Modelling Reference Evapotranspiration using a Novel Regression-based Method: Radial basis M5 Model Tree

doi:10.21203/rs.3.rs-208205/v1

In the current study, an ability of a novel regression-based method is evaluated in modelling daily reference evapotranspiration (ET₀), which is an important issue in water resources management plans and helps farmers in irrigation planning. The method was developed by hybridizing radial basis function and M5 model tree and called as radial basis M5 model tree (RM5Tree). The radial-based kernel function was used to control the input variables in modelling process of M5 model tree. The new model results were compared with traditional M5 model tree (M5Tree), response surface method (RSM) and two neural networks (multi-layer perceptron neural networks, MLPNN & radial basis function neural network, RBFNN) with respect to several statistical indices. Daily climatic data (relative humidity, RH, solar radiation, SR, wind speed, air temperature, T) recorded at three stations in Turkey, Mediterranean Region, were used. The effect of each weather data on ET₀ was also investigated by utilizing three different input scenarios with various combinations of input variables. On the whole, the RM5Tree provided the best results (NSE > 0.997) followed by the MLPNN (NSE > 0.990), and M5Tree (NSE > 0.945) in modelling daily ET₀. The SR was observed as the most effective input parameter on ET₀ which was followed by the T and RH. However, the findings of the third modelling scenario revealed that taking into account of all variables would considerably increase models’ accuracies for the three stations.

Climatology

Reference evapotranspiration

Modelling

Radial basis M5 model tree

Response surface method

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Table 1

Daily statistical characteristics of the dependent (T, W, SR & RH) and independent (ET₀) parameters in the three different meteorological stations in Turkey
Parameter	Station	No. data	Min	Max	Mean	Std.	Cv	Correlation*
Air temperature (C)	Adana	9959	0.40	35.40	19.93	7.25	0.36	0.76
	Antalya	7743	1.70	37.50	18.49	7.10	0.38	0.75
	Isparta	8734	-10.45	29.35	12.02	7.93	0.66	0.82
Wind speed (m/s)	Adana	9959	0.00	33.30	1.49	0.87	0.58	0.27
	Antalya	7743	0.00	13.00	2.80	1.55	0.56	0.27
	Isparta	8734	0.00	10.90	1.94	1.33	0.69	0.07
solar radiation (langley)	Adana	9959	3.00	726.15	342.95	152.02	0.44	0.92
	Antalya	7743	3.60	797.40	411.84	179.18	0.44	0.83
	Isparta	8734	7.70	731.80	321.00	140.14	0.44	0.91
Relative humidity (%)	Adana	9959	13.00	97.70	65.67	13.87	0.21	-0.32
	Antalya	7743	14.00	98.00	63.69	16.84	0.26	-0.68
	Isparta	8734	19.30	96.70	61.16	13.62	0.22	-0.79
ET0 (mm)	Adana	9959	-0.36	11.43	3.04	1.63	0.54	1.00
	Antalya	7743	-0.31	15.02	4.08	2.40	0.59	1.00
	Isparta	8734	-0.37	8.19	2.67	1.67	0.62	1.00
*Note: Correlations have been between the independent values and dependent value for each station

Table 2

Modelling scenarios based on the input variables for the applied models
Scenario	No. of input parameters	Category	Input parameter(s)
I	1	i	T_mean
		ii	W
		iii	SR
		iv	RH
II	2	v	T_mean, SR
II	3	vi	T_mean, SR, RH
III	4 (all)	vii	T_mean, SR, RH, W

