Abstract
Multivariate models with multiple linear regression (MLR), artificial neural network (ANN), and k-nearest neighbors (KNN) were developed to predict the modulus of rupture of Pinus sylvestris structural timber. The aim of this study was to develop and compare these models obtained from resonance and ultrasound tests, static modulus of elasticity tests, and different measured wood feature. Resonance tests were performed in the three vibration modes (edgewise, flatwise, and longitudinal) to obtain the fundamental resonant frequencies. To compare the goodness-of-fit of the different models, the 10-fold cross-validation method was used, which proved to be an adequate strategy to avoid overfitting. The variable with the best predictive capacity of the modulus of rupture was knottiness. The error was notably lower in the multivariate than the univariate models. The ANN and KNN algorithms showed no improvement over the MLR. The most suitable MLR for prediction of the modulus of rupture was the model with four variables: knottiness, edgewise dynamic modulus of elasticity, velocity of ultrasounds, and longitudinal resonant frequency.
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Modulus of Rupture Prediction in Pinus sylvestris with Multivariate Models Constructed with Resonance, Ultrasound, and Wood Heterogeneity Variables
Álvaro Fernández-Serrano * and Antonio Villasante
Multivariate models with multiple linear regression (MLR), artificial neural network (ANN), and k-nearest neighbors (KNN) were developed to predict the modulus of rupture of Pinus sylvestris structural timber. The aim of this study was to develop and compare these models obtained from resonance and ultrasound tests, static modulus of elasticity tests, and different measured wood feature. Resonance tests were performed in the three vibration modes (edgewise, flatwise, and longitudinal) to obtain the fundamental resonant frequencies. To compare the goodness-of-fit of the different models, the 10-fold cross-validation method was used, which proved to be an adequate strategy to avoid overfitting. The variable with the best predictive capacity of the modulus of rupture was knottiness. The error was notably lower in the multivariate than the univariate models. The ANN and KNN algorithms showed no improvement over the MLR. The most suitable MLR for prediction of the modulus of rupture was the model with four variables: knottiness, edgewise dynamic modulus of elasticity, velocity of ultrasounds, and longitudinal resonant frequency.
DOI: 10.15376/biores.17.1.1106-1119
Keywords: Cross-validation; k-Nearest neighbors; Neural network; Non-destructive testing; Resonant frequency
Contact information: Department of Agricultural and Forest Engineering, University of Lleida, Av. Rovira Roure, 191, 25198 Lleida, Spain; *Corresponding author: alvaro.fernandezserrano@udl.cat
INTRODUCTION
Resonant frequencies have been employed to determine bending strength of Pinus sylvestris wood. Most of the works (Arriaga et al. 2012; Hassan et al. 2013; Aira et al. 2019) only used one vibration mode (edgewise, flatwise, or longitudinal) to predict the modulus of rupture (MOR). Only a few used more than one vibration mode (Hassan et al. 2013; Arriaga et al. 2014; Dahlen et al. 2018). Likewise, few studies have used a combination of vibration and ultrasound techniques (Halabe et al. 1997; Hassan et al. 2013). Some authors have analysed the effect on the MOR of features of sawn timber such as knots (Arriaga et al. 2014; França et al. 2019; Villasante et al. 2019), the slope of grain (Arriaga et al. 2014), or the rate of growth (Martins et al. 2017; França et al. 2019).
Some studies have taken into account the combined effect of different variables, while using multiple linear regression (MLR) for MOR prediction, with different pine species (Arriaga et al. 2012; Martins et al. 2017; Villasante et al. 2019). Usually, the coefficient of determination (R2) is used to compare the model’s capacity to predict MOR. However, some authors proposed use of the root-mean-square error (RMSE) as a more suitable measure of the fit of the model (Pommier et al. 2013; Yang et al. 2017; Villasante et al. 2019).
Most of the works consulted employed linear regressions, though some authors proposed nonlinear regressions based on artificial neural networks (ANNs). In some cases, prediction of the mechanical properties of wood through ANN has given good results (Mansfield et al. 2007; García Esteban et al. 2009; Fathi et al. 2020); however, in others no improvements over MLR-based models were found when using ANN-based models (Tanaka et al. 1996; Villasante et al. 2019)
Multivariate analysis can be affected by overfitting, producing misleading models valid only for the used dataset. To avoid overfitting (Lever et al. 2016), a few authors employed the cross-validation method in studies on pine (Villasante et al. 2019), chestnut (Vega et al. 2012), poplar (Fathi et al. 2020), and oak (Faydi et al. 2017).
