Seventh International Conference on Hybrid Intelligent Systems
Fuzzy and Neuro-Fuzzy Modeling for Total Volume study of Eucalyptus sp Ricardo M. de A. Silva¹, Adriano R. de Mendonça², Fillipe G. Brandão¹, Danilo M. Pires¹, Gleimar B. Baleeiro¹, Alexandre A. S. e Oliveira¹, Felipe L. Valentim¹, Natalino Calegário² ¹Grupo de Otimização e Inteligência Computacional – GOIC Departamento de Ciência da Computação – Universidade Federal de Lavras, CP 3037, Campus Universitário, Lavras, CEP 37200-000, MG, Brazil. 2
Departamento de Ciências Florestais - Universidade Federal de Lavras, CP 3037, Campus Universitário, Lavras, CEP 37200-000, MG, Brazil
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[email protected] these authors, the spread of Spurr model [8] should be explained by its adjustment facility, despite of its imprecise volume estimative for short trees. Pursuant to Couto et. al [5], the Spurr model [8] has presented better results to estimate volume and dry weight (overbark and underbark) of Pinus taeda six years old trees. Veiga et. al [9], studying equations to estimate volume of Acacia mangium Willd tree, have concluded that the best model was the Meyer model, followed by logarithmic form of Schumacher´s model. Modeling volume of Pinus oocarpa trees in different ages and thinning regime, Machado et. al [4] showed that the Schumacher and Hall model presents satisfactory residuals graphic distribution, although the adjusted R2 value to relative standard error had not been reached the best score. In short, due to all characteristics explained above, the Schumacher-Hall and Spurr model were selected as reference to volume estimative in this work. Therefore, the objective of this work consists in to evaluate fuzzy and neuro-fuzzy modeling for total volume study of Eucalyptus sp and compare them with the classical models.
Abstract Eucalyptus is the most valuable cultivated forest genus in Brazil nowadays. Modeling eucalypts volume has been important for foresters in recent years due to strong site and genetic variations, management regimes and multiple products generated from these plantations. At this work, fuzzy and neuro-fuzzy models were developed to represent the volume pattern of eucalypts clonal stands from Brazilian coast region. Likewise in other scientific field, these types of modeling methodologies showed to be precise and accurate.
1. Introduction
Due to wood supply reduction from indigenous forest, efforts have been made to implant them in order to attend consume and forest industry demands. The volume is one of most important information to forest stock in a region, because it is the base for forest inventory planning and execution, which are essentials for sustentable management of forest resource [1]. Several volume equations were just developed for forest planted in the Brazil with fast growing genus such as Eucalyptus [2][3] and Pinus [4][5]. The most common procedure use to estimate the volume per tree is the application of equations where the volume is a dependent variable, while the independent variables are usually represented by diameter at breast height (DBH) and total height (H). According to Campos et. al [6], Schumacher and Hall model [7] has been the most spread model due to some of its statistics characteristics, such as its usual unbiased estimates. Further, in conformity to
0-7695-2946-1/07 $25.00 © 2007 IEEE DOI 10.1109/HIS.2007.64
2. Materials and Methods The collected data corresponds to planting area located in the municipal district of Caravelas, Bahia, Brazil. We used a stand with area equals to 4.31 hectares of Eucalyptus sp planting 16 years old. The initial spacing in the planting was of 3x3 meters with a selective thinning 8.6 years age, resulting in 250 trees per hectare in the cutting cycle final. To get the data used in this work, we made a scaling of 40 trees to adjust the models. Using a
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caliper, two orthogonal measures were gotten with diameter in height of 1,30 meters and diameters at 0%, 1%, 2%, 3%, 4%, 5%, 10%, 15%, 25%, ..., and 95 % of total height. The calculus of volume of the overbark sections were made using the Smalian method:
memberships functions for total height and 25 fuzzy rules. The system used the “Sugeno” implication operator [15], by the easiness in adapt the technique of neural networks in the construction of a fuzzy logic systems. The neuro-fuzzy system treated the problem using the same variables of the Fuzzy System proposed: DBH (cm), Total Height (m) and Total Volume (m3). The fuzzy sets defined for the inputs variables are presented on the Figure 2.
g + g2 vi = 1 l , where vi = ith section’s volume; 2 g1= section area of extremity 1; g2= section area of extremity 2; and l = section length. After the section volume calculus, it was made the sum of the sections and gotten the tree total volume.
3. Evaluated Models
In the literature, there are many models that express the total volume of the trees. This work has used the two models most spread out among foresters: Schumacher-Hall and Spurr model. Author Model Schumacher and V = β 0 .DBH β1 .H β 2 .ε Hall [7] Spurr [8] V = β + β DBH 2 H + ε 0
1
where: V=total volume(m³); DBH (diameter at breast height)=diameter (cm) at 1,3 meters; H=total height (m); βi =regression parameters; ε =random error.