Table 3

Comparative results of different modeling approaches for Isparta Station in training and testing phases
	Models	Structure	Training						Testing
	Models	Structure	MAE (mm)	RMSE (mm)	NES	R²	d	Max(RE)	MAE (mm)	RMSE (mm)	NES	R²	d	Max(RE)
	Category i, Input variables, T_mean (^oC)
Scenario I	M5Tree	--	0.675	0.862	0.733	0.856	0.918	4.291	0.782	0.975	0.659	0.825	0.898	3.409
	RSM	--	0.717	0.896	0.712	0.844	0.909	4.547	0.806	0.999	0.642	0.823	0.895	3.746
	RM5Tree	RF=5	0.674	0.861	0.733	0.856	0.918	4.296	0.782	0.975	0.659	0.825	0.898	3.409
		RF=10	0.688	0.873	0.726	0.852	0.915	4.326	0.779	0.968	0.664	0.828	0.900	3.505
		RF=20	0.688	0.873	0.726	0.852	0.915	4.348	0.779	0.968	0.664	0.828	0.899	3.480
		RF=50	0.688	0.873	0.726	0.852	0.915	4.349	0.779	0.968	0.664	0.828	0.899	3.482
	Category ii, Input variables, W(m/s)
	M5Tree	--	1.342	1.583	0.101	0.317	0.412	6.060	1.401	1.688	-0.022	0.267	0.472	4.996
	RSM	--	1.422	1.639	0.035	0.188	0.257	5.786	1.408	1.658	0.014	0.164	0.279	4.556
	RM5Tree	RF=5	1.341	1.579	0.105	0.323	0.415	6.049	1.406	1.694	-0.029	0.261	0.466	4.986
		RF=10	1.342	1.583	0.100	0.317	0.412	6.064	1.402	1.690	-0.025	0.264	0.471	4.990
		RF=20	1.340	1.581	0.103	0.320	0.416	5.963	1.400	1.688	-0.022	0.267	0.472	4.985
		RF=50	1.340	1.581	0.103	0.320	0.415	6.011	1.402	1.690	-0.024	0.265	0.471	4.984
	Category iii, Input variables, SR(langley)
	M5Tree	--	0.423	0.594	0.873	0.935	0.965	3.841	0.724	1.147	0.528	0.868	0.904	7.286
	RSM	--	0.505	0.672	0.838	0.915	0.954	4.230	0.601	0.814	0.762	0.914	0.945	3.912
	RM5Tree	RF=5	0.495	0.666	0.841	0.917	0.955	4.333	0.530	0.686	0.831	0.927	0.957	3.242
		RF=10	0.572	0.734	0.806	0.898	0.944	3.973	0.582	0.720	0.814	0.914	0.951	2.866
		RF=20	0.415	0.586	0.881	0.937	0.975	4.041	0.525	0.679	0.835	0.927	0.958	3.253
		RF=50	0.415	0.586	0.881	0.937	0.975	4.042	0.524	0.678	0.835	0.927	0.958	3.255
	Category iv, Input variables, RH_mean (%)
	M5Tree	--	0.740	0.958	0.671	0.819	0.894	3.574	0.851	1.099	0.566	0.762	0.850	4.202
	RSM	--	0.753	0.970	0.662	0.814	0.891	3.664	0.854	1.099	0.566	0.762	0.847	4.116
	RM5Tree	RF=5	0.740	0.958	0.670	0.819	0.894	3.573	0.851	1.099	0.566	0.762	0.850	4.205
		RF=10	0.741	0.959	0.670	0.818	0.894	3.547	0.850	1.098	0.567	0.763	0.850	4.215
		RF=20	0.741	0.959	0.670	0.818	0.894	3.550	0.850	1.098	0.567	0.763	0.850	4.212
		RF=50	0.741	0.959	0.670	0.818	0.894	3.550	0.850	1.098	0.567	0.763	0.850	4.208
Scenario II	Category v, Input variables, T_mean (^oC), SR(langley)
	M5Tree	--	0.214	0.331	0.961	0.980	0.990	2.766	0.501	0.729	0.809	0.935	0.956	9.957
	RSM	--	0.333	0.458	0.925	0.962	0.980	3.736	0.473	0.600	0.871	0.966	0.971	2.601
	RM5Tree	RF=5	0.159	0.256	0.976	0.988	0.994	2.555	0.459	0.566	0.891	0.967	0.974	2.614
		RF=10	0.149	0.243	0.979	0.989	0.995	2.274	0.479	0.618	0.863	0.965	0.969	2.597
		RF=20	0.138	0.236	0.980	0.990	0.995	2.377	0.494	0.660	0.844	0.961	0.966	3.198
		RF=50	0.154	0.254	0.977	0.988	0.994	2.561	0.455	0.551	0.891	0.967	0.974	2.571
	Category vi, Input variables, T_mean (^oC), SR (langley), RH_mean (%)
	M5Tree	--	0.130	0.212	0.984	0.992	0.996	3.098	0.378	0.516	0.904	0.959	0.976	2.923
	RSM	--	0.266	0.370	0.951	0.975	0.987	2.621	0.342	0.445	0.929	0.978	0.983	1.832
	RM5Tree	RF=5	0.115	0.186	0.988	0.994	0.997	1.831	0.327	0.422	0.936	0.973	0.984	1.616
		RF=10	0.098	0.161	0.991	0.995	0.998	1.742	0.279	0.398	0.943	0.978	0.987	2.967
		RF=20	0.094	0.150	0.992	0.996	0.998	1.743	0.230	0.331	0.961	0.986	0.991	2.869
		RF=50	0.084	0.134	0.994	0.997	0.998	1.522	0.215	0.320	0.963	0.986	0.991	3.013
Scenario III	Category vii, Input variables, T_mean (^oC),SR(langley), RH_mean(%),W(m/s)
	M5Tree	--	0.048	0.068	0.998	0.999	1.000	0.590	0.067	0.105	0.996	0.998	0.999	1.330
	RSM	--	0.062	0.087	0.997	0.999	0.999	0.699	0.069	0.096	0.997	0.999	0.999	0.632
	RM5Tree	RF=5	0.048	0.069	0.998	0.999	1.000	1.156	0.072	0.112	0.996	0.998	0.999	1.025
		RF=10	0.049	0.069	0.998	0.999	1.000	0.622	0.083	0.143	0.993	0.996	0.998	1.380
		RF=20	0.039	0.054	0.999	0.999	1.000	0.378	0.043	0.063	0.999	0.999	1.000	0.526
		RF=50	0.038	0.053	0.999	1.000	1.000	0.372	0.038	0.051	0.999	1.000	1.000	0.327

