Fuzzy and Neuro-Fuzzy Modeling for Total Volume

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. da S. e Oliveira¹, Felipe L. Valentim¹, Natalino Calegario², Geraldo R. Mateus3 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

3

Departamento de Ciência da Computação – Universidade Federal de Minas Gerais, Belo Horizonte, CEP 31270-010, MG, Brazil.

[email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected]

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 to attend consume and forest industry demands. The volume is one of most important information to forest stock in a region, because they are 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 used 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 [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, among others. Also in conformity to these authors, the spread Spurr [8] model should be explained due to its adjustment facility, despite of its imprecise volume estimative for short trees. Pursuant to [5], the Spurr model [8] has presented better results to estimate different models 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, concluded that the best model was the Meyer model, followed by logarithmic form Schumacher model. Modeling volume of Pinus oocarpa trees in different ages and thinning regime, [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.

2. Materials and Methods The collected data corresponds to the planting area located in the municipal district of Caravelas, Bahia, Brazil. It has used a stand with area equals to 4.31 hectares Eucalyptus sp planting 16 years old. The initial spacing used in the planting was of 3x3 m 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, it was made a scaling of 40 trees to adjust the models. Using a 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%, ..., e 95 % of total height. The calculus of volume of the overbark sections was made using the Smalian method:

 g + g2  vi =  1 .l  2  where: vi = ith section volume; g1= extremity 1 section area; g2= extremity 2 section area; 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 trees total volume. In this work they were used two models, which are much spread out among foresters. Author Schumacher and Hall [7] Spurr [8]

Model

V = β 0 .DBH β1 .H β 2 .ε V = β 0 + β1DBH 2 H + ε

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 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 “Mim” and ‘max’ composition method, because they are intuitive, widely accept and better represent the human experience [11]. The specified inference rules and system variables membership functions are presented by Table 1 and Figure 1, respectively. Table 1. Logic Fuzzy System Inference Rules

Rule

IF (DBH - cm)

AND (Total Height - m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

very high high small medium very small medium small high very small very high small small very small high

medium medium small medium very small small medium high medium high very small high small very high

THEN (Total Volume – m3) very high high small medium very small medium small very high small very high small medium very small very high

5. Neuro-fuzzy 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 method “backpropagation” and the method of the minimum squares. The Neuro-Fuzy System was configured with the following operators: 'AndMethod' = 'prod', 'OrMethod' = 'probor', 'ImpMethod' = 'min', 'AggMethod' = 'max', 'DefuzzMethod' = 'wtaver', in agreement with a Fuzzy Inference System. Starting from neural networks with 75 nodes, they were defined 75 linear parameters, 20 non-linear parameters, 5 pertinence functions for DHB, 5 pertinence functions for total height and 25 fuzzy rules. The system used the implication operator “Sugeno” [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.

The inference rules that relate the variables membership functions and the output linear functions are represented in the Table 2.

The inference rules that relate the variables membership functions and the output linear functions are represented in the table 2.

DBH (cm)

DBH (cm)

Total Height (m) Total Height (m) Figure 2. DBH and Total Height Variables Membership Functions for Neuro-Fuzzy System. Total Height (m)

Table 2. Neuro-Fuzzy System Inference Rules.

Rule

Total Volume (m3) Figure 1. DBH, Total Height and Total Volume variables Membership Functions.

The neuro-fuzzy system has 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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

IF (DBH cm) is in MF 1 is in MF 1 is in MF 1 is in MF 1 is in MF 1 is in MF 2 is in MF 2 is in MF 2 is in MF 2 is in MF 2 is in MF 3 is in MF 3 is in MF 3 is in MF 3 is in MF 3 is in MF 4 is in MF 4 is in MF 4 is in MF 4

AND (Total Height - m) is in MF 1 is in MF 2 is in MF 3 is in MF 4 is in MF 5 is in MF 1 is in MF 2 is in MF 3 is in MF 4 is in MF 5 is in MF 1 is in MF 2 is in MF 3 is in MF 4 is in MF 5 is in MF 1 is in MF 2 is in MF 3 is in MF 4

THEN (Volume – m3) is in MF 1 is in MF 2 is in MF 3 is in MF 4 is in MF 5 is in MF 6 is in MF 7 is in MF 8 is in MF 9 is in MF 10 is in MF 11 is in MF 12 is in MF 13 is in MF 14 is in MF 15 is in MF 16 is in MF 17 is in MF 18 is in MF 19

Rule

20 21 22 23 24 25

IF (DBH cm) is in MF 4 is in MF 5 is in MF 5 is in MF 5 is in MF 5 is in MF 5

AND (Total Height - m) is in MF 5 is in MF 1 is in MF 2 is in MF 3 is in MF 4 is in MF 5

THEN (Volume – m3)

Bias (B) n

B=

i =1

Forty sample trees were used to the models adjustment. The equations were compared considering the correlation coefficient between observed values and estimated values by adjusted equations (r) and the relative standard error (SYX (%)). 2

