Adjusting the growth curve of sugarcane varieties using nonlinear models

Jane, Sérgio Alberto; Fernandes, Felipe Augusto; Silva, Edilson Marcelino; Muniz, Joel Augusto; Fernandes, Tales Jesus; Pimentel, Guilherme Vieira

doi:10.1590/0103-8478cr20190408

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CROP PRODUCTION • Cienc. Rural 50 (3) • 2020 • https://doi.org/10.1590/0103-8478cr20190408 copy

Adjusting the growth curve of sugarcane varieties using nonlinear models

Ajuste da curva de crescimento de variedades de cana-de-açúcar utilizando modelos não lineares

Authorship SCIMAGO INSTITUTIONS RANKINGS

ABSTRACT:

Assessing sugarcane (Saccharum spp.) stalk growth helps to adequately manage the phenological stages of the crop. The aim of this study was to describe the height-growth curve of four sugarcane varieties (RB92579, RB93509, RB931530 and SP79-1011), in irrigated plant-cane and ratoon cane plantations, using the Logistic and Gompertz nonlinear models, while considering all deviations from assumptions. The model parameters were estimated based on the least squares method using the Gauss-Newton algorithm. To select the most suitable model, nonlinear measures, adjusted coefficient of determination (R² _adj), residual standard deviation (RSD), and corrected Akaike information criterion (AICc) were used. Based on the best models, stalk height growth rates and crop phenological stages were determined using critical points. All tests were performed in the free software environment for statistical computing and graphics, R. In general, the Logistic and Gompertz models without AR(1) better described the plant-cane and ratoon cane stalk height, respectively. All varieties showed early growth, and the RB92579 variety presented higher rates in both cycles.

Key words:
logistic model; gompertz model; growth rates

Authorship

Sérgio Alberto Jane

Faculdade de Ciências Agrárias, Universidade Lúrio (UniLúrio), Nampula, Moçambique.Universidade Lúrio (UniLúrio)MoçambiqueNampula, MoçambiqueFaculdade de Ciências Agrárias, Universidade Lúrio (UniLúrio), Nampula, Moçambique.

Programa de Pós-graduação em Estatística e Experimentação Agropecuária, Departamento de Estatística (DES), Universidade Federal de Lavras (UFLA), Lavras, MG, Brasil.Universidade Federal de Lavras (UFLA)BrazilLavras, MG, BrazilPrograma de Pós-graduação em Estatística e Experimentação Agropecuária, Departamento de Estatística (DES), Universidade Federal de Lavras (UFLA), Lavras, MG, Brasil.

https://orcid.org/0000-0001-9666-8117

Felipe Augusto Fernandes

https://orcid.org/0000-0003-2552-3433

Edilson Marcelino Silva

https://orcid.org/0000-0002-2800-3495

Joel Augusto Muniz ^*E-mail: joamuniz@ufla.br. *Corresponding Author

Departamento de Estatística (DES), Universidade Federal de Lavras (UFLA), 37200-000, Lavras, MG, Brasil. Universidade Federal de Lavras (UFLA)BrazilLavras, MG, BrazilDepartamento de Estatística (DES), Universidade Federal de Lavras (UFLA), 37200-000, Lavras, MG, Brasil.

https://orcid.org/0000-0002-1069-4136

Tales Jesus Fernandes

https://orcid.org/0000-0002-1457-9653

Guilherme Vieira Pimentel

Departamento de Agricultura (DAG), Universidade Federal de Lavras (UFLA), Lavras, MG, Brasil.Universidade Federal de Lavras (UFLA)BrazilLavras, MG, BrazilDepartamento de Agricultura (DAG), Universidade Federal de Lavras (UFLA), Lavras, MG, Brasil.

https://orcid.org/0000-0001-9849-6427

E-mail: joamuniz@ufla.br. ^*Corresponding Author

DECLARATION OF CONFLICT OF INTERESTS

The authors declare no conflict of interest. The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; and in the decision to publish the results.

AUTHORS’ CONTRIBUTIONS

All authors contributed equally for the conception and writing of the manuscript. All authors critically revised the manuscript and approved of the final version.

SCIMAGO INSTITUTIONS RANKINGS

Figures | Tables | Formulas

Figures (3)
Tables (2)
Formulas (1)

Thumbnail

Figure 1
Graphs of the residuals of the fit of the Logistic (a, b, c, and d) and Gompertz (e, f, g, and h) models and of the Logistic (i, j, k, and l) and Gompertz (m, n, o, and p) models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles, respectively.

Thumbnail

Figure 2
Graphs of the partial autocorrelation function of the residuals of the fit of the Logistic (a, b, c, and d) and Gompertz (e, f, g, and h) models and of the Logistic (i, j, k, and l) and Gompertz (m, n, o, and p) models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles, respectively.

Thumbnail

Figure 3
Graphs of the fit of the best models adjusted to the average sugarcane stalk height. Logistic model (1A) for plant-cane. Gompertz (varieties RB92579, RB931530. and SP79-1011) and Logistic (RB93509) models for ratoon cane (1C). and respective stalk growth rates (1B and 1D).

Thumbnail

Table 1
P-values for the Shapiro-Wilk (SW), Breusch-Pagan (BP), and Durbin-Watson (DW) tests, intrinsic nonlinearity (IN), parametric effect (PE), adjusted coefficient of determination (

R_{a d j}^{2}

), residual standard deviation (RSD), and corrected Akaike information criterion (AICc) of the Logistic and Gompertz models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles.

Thumbnail

Table 2
Estimates of the parameters (A, B, and k), standard error of the estimate in parentheses, and autoregressive parameters (ϕ ₁ ) of the Logistic and Gompertz models adjusted to the average stalk height (m) of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles.