4.Fuzzy System To develop the logic fuzzy system using the Mamdani’s direct method, membership functions (MF) were built to each variable and IF-THEN type inference rules. The membership functions were defined in five categories related to the three variables: DBH (cm), Total Height (m) and Total Volume (m3). They were classified as very small, small, medium, high and very high. After that, it was specified a set of inference rules to relate the membership functions of each variable in order to build the inference mechanism [10]. In the premise part there are two variables: DBH and Total Height, and in the consequent part there is the Total Volume variable. The system has utilized the implication operator ‘min’ and ‘max’ composition method, because they are intuitive, widely accept and better represent the human experience [11]. The specified inference system variables membership functions are presented by Figure 1.
Figure 1. DBH, Total Height and Total Volume variables Membership Functions.
The neuro-fuzzy system treated the problem using the same variables of the Fuzzy System proposed: DBH (cm), Total Height (m) and Total Volume (m3). The fuzzy sets defined for the inputs variables are presented on the Figure 2.
5. Neuro-fuzzy System To develop the Neuro-Fuzzy System it was used the data base referring to forty trees. In this case, the membership functions and the set of inference rules were defined through a neuronal network [12][13][14]. The networks were trained with the hybrid method characterized by combination of the backpropagation and the minimum squares methods. Starting from neural networks with 75 nodes, they were defined 75 linear parameters, 20 non-linear parameters, 5 memberships functions for DHB, 5
Figure 2. DBH and Total Height NeuroFuzzy.
6. Models Evaluation Forty sample trees were used to the models adjustment. The equations were compared
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The precision of the measures on the tested equations are showed in the Table 3. Analyzing them, it can be verified the best adjustment degree to Neuro-Fuzzy model, because it presented superior r values and lower relative standard error (SY.X (%)), followed by the Schumacher-Hall, Spurr and Fuzzy models.
considering the correlation coefficient between observed and estimated values by adjusted equations (r) and relative standard error (SYX (%)). 2 ∧ S yx = Y − Y ( n − p − 1) ⋅ 100 ∧
where: SYX (%)) = Relative standard error; Y = total volumes (m³) estimated by equation; Y = total
Table 3. Precision Measures to the tested equations.
Model r Syx (%) Spurr 0.9760 9.60 Schumacher 0.9767 8.65 and Hall Fuzzy 0.9636 11.27 Neuro-Fuzzy 0.9999 0.22 The Figure 3 presents graphically the residual distribution in the total volume estimative. Analyzing this figure, it is possible to observe that Spurr and Fuzzy models do not present biased estimative. The residual distribution has values in the range of +/-15%. The Schumacher model presents fellow characteristics to Spurr and Fuzzy models, differentiating the residual variation gap, being from -15% to + 15%. The Neuro-Fuzzy residual variation has presented the best residual distribution, with values near to zero (0). The table 4 presents the “bias” (B), average of absolute differences (AAD) and standard deviations of the differences (SDD) statistics for the total volume. The table 5 shows the order assigned to the volumes estimative referring to the commercial diameters of 7 and 28 cm, based on statistics from Table 4. For instance, the Spurr equation has a bias (B) equals to 3,03 x 10-16 (Tabela 4). When this value is compared with B associated to the other models, the order assigned to this equation was 1. This value means that, considering B, the Spurr equation have the best estimative compared to the others evaluated models, followed, by the order, by Neuro-Fuzzy (2), Schumacher e Hall (3) e Fuzzy (4) models.
volumes (m³) observed; Y = average total volumes; n = number of observations; p = number of parameters (considering 0 (zero) to fuzzy and neurofuzzy, because they are nonparametric models). We made a residue graphic analysis. The residual values utilized in graphics construction are expressed by: Residuals (%) =
∧ 100 Y − Y Y
Also, we made complementary tests through the following statistics: bias (B); Average absolute differences (AAD); and Standard Deviations of the Differences (SDD). Bias (B) n ∧ n B = ∑ Yi − ∑ Y i n i =1 i =1
Average of the Absolute Differences (AAD) ∧ n AAD= ∑ Yi − Y i n i=1
Standard Deviation of the Differences (SDD)
SD D =
2 n n ∑ di2 − ∑ di i=1 i=1
n
(n − p)
Start from statistics analysis B, AAD and SDD, the order function was applied considering the precision degree, attributing weights from 1 to 4, according to the results of the statistics obtained from each equation and with the minimal commercial diameter in question [16][17]. It was considered the model most accurate that results the minimal sum in the notes to total volume
∧
[16][17], where: d i = Yi − Yi .
Table 4 - “bias” (B), average of absolute differences (AAD) and standard deviations of the differences (SDD) to the total volume.
The positives and negatives values of B statistic indicate sub-estimative and super-estimative, respectively. The least values from three tested statistics indicates that the equation presents best precision to the objective of the work.