∧   Y − Y   S yx =  n − p −1

Y

.100

Where: SYX = Estimative standard error (m³); SYX (%)) = Relative standard error; ∧

Y = estimated by equation total volumes (m³);

n n

∧

∑Yi −Y i i=1

n

Standard Deviation of the Differences (SDD)

SDD =

2  n  n  2  d − d   ∑ i i ∑  i =1   i=1

 n   n− p

Start from statistics analysis B, AAD and SDD, the function ordination was proceeded considering the precision degree, attributing weights from 1 to 4, according to the results of the statistics gotten to each equation and with the minimal commercial diameter in question [16;17]. It was considered the most accurate, the model that results the minimal sum in the score to total volume [16;17]. Where:

∧   d i =  Yi − Yi  .  

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.

Y = observed total volumes (m³); Y = total volumes average; n = number of observations; p = number of parameters (considering 0 (zero) to fuzzy and neuro-fuzzy, because they are nonparametric models). A residue graphic analysis was made. The residual values utilized in graphics construction are expressed by: ∧

Residuals (%) =

i

i =1

Average of the Absolute Differences (AAD)

AAD=

S yx

∧

i

is in MF 20 is in MF 21 is in MF 22 is in MF 23 is in MF 24 is in MF 25

6. Models Evaluation

S yx (%) =

n

∑Y − ∑Y

Y −Y 100 Y

Also, complementary tests were realized through the following statistics: bias (B); Average of absolute differences (AAD); and Standards Deviations of the Differences (SDD).

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 (1) V = 0,0718437 + 0,0000292.DBH 2 H (2) The precision measures to the tested equations are showed in the Table 3. Analyzing the precision measures, it can be verified the best adjustment degree to Neuro-Fuzzy model, because presented the superior r values and lower relative standard error (SY.X (%)), followed by the Schumacher-Hall, Spurr and Fuzzy models.

Table 3. Precision Measures to tested equations.

8.65

0.9636 0.9999

11.27 0.22

The Figure 3 presents graphically the residual distribution of the total volume estimative. Analyzing the Figure 3 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% to +15%. It is also possible to observe a outlier presence in Spurr model. The Schumacher model presents fellow characteristics to Spurr and Fuzzy models, differentiating the residual variation range, being from -15% to + 15%. The Neuro-Fuzzy residual variaton 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 standards deviations of the differences (SDD) statistics to estimate the total volume. The Table 5 shows the score assigned to the volumes estimative referring to the commercial diameters of 7 and 28 cm, based on statistics from Table 4. As example, the Spurr equation has the bias (B) the value equals 3,03 x 10-16 (Tabela 4). When this value is compared with B associed to the other models, the score assigned to this equation was 1 (one). 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 (Score 2), Schumacher e Hall (Score 3) e Fuzzy (Score 4) models. Table 4 - “bias” (B), average of absolute differences (AAD) and standards deviations of the differences (SDD) statistics estimating the total volume.

Model 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 4 1.29x10-6 0.0019 0.0233 1 = Spurr, 2 = Schumacher and Hall, 3 = Fuzzy and 4 = Neuro-Fuzzy.

Residuals (%)

0.9767

40 35 30 25 20 15 10 5 0 -5 0,0 -10 -15 -20 -25 -30 -35 -40

0,5

1,0

1,5

2,0

2,5

3,0

2,5

3,0

Estimated Volume (m³)

Schumacher e Hall

Residuals (%)

Syx (%) 9.60

40 35 30 25 20 15 10 5 0 -5 0,0 -10 -15 -20 -25 -30 -35 -40

0,5

1,0

1,5

2,0

Estimated Volume (m³)

Fuzzy

Residuals (%)

r 0.9760

40 35 30 25 20 15 10 5 0 -5 0,0 -10 -15 -20 -25 -30 -35 -40

0,5

1,0

1,5

2,0

2,5

3,0

Estimated Volume (m³) NeuroFuzzy

Residuals (%)

Model Spurr Schumacher e Hall Fuzzy Neuro-Fuzzy

Spurr

40 35 30 25 20 15 10 5 0 -5 0,0 -10 -15 -20 -25 -30 -35 -40

0,5

1,0

1,5

2,0

2,5

3,0

Estimated Volume (m³)

Figure 3 - Distribution of the residues of the volume, in percentage, in function for estimate of total volume.

Tabela 5 – Score assigned to the total volume estimatives, start from statistics on Table 4.

Model 1 2 3 4

B 1 3 4 2

AAD 3 2 4 1

SDD 3 2 4 1

Total 7 7 12 4

Following the presented reasoning and analyzing the data on Tables 4 and 5, it was verified that the Neuro-Fuzy model presents the best results to total volume estimative, followed by Schumacher e Hall [7] and Spurr models. This results, generally, corroborates with the results previously founded, that is, Syx (%) and the residual graphic analyzis (Figura 3).

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.

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