EVALUATION OF NON-LINEAR TAPER EQUATIONS FOR PREDICTING THE DIAMETER OF EUCALYPTUS TREES

Souza, Guilherme Silverio Aquino de; Cosenza, Diogo Nepomuceno; Araújo, Ana Carolina da Silva Cardoso; Pimenta, Lucas Veiga Ayres; Souza, Ramon Barreto; Almeida, Filipe Monteiro; Leite, Helio Garcia

doi:10.1590/1806-90882018000100002

Guilherme Silverio Aquino de Souza ^**Corresponding author.

Universidade Federal de Viçosa, Programa de Pós-Graduação em Ciência Florestal, Viçosa, Minas Gerais -Brasil. E-mail: <guilhermesas.eng@gmail.com> and <dncosenza@gmail.com>.

Diogo Nepomuceno Cosenza

Universidade Federal de Viçosa, Programa de Pós-Graduação em Ciência Florestal, Viçosa, Minas Gerais -Brasil. E-mail: <guilhermesas.eng@gmail.com> and <dncosenza@gmail.com>.

Ana Carolina da Silva Cardoso Araújo

Universidade Federal dos Vales do Jequitinhonha e Mucuri, Doutorado em Ciência Florestal , Diamantina, Minas Gerais-Brasil. E-mail: <anascaraujo@gmail.com>.

Lucas Veiga Ayres Pimenta

Universidade Federal de Viçosa, Programa de Pós-Graduação em Genética e Melhoramento, Viçosa, Minas Gerais -Brasil. E-mail: <lucasfip@yahoo.com.br>.

Ramon Barreto Souza

Universidade Federal de Viçosa, Graduado em Engenharia Florestal, Viçosa, Minas Gerais -Brasil. E-mail: <rabarretoso@gmail.com> and <filipema@gmail.com>.

Filipe Monteiro Almeida

Universidade Federal dos Vales do Jequitinhonha e Mucuri, Doutorado em Ciência Florestal , Diamantina, Minas Gerais-Brasil. E-mail: <anascaraujo@gmail.com>.

Helio Garcia Leite

Universidade Federal de Viçosa, Departamento de Engenharia Florestal, Viçosa, Minas Gerais -Brasil. E-mail: <hgleite@ufv.br>.

*Corresponding author.

Table 1 Parameter estimates of four non-linear taper models for comparison with three eucalyptus genotypes.

Table 2 Fit statistics of four non-linear models for three eucalyptus genotypes.

Model	Genotype	Fit statistics
		r_ŶY	s_yx (%)	Bias (%)	Bias (cm)	s²_bias	MAD (cm)
Garay (1979)Garay L. Tropical forest utilization system. VIII A taper model for the entire stem profile, including buttressing. Seattle: Coll. Forest Res., Inst. Forest Prod. Univ. Wash., 1979. 64p. (Contribuition, 36).	G1	0.995	4.577	0.080^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.011^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.396	0.462
	G2	0.995	4.890	-0 . 3 5 2 ^* * significantly different from zero at 5% probability by one sample t-test;	0.043^* * significantly different from zero at 5% probability by one sample t-test;	0.360	0.460
	G3	0.994	5.363	-0 . 3 4 9 ^* * significantly different from zero at 5% probability by one sample t-test;	0.044^* * significantly different from zero at 5% probability by one sample t-test;	0.447	0.513
Kozak (1988)Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988;18:1363 8.	G1	0.997	3.934	0.017^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.002^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.293	0.395
	G2	0.996	4.551	0.018^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.002^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.313	0.417
	G3	0.994	4.995	0.037^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.005^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.389	0.460
Max and Burkhart (1976)Max TA, Burkhart HE. Segmented polynomial regression applied to taper equations. Forest Science, 1976;22(33):283 9.	G1	0.996	4.195	0.092^* * significantly different from zero at 5% probability by one sample t-test;	0. 013^* * significantly different from zero at 5% probability by one sample t-test;	0.333	0.424
	G2	0.995	4.883	-0 . 3 7 2 ^* * significantly different from zero at 5% probability by one sample t-test;	0.046^* * significantly different from zero at 5% probability by one sample t-test;	0.359	0.465
	G3	0.994	5.085	-0 . 3 5 6 ^* * significantly different from zero at 5% probability by one sample t-test;	0.044^* * significantly different from zero at 5% probability by one sample t-test;	0.402	0.487
Demaerschalk (1972)Demaerschalk JP. Converting volume equations to compatible taper equations. Forest Science, 1972;18(3):241-5.	G1	0.995	4.833	0.030^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.004^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.442	0.447
	G2	0.993	5.795	-0 . 0 9 7 ^* * significantly different from zero at 5% probability by one sample t-test;	0.012^* * significantly different from zero at 5% probability by one sample t-test;	0.508	0.504
	G3	0.992	6.143	0.001^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.001^ns ns significantly equal to zero at 5% probability by one sample t-test.	0.589	0.520

Table 3 Validation statistics for individual tree volume (m3 per tree) by diameter at breast height classes.

