Differences of Calculated Results - Revision history

Rosenhagen: /* Other skill definitions */

2022-05-16T08:25:08Z

Other skill definitions

← Older revision		Revision as of 08:25, 16 May 2022
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	Literature:		Literature:
	# Murphy, Allan H. (1988) "Skill Scores Based on the Mean Square Error and Their Relationship to the Correlation Coefficient". Monthly Weather Review, Dec. 1988, ~~Seiten~~ 2417 - 2424.		# Murphy, Allan H. (1988) "Skill Scores Based on the Mean Square Error and Their Relationship to the Correlation Coefficient". Monthly Weather Review, Dec. 1988, pages 2417 - 2424.

	Computation is performed for categories '''KC''' and '''KN'''. Skill S4 according to Murphy (1988, equation 4) can be characterised as follows:		Computation is performed for categories '''KC''' and '''KN'''. Skill S4 according to Murphy (1988, equation 4) can be characterised as follows:
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	* 0.0 indicates that the mean value of the reference data models the (reference) data as good as the variant data, because both "models" show the same mean square error (MSE);		* 0.0 indicates that the mean value of the reference data models the (reference) data as good as the variant data, because both "models" show the same mean square error (MSE);
	* negative skill indicates, that the mean value of the reference data models the (reference) data better than the variant data do;		* negative skill indicates, that the mean value of the reference data models the (reference) data better than the variant data do;
	* ''bias'' is taken into account..		* ''bias'' is taken into account.

			====Willmott Skill (1981)====

			Literature:
			# Willmott, Cort J. (1981) "On the validation of models". Physical Geography, pages 184–194.

			Computation is performed for categories '''KC''' and '''KN'''. Skill according to Willmott (1981, equation 9) can be characterised as follows:
			* 1.0 indicates perfect fit;
			* 0.0 is the lower limit, for example resulting from phase shift by 180°;
			* ''bias'' is taken into account.

	----		----

imported>Lang Guenther: /* Quantile Q99 */ Murphy Skill 4

2018-08-27T08:29:47Z

Quantile Q99: Murphy Skill 4

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 # in case that there are less than 32 valid data <math>d_{\rm{Q99}} = \rm{invalid}</math> is set.
 Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q99}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.
 ----

imported>Lang Guenther: /* Root mean square error (RMSE) according to Taylor */ Taylor Skill 4 and 5

2018-08-27T08:01:41Z

Root mean square error (RMSE) according to Taylor: Taylor Skill 4 and 5

← Older revision		Revision as of 08:01, 27 August 2018
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	Definition of RMSE according to Taylor (2001) is mathematically identical to the standard equation for RMSE but computational results may differ slightly due to numerical roundoff.		Definition of RMSE according to Taylor (2001) is mathematically identical to the standard equation for RMSE but computational results may differ slightly due to numerical roundoff.

			====Taylor Skill 4====

			Skill S4 is computed according to Taylor (2001, equation 4) from correlation and normalized variance.
			Computation is performed for categories '''KC''' and '''KN'''. A valid S4 is obtained as long as there exists at least one valid difference <math>d_i</math>. S4 can be characterised as follows:
			* S4 = 1.0 indicates perfect fit;
			* S4 = 0.0, if correlation R = -1.0 or variance of the variant tends towards 0.0 or infinity;
			* S4 is linear with respect to R (at constant variance);
			* For zero variance S4 is not defined;
			* Bias between variant and reference has no influence on S4.
			S4 ''punishes'' deviations in ''Pattern RMS'', and is more ''tolerant'' with respect to deviations in R. See Taylor (2001, figure 10).

			====Taylor Skill 5====

			Skill S5 according to Taylor (2001, euqation 5) is computed from correlation and normalized variance.
			Computation is performed for categories '''KC''' and '''KN'''.

			In contrast to S4 deviations with respect to ''Pattern RMS'' and correlation are treated (''punished'') similarly. See Taylor (2001, figure 11).