The aim of this study was to develop and compare multivariate (linear, and non-linear) models for MOR prediction of a batch of Pinus sylvestris timber. These models were developed based on the values of different variables obtained in resonance and ultrasound tests and different measured features (blue stain, rate of growth, slope of grain, resin pockets, knottiness, and wanes). Models were compared using RMSE values obtained with the cross-validation method. This study also aimed to find the variables that best explained the MOR. Grading of Pinus sylvestris timber from Montsec mountains (Spain) was not an objective of this work.
EXPERIMENTAL
Materials
Analyses were undertaken with samples of Pinus sylvestris from Montsec Range (NE of Spain) obtained from a local sawmill (Lleida, Spain). A total of 69 sawn wood samples were obtained by random sampling of timber stored at the sawmill. The nominal sample size was 70 mm × 100 mm × 2000 mm with different growth rings orientations. At the laboratory, 12 samples with bark pockets or rot were rejected, leaving a final total of 57 samples. The samples were stored in the test laboratory and subjected to periodic moisture content analyses using a wood moisture meter (Hydromette HB 30, GANN, Gerlingen, Germany) following EN 13183-2 (2002). The samples were removed from the drying process when they reached a constant weight (± 0.1% in 6 h), in accordance with EN 408:2011+A1 (2012). Final moisture was calculated using the oven dry method following EN 13183-1 (2002), with 20-mm-thick slices obtained from similar specimens that were not included in the test batch.
Methods
Resonance tests
The resonance tests were conducted following ASTM E1876‑15 (2015). The specimens were suspended using elastic cords situated in the nodes corresponding to the fundamental flexural resonance (Fig. 1) and were tapped in the centre of the edge, face, and end to obtain the three vibration modes: edgewise, flatwise, and longitudinal, respectively. A 22.7 g impulser was used, consisting of a 230 mm long × 4 mm diameter wooden handle with a 26-mm diameter glass marble at the tip. The vibration was registered using a microphone with cardioid polar pattern and frequency range of 20 Hz to 20 kHz (Rode NT-USB, Rode Microphones, Silverwater, Australia). The signal obtained was recorded and analysed with Audacity® (2015) (Audacity Team, Pittsburgh, PA, USA). A project sampling rate of 384 kHz and resolution of 24 bits were used for the recording. The frequency spectrum was analysed with the Hann function to obtain the fundamental resonant frequencies for the longitudinal (fLV), flatwise (fFV), and edgewise (fEV) vibration modes.
Fig. 1. Scheme of the resonant frequencies tests
On the basis of the frequencies, the corresponding MOEdyn were obtained through the following Eqs. 1 through 3 (Weaver et al. 1990),
where MOELV, MOEFV, and MOEEV are the dynamic modulus of elasticity (all expressed in MPa) for each type of vibration (longitudinal, flatwise, and edgewise, respectively), fLV, fFV, and fEV are the fundamental resonant frequencies (kHz) for each type of vibration, p is the density (g·mm-3) of the sample, and L, h, and b are the actual length, width, and thickness (all expressed in mm) of each sample, respectively. The effect of shear in the MOEdyn calculation was not taken into account because it had a reduced influence, as the ratios L/b and L/h were equal to or greater than 20 (Arriaga et al. 2014).
Ultrasound tests
The ultrasound tests were performed with a Sylvatest Duo ultrasound device with a frequency of 22 kHz (Concept Bois Structure, Montandon, France). The transducers were placed at the ends of the samples. Information about species, length, and MC of each sample had to be introduced in the ultrasounds device. After the test, the device’s screen showed the velocity of ultrasound waves (VU), the attenuation of the ultrasonic wave (peakU), the modulus of elasticity by ultrasound (MOEU), and the modulus of rupture by ultrasound (MORU).
Features and biological degradations
The samples were measured and weighed to obtain their dimensions and density following EN 408:2011+A1 (2012). The blue stain (BS), rate of growth (RG), slope of grain (SLG), and resin pockets were also measured in accordance with the procedure outlined in EN 1309-3 (2018).
Two different procedures were used for knot measurement. The first procedure (Fig. 2a) was based on the knot area ratio (KAR). The second one (Fig. 2b) was based on Annex A of EN 1309-3 (2018). The KAR indicates the proportion of the cross-section occupied by knots (Walker 1993), while MKAR (Fig. 2a) indicates the proportion of the margin cross-section (outer quarters) occupied by knots (Lam et al. 2005). The MKAR in compression (MKARC) refers to the outer quarter subjected to compression, and the MKAR in tension (MKART) denotes to the outer quarter subjected to tension. The KAR-based measurements were made in the centre third of the length of the sample, in the section where the KAR values were higher. 