Note: the bold numbers are the best statistics among others; RMSE and MAE values are in mm

Table 4

Comparative results of different modeling approaches for Antalya station in training and testing phases
	Models		Structure		Training												Testing
					MAE (mm)		RMSE (mm)		NES		R²		d		Max(RE)		MAE (mm)		RMSE (mm)		NES		R²		d		Max(RE)
	Category i, Input variables, T_mean (^oC)
Scenario I	M5Tree		--		1.109		1.407		0.644		0.803		0.882		6.043		1.238		1.561		0.594		0.780		0.873		8.160
	RSM		--		1.156		1.443		0.626		0.791		0.874		6.423		1.240		1.549		0.600		0.783		0.875		8.260
	RM5Tree		RF=5		1.110		1.393		0.649		0.818		0.875		6.364		1.234		1.548		0.600		0.784		0.876		8.349
			RF=10		1.104		1.388		0.651		0.819		0.876		6.298		1.224		1.540		0.604		0.785		0.875		8.387
			RF=20		1.104		1.388		0.651		0.819		0.876		6.276		1.225		1.541		0.604		0.786		0.876		8.366
			RF=50		1.102		1.387		0.652		0.820		0.877		6.292		1.226		1.544		0.602		0.784		0.875		8.377
	Category ii, Input variables, W(m/s)
	M5Tree		--		1.788		2.200		0.131		0.361		0.475		9.819		1.849		2.321		0.101		0.364		0.445		10.529
	RSM		--		1.819		2.220		0.115		0.338		0.438		9.060		1.859		2.323		0.099		0.368		0.427		10.481
	RM5Tree		RF=5		1.766		2.160		0.127		0.357		0.462		9.723		1.851		2.320		0.102		0.368		0.439		10.904
			RF=10		1.765		2.157		0.129		0.360		0.467		9.955		1.850		2.316		0.106		0.372		0.446		10.326
			RF=20		1.765		2.157		0.129		0.360		0.467		9.805		1.849		2.316		0.105		0.372		0.446		10.545
			RF=50		1.766		2.156		0.130		0.362		0.470		9.843		1.849		2.316		0.105		0.372		0.447		10.175
	Category iii, Input variables, SR(langley)
	M5Tree		--		0.869		1.211		0.737		0.858		0.919		6.968		1.034		1.441		0.654		0.810		0.890		8.680
	RSM		--		1.005		1.342		0.677		0.822		0.895		8.122		0.963		1.330		0.705		0.841		0.903		7.319
	RM5Tree		RF=5		0.951		1.272		0.678		0.823		0.896		8.206		0.950		1.312		0.713		0.845		0.907		7.095
			RF=10		0.949		1.271		0.678		0.824		0.896		8.183		0.955		1.324		0.708		0.842		0.904		7.209
			RF=20		0.948		1.270		0.679		0.824		0.896		8.152		0.952		1.320		0.709		0.843		0.905		7.239
			RF=50		0.947		1.270		0.679		0.824		0.896		8.145		0.953		1.319		0.710		0.843		0.905		7.263
	Category iv, Input variables, RH_mean(%)
	M5Tree		--		1.411		1.665		0.502		0.708		0.812		5.792		1.548		1.841		0.434		0.665		0.783		7.775