Thumbnail

Figure 1 Graphs of the residuals of the fit of the Logistic (a, b, c, and d) and Gompertz (e, f, g, and h) models and of the Logistic (i, j, k, and l) and Gompertz (m, n, o, and p) models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles, respectively.

Figure 2 Graphs of the partial autocorrelation function of the residuals of the fit of the Logistic (a, b, c, and d) and Gompertz (e, f, g, and h) models and of the Logistic (i, j, k, and l) and Gompertz (m, n, o, and p) models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles, respectively.

Figure 3 Graphs of the fit of the best models adjusted to the average sugarcane stalk height. Logistic model (1A) for plant-cane. Gompertz (varieties RB92579, RB931530. and SP79-1011) and Logistic (RB93509) models for ratoon cane (1C). and respective stalk growth rates (1B and 1D).

Table 1 P-values for the Shapiro-Wilk (SW), Breusch-Pagan (BP), and Durbin-Watson (DW) tests, intrinsic nonlinearity (IN), parametric effect (PE), adjusted coefficient of determination ( $R_{a d j}^{2}$ ), residual standard deviation (RSD), and corrected Akaike information criterion (AICc) of the Logistic and Gompertz models adjusted to the average stalk height of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles.

Models	Varieties	SW	BP	DW	IN	PE	$R_{a d j}^{2}$	RSD	AICc
--------------------------------------------------------------------------------Plant-cane-------------------------------------------------------------------------
Logistic	RB92579	0.053	0.087	0.040	0.056	0.106	0.994	0.087	-28.087
	RB93509	0.583	0.020	0.048	0.032	0.062	0.998	0.044	-51.511
	RB931530	0.886	0.026	0.020	0.074	0.162	0.987	0.097	-24.520
	SP79-1011	0.169	0.013	0.020	0.077	0.143	0.992	0.089	-27.653
Gompertz¹	RB92579	0.605	0.779	0.076	0.115	0.212	0.984	0.114	-17.998
	RB93509	0.067	0.472	0.244	0.081	0.152	0.991	0.089	-35.623
	RB931530	0.999	0.070	0.650	0.099	0.250	0.986	0.104	-23.498
	SP79-1011	0.018	0.808	0.290	0.124	0.238	0.981	0.140	-15.819
-------------------------------------------------------------------------------Ratoon cane -----------------------------------------------------------------------
Logistic	RB92579	0.659	0.015	0.886	0.046	0.110	0.989	0.093	-26.043
	RB93509	0.774	0.549	0.046	0.035	0.084	0.994	0.065	-38.066
	RB931530	0.091	0.159	0.046	0.052	0.123	0.985	0.080	-31.312
	SP79-1011	0.312	0.443	0.082	0.051	0.123	0.986	0.086	-28.716
Gompertz	RB92579	0.165	0.732	0.118	0.031	0.099	0.995	0.067	-37.051
	RB93509	0.193	0.456	0.044	0.036	0.115	0.994	0.068	-36.638
	RB931530	0.046	0.527	0.184	0.034	0.109	0.993	0.056	-43.190
	SP79-1011	0.497	0.069	0.126	0.035	0.116	0.993	0.063	-39.460

¹model with AR(1) errors

Table 2 Estimates of the parameters (A, B, and k), standard error of the estimate in parentheses, and autoregressive parameters (ϕ ₁ ) of the Logistic and Gompertz models adjusted to the average stalk height (m) of four sugarcane varieties (RB92579, RB93509, RB931530, and SP79-1011), produced in the plant-cane and ratoon cane cycles.

Models	Varieties	------------------------------------------Estimates of the parameters -----------------------------------
		A	B	k	ϕ ₁
------------------------------------------------------------------------------Plant-cane---------------------------------------------------------------------------
Logistic	RB92579	2.685 (0.039)	136.491 (1.919)	0.039 (0.002)
	RB93509	2.266 (0.019)	128.913 (1.159)	0.037 (0.001)
	RB931530	2.166 (0.049)	145.670 (3.281)	0.031 (0.003)
	SP79-1011	2.319 (0.045)	143.597 (2.550)	0.038 (0.003)
Gompertz¹	RB92579	2.711 (0.114)	121.752 (4.857)	0.026 (0.004)	0.634
	RB93509	2.300 (0.074)	114.051 (3.806)	0.025 (0.003)	0.683
	RB931530	2.262 (0.093)	127.230 (5.076)	0.019 (0.003)	0.441
	SP79-1011	2.336 (0.107)	128.017 (5.240)	0.026 (0.005)	0.549
----------------------------------------------------------------------------- Ratoon cane ------------------------------------------------------------------------
Logistic	RB92579	2.813 (0.043)	101.700 (2.458)	0.027 (0.002)
	RB93509	2.566 (0.031)	111.800 (1.937)	0.027 (0.001)
	RB931530	2.113 (0.036)	100.400 (2.782)	0.027 (0.002)
	SP79-1011	2.275 (0.040)	104.300 (2.829)	0.027 (0.002)
Gompertz	RB92579	2.874 (0.036)	80.199 (1.626)	0.019 (0.001)
	RB93509	2.636 (0.039)	89.671 (1.881)	0.018 (0.001)
	RB931530	2.160 (0.029)	79.395 (1.789)	0.019 (0.001)
	SP79-1011	2.334 (0.034)	82.920 (1.907)	0.018 (0.001)

¹model with AR(1) errors.

R_{a d j}^{2}

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[1] E-mail: joamuniz@ufla.br. ^*Corresponding Author