Fig. 2. Scheme for the measurement of knottiness: a) Cross-section, KAR, MKAR, MKARC, and MKART). The dotted areas represent knots; b) Measure of knots following Annex A of EN 1309 3 (2018)
The knottiness was also measured following the procedure set out in Annex A of EN 1309-3 (2018), measuring each knot in the direction perpendicular to the length of the sample (Fig. 2b). For the variable knotT, the sum of the measurements of knots in the four faces was calculated. For the variable knot1/3, only the knots situated in the centre third of length of the sample were included.
Two procedures were also followed to measure the wanes. The procedure described in EN 1309-3 (2018) was performed first (see Fig. 3). The wane was evaluated as the highest ratio of the dimension of the wane on the face with respect to the complete dimension of the face (WNF). The same procedure was performed with respect to the edge (WNE). The mean of these two values was also calculated (WNM). These same measurements were also taken exclusively in the centre third of the length of the sample (WNF1/3, WNE1/3, and WNM1/3). The second procedure was to measure the wanes with the loss of volume with respect to the theoretical volume without wanes (WNVOL).
Static bending test
The samples were subjected to a four point static bending test following EN 408:2011+A1 (2012). This test was performed to obtain the global static modulus of elasticity (MOES) and the MOR. The tests were completed with a 50-kN universal testing machine (Cohiner, Lleida, Spain) and data acquisition with LabVIEW 7.1 (National Instruments, Austin, TX, USA). The distance between supports was 1.8 m and the distance between load points was 0.6 m. A displacement transducer was placed at the midpoint between the two supports. After the bending test, a 20-mm-thick slice was obtained close to the area of rupture to determine moisture content using the oven dry method following EN 13183-1 (2002).
Fig. 3. Measure of wanes following EN 1309-3 (2018)
Statistical analyses
Simple linear regressions (SLR) were developed to predict the MOR from each of the studied variables. The significance level for each variable was calculated with F-test. In the present work, the RMSE (Eq. 4) was used to compare the fit of the MOR prediction models.
(4)
where is the predicted value from non-destructive testing (NDT), MORi is the observed value with EN 408:2011+A1 (2012), and n the number of observations.
To evaluate the usefulness of a prediction model, it is more important to know the precision of the values that it generates (RMSE) than to quantify the variability (R2) of the predicted values (Mansfield et al. 2007; Alexander et al. 2015). The R2 value obtained with the whole dataset was used exclusively to compare the results with those of other authors. The MOR prediction was also carried out using multivariate models with different algorithms: multiple linear regression (MLR), artificial neural network (ANN), and k-nearest neighbors (KNN). For the multivariate algorithms, the variables that did not present a statistically significant SLR were discarded. Similarly, only one knottiness and one wane variable were selected.
For the MLR, a greedy selection method was applied using the Akaike information metric to reduce the number of variables (Frank et al. 2016). A stepwise regression method was then applied, removing one-by-one the variables that caused the lowest increase in the RMSE until a univariate model was obtained. The predictive capacity of the three multivariate algorithms was compared for each group of variables.
The ANN prediction model was generated with the WEKA software (Waikato University, version 3.6, Hamilton, New Zealand). A multilayer perceptron was used with sigmoid nodes and learning by backpropagation (Frank et al. 2016). The ANN parameters were adjusted in a prior test (learning rate, training time, momentum applied to the weights, and number of neurons).
Finally, a prediction model was also generated using a KNN model with the IBk algorithm in WEKA software. The KNN model is a non-parametric classification method that constructs a regression from the weighted average of the values of KNN (Frank et al. 2016). The KNN parameters were adjusted in a prior test (number of neighbor values and distance weighting method).
The prediction models generated for a dataset can be affected by overfitting. This effect appears when the model is fitted even to the noise of the sample, confusing the noise with the internal structure of the model (Lever et al. 2016). For small datasets, the K-fold cross-validation can help avoid overfitting (Lever et al. 2016). In this respect, according to Tetko et al. (1995), the cross-validation method avoids overfitting in complex predictive models.