	====Number of valid reference data====		====Number of valid reference data====

imported>Lang Guenther: /* Deviation of means (Bias) */ RMSE according to Taylor (2001) added

2018-08-27T07:32:10Z

Deviation of means (Bias): RMSE according to Taylor (2001) added

← Older revision		Revision as of 07:32, 27 August 2018
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	#: <math>E = \sqrt{\bar{E}^2+E'^2}</math>;		#: <math>E = \sqrt{\bar{E}^2+E'^2}</math>;
	Computation is performed for categories '''KC''' und '''KN'''. A valid <math>\bar{E}</math> is obtained as long as there exists at least one valid difference <math>V(r_i,f_i)</math>. Biases for different locations may be computed from data sets of different size. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid Taylor data''.		Computation is performed for categories '''KC''' und '''KN'''. A valid <math>\bar{E}</math> is obtained as long as there exists at least one valid difference <math>V(r_i,f_i)</math>. Biases for different locations may be computed from data sets of different size. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid Taylor data''.

			====Root mean square error (RMSE) according to Taylor====

			RMSE is computed according to Taylor (2001, equation 3) from ''Pattern RMS'' and ''Bias''. Computation is performed for categories '''KC''' and '''KN'''. A valid RMSE is obtained as long as there exists at least one valid difference <math>d_i</math>. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

			Definition of RMSE according to Taylor (2001) is mathematically identical to the standard equation for RMSE but computational results may differ slightly due to numerical roundoff.

	====Number of valid reference data====		====Number of valid reference data====

imported>Lang Guenther: NOATOLINKS, NOATOLINKSTARGET added

2018-08-27T07:11:13Z

NOATOLINKS, NOATOLINKSTARGET added

← Older revision		Revision as of 07:11, 27 August 2018
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	[[de:Differenzen der Berechnungsergebnisse]]		[[de:Differenzen der Berechnungsergebnisse]]
			__NOAUTOLINKS__
			__NOAUTOLINKTARGET__

	==Introduction==		==Introduction==

imported>Lang Guenther: /* Mean deviation */ RMSE

2018-08-27T07:10:24Z

Mean deviation: RMSE

imported>Lang Guenther: sction percentiles deleted

2015-05-07T13:10:37Z

sction percentiles deleted

← Older revision		Revision as of 13:10, 7 May 2015
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	# in case that there are less than 32 valid data <math>d_{\rm{Q99}} = \rm{invalid}</math> is set.		# in case that there are less than 32 valid data <math>d_{\rm{Q99}} = \rm{invalid}</math> is set.
	Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q99}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.		Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q99}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

	~~===Percentiles===~~

	----		----

imported>Lang Guenther: /* Quantile Q95 */ Q99 added

2015-05-07T13:09:55Z

Quantile Q95: Q99 added

← Older revision		Revision as of 13:09, 7 May 2015
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	# in case that there are less than 32 valid data <math>d_{\rm{Q95}} = \rm{invalid}</math> is set.		# in case that there are less than 32 valid data <math>d_{\rm{Q95}} = \rm{invalid}</math> is set.
	Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q95}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.		Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q95}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

			====Quantile Q99====

			Quantile <math>p=0.99</math> is computed for all valid differences <math>d_i</math>. In other words: we obtain a specific value
			for <math>d_i</math>, which is deceeded by 99 % of all events but exceeded by just 1 % of all events:
			# if <math>n \cdot p</math> integer: <math>d_{\rm{Q99}} = 0.5\left( s_{n \cdot p}+s_{n \cdot p+1}\right)</math>;
			# if <math>n \cdot p</math> real: <math>d_{\rm{Q99}} = s_{\lceil n \cdot p \rceil}</math>;
			# in case that there are less than 32 valid data <math>d_{\rm{Q99}} = \rm{invalid}</math> is set.
			Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q99}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

	===Percentiles===		===Percentiles===

imported>Lang Guenther: /* Quantile Q05 */ Q95 added

2015-05-07T13:08:38Z

Quantile Q05: Q95 added

← Older revision		Revision as of 13:08, 7 May 2015
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	# in case that there are less than 32 valid data <math>d_{\rm{Q05}} = \rm{invalid}</math> is set.		# in case that there are less than 32 valid data <math>d_{\rm{Q05}} = \rm{invalid}</math> is set.
	Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q05}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.		Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q05}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

