Use of mathematical models to estimate the total leaf area of banana

The objective of this work was to adjust and validate a mathematical equation to estimate the total leaf area of banana plants of the cultivars Maça BRS Princesa, Prata BRS Platina, Prata Gorutuba and Prata-Anã together, through the linear dimensions of the third leaf and the total number of leaves. For that, the linear first degree and power models were tested, where was used a dependent variable (y) the total leaf area (TLA) an independent variable (x) the multiplication of the length with the total number of leaves (LN), multiplication of the width with the total number of leaves (WN), and multiplying the length by the width with the total number of leaves (LWN). Analysis of covariance was used to test the possibility of using a single equation to estimate the total leaf area of the analyzed cultivars. For all equations, the mean absolute error, the root mean square error and the Willmott d index were obtained as validation criteria. The power model equation TLA = 3.9292(WN)^1.5851 can be used simply and quickly to determine the total leaf area of plants of the four cultivars under study.

Index terms
Musa spp.; non-destructive method; linear dimension; validation

Authorship

João Guilherme Viana Vieira João Guilherme Viana Vieira E-mail: joao5310b@gmail.com

Student in agronomy, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: joao5310b@gmail.com Universidade Federal do Espírito SantoBrasilAlegre, ES, Brasil Student in agronomy, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: joao5310b@gmail.com

http://orcid.org/0000-0002-2437-5320

Moises Zucoloto

PhD in Phytotechnics, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: moises.zucoloto@ufes.br Universidade Federal do Espírito SantoBrasilAlegre, ES, Brasil PhD in Phytotechnics, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: moises.zucoloto@ufes.br

http://orcid.org/0000-0003-0539-4750

Vinicius de Souza Oliveira

PhD student in Agronomy, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email:souzaoliveiravini@gmail.com Universidade Federal do Espírito SantoBrasilAlegre, ES, Brasil PhD student in Agronomy, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email:souzaoliveiravini@gmail.com

http://orcid.org/0000-0003-4068-1587

Fábio Luiz de Oliveira

PhD in Phytotechnics, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: fabio.oliveira.2@ufes.br Universidade Federal do Espírito SantoBrasilAlegre, ES, Brasil PhD in Phytotechnics, Universidade Federal do Espirito Santo, Alegre-ES, Brazil. Email: fabio.oliveira.2@ufes.br

http://orcid.org/0000-0002-1711-6988

João Guilherme Viana Vieira E-mail: joao5310b@gmail.com

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (1)
Tables (3)
Formulas (11)

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Figure 1
Power model equation fitted using total leaf area (TLA) in function of the product of the width with the total number of leaves (WN) and coefficient of determination (R2) to estimate the total area of leaves of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã.

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Table 1
Banana cultivars and their respective number of plants and number of leaves used for modeling.

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Table 2
Presentation of the p value in the analysis of covariance of the linear coefficients ( and angular coefficient ( of the equations of model linear first degree and power fitted with the total leaf area (TLA) in function of the product of the length with the total number of leaves (LN), product of the width with the total number of leaves (WN) and product of the length with the width and the total number of leaves (LWN) of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã.

Thumbnail

Table 3
Adjusted equations of the model linear first degree and power for estimating the total leaf area jointly of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã, in addition to the coefficient of determination (R2), mean absolute error (MAE), root mean square error (RMSE), Willmott index (d) and Lin's concordance correlation coeficiente (CCC).

(4)

(5)

(6)

(7)

(8)

(9)

Figure 1 Power model equation fitted using total leaf area (TLA) in function of the product of the width with the total number of leaves (WN) and coefficient of determination (R2) to estimate the total area of leaves of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã.

Table 1 Banana cultivars and their respective number of plants and number of leaves used for modeling.

Cultivar	Number of plants	Number of Leaves
Manzano BRS Princesa	20	164
Chunkey BRS Platina	18	144
Chunkey Gorutuba	9	46
Chunkey-Anã	10	54

Table 2 Presentation of the p value in the analysis of covariance of the linear coefficients ( and angular coefficient ( of the equations of model linear first degree and power fitted with the total leaf area (TLA) in function of the product of the length with the total number of leaves (LN), product of the width with the total number of leaves (WN) and product of the length with the width and the total number of leaves (LWN) of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã.