The 10-fold cross-validation method (Hashim et al. 2016; Faydi et al. 2017; Villasante et al. 2019) was used to calculate the RMSE of the models. The samples were randomly distributed into ten groups or folds, using each group to validate the model generated with the remaining nine. Using the WEKA software, this procedure was repeated 5 times to obtain 50 RMSE values of each model (MLR, ANN, and KNN). The nonparametric Kruskal-Wallis test was used to detect significant differences between the RMSEs of the models. In those cases in which significant differences were detected, a post hoc analysis was performed with Dunn’s test and Bonferroni adjustment (Brunetti et al. 2016). A significance level of 0.05 was applied. The analysis of significant differences was performed using R-software (The R Foundation, version 3.6.1, Vienna, Austria).
RESULTS AND DISCUSSION
The values obtained for the measured variables are shown in Table 1. The mean MOR value (40.0 MPa) was similar to those obtained in previous studies (between 35 and 45 MPa) for Pinus sylvestris (Arriaga et al. 2012; Villasante et al. 2019) and other pine species, including Pinus radiata (Arriaga et al. 2014), Pinus nigra ( Íñiguez González et al. 2007; Arriaga et al. 2012), and southern pine (Yang et al. 2017).
Simple Linear Regression
For most of the variables in this study, a statistically significant correlation, though weak, was found for the MOR prediction while using a SLR (Table 1). In this work, the R2 value was used only to facilitate a comparison with results obtained by other authors in previous studies. The R2 values for the variables of knottiness were similar or slightly higher than those observed in previous studies with different pine species (Conde García et al. 2007; Arriaga et al. 2014; França et al. 2019). In addition, the R2 values for the different MOEdyn were similar to those found by other authors in different pine species (Pommier et al. 2013; Arriaga et al. 2014; França et al. 2019).
Table 2 contains the RMSE results obtained through the 10-fold cross-validation method with 5 repetitions. The variables that did not present a statistically significant SLR were excluded. In this sense, the 29 initial variables were reduced to 15.
For knot‑based MOR prediction, no significant differences were observed between the RMSE of the different measurement methods employed (KAR based or EN 1309-3 (2018)-based), except for MKARC that gave a worse predictive capacity than the rest (Table 1). Conde Garcia et al. (2007) in Pinus sylvestris and Pinus nigra and França et al. (2019) in southern pine also found small differences in the R2 values between different knottiness variables and MOR. In the present study, the variable knotT was chosen because it is easier to measure than any of the KAR-based variables. With respect to the different methods used to measure the influence of wanes, no statistically significant differences were observed between the RMSE values in MOR prediction (Table 1). The variable WNM was chosen because it was easier to measure than the loss of volume.
Table 1. Summary of the Study Variables (Regression with the Whole Dataset)
With respect to the variables included in the group of features and biological degradations, it was found that knotT and RG provided the lower RMSE values in MOR prediction (Table 2). Various authors have also found in different conifers that knottiness is a good MOR predictor, better than density (Conde García et al. 2007) and better than SLG (Zhou and Smith 1991). According to Zhou and Smith (1991), RG was also a good MOR predictor. However, other studies conducted with different pine species found that density was a better MOR predictor than other characteristics such as knottiness and RG (Carballo et al. 2009; França et al. 2019). The RMSE values obtained for RG (12.8 MPa) and for density (14.9 MPa) are higher than those obtained by Hautamäki et al. (2014) in Pinus sylvestris (9.8 MPa and 10.2 MPa, respectively). These differences may be due to the heterogeneity of timber from such distant sources. It is notable, however, that the RMSE of knotT (10.6 MPa) was practically the same as that obtained by Hautamäki et al. (2014) for KAR (10.4 MPa).
Table 2. RMSE of the Simple Linear Regressions for MOR Prediction (10-fold cross-validation, 5 repetitions)
In the simple vibration variables for MOR prediction, no statistically significant differences were found between the RMSE values obtained from VU and the three resonance frequencies (fLV, fFV, and fEV). In the compound variables, it was found that the ultrasound-based variables (MOEU and MORU) had the highest errors, while MOES and MOEEV gave the best MOR predictions. The RMSE value obtained for MOR prediction with MOES (11.71 MPa) was similar to those obtained by Pommier et al. (2013) in Pinus pinaster (13.9 MPa), Yang et al. (2017) in southern pine (14.65 MPa), and Hautamäki et al. (2014) in Pinus sylvestris (8.20 MPa). The RMSE values for MOEdyn calculated on the basis of SLR with the resonance tests (MOELV, MOEFV, and MOEEV) were between 11.66 MPa and 12.11 MPa, which were similar to those obtained by Pommier et al. (2013) and by Yang et al. (2017) in different pine species (12.5 MPa and 14.65 MPa, respectively).