			====Quantile Q95====

			Quantile <math>p=0.95</math> is computed for all valid differences <math>d_i</math>. In other words: we obtain a specific value
			for <math>d_i</math>, which is deceeded by 95 % of all events but exceeded by 5 % of all events:
			# if <math>n \cdot p</math> integer: <math>d_{\rm{Q95}} = 0.5\left( s_{n \cdot p}+s_{n \cdot p+1}\right)</math>;
			# if <math>n \cdot p</math> real: <math>d_{\rm{Q95}} = s_{\lceil n \cdot p \rceil}</math>;
			# in case that there are less than 32 valid data <math>d_{\rm{Q95}} = \rm{invalid}</math> is set.
			Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q95}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

	===Percentiles===		===Percentiles===

imported>Lang Guenther: /* Quantile Q01 */ Q05 added

2015-05-07T13:07:30Z

Quantile Q01: Q05 added

← Older revision		Revision as of 13:07, 7 May 2015
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	Quantile <math>p=0.01</math> is computed for all valid differences <math>d_i</math>. In other words: we obtain a specific value		Quantile <math>p=0.01</math> is computed for all valid differences <math>d_i</math>. In other words: we obtain a specific value
	for <math>d_i</math>, which is deceeded ~~for~~ just 1 % of all events but exceeded ~~for~~ 99 % of all events:		for <math>d_i</math>, which is deceeded by just 1 % of all events but exceeded by 99 % of all events:
	# if <math>n \cdot p</math> integer: <math>d_{\rm{Q01}} = 0.5\left( s_{n \cdot p}+s_{n \cdot p+1}\right)</math>;		# if <math>n \cdot p</math> integer: <math>d_{\rm{Q01}} = 0.5\left( s_{n \cdot p}+s_{n \cdot p+1}\right)</math>;
	# if <math>n \cdot p</math> real: <math>d_{\rm{Q01}} = s_{\lceil n \cdot p \rceil}</math>;		# if <math>n \cdot p</math> real: <math>d_{\rm{Q01}} = s_{\lceil n \cdot p \rceil}</math>;
	# in case that there are less than 32 valid data <math>d_{\rm{Q01}} = \rm{invalid}</math> is set.		# in case that there are less than 32 valid data <math>d_{\rm{Q01}} = \rm{invalid}</math> is set.
	Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q01}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.		Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q01}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

			====Quantile Q05====

			Quantile <math>p=0.05</math> is computed for all valid differences <math>d_i</math>. In other words: we obtain a specific value
			for <math>d_i</math>, which is deceeded by 5 % of all events but exceeded by 95 % of all events:
			# if <math>n \cdot p</math> integer: <math>d_{\rm{Q05}} = 0.5\left( s_{n \cdot p}+s_{n \cdot p+1}\right)</math>;
			# if <math>n \cdot p</math> real: <math>d_{\rm{Q05}} = s_{\lceil n \cdot p \rceil}</math>;
			# in case that there are less than 32 valid data <math>d_{\rm{Q05}} = \rm{invalid}</math> is set.
			Computation is performed for categories '''KC''' and '''KN'''. <math>d_{\rm{Q05}}</math> may be computed for different data set sizes at different locations. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

	===Percentiles===		===Percentiles===

← Older revision		Revision as of 07:10, 27 August 2018
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	For data generated by		For data generated by
	* [[Mathematical Models for Coastal Areas and Estuaries\|mathematical models]] (model results), or		* [[Mathematical Models for Coastal Areas and Estuaries\|mathematical models]] ([[model]] results), or
	* [[Analysis of Calculated Results\|analysis of calculated results]] (characteristic numbers), or		* [[Analysis of Calculated Results\|analysis of calculated results]] (characteristic numbers), or
	* measured data (observational data)		* measured data (observational data)
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	For geophysical data categories K1, KC and KN are of significance. Examples:		For geophysical data categories K1, KC and KN are of significance. Examples:
	* Category K1: topography/bathymetry <math>h(x,y,z,t_1)</math> for a specific instant in time;		* Category K1: topography/bathymetry <math>h(x,y,z,t_1)</math> for a specific instant in time;
	* Category KC: water level <math>\eta(x,y,z,t_i)</math> at discrete times <math>t_i</math> with constant time step, e. g. computed by a mathematical model;		* Category KC: [[water level]] <math>\eta(x,y,z,t_i)</math> at discrete times <math>t_i</math> with constant time step, e. g. computed by a mathematical model;
	* Category KN: tidal high water <math>\eta^{\rm{HW}}(x,y,z,t_i)</math> for times <math>t_i</math> at ''non-equidistant'' time intervals, e.g. derived from a water level time serie.		* Category KN: tidal high water <math>\eta^{\rm{HW}}(x,y,z,t_i)</math> for times <math>t_i</math> at ''non-equidistant'' time intervals, e.g. derived from a water level time serie.

