Semi-empirical distributions for spell lengths and generated variables Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 10
43 Semi-Parametric: Long Ashton Research Station Weather Generator (LARS-WG) Generates a series of dry and wet spells Semi-empirical distributions for spell lengths and generated variables Precipitation values independent from previous values Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 10
44 Non-Parametric: K-Nearest-Neighbor (KNN) resampling Measurement data X KDE/ parametric distributions F doy Transformation to Standard-Normal Φ 1 (F doy (X)) Resample Scenario perturbation Re-transformation ˆX = F 1 (Φ(Y )) doy Synthetic time series ˆX Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 11
45 Entropy Comparison 3
46 Shannon-entropy: chaos, disorder and uncertainty Measure of information gained by observing a quantity, given its distribution Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 13
47 Shannon-entropy: chaos, disorder and uncertainty Measure of information gained by observing a quantity, given its distribution Even distribution: not much known a priori high information gain high entropy Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 13
48 Shannon-entropy: chaos, disorder and uncertainty Measure of information gained by observing a quantity, given its distribution Even distribution: not much known a priori high information gain high entropy Narrow distribution: some values more likely low information gain low entropy Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 13
49 1D-Entropy: 10 classes Classes are fixed to equal occurrence in calibration set Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 14
50 1D-Entropy: 10 classes Classes are fixed to equal occurrence in calibration set Black bar is the validation set Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 14
51 1D-Entropy: 10 classes Classes are fixed to equal occurrence in calibration set Black bar is the validation set Consistent for all stations Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 14
52 Towards 2D-entropy: bivariate distributions Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 15
53 2D-Entropy: 10 Classes per variable No big differences in the mean Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 16
54 2D-Entropy: 10 Classes per variable No big differences in the mean Order varies from station to station Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 16
55 2D-Entropy: 10 Classes per variable No big differences in the mean Order varies from station to station 2D entropy has connection to dependence Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 16
56 2D-Entropy: 10 Classes per variable No big differences in the mean Order varies from station to station 2D entropy has connection to dependence Copula model has worst performance (Problem with Zavizan and Rennes?) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 16
57 Characterizing order within episodes Separate precipitation time series into episodes of fixed length and minimum precipitation sum Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 17
58 Characterizing order within episodes Separate precipitation time series into episodes of fixed length and minimum precipitation sum Divide each precipitation value by the precipitation sum of each episode Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 17
59 Characterizing order within episodes Separate precipitation time series into episodes of fixed length and minimum precipitation sum Divide each precipitation value by the precipitation sum of each episode Treat each such precipitation share as a probability p t in the entropy equation Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 17
60 Characterizing order within episodes Separate precipitation time series into episodes of fixed length and minimum precipitation sum Divide each precipitation value by the precipitation sum of each episode Treat each such precipitation share as a probability p t in the entropy equation Normalize entropies (to enable lumping with entropies for different episode length) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 17
61 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
62 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even VG, KNN and WeatherCop are similar in their tendency towards too high entropies Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
63 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even VG, KNN and WeatherCop are similar in their tendency towards too high entropies LARS samples serially independent too many uneven episodes Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
64 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even VG, KNN and WeatherCop are similar in their tendency towards too high entropies LARS samples serially independent too many uneven episodes Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
65 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even VG, KNN and WeatherCop are similar in their tendency towards too high entropies LARS samples serially independent too many uneven episodes Negative autocorrelation in LARS: look up Negative Correlation introduced by success Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
66 Results: entropies of strong precipitation episodes SDSM generates episodes that are too even VG, KNN and WeatherCop are similar in their tendency towards too high entropies LARS samples serially independent too many uneven episodes Negative autocorrelation in LARS: look up Negative Correlation introduced by success Reality lies between the models! Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 18
67 Conclusions 4
68 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
69 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
70 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
71 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked SDSM and LARS-WG allow too little or any order too many lonely peaks Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
72 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked SDSM and LARS-WG allow too little or any order too many lonely peaks KNN, VG, WeatherCop lack low-entropy events not enough intense short-term precipitation events Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
73 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked SDSM and LARS-WG allow too little or any order too many lonely peaks KNN, VG, WeatherCop lack low-entropy events not enough intense short-term precipitation events Onwards What influences episode entropy? (larg-scale predictors?) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
74 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked SDSM and LARS-WG allow too little or any order too many lonely peaks KNN, VG, WeatherCop lack low-entropy events not enough intense short-term precipitation events Onwards What influences episode entropy? (larg-scale predictors?) How to model for realistic episode entropies? (non-stationary models?) Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
75 Conclusions Entropy exposes disorder not visible via statistical moments or measures of dependence... but it matters how entropy is calculated (choice of classes) Observed rain episodes vary from ordered to peaked SDSM and LARS-WG allow too little or any order too many lonely peaks KNN, VG, WeatherCop lack low-entropy events not enough intense short-term precipitation events Onwards What influences episode entropy? (larg-scale predictors?) How to model for realistic episode entropies? (non-stationary models?) How do impact models react on higher/lower episode entropy? Dirk Schlabing & András Bárdossy, Department of Hydrology and Geohydrology, Universität Stuttgart: Entropy Comparison 20
76 Dirk Schlabing & András Bárdossy Department of Hydrology and Geohydrology, Universität Stuttgart Telefon Fax
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