Table 1 Unit and measuring range of sensor
Name of sensor
|
Measuring unit
|
Psychrometer
|
%
|
Heliograph
|
Minute
|
Anemometer
|
0.3 to 50 m/s
|
Wind direction
|
0° to 360°
|
Pyranometer
|
0…1400 W/m2 (Max 2000)
|
Albedometer
|
-2000 to 2000 W/m2
|
Air temperature
|
-30°C to 70°C
|
Soil temperature
|
-50°C to 50°C
|
Evaporation pan
|
Mm of water
|
Rain gauge
|
mm of water (resolution 0.1mm)
|
Table 2 Daily statistical parameters of data set
Data set
|
Unit
|
Xmin
|
Xmax
|
Xmean
|
Sx
|
CV (Sx/Xmean)
|
Tmin
Tmax
Tmean
RH
Ws
Sd
GR
|
°C
°C
°C
(%)
(m/s)
(h)
(mm)
|
-4.30
6.98
3.87
21.50
0.00
0.00
9.72
|
26.29
48.16
37.23
95.66
28.94
14.10
1791.04
|
11.68
27.28
19.48
59.69
6.66
7.21
969.52
|
6.87
8.89
7.53
14.39
3.81
4.14
446.08
|
0.59
0.33
0.39
0.24
0.57
0.57
0.46
|
Table 3 Parameters used for FFNN with one hidden layer
Hidden layer transfer Function
|
tangent sigmoid transfer function (tansig)
|
output layer transfer Function
|
Linear transfer function (purelin)
|
Training function
|
Levenberg-Marquardt
|
Maximum number of epochs to train
|
1000
|
Maximum validation failures
|
6
|
Minimum performance gradient
|
1e-7
|
Initial mu
|
0.001
|
mu decrease factor
|
0.1
|
mu increase factor
|
10
|
Maximum mu
|
1e10
|
Maximum time to train in seconds
|
Inf
|
Table 4 The R2 coefficient, RMSE and E criteria for estimation of ET and different variables input also optimum FFNN architect
Model
|
Input
|
Neurons
|
Training phase
|
Testing Phase
|
R2
|
RMSE
|
EF
|
R2
|
RMSE
|
EF
|
FFNN1
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
18
|
0.9903
|
0.2338
|
0.9901
|
0.9918
|
0.2389
|
0.9898
|
FFNN2
|
Tmax,Tmean,(Tmax-Tmin),RH,I,Ws,GR
|
19
|
0.9903
|
0.2332
|
0.9902
|
0.9921
|
0.2342
|
0.9902
|
FFNN3
|
Tmin,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
13
|
0.9905
|
0.2308
|
0.9904
|
0.9917
|
0.2368
|
0.9900
|
FFNN4
|
Tmin, Tmax,(Tmax-Tmin),RH,I,Ws,GR
|
19
|
0.9903
|
0.2336
|
0.9901
|
0.9920
|
0.2378
|
0.9899
|
FFNN5
|
Tmin, Tmax, Tmean,RH,I,Ws,GR
|
11
|
0.9899
|
0.2376
|
0.9898
|
0.9916
|
0.2393
|
0.9897
|
FFNN6
|
Tmin, Tmax, Tmean, (Tmax-Tmin),I,Ws,GR
|
12
|
0.9782
|
0.3481
|
0.9781
|
0.9859
|
0.3102
|
0.9828
|
FFNN7
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,Ws,GR
|
19
|
0.9883
|
0.2566
|
0.9881
|
0.9900
|
0.2536
|
0.9885
|
FFNN8
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,GR
|
14
|
0.9399
|
0.5799
|
0.9393
|
0.9603
|