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	Maximum difference is determined using ''absolute'' value (modulus) in combination with sign preservation:		Maximum difference is determined using ''absolute'' value (modulus) in combination with sign preservation:
	# At first all differences <math>d_i</math> will be computed as indicated above;		# At first all differences <math>d_i</math> will be computed as indicated above;
	# Out of all valid data index <math>i^\max</math> is determined in such a way that <math>\left\|d_i\right\|</math> is maximal		# Out of all valid data [[index]] <math>i^\max</math> is determined in such a way that <math>\left\|d_i\right\|</math> is maximal
	#:<math>d^\max = d_{i^\max}</math>		#:<math>d^\max = d_{i^\max}</math>
	#:is equal to the ''maximum difference'' according to this definition; this value can be negative, positive or zero;		#:is equal to the ''maximum difference'' according to this definition; this value can be negative, positive or zero;
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	# In case all <math>d_i</math> are invalid <math>d^{\rm{abw}} = \rm{invalid}</math> will be set.		# In case all <math>d_i</math> are invalid <math>d^{\rm{abw}} = \rm{invalid}</math> will be set.
	Computation is performed for categories '''KC''' and '''KN'''. A valid <math>d^{\rm{abw}}</math> is obtained as long there exists at least one valid difference <math>d_i</math>. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.		Computation is performed for categories '''KC''' and '''KN'''. A valid <math>d^{\rm{abw}}</math> is obtained as long there exists at least one valid difference <math>d_i</math>. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

			====Root mean square error (RMSE)====

			RMSE is computed for all valid differences:
			# At first all differences <math>d_i</math> will be computed as indicated above;
			# From all valid differences the mean deviation is computed from the well known definition for RMSE;
			# In case all <math>d_i</math> are invalid RMSE = '''invalid''' will be set.
			Computation is performed for categories '''KC''' and '''KN'''. A valid RMSE is obtained as long there exists at least one valid difference <math>d_i</math>. During visualization [[NCPLOT]] enables filtering using ancillary variable ''Number of valid differences''.

	====Number of valid differences====		====Number of valid differences====
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	# The number of valid differences is computed as		# The number of valid differences is computed as
	#:<math>N_{\rm{ord}}=\sum_{i\in I}P(d_i)</math>;		#:<math>N_{\rm{ord}}=\sum_{i\in I}P(d_i)</math>;
	Computation is performed for categories '''KC''' and '''KN'''. In program [[NCPLOT]] quantities like ''maximum difference'', ''minimum difference'', ''mean value'' as well as ''mean deviation'' can be visualised using this variable as a filter.		Computation is performed for categories '''KC''' and '''KN'''. In program [[NCPLOT]] quantities like ''maximum difference'', ''minimum difference'', ''mean value'' as well as ''mean deviation'' can be visualised using this variable as a [[filter]].
	In that way visualizations can be created where the results are shown for points only where e. g. either the maximum number of events or a specific number of events occurred.		In that way visualizations can be created where the results are shown for points only where e. g. either the maximum number of events or a specific number of events occurred.

	===Data for a Taylor diagram===		===Data for a Taylor diagram===

	Taylor diagrams provide "''a concise statistical summary of how well patterns match each other in terms of their correlation, their root-mean-square difference and the ratio of their variances''." Additional information such as bias can be added to the conventional Taylor diagram. The Taylor diagram provides a graphical framework that allows a suite of variables from a variety of (say) one or more models or reanalyses to be compared to reference data. The reference data can be observationally based (eg, reanalysis) or to another model or a control run.		Taylor diagrams provide "''a concise statistical summary of how well patterns match each other in terms of their correlation, their root-mean-square difference and the ratio of their variances''." Additional information such as [[bias]] can be added to the conventional Taylor diagram. The Taylor diagram provides a graphical framework that allows a suite of variables from a variety of (say) one or more models or reanalyses to be compared to reference data. The reference data can be observationally based (eg, [[reanalysis]]) or to another model or a control run.

	Literature and further informations:		Literature and further informations: