Mathematical Modeling for Leaf Area Estimation From Papaya Seedlings ‘Golden THB’

The aim of this study was to select the most suitable model for leaf area estimation from papaya seedlings cv. ‘Golden THB’ using linear dimensions of leaves with unilobular and trilobular morphology. It was used leaves of 60 seedlings with 30 days after sowing produced in nursery of the Fazenda Santa Teresinha which belongs to company Caliman Agrícola S.A., in the municipality of Linhares, state of Espírito Santo, in March 2016. The measurement of the length (L) was performed along the midrib, the maximum width (W) of the leaf blade, the product of the length by the width (LW) and the observed leaf area (OLA). From these results, first degree and power linear regression models was adjusted. From the proposed regression models, the validation was performed with a leaves sample of 60 seedlings produced in June 2016, obtaining, thus, the estimated leaf area (ELA). The following criteria were used to choose the best model: the highest coefficient of determination (R2), the values do not significant of the comparison of means of OLA and ELA and values of MAE and RMSE closer to zero. The leaf area estimation from papaya seedlings cv. ‘Golden THB’ can be represented through equation ELA = -0.402619 + 0.612525(LW) for trilobular leaves and through equation ELA = 0.623355 + 0.610552(LW) for unilobular leaves.


Introduction
Papaya (Carica papaya L.) cv.'Golden THB' is characterized by great planting uniformity, vigorous plants and high yield, whose destination is mainly to external Market (Serrano & Cattaneo, 2010).
Knowing the leaf area is fundamental to evaluate the plants growth and development, being important in works involving physiology, photosynthesis efficiency, perspiration and behavior related to fertilizer and irrigation (Blanco & Folegatti, 2005).
The leaf area may be measured by direct or indirect method, depending on the objective of the study.The direct methods are destructive because the plant leaves are removed, and this method, mostly, is expensive for requesting specific equipment.The indirect methods are non-destructive, allowing successive leaf area estimation, and less costly (Norman & Campbell, 1989).
One of the non-destructive and indirect methods to estimate leaf area is through mathematical equations from linear dimensions as leaf length and width, and both dimensions in combination, whose high degree of accuracy is shown in most cases (Gamiely, Randle, Milks, & Smittle, 1991;Blanco & Folegatti, 2005).
Mathematical models that aim the indirect leaf area estimation have been used for different plant species as cocoa (Asomaning & Lockard, 1963), Cucumis sativus L. (Cho, S. Oh, M. M. Oh, & Son, 2007), Vicia faca L. (Peksen, 2007), Tabebuia and Handroanthus (Monteiro et al., 2017), colza (Tian et al., 2017) and Coffea canephora (Schmildt, Amaral, Santos, & Schmildt, 2015;Espindula et al., 2018).Methods have been described to estimate leaf area of papaya from adult plants, as mentioned by Campostrini and Yamanishi (2001)   (2) For the data validation, a new sample with 287 leaves was used, being 144 trilobular leaves and 143 unilobular leaves of 60 seedlings with 30 days after sowing produced in June 2016.The variables L, W, LW and OLA were measured according to previously proposed methodology.The estimated leaf area (ELA, in cm 2 ) was obtained replacing all the values of L, W and LW in the obtained equation for modeling.A simple linear regression for each proposed model was generated, as well as the respective coefficient of determination (R 2 ), where ELA was the dependent variable and OLA was the independent variable.The means of OLA and ELA were compared by Student t-test at 5% probability level.The mean absolute error (MAE) and root mean square error (RMSE) were determined by the following equations: The choice of the best mathematical model that estimates the leaf area from papaya seedlings cv.'Golden THB' as a function of the length (L) along the midrib, the maximum width (W) of the leaf blade or the product of the length by the width (LW) considered the value of coefficient of determination (R 2 ) closest to the unit, the values do not significant of the comparison of means of OLA and ELA and values of MAE and RMSE closer to zero.The statistical analyses were perfor¨using R software (R Core Team, 2018) with scripts developed by data package ExpDes.ptversion 1.2 (Ferreira et al., 2018).

Results and Discussion
In table 1, it can be observed that in relation to the trilobular leaves used for the modeling, the value of the length (L) ranged from 1.600 to 6.200 cm, with a mean of 4.113 cm.The width (W) varied from 1.400 to 5.700 cm, average of 3.659 cm.The product of length and width (LW) varied from 2.240 to 35.340 cm 2 with an average of 15.958 cm 2 and the leaf area observed (OLA) varied from 1.200 to 20.700 cm 2 with a mean of 9.372 cm 2 .For the unilobular leaves L values varied from 2.100 to 5.300 cm with an average of 3,446 cm.The W ranged from 1.700 to 4.500 cm with a mean of 2.703 cm.LW ranged from 3.780 to 23.850 cm 2 with a mean of 9.749 cm 2 .OLA ranged from 1.900 to 15.100 cm 2 with an average of 6.576 cm 2 .All variables of the leaf sample used for validation presented values close to those used for modeling, and this practice is recommended by Levine, Berenson, Krehbiel, and Stephan (2012), since the values used for the validation should not extrapolate those used for the modeling.
In relation to the coefficient of variation (CV) of the trilobular and unilobular leaves samples, used in modeling, it was observed that the values ranged from 21.98 to 46.75%, whose values are classified as high and very high, according to Pimentel-Gomes (2009).However, these values are recommended in works that aim the leaf area modeling for characterizing different plant growth stages (Pezzini et al., 2018).
Table 1.Minimum, maximum and mean coefficient of variation (CV) of the variables length (L), width (W), product of the length by the width (LW) and observed leaf area (OLA) for papaya seedlings trilobular and unilobular leaves cv.'Golden THB' The accuracy of the leaf area estimation depends on the equation model used (Borghezan, Gavioli, Pit, & Silva, 2010).According to Tsialtas, Koundouras, and Zioziou (2008), in a few cases the equations may be used to estimate the leaf area of leaves with different morphologies, however, the adjusts do not always show efficiency when a high degree of accuracy is desirable.Thus, obtaining individual equations for papaya seedlings leaves cv.'Golden THB' with trilobular and unilobular shape become necessary.
When we analyzed the behavior of the first degree linear model for the trilobular leaves we saw that the lowest value of R 2 was obtained using W as independent variable and the highest value was used for LW.In relation to the behavior of equations with quadratic adjustment and power for trilobular leaves, the lowest value of R 2 was observed based on W, and the highest value was used as an independent variable (Table 2).Although the largest values of R 2 for quadratic and power adjustments were observed using L as the independent variable, the values were not very different from those found on the basis of LW as an independent variable.Montero, Juan, Cuesta, and Brasa (2000), studying non-destructives methods for leaf area estimation of Vitis vinifera L., verified that the use of only one variable such as the width, for instance, shows an inconstant method with the vegetative growth, being necessary making adjusts for different phenological stages.
Thus, models used to determine leaf area that takes into consideration only one linear dimension show lower degree of efficiency, being used only in a few cases.Thus, equations based on the set of dimensions and several leaves size, such as the product of the length by the width, are more desired for showing better adjusts for leaf area estimation (Espindula et al., 2018).
For the behavior of the proposed models for the unilobular leaves (Table 2), it was observed that the highest values of R 2 were achieved using LW as independent variable and the lowest values were obtained based on W as independent variable for all the equations.Schmildt et al. (2015), studying allometric model for leaf area estimation of Coffea canephora, also found higher values of R 2 using LW as independent variable, verifying that this characteristic better represents the modeling for this species and shows better adjust in the first degree linear model.Therefore, based on the R2 value of the mathematical models and the validation equations closest to the unit, the non-significant valuesof the comparison of the means of ELA and OLA, besides the values of MAE and RMSE closer to zero, the models of linear equation of first degree, quadratic and power using LW as independent variable are the most suitable to estimate leaf area of papaya seedlings of cv.'Golden THB' for trilobular and unilobular shaped sheets, attesting to a high degree of accuracy and efficiency.However, due to the ease of the calculations, the first degree linear model, represented by the ELA = -0.402619+ 0.612525 (LW) and ELA = 0.623355 + 0.610552 (LW) equations for trilobular and unilobular leaves, respectively, is recommended.

Conclusion
The leaf area estimation from papaya seedling cv.'Golden THB' can be determined with accuracy by the first degree linear model taking into consideration the product of the length by the width for trilobular leaves through equation ELA = -0.402619+ 0.612525(LW) and for unilobular leaves through equation ELA = 0.623355 + 0.610552(LW).

Table 3 .
Observed leaf area (OLA) and estimated leaf area (ELA) of first degree, quadratic and power linear equations for the independent variables length (L), width (W), product of the length by the width (LW), beyond the p value, mean absolute error (MAE) and root mean square error (RMSE) for papaya seedlings trilobular and unilobular leaves cv.'Golden THB' used in validation Note.*P values higher than 0.05 indicate that the observed leaf area (OLA) and the estimated leaf area (ELA) do not differ by Student t-test.