	RSM		--		1.467		1.707		0.476		0.690		0.798		6.580		1.584		1.851		0.429		0.660		0.772		7.257
	RM5Tree		RF=5		1.398		1.638		0.497		0.705		0.803		5.800		1.547		1.821		0.446		0.673		0.781		6.806
			RF=10		1.389		1.627		0.504		0.710		0.807		5.850		1.547		1.828		0.443		0.670		0.783		7.229
			RF=20		1.386		1.626		0.505		0.711		0.808		5.902		1.545		1.826		0.443		0.671		0.785		7.134
			RF=50		1.386		1.625		0.506		0.711		0.809		5.885		1.546		1.827		0.443		0.671		0.784		7.114
Scenario II	Category v, Input variables, T_mean (^oC), SR(langley)
	M5Tree		--		0.446		0.683		0.916		0.957		0.978		4.808		0.966		1.338		0.701		0.864		0.926		6.964
	RSM		--		0.775		1.042		0.805		0.897		0.943		5.841		0.832		1.122		0.790		0.893		0.943		7.013
	RM5Tree		RF=5		0.362		0.597		0.936		0.967		0.983		5.845		0.864		1.146		0.781		0.885		0.938		7.379
			RF=10		0.293		0.493		0.956		0.978		0.989		5.156		0.826		1.098		0.799		0.900		0.947		6.202
			RF=20		0.292		0.495		0.956		0.978		0.989		5.448		0.818		1.101		0.798		0.903		0.948		5.755
			RF=50		0.273		0.465		0.961		0.980		0.990		5.448		0.817		1.099		0.799		0.903		0.948		5.809
	Category vi, Input variables, T_mean (^oC), SR (langley), RH_mean(%)
	M5Tree		--		0.142		0.242		0.989		0.995		0.997		2.720		0.370		0.607		0.939		0.970		0.984		3.699
	RSM		--		0.257		0.409		0.970		0.985		0.992		3.689		0.297		0.469		0.963		0.982		0.991		3.224
	RM5Tree		RF=5		0.138		0.253		0.988		0.994		0.997		2.774		0.399		0.580		0.944		0.973		0.985		4.305
			RF=10		0.124		0.214		0.992		0.996		0.998		3.312		0.309		0.470		0.963		0.982		0.990		3.391
			RF=20		0.111		0.192		0.993		0.997		0.998		2.068		0.294		0.462		0.964		0.983		0.991		3.180
			RF=50		0.101		0.164		0.995		0.998		0.999		1.694		0.300		0.469		0.963		0.983		0.991		3.051
Scenario III	Category vii, Input variables, T_mean (^oC),SR(langley), RH_mean(%),W(m/s)
	M5Tree		--		0.050		0.074		0.999		1.000		1.000		0.688		0.064		0.110		0.998		0.999		0.999		1.165
	RSM		--		0.066		0.097		0.998		0.999		1.000		0.832		0.074		0.108		0.998		0.999		1.000		0.870
	RM5Tree	RF=5		0.049		0.073		0.999		1.000		1.000		0.728		0.067		0.113		0.998		0.999		0.999		1.405
		RF=10		0.050		0.072		0.999		1.000		1.000		0.855		0.068		0.104		0.998		0.999		1.000		1.077
		RF=20		0.037		0.054		0.999		1.000		1.000		0.333		0.045		0.068		0.999		1.000		1.000		0.562
		RF=50		0.031		0.044		1.000		1.000		1.000		0.297		0.038		0.052		1.000		1.000		1.000		0.428