Centro		Garay (1979)Garay L. Tropical forest utilization system. VIII A taper model for the entire stem profile, including buttressing. Seattle: Coll. Forest Res., Inst. Forest Prod. Univ. Wash., 1979. 64p. (Contribuition, 36).		Kozak (1988)Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988;18:1363 8.		Max and Burkhart (1976)Max TA, Burkhart HE. Segmented polynomial regression applied to taper equations. Forest Science, 1976;22(33):283 9.		Demaerschalk 1972)Demaerschalk JP. Converting volume equations to compatible taper equations. Forest Science, 1972;18(3):241-5.
de Classe	n
(DBH)		Bias	RMSE	Bias	RMSE	Bias	RMSE	Bias	RMSE
		(%)	(%)	(%)	(%)	(%)	(%)	(%)	(%)
G1:
1	12	0.647	7.292	0.308	7.617	0.682	7.577	0.310	8.539
2	12	0.841	7.976	0.801	7.943	0.807	7.807	1.390	8.929
3	12	1.103	7.539	0.492	6.924	1.137	7.091	0.933	7.891
4	15	0.267	7.735	0.020	6.098	0.231	6.586	0.007	7.993
5	12	1.648	7.570	1.601	5.799	1.687	5.588	1.6930	8.419
6	7	1.868	9.011	1.462	7.855	1.828	8.020	1.941	9.888
G2:
1	11	8.156	18.306	3.619	10.873	8.220	18.172	9.060	15.170
2	13	5.593	12.661	1.498	8.749	5.625	12.111	0.623	13.346
3	14	2.898	12.246	1.337	11.619	2.908	11.937	0.083	14.183
4	14	1.281	8.606	0.618	7.073	1.880	8.332	1.251	7.399
5	14	0.619	8.107	0.536	7.007	0.621	6.302	0.605	7.938
6	7	1.140	5.262	0.210	4.527	1.144	5.686	0.946	4.7470
G3:
1	12	8.661	14.138	1.021	9.108	8.671	14.100	1.612	10.311
2	11	8.019	11.649	2.482	8.391	8.001	11.769	2.677	9.093
3	16	1.583	9.321	0.796	8.286	1.558	8.802	1.389	12.167
4	13	0.483	10.442	0.417	9.570	0.405	9.118	0.381	9.520
5	13	0.228	9.044	0.095	7.271	0.266	7.189	0.120	7.482
6	7	2.479	9.420	0.706	8.518	2.527	9.497	0.048	9.054

Table 4 Parameter estimates of Kozak (1988) taper model for comparison of model equality of the genotypes and F value for the identity test.

Treatments	Estimates of Reduced Model Parameters
Treatments	α₁	α₂	α₃	β₁	β₂	β₃	β₄	β₅	F(H₀)
G1×G2 ×G3	1.126417	0.861247	1.005866	0.11343	0.05241	0.214200	0.180505	0.01327	14.58^* * Significant at 5% probability.
G1 × G2	1.154 825	0.843323	1.007321	0.16495	0.07089	0.497001	0.099817	0.06878	43.26^* * Significant at 5% probability.
G1 × G3	0.986572	0.936553	1.0191	0.13347	0.08122	0.520698	0.098094	0.16346	23.48^* * Significant at 5% probability.
G3 × G5	1.186154	0.836148	1.006253	0.16368	0.04264	0.02765	0.279462	0.00504	30.27^* * Significant at 5% probability.

\begin{matrix} Kozak (1988) : \\ d = α_{0} {DBH}^{α_{1}} α_{2} DBH {(\frac{1 - \sqrt{Z}}{1 - \sqrt{p}})}^{β_{1} Z^{2} +} \\ β_{2} \ln (Z + 0, 001) + β_{3} \sqrt{Z} + β_{4} e^{Z} + β_{5} {D B H . H}^{- 1} \\ + ε \end{matrix}

\begin{matrix} Max and Burkhart (1976) : \\ d = DBH \sqrt{β_{1} (Z - 1) + β_{2} (Z^{2} - 1) + β_{3} {(α_{1} - Z)}^{2} I_{1} + β_{4} {(α_{2} - Z)}^{2} I_{2}} + ε \end{matrix}