0.5046
|
0.9544
|
FFNN9
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws
|
14
|
0.9676
|
0.4245
|
0.9675
|
0.9748
|
0.4032
|
0.9709
|
FFNN10
|
Tmean,RH,I,Ws,GR
|
16
|
0.9895
|
0.2435
|
0.9893
|
0.9907
|
0.2513
|
0.9887
|
FFNN11
|
Tmean,RH,Ws,GR
|
19
|
0.9875
|
0.2656
|
0.9873
|
0.9892
|
0.2623
|
0.9877
|
FFNN12
|
Tmean,RH,I,Ws
|
11
|
0.9672
|
0.4265
|
0.9671
|
0.9716
|
0.4124
|
0.9695
|
FFNN13
|
RH,I,Ws,GR
|
8
|
0.9217
|
0.6593
|
0.9215
|
0.9465
|
0.5533
|
0.9452
|
FFNN14
|
Tmean, ,RH,Ws
|
19
|
0.9172
|
0.6770
|
0.9172
|
0.9165
|
0.6845
|
0.9161
|
FFNN15
|
Tmean,RH
|
14
|
0.8528
|
0.9047
|
0.8522
|
0.8966
|
0.7745
|
0.8926
|
FFNN16
|
Tmean,Ws
|
7
|
0.8326
|
0.9633
|
0.8324
|
0.8520
|
0.9224
|
0.8477
|
FFNN17
|
RH,Ws
|
20
|
0.7771
|
1.1120
|
0.7767
|
0.8439
|
0.9488
|
0.8388
|
Table 5 The R2 coefficient, RMSE and E criteria for estimation of ET0 and different variables input also optimum RBF architect
Model
|
Input combination
|
|
Training phase
|
Test phase
|
Spread
|
R2
|
RMSE
|
EF
|
R2
|
RMSE
|
EF
|
RBF1
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
1187.55
|
0.9911
|
0.2215
|
0.9911
|
0.9909
|
0.2406
|
0.9896
|
RBF2
|
Tmax,Tmean,(Tmax-Tmin),RH,I,Ws,GR
|
1187.55
|
0.9910
|
0.2238
|
0.9910
|
0.9910
|
0.2382
|
0.9898
|
RBF3
|
Tmin,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
1385.47
|
0.9906
|
0.2279
|
0.9906
|
0.9910
|
0.2377
|
0.9899
|
RBF4
|
Tmin, Tmax,(Tmax-Tmin),RH,I,Ws,GR
|
1385.47
|
0.9907
|
0.2265
|
0.9907
|
0.9910
|
0.2378
|
0.9899
|
RBF5
|
Tmin, Tmax, Tmean,RH,I,Ws,GR
|
1385.47
|
0.9907
|
0.2270
|
0.9907
|
0.9911
|
0.2374
|
0.9899
|
RBF6
|
Tmin, Tmax, Tmean, (Tmax-Tmin),I,Ws,GR
|
791.70
|
0.9805
|
0.3284
|
0.9805
|
0.9842
|
0.3216
|
0.9815
|
RBF7
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,Ws,GR
|
1187.55
|
0.9890
|
0.2466
|
0.9890
|
0.9901
|
0.2445
|
0.9893
|
RBF8
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,GR
|
593.77
|
0.9456
|
0.5489
|
0.9456
|
0.9549
|
0.5076
|
0.9539
|
RBF9
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws
|
1781.32
|
0.9753
|
0.3696
|
0.9753
|
0.9616
|
0.4729
|
0.9600
|
RBF10
|
Tmean,RH,I,Ws,GR
|
791.70
|
0.9907
|
0.2267
|
0.9907
|
0.9901
|
0.2530
|
0.9885
|
RBF11
|
Tmean, RH, Ws, GR
|
593.77
|
0.9886
|
0.2514
|
0.9886
|
0.9892
|
0.2551
|
0.9884
|
RBF12
|
Tmean,RH,I,Ws
|
1583.40
|
0.9704
|
0.4047
|
0.9704
|
0.9699
|
0.4298
|
0.9669
|
RBF13
|
RH,I,Ws,GR
