Bamboo is widely distributed in tropical, subtropical, and some temperate zones of the planet (Du et al., 2018). This forest resource has attracted interest due to several ecological benefits such as its high potential for carbon sequestration, erosion control, water harvesting, and biodiversity conservation (Ceccon & Gómez-Ruiz, 2019). Its wide distribution, rapid growth rate, and excellent mechanical properties of its culms make bamboo a suitable material for multiple uses, among which construction stands out (Liese et al., 2015). The structural use of bamboo in its original form as it is found naturally in ready-to-use mode makes it a low-cost raw material in civil engineering works. In this sense, Trujillo (2016) indicated that external diameter and wall thickness are the most important dimensional properties for the use of bamboo culms in structural design, while the taper is the most relevant geometric property for the same purpose, so its study and analysis by taxon is essential.

Although more than 1,650 bamboo species have been recorded on the planet (Ruiz-Sánchez et al., 2021), not all of them can be used as a structural material in construction, so interest is focused on some genera that reach the largest dimensions in height and wall thickness, such as Arundanaria, Bambusa, Cephalostachyum, Dendrocalamus, Gigantochloa, Melocanna, Phyllostachys, Schizostachyum, Guadua and Chusqea (Correal, 2020). In Mexico, bamboo is naturally distributed in three main regions: (1) the western region, which includes the states of Michoacán, Colima, and Jalisco; (2) the southern region, which only includes the state of Chiapas; and (3) the central region, which includes the states of Puebla and Veracruz (Cedeño-Valdiviezo & Irogoyen-Castillo, 2011). It is possible to find natural stands of Guadua aculeata Rupr. bamboo, a native Mexican species that due to the commercial dimensions it reaches and its physical-mechanical properties, are frequently used in housing construction, mainly in the Totonaca culture region (Hernández & Torres, 2020). Important commercial plantations of the exotic bamboos Guadua angustifolia Kunth and Bambusa oldhamii Munro have also been established (Muñoz-Flores et al., 2021).

The species B. oldhamii, G. aculeata and G. angustifolia can reach heights of over 20 m and diameters over wall at breast height less than 20 cm in the case of the genus Guadua. Due to these characteristics, these taxa are labeled as giant bamboo (Toledo-Bruno et al., 2017). Their culms are used in rural constructions, furniture manufacturing, and handicrafts, and as tutors for tomato and vine crops (Camarillo-Cuenca et al., 2020). Recently, a large part of bamboo production has been absorbed by the construction industry, since in the last decade large tourism projects have been developed in which bamboo is considered the main material and structural element (Lorenzo et al., 2017). This allows for resilient constructions that are in harmony with the environment, while they function as carbon stores where this will remain sequestered for many years before being released into the atmosphere (Yadav & Mathur, 2021), an aspect that can contribute to reducing the phenomenon of global warming and climate change.

Despite the aforementioned importance, studies aimed at modeling the taper, predicting the external diameter, the wall thickness and therefore internal diameter, and estimating the distribution by type of product in bamboo culms are scarce both abroad and in Mexico. For this reason, there is a need to develop quantitative tools in the form of specific mathematical systems to determine the distribution of structural products for bamboo stands in both natural forests and commercial plantations. To address this issue, the approach used in timber-yielding tree species can be used, which consists of modeling the diameter profile using taper functions, whose mathematical expressions simulate a progressive thinning from the base to the tip of the stem (Ulak et al., 2022).

Strictly speaking, bamboos are not trees, but the shape of the culm is similar to that of a stem as its diameter also decreases as height increases and its value is zero when full height is reached (Inoue et al., 2021), so its shape could be adequately represented by taper models. The advantage of using this type of model is that an expression can be derived that allows estimating the commercial height when a given commercial diameter over wall of interest is predefined at the tip of the culm and vice versa (Tamarit-Urias et al., 2014). Concerning this topic, Zhang et al. (2021), and Ulak et al. (2022) concentrate different types of taper models to describe the stem profile of tree species, from which also expressions of total and commercial volume can be derived, making it possible to develop fully compatible volume estimation systems. Among the wide variety of taper models available, the segmented type such as that of Fang et al. (2000) and those of variable exponent such as those of Kozak (1997) are the most used in tree species.

The former is relevant in the case of bamboo harvesting because, unlike trees, where what is marketed is their volume, bamboo culms are marketed in linear meters mainly in the construction market and industry, where a minimum commercial diameter over wall at the tip is commonly required in purchase-sale operations. The main aim of this study was to generate a mathematical system to perform the distribution by structural product of bamboo culms. The specific ones were: (1) to evaluate the quality of fit of four taper models to describe the diameter profile over and under wall, (2) from the model with the best predictive capacity derivate an expression that estimates the commercial height, and 3) a function to modeling the wall thickness in the bamboo species B. oldhamii, G. aculeata, and G. angustifolia from commercial plantations of the first and third species, while the second corresponds to natural stands in the northeast of the state of Puebla, Mexico. The fundamental assumption that was taken as a basis was that at least one taper model can be applied to describe the diameter over wall profile of culms of the aforementioned species with a low estimation error.

The values of the goodness-of-fit statistics of the fitted taper models explained from 86.84% to 95.14% of the observed variability of the taper of the bamboo taxa studied; the estimation error ranged from 0.8263 cm to 1.3599 cm (Table 3). The rating system implemented revealed that model 1 presented the best predictive capacity; in particular, it had the highest R²_adj value of 95.14%, as well as the lowest RMSE value of 0.8263 cm. Model 3 ranked second in plausibility. Such evidence led to selecting model 1 as the appropriate one to fit the data of over and under wall taper of the bamboo culms for the species studied. Model 1 has the additional advantage that the parameters α₀, α₁, and α₂ correspond to the total culm volume (v) model known as the Schumacher-Hall (v=α₀Dow^α1 H^α2) (Fang & Bailey, 1999). Therefore, this model can additionally be useful to estimate the respective apparent volume and infer other relevant variables such as biomass and carbon content. In contrast, model 4 obtained the highest score, which makes it the least appropriate to describe the diameter profiles of the studied taxa.