\begin{matrix} r_{\hat{YY}} = \frac{n^{- 1} \sum_{i = l}^{n} ({\hat{Y}}_{i -} {\hat{Y}}_{m}) (Y_{i -} \bar{Y})}{\sqrt{[n^{- 1} \sum_{i = l}^{n} ({\hat{Y}}_{i -} {\hat{Y}}_{m})] [n^{- 1} \sum_{i = l}^{n} (Y_{i -} \bar{Y})]}} RMSE \sqrt{n^{- 1} \sum_{i = l}^{n} {(Y_{i -} {\hat{Y}}_{i})}^{2}} \\ \begin{matrix} Bias = n^{- 1} \sum_{i = l}^{n} ({\hat{Y}}_{i -} Y_{i}) & MAD = n^{- 1} \sum_{i = l}^{n} |{\hat{Y}}_{i -} Y_{i}| \\ s_{yx} = \sqrt{{(n - p)}^{- 1} \sum_{i = l}^{n} {(Y_{i -} {\hat{Y}}_{i})}^{2}} & s_{bias}^{2} = \frac{Σ {[bias - (Y_{i -} Y_{i})]}^{2}}{n - 1} \end{matrix} \end{matrix}

\begin{matrix} F (H_{0}) = \frac{{SQ}_{R (H_{0})} \cdot {(pH - p)}^{- 1}}{{SQ}_{Res} (C) \cdot {(n - pH)}^{- 1}} \\ {SQ}_{R (H 0)} = {SQ}_{Par} (R) - {SQ}_{Par} (C); \\ {SQ}_{Par} (C) = Σ y_{i}^{2} - Σ Σ {(y_{h} - w_{h})}^{2} \\ {SQ}_{Par} (R) = y_{r}^{2} - Σ {(y_{r} - w_{r})}^{2} \\ {GL}_{R (H 0)} = {GL}_{C} - {GL}_{R} \\ {GL}_{Res} = n - p . H \\ F_{tab} = F_{α} ({GL}_{R (H 0)} {; G L}_{Res}) \end{matrix}

[1] *Corresponding author.

Brasil

Brasil

EVALUATION OF NON-LINEAR TAPER EQUATIONS FOR PREDICTING THE DIAMETER OF EUCALYPTUS TREES

AVALIAÇÃO DE FUNÇÕES DE AFILAMENTO NÃO-LINEARES PARA ESTIMAÇÃO DE DIÂMETROS COMERCIAIS EM ÁRVORES DE EUCALIPTO

ABSTRACT

Models	Parameters	Genotypes
		G1	G2	G3
	β₀	1.107340^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.190890^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.198089^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
Garay (1979)Garay L. Tropical forest utilization system. VIII A taper model for the entire stem profile, including buttressing. Seattle: Coll. Forest Res., Inst. Forest Prod. Univ. Wash., 1979. 64p. (Contribuition, 36).	β₁	0.414265^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.296575^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.246004^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₂	0.895645^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.961870^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.982824^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₃	0.525989^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.275107^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.232351^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	α₀	0.743504^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.303963^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.085063^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	α₁	1.079283^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.781925^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.888015^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	α₂	0.995487^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.009075^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.003484^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
Kozak (1988)Kozak A. A variable exponent taper equation. Canadian Journal of Forest Research. 1988;18:1363 8.	β₁	-0.307557^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.388695^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.047764^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₂	-0.091382^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.028682^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.047157^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₃	0.741395^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.149235^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.037372^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₄	0.126891^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.436131^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.231334^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₅	-0.372076^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.092661^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.032447^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	α₁	0.905747^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.917845^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.866042^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	α₂	0.026687^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.063157^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.069662^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
Max and Burkhart (1976)Max TA, Burkhart HE. Segmented polynomial regression applied to taper equations. Forest Science, 1976;22(33):283 9.	β₁	-4.245083^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-5.185282^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-4 . 1 8 1 6 3 ^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₂	1.910249^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	2.442356^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	1.867851^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₃	-1.550061^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-2.074024^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-1 . 8 0 3 6 9 ^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₄	541.005660^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	87.561162^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	89.320141^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₀	0.030710^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.081341^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.034373^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
Demaerschalk (1972)Demaerschalk JP. Converting volume equations to compatible taper equations. Forest Science, 1972;18(3):241-5.	β₁	1.011851^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.958780^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.924798^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₂	-0.327240^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.673794^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	-0.547186^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.
	β₃	0.632890^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.658029^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.	0.592764^* * Significant at 5% probability; βk, αk = Estimated Model Parameters.