|
593.77
|
0.9300
|
0.6224
|
0.9300
|
0.9400
|
0.5873
|
0.9382
|
RBF14
|
Tmean, ,RH,Ws
|
1385.47
|
0.9214
|
0.6599
|
0.9214
|
0.9140
|
0.6941
|
0.9137
|
RBF15
|
Tmean,RH
|
791.70
|
0.8569
|
0.8902
|
0.8569
|
0.8915
|
0.7834
|
0.8901
|
RBF16
|
Tmean,Ws
|
791.70
|
0.8400
|
0.9413
|
0.8400
|
0.8480
|
0.9279
|
0.8458
|
RBF17
|
RH,Ws
|
791.70
|
0.7779
|
1.1089
|
0.7779
|
0.8434
|
0.9441
|
0.8404
|
Table 6 Used parameters in gene expression programming (GEP)
Number of chromosomes
|
30
|
head size
|
8
|
number of genes
|
3
|
linking function
|
Addition
|
fitness function error type
|
RMSE
|
mutation rate
|
0.044
|
inversion rate
|
0.1
|
IS transposition
|
0.1
|
RIS transposition
|
0.1
|
one-point recombination rate
|
0.3
|
wo-point recombination rate
|
0.3
|
gene recombination rate
|
0.1
|
Gene transposition rate
|
0.1
|
Table 7 The R2 coefficient, RMSE and E criteria for estimation of ET0 and different variables input also optimum GEP model
Model
|
Input combination
|
Training phase
|
Test phase
|
R2
|
RMSE
|
EF
|
R2
|
RMSE
|
EF
|
GEP1
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
0.8959
|
0.7732
|
0.8920
|
0.9190
|
0.6945
|
0.9136
|
GEP2
|
Tmax,Tmean,(Tmax-Tmin),RH,I,Ws,GR
|
0.9075
|
0.7227
|
0.9057
|
0.9323
|
0.6251
|
0.9300
|
GEP3
|
Tmin,Tmean, (Tmax-Tmin),RH,I,Ws,GR
|
0.9026
|
0.7361
|
0.9021
|
0.9300
|
0.6652
|
0.9208
|
GEP4
|
Tmin, Tmax,(Tmax-Tmin),RH,I,Ws,GR
|
0.8355
|
0.9692
|
0.8303
|
0.8627
|
0.9407
|
0.8416
|
GEP5
|
Tmin, Tmax, Tmean,RH,I,Ws,GR
|
0.8415
|
0.9721
|
0.8294
|
0.9033
|
0.8151
|
0.8810
|
GEP6
|
Tmin, Tmax, Tmean, (Tmax-Tmin),I,Ws,GR
|
0.9393
|
0.5804
|
0.9392
|
0.9629
|
0.4687
|
0.9607
|
GEP7
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,Ws,GR
|
0.9664
|
0.4323
|
0.9663
|
0.9762
|
0.3795
|
0.9742
|
GEP8
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,GR
|
0.8636
|
0.8695
|
0.8635
|
0.9194
|
0.6925
|
0.9141
|
GEP9
|
Tmin,Tmax,Tmean, (Tmax-Tmin),RH,I,Ws
|
0.9353
|
0.6081
|
0.9332
|
0.9537
|
0.5604
|
0.9438
|
GEP10
|
Tmean,RH,I,Ws,GR
|
0.9085
|
0.7138
|
0.9080
|
0.9275
|
0.6435
|
0.9258
|
GEP11
|
Tmean,RH,Ws,GR
|
0.9606
|
0.4830
|
0.9579
|
0.9775
|
0.3701
|
0.9755
|
GEP12
|
Tmean,RH,I,Ws
|
0.9420
|
0.5885
|
0.9374
|
0.9597
|
0.5045
|
0.9544
|
GEP13
|
RH,I,Ws,GR
|
0.8560
|
0.9004
|
0.8536
|
0.9236
|
0.7069
|
0.9105
|
GEP14
|
Tmean, ,RH,Ws
|
0.8388
|
0.9563
|
0.8349
|
0.8797
|
0.8406
|
0.8735
|
GEP15
|