Modelo B AAD SDD 1 3.03x10-16 0.1078 1.0222 2 0.0019 0.0990 0.9090 3 0.0366 0.1399 1.2150 0.0019 0.0233 4 1.29x10-6 1 = Spurr, 2 = Schumacher and Hall, 3 = Fuzzy and 4 = Neuro-Fuzzy.
7. Results and Discussion The estimated equations to the Spurr (1) and Schumacher e Hall (2) models were: V = 0,00000701408.DBH1,71426 .H1,67735 2
V = 0,0718437 + 0,0000292.DBH H
(1) (2)
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[2] MCTAGUE, J. P.; BATISTA, J. L. F.; STEIN, L.. Equações de vol. total, vol. comercial e forma do tronco de Eucalyptus em SP e RJ. IPEF, n. 41/42, p. 56-63, 1989. [3] GUIMARÃES, D.; LEITE, H. G.. Influência no no. de árvore na determinação da equação vol. para Eucalyptus grandis. Scientia forestalis, n. 50, p. 37-42. 1996. [4] MACHADO, S. A., CONCEIÇÃO, M. B., FIGUEIREDO, D. J. Modelagem do volume individual para diferentes idades e regimes de desbaste em plantações de Pinus oocarpa. Revista Ciêncas Exatas e Naturais. Guarapuava, v.4, n.2, p.185 - 197, 2002. [5] COUTO, H. T. Z. do; VETTORAZZO, S. C.. Seleção de equações de volume e peso seco comercial para Pinus taeda. Cerne. Lavras, v.5, n.1, p.69-80, 1999. [6] CAMPOS, J. C. C.; LEITE, H. G.. Mensuração florestal: perguntas e respostas. Viçosa: UFV, 2002. 407p. [7] SCHUMACHER, F.X.; HALL, F.dos. S. Logarithmic expression of timber-tree volume. Journal of Agric. Research, Washington, v.47, n.9, p.719-734, 1933.
Figure 3 – Residues distribution of the volume, in percentage, to estimate total volume (m3). Tabela 5 – Order assigned to the total volume estimatives, start from statistics on Table.
[8] SPURR, S.H. Forest inventory. New York: Ronald Press, 1952. 476p. [9] VEIGA, R. A. de A.; CARVALHO, C. M. de; BRASIL, M. A. M.. Determinação de equações de volume para árvores de Acacia mangium Willd. Cerne. Lavras, v.6, n.1, p.103-107, 2000. [10] TANAKA, K. An Introd. to Fuzzy Logic for Practical Applications. Kanazawa, Japão. Kanazawa Univ., 1991
Modelo B MD DPD Total 1 1 3 3 7 2 3 2 2 7 3 4 4 4 12 4 2 1 1 4 Following the presented reasoning and analyzing the data on Tables 4 and 5, we verified that the Neuro-Fuzy model presents the best results to total volume estimative, followed by Schumacher-Hall [7] and Spurr models. This results, generally, correspond with that founded earlier, that is, Syx (%) and the residual graphic analyses (Figure 3).
[11] DRIANKOV, D.; HELLENDOORN, H.; REINFRANK, M. An introduction to Fuzzy control. New York: Springer-Verlag, 1993. 316 p. [12] JANG, J. S. R. ANFIS: Adaptive-Net.-Based Fuzzy Inference System. IEEE Trans. on Systems, Man, and Cybernetics, New York, v. 23, n. 3, p. 665-685, 1993. [13] JANG, J. S. R.; SUN, C. T. Neuro-Fuzzy modeling and control. Proceedings of the IEEE, New York, v. 83, n. 3, p. 378-406, Mar. 1995. [14] ALMEIDA, M. R. A. Sistema híbrido Neuro-Fuzzygenético para mineração automática de dados. Rio de Janeiro : PUC. Dept. de Engenharia Elétrica, 2004. 112 p.
8. Conclusion The neuro-fuzzy model presented the better accuracy among tested models, including traditional models used in forestry scientific community (such as Spurr and Schumacher-Hall models). Even the fuzzy model had been presented worst result in comparison to other models, it has potential to be improved and this way, to get more accurated total volumes estimative.
[15] SUGENO, M. Industrial applications of fuzzy control. Elsevier Science Pub. Co., 1985. [16] LIMA, F. Análise de funções de “taper” destinadas à avaliação de multiprodutos de árvores de Pinus elliottii. 1986. 79p. Dissertação (Mestrado em Ciência Florestal) - Universidade Federal de Viçosa, Viçosa,1986.
9. References
[17] MENDONÇA, A. R. de; SILVA, G. F. da; OLIVEIRA, J. T. da S.; NOGUEIRA, G. S.. Avaliação de funções de afilamento de fustes de Eucalyptus sp. para multiprodutos. Cerne, v.13, n.1, p.71-82, 2007
[1] CLUTTER, J.L., FORTSON, J.C., PIENAAR, L.V., BRISTER, G.H., BAILEY, R.L., 1983. Timber Manag.: A Quantitative Approach. Krieger, Malabar, FL, p. 333.
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