Model	Cultivar	Covariant	p Value
Model	Cultivar	Covariant
(4) $T L A = {\hat{β}}_{0} + {\hat{β}}_{1} L$	Princesa	Platina	0.053	<0.05
	Princesa	Gorutuba	0.349	0.279
	Princesa	Chunkey-Anã	0.291	0.174
	Platina	Gorutuba	<0.05	<0.05
	Platina	Chunkey-Anã	<0.05	<0.05
	Gorutuba	Chunkey-Anã	0.937	0.916
(5) $T L A = {\hat{β}}_{0} + {\hat{β}}_{1} W$	Princesa	Platina	0.505	0.907
	Princesa	Gorutuba	0.136	<0.05
	Princesa	Chunkey-Anã	0.247	0.059
	Platina	Gorutuba	0.055	<0.05
	Platina	Chunkey-Anã	0.102	0.065
	Gorutuba	Chunkey-Anã	0.510	0.494
(6) $T L A = {\hat{β}}_{0} + {\hat{β}}_{1} L W N$	Princesa	Platina	0.132	0.193
	Princesa	Gorutuba	0.697	0.261
	Princesa	Chunkey-Anã	0.924	0.279
	Platina	Gorutuba	0.106	0.068
	Platina	Chunkey-Anã	0.151	0.055
	Gorutuba	Chunkey-Anã	0.632	0.675
(7) $T L A = {\hat{β}}_{0} {(L N)}^{{\hat{β}}_{1}}$	Princesa	Platina	0.184	0.172
	Princesa	Gorutuba	0.071	0.073
	Princesa	Chunkey-Anã	0.115	0.120
	Platina	Gorutuba	<0.05	<0.05
	Platina	Chunkey-Anã	0.003	0.003
	Gorutuba	Chunkey-Anã	0.543	0.541
(8) $T L A = {\hat{β}}_{0} {(W N)}^{{\hat{β}}_{1}}$	Princesa	Platina	0.416	0.433
	Princesa	Gorutuba	0.073	0.065
	Princesa	Chunkey-Anã	0.141	0.120
	Platina	Gorutuba	0.058	0.056
	Platina	Chunkey-Anã	0.131	0.122
	Gorutuba	Chunkey-Anã	0.571	0.576
(9) $T L A = {\hat{β}}_{0} {(L W N)}^{{\hat{β}}_{1}}$	Princesa	Platina	0.416	0.433
	Princesa	Gorutuba	0.079	0.073
	Princesa	Chunkey-Anã	0.529	0.493
	Platina	Gorutuba	<0.05	<0.05
	Platina	Chunkey-Anã	0.081	0.075
	Gorutuba	Chunkey-Anã	0.200	0.203

^*
p Values greater than 0.05 indicate that the linear coefficients ( and the angular coefficient ( do not differ between the equations by Student's t-test.

Table 3 Adjusted equations of the model linear first degree and power for estimating the total leaf area jointly of the banana cultivars Manzano BRS Princesa, Chunkey BRS Platina, Chunkey Gorutuba, and Chunkey-Anã, in addition to the coefficient of determination (R2), mean absolute error (MAE), root mean square error (RMSE), Willmott index (d) and Lin's concordance correlation coeficiente (CCC).

Equation $T L A = - 12141.89057 + 0.81988 (L W N)$ $T L A = 3.9292 {(W N)}^{1.5851}$	R²	MAE	RMSE	d	CCC
	0.9097	97896.43	106193.1	0.0798	0.0258
	0.8251	25162.84	37341.83	0.8508	0.7366

M A E = \frac{\sum_{i = 1}^{n} |E T L A - T L A|}{n}

R M S E = \sqrt{\frac{\sum_{i = 1}^{n} {(E T L A - T L A)}^{2}}{n}}

d = 1 - [\frac{\sum_{i = 1}^{n} {(E T L A - T L A)}^{2}}{\sum_{i = 1}^{n} {(|E T L A - \bar{T L A}| + |T L A - \bar{T L A}|)}^{2}}]

(4)

T L A = {\hat{β}}_{0} + {\hat{β}}_{1} L

(5)

T L A = {\hat{β}}_{0} + {\hat{β}}_{1} W

(6)

T L A = {\hat{β}}_{0} + {\hat{β}}_{1} L W N

(7)

T L A = {\hat{β}}_{0} {(L N)}^{{\hat{β}}_{1}}

(8)

T L A = {\hat{β}}_{0} {(W N)}^{{\hat{β}}_{1}}

(9)

T L A = {\hat{β}}_{0} {(L W N)}^{{\hat{β}}_{1}}

T L A = - 12141.89057 + 0.81988 (L W N)

T L A = 3.9292 {(W N)}^{1.5851}

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Use of mathematical models to estimate the total leaf area of banana

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[1] João Guilherme Viana Vieira E-mail: joao5310b@gmail.com