Tmean,RH
|
0.8136
|
1.0598
|
0.7972
|
0.8635
|
0.9331
|
0.8441
|
GEP16
|
Tmean,Ws
|
0.7769
|
1.1312
|
0.7689
|
0.8057
|
1.0588
|
0.7993
|
GEP17
|
RH,Ws
|
0.6973
|
1.3112
|
0.6895
|
0.8062
|
1.1224
|
0.7744
|
Table 8 The R2 coefficient, RMSE and E criteria for the best combination of inputs
Model
|
Training
|
|
Testing
|
R2
|
RMSE (mm)
|
E
|
|
R2
|
RMSE (mm)
|
E
|
FFNN2
|
0.9903
|
0.2332
|
0.9902
|
|
0.9921
|
0.2342
|
0.9902
|
RBF-NN5
|
0.9907
|
0.2270
|
0.9907
|
|
0.9911
|
0.2374
|
0.9899
|
GEP11
|
0.9606
|
0.4830
|
0.9579
|
|
0.9775
|
0.3701
|
0.9755
|
Table 9 Single factor ANOVA results for the best combination of inputs
Source of Variation
|
F
|
P-value
|
Fcrit
|
Variation among groups
|
Between actual and FFNN2
|
0.171751
|
0.678682
|
3.854264
|
Insignificant
|
Between actual and RBF-NN5
|
0.101036
|
0.750681
|
3.854264
|
Insignificant
|
Between actual and GEP11
|
0.126406
|
0.72229
|
3.854264
|
Insignificant
|
Table 10 The R2 coefficient, RMSE and E criteria for the best combination of inputs
Statistic
|
FFNN2
|
RBF-NN5
|
GEP11
|
Minimum
|
-0.8840
|
-1.2231
|
-0.8671
|
Maximum
|
1.4199
|
1.5204
|
1.9343
|
1st Quartile
|
-0.0681
|
-0.0726
|
-0.1503
|
Median
|
0.0606
|
0.0250
|
-0.0055
|
3rd Quartile
|
0.2091
|
0.1742
|
0.2176
|
Mean
|
0.0713
|
0.0548
|
0.0630
|
Table 11 The R2 coefficient, RMSE and E criteria for the optimum combination of inputs
Model
|
Training
|
|
Testing
|
R2
|
RMSE (mm)
|
E
|
|
R2
|
RMSE (mm)
|
E
|
FFNN
|
0.9875
|
0.2656
|
0.9873
|
|
0.9892
|
0.2623
|
0.9877
|
RBF
|
0.9886
|
0.2514
|
0.9886
|
|
0.9892
|
0.2551
|
0.9884
|
GEP
|
0.9606
|
0.4830
|
0.9579
|
|
0.9775
|
0.3701
|
0.9755
|
Table 12 Single factor ANOVA results for the optimum combination of inputs
Source of Variation
|
F
|
P-value
|
Fcrit
|
Variation among groups
|
Between Observed and FFNN11
|
0.101466
|
0.750169
|
3.854264
|
Insignificant
|
Between Observed and RBF-NN11
|
0.119424
|
0.72976
|
3.854264
|
Insignificant
|
Between Observed and GEP11
|
0.126406
|
0.72229
|
3.854264
|
Insignificant
|
Table 13 Descriptive statistic of prediction errors for the optimum combination of inputs
Statistic
|
FFNN11
|
RBF-NN11
|
GEP11
|
Minimum
|
-0.6918
|
-0.7073
|
-0.8671
|
Maximum
|
1.5230
|
1.3700
|
1.9343
|
1st Quartile
|
-0.1227
|
-0.0952
|
-0.1503
|
Median
|
0.0119
|
0.0221
|
-0.0055
|
3rd Quartile
|
0.2228
|
0.2111
|
0.2176
|
Mean
|
0.0548
|
0.0599
|
0.0630
|