Multiple Linear Regression
Including different variables should improve the fit of the model. A comparison of the RMSE values obtained in the univariate and multivariate linear regressions is shown in Table 3. To show overfitting, the RMSE calculated with the whole dataset, without cross-validation, is also provided in the table. It can be seen that this overfitted RMSE decreases constantly as the number of variables increases. However, this decrease is the result of the model being fitted to the noise of the dataset and therefore becoming an overfitted model. This can be verified in the fact that the RMSE calculated with 10-fold cross-validation initially decreases as variables are added to the model, but then subsequently increases. These results confirm that the cross-validation method is a good strategy to avoid overfitting.
Table 3. RMSE for MOR Prediction through MLR and SLR with 10-fold cross-validation and 5 Repetitions
Starting with a SLR with knotT, the inclusion of a new variable in the model resulted in important RMSE decrease in MOR prediction, reaching 17.9% in the model with 4 variables. However, the models with 5 or 6 variables added small decreases in the RMSE that were below 1% (with no statistically significant differences). With 7 or more variables the RMSE values began to increase indicating that these new variables added to the model were redundant. The model that obtained the lowest RMSE value was the MLR with 6 variables (8.56 MPa). However, the MLR model with 4 variables (knotT, VU, fLV, and MOEEV) was the simplest model that presented an RMSE (8.69 MPa) significantly lower than the SLR.
Other authors have also found improvements in MLR compared to SLR models in MOR prediction in pine. Arriaga et al. (2012, 2014) found that MOR prediction on the basis of MOELV or MOEEV improved when including in the model a knottiness variable. The results obtained by Grazide et al. (2015) showed that multivariate models based on knot variables and the MOES increased the R2 value 27% compared to the SLR based only on the MOES. Similarly, while working with RG, MOELV, and knottiness, França et al. (2019) found that the univariate MOR prediction models obtained lower R2 values than the MLR models that included two of these variables. The best result was obtained with those three variables for the MLR. Villasante et al. (2019) found that the lowest RMSE value in MOR prediction was obtained with a MLR that included 5 variables (velocity of resonant frequency, concentrated knot diameter ratio, attenuation of the ultrasonic wave, penetration depth, and withdrawal force of a screw).
Fig. 4. RMSE for MOR prediction through MLR, ANN, and KNN algorithms with 10-fold cross-validation and 5 repetitions; mean values and 95% confidence intervals (ns: non-significant differences)
Nonlinear Multivariate Models
It was found that the ANN models with a lower number of variables gave a lower error when using fewer neurons. The same was found by Tetko et al. (1995). The best predictions were obtained with an ANN based on a single hidden layer of 3 to 5 neurons. No statistically significant differences were found in MOR prediction between the MLR- and ANN-based models for any of the eight groups of variables studied (Fig. 4). These results concur with those of Villasante et al. (2019) with Pinus sylvestris and Tanaka et al. (1996) with Cryptomeria japonica. However, other studies have reported better predictions with ANN than MLR, including Mansfield et al. (2007) with Tsuga heterophyla and Fathi et al. (2020) with poplar. These discrepancies may be attributable to the fact that both these studies worked with small clear samples whereas the present study was conducted using structural size samples.
For the KNN-based models, the number of neighbor values employed were between 4 and 6. The inverse of the distance was used as distance weighting method. No improvement was observed in MOR prediction when using the KNN algorithm, with no statistically significant differences detected between the MLR and KNN results (Fig. 4).
CONCLUSIONS
- The variable with the lowest modulus of rupture (MOR) prediction error was knottiness. Despite the complexity required to obtain the knot area ratio (KAR) value, this method was unable to significantly decrease the error obtained when using the simpler method described in Annex A of EN 1309‑3 (2018).
- The prediction error of the best univariate model, based on knottiness, was clearly improvable through the incorporation of other variables. The model with 4 variables (knotT, VU, fLV, and MOEEV) was the most suitable, as adding more variables only decreased the error slightly and non-significantly.
- Performing tests in different vibration modes proved to be better than when using a single vibration mode test, as the best model combined a transverse and a longitudinal variable.
- The usage of a cross-validation method also proved to be a suitable strategy to avoid overfitting
.in multivariate models based on non-destructive testing (NDT) to determine bending strength in timber. - The nonlinear artificial neural network (ANN)- or k-nearest neighbors (KNN)-based models obtained similar results to those with multiple linear regression (MLR). No statistically significant differences were detected between these models in terms of MOR predictive capacity.
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Article submitted: July 9, 2021; Peer review completed: November 30, 2021; Revised version received and accepted: December 15, 2021; Published: December 17, 2021.
DOI: 10.15376/biores.17.1.1106-1119