Note: the bold numbers are the best statistics among others; RMSE and MAE values are in mm

Table 5

Comparative results of different modeling approaches for Adana station in training and testing phases
	Models	Structure	Training						Test
	Models	Structure	MAE (mm)	RMSE (mm)	NES	R²	d	Max(RE)	MAE (mm)	RMSE (mm)	NES	R²	d	Max(RE)
	Category i, Input variables, T_mean (^oC)
Scenario I	M5Tree	--	0.809	1.016	0.595	0.771	0.860	4.625	0.875	1.102	0.580	0.762	0.850	7.161
	RSM	--	0.830	1.034	0.580	0.762	0.852	4.575	0.876	1.100	0.581	0.763	0.848	7.276
	RM5Tree	RF=5	0.804	1.007	0.586	0.769	0.860	4.622	0.861	1.084	0.594	0.771	0.853	7.209
		RF=10	0.803	1.006	0.587	0.770	0.860	4.669	0.864	1.091	0.588	0.767	0.852	7.260
		RF=20	0.802	1.005	0.587	0.770	0.860	4.660	0.867	1.094	0.586	0.766	0.852	7.259
		RF=50	0.805	1.007	0.586	0.769	0.859	4.595	0.864	1.088	0.591	0.769	0.852	7.229
	Category ii, Input variables, W(m/s)
	M5Tree	--	1.181	1.454	0.171	0.414	0.540	6.191	1.468	1.798	-0.118	0.234	0.483	7.819
	RSM	--	1.196	1.466	0.157	0.397	0.511	5.930	1.554	4.421	-5.762	-0.004	0.133	9.890
	RM5Tree	RF=5	1.193	1.462	0.162	0.402	0.515	6.090	1.548	1.906	-0.257	0.240	0.494	9.487
		RF=10	1.180	1.453	0.172	0.415	0.541	6.191	1.469	1.796	-0.116	0.225	0.479	9.198
		RF=20	1.214	1.488	0.132	0.367	0.499	5.925	1.607	1.979	-0.355	0.239	0.493	10.665
		RF=50	1.180	1.453	0.172	0.414	0.541	6.191	1.477	1.814	-0.138	0.225	0.482	9.500
	Category iii, Input variables, SR(langley)
	M5Tree	--	0.418	0.576	0.870	0.933	0.964	4.085	0.640	0.779	0.790	0.928	0.946	4.475
	RSM	--	0.465	0.625	0.847	0.920	0.957	4.382	0.629	0.745	0.808	0.937	0.950	4.878
	RM5Tree	RF=5	0.414	0.556	0.872	0.933	0.971	4.392	0.628	0.744	0.808	0.937	0.950	4.914
		RF=10	0.413	0.556	0.873	0.933	0.971	4.375	0.630	0.746	0.807	0.938	0.950	4.569
		RF=20	0.413	0.556	0.873	0.933	0.971	4.385	0.632	0.748	0.806	0.938	0.950	4.495
		RF=50	0.414	0.557	0.873	0.933	0.971	4.402	0.626	0.743	0.809	0.937	0.950	4.979
	Category iv, Input variables, RH_mean(%)
	M5Tree	--	1.169	1.405	0.225	0.475	0.582	6.056	1.308	1.591	0.124	0.429	0.604	6.465
	RSM	--	1.282	1.490	0.129	0.359	0.476	6.807	1.344	1.583	0.132	0.393	0.519	8.842
	RM5Tree	RF=5	1.166	1.376	0.194	0.464	0.567	5.550	1.313	1.552	0.167	0.449	0.594	6.712
		RF=10	1.136	1.360	0.213	0.486	0.597	6.203	1.303	1.569	0.148	0.439	0.599	6.215
		RF=20	1.164	1.373	0.197	0.467	0.568	5.903	1.312	1.550	0.169	0.453	0.597	6.447
		RF=50	1.163	1.373	0.198	0.468	0.569	5.795	1.311	1.550	0.168	0.452	0.597	6.686
Scenario II	Category v, Input variables, T_mean (^oC), SR(langley)
	M5Tree	--	0.246	0.386	0.942	0.970	0.985	3.486	0.601	0.762	0.799	0.929	0.950	4.726
	RSM	--	0.393	0.538	0.886	0.942	0.969	3.960	0.561	0.657	0.850	0.951	0.962	5.101
	RM5Tree	RF=5	0.395	0.541	0.885	0.941	0.969	3.891	0.554	0.652	0.853	0.950	0.962	5.237
		RF=10	0.172	0.286	0.968	0.984	0.992	2.696	0.559	0.654	0.852	0.952	0.962	4.943
		RF=20	0.164	0.275	0.970	0.985	0.992	2.374	0.553	0.651	0.849	0.953	0.962	4.776
		RF=50	0.156	0.267	0.972	0.986	0.993	2.770	0.572	0.674	0.843	0.952	0.961	4.476
	Category vi, Input variables, T_mean (^oC), SR (langley), RH_mean(%)
	M5Tree	--	0.154	0.241	0.977	0.989	0.994	1.610	0.419	0.581	0.883	0.954	0.971	4.726
	RSM	--	0.238	0.339	0.955	0.977	0.988	2.435	0.313	0.424	0.938	0.980	0.985	3.061
	RM5Tree	RF=5	0.122	0.196	0.985	0.992	0.996	1.659	0.331	0.445	0.932	0.971	0.983	3.389
		RF=10	0.098	0.164	0.989	0.995	0.997	2.517	0.305	0.420	0.939	0.979	0.985	3.117
		RF=20	0.080	0.124	0.994	0.997	0.998	1.380	0.167	0.325	0.963	0.984	0.991	5.310
		RF=50	0.083	0.142	0.992	0.996	0.998	1.802	0.273	0.389	0.948	0.979	0.987	3.069
	Category vii, Input variables, T_mean (^oC),SR(langley), RH_mean(%),W(m/s)
Scenario III	M5Tree		0.079	0.135	0.993	0.996	0.998	1.860	0.175	0.399	0.945	0.973	0.986	9.234
	RSM		0.037	0.055	0.999	0.999	1.000	0.652	0.071	0.533	0.902	0.952	0.976	11.300
	RM5Tree	RF=5	0.045	0.071	0.998	0.999	1.000	0.973	0.095	0.165	0.991	0.996	0.998	2.779
		RF=10	0.056	0.086	0.997	0.999	0.999	0.791	0.106	0.199	0.986	0.993	0.997	7.233
		RF=20	0.035	0.054	0.999	0.999	1.000	0.977	0.042	0.134	0.994	0.997	0.998	4.784
		RF=50	0.034	0.050	0.999	1.000	1.000	0.645	0.038	0.122	0.996	0.998	0.998	3.023

Note: the bold numbers are the best statistics among others; RMSE and MAE values are in mm

Table 6

Comparing the results of the best RM5Tree model with the best structures of well-known Radial basis neural network (RBFNN) and Multi-Layer Perceptron Neural Network (MLPNN) in the testing phases for the three meteorological stations
Station	Best Scenario	Best structure	Statistics
Station	Best Scenario	Best structure	MAE (mm)	RMSE (mm)	d	NSE
Isparta	III	RM5Tree(4,50,1)	0.038	0.051	1.000	0.999
		RBNN(4,50,1)	0.239	0.338	0.990	0.959
		MLPNN(4,7,1)	0.085	0.116	0.999	0.995
Antalya	III	RM5Tree(4,50,1)	0.038	0.052	1.000	1.000
		RBNN(4,50,1)	0.302	0.540	0.987	0.951
		MLPNN(4,7,1)	0.125	0.193	0.998	0.994
Adana	III	RM5Tree(4,50,1)	0.054	0.080	0.999	0.998
	II	RBNN(4,50,1)	0.333	0.468	0.981	0.924
	III	MLPNN(4,8,1)	0.063	0.171	0.997	0.990

Modelling Reference Evapotranspiration using a Novel Regression-based Method: Radial basis M5 Model Tree

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