The system of additive equations fitted through the WNSUR method and under the dummy variable technique showed non-significance in some parameters, so it is inferred that these are common, and therefore it is feasible to use the same values to describe profile culm. It was determined that the best combination of the system that achieves the additivity property, provides superior goodness-of-fit statistics, and has the best logical graphic behavior is defined as duw=dow-wt. Table 4 shows the statistics of the fitting, e.g., for dow, the model explained on average 97.7% (R²_adj) of the observed variability with an estimation error < 0.186 cm (RMSE), which confirms that model 1 is suitable for describing the over wall taper of culms at the bamboo taxon level. In the adjustment of the final structure of additive equations system with indicator variables, all estimated parameters were highly significant (p<0.05), highlighting the robustness of the model to predict this response variable (Table 5).

The assumption made that at least one taper model is applicable to describe the culm diameter profile of the studied species was confirmed. Based on bias values, model 1 tends to slightly underestimate the over wall taper in all three species. As for the modeling of wt using Eq. 1, on average an explanation of variability of 91.8% (R²_adj) was obtained with an estimation error < 0.134 cm (RMSE) (Table 4).

Regarding the values of the DW statistic, the autocorrelation correction with the CAR(2) structure minimized the effect of correlated errors, although to a lesser extent for dow which presented the lowest value (Table 4). The correction of heteroscedasticity for dow, wt, and duw determined the weighting factor that provided the best graphical behavior of the residuals vs. the predicted ones was the one corresponding to the variable Dow^0.5, the residuals compared to those predicted for dow, wt, and duw by taxon show a pattern with a random distribution trend (Fig. 3).

The estimated dow presented errors smaller than ±5 mm by diameter class for bamboo species; for the dow_min of commercial interest, the error was less than ±2 mm (Figs. 4a-c). The errors in the estimation of dow as a function of h presented acceptable behavior along the culm, the errors remained within ±5 mm (Figs. 4d-f). This indicates that the model accurately represents the culm profile of the species and the proposed dow_min for the distribution of structural products (Fig. 5).

The estimation of dow at different heights with model 1, and modeling duw presents a logical behavior because they reproduce profiles similar to those observed in the taxa studied (Fig. 5). In this sense, the system of additive equations is consistent because a decrease in the diameter over and under wall is maintained, especially in the upper two thirds of the height (Fig. 5). The same situation occurs for wall thickness prediction. Additionally, the additivity property is fulfil led, where the duw results from the difference between the dow and the wt. It is also guaranteed that the predicted duw will never be greater than the dow. In addition, dow, and duw faithfully adhere to the different growth habit of each species since the dimensions and level of conicity vary by taxon. In this analysis, G. angustifolia is comparatively more conical (Fig. 5c), followed by G. aculeata, which implies that they have a larger base that contributes to both having greater stability. B. oldhamii presented a more homogeneous taper compared to the other two species, which makes it special for obtaining products of better decorative and artistic quality. It was found that the equation for duw tends to produce slightly overestimated values for G. angustifolia and G. aculeata.

Equation solved to h from model 1 was useful for estimating the commercial height at a minimum diameter over wall (dow_min) given at the tip of the culm (Table 2), which in turn is important to be able to carry out the distribution of culm length by product type and to derive the respective table with the classification of the commercial use of culms by product.

When a dow_min ≥ 6 cm was set for the main use of the culm in structures for the construction industry and the rest of the culm diameter (dow_min < 6 cm) for other uses, slight differences were obtained in the estimated culm heights (lengths) for the species studied, which is indicative that the taper varies among taxa (Fig. 5).

The culm length distribution system by type of product can be adapted to other types of commercial uses and destinations by defining the minimum commercial diameter over wall of interest at the tip of the culm and then estimating the corresponding commercial height or length. The system allows knowing the linear meters (culm length) for some logical combination of Dow and H. The primary product with dow_min ≥ 6 cm that was destined for structural use can be in the form of beams, columns, or morillos. The fact that G. angustifolia and B. oldhamii are the species that provide the most primary product highlights their potential for the establishment of commercial forest plantations.

Of the taper models used, the Fang & Bailey (1999) one presented the best behavior for predicting the culm diameter profile of the three bamboo taxa studied. This result contrasts with that of Sanquetta et al. (2015), who determined that the description of such a profile in culms of B. oldhamii and Bambusa vulgaris Schrad. ex J. C. Wendl. can be done with a polynomial taper model with variable exponent, where B. oldhamii comparatively presented the best graphical behavior of the residuals vs. the predicted ones; likewise, they report that another taper model that was plausible for the same purpose was a fifth-order polynomial one. In the present study, model 3 of Demaerschalk (1972) was also plausible (Table 3).

The three bamboo species studied shared the parameters α0 y α2, in the model used to describe the dow and duw, while G. aculeata and G. angustifolia also shared α1. As for the wt modeling the two Guadua species shared δ1, while B. oldhamii shared δ4 with G. angustifolia (Table 5).

In general, the values of the DW statistic were higher for wt than for duw; wt had the value closest to 2, which shows that the autocorrelation correction was adequate. In all variables, the value of this statistic was greater than 1.609, except for dow, which was 1.237. However, when the autoregressive error structure was not introduced, this statistic in no case exceeded the value of 0.95, from which it can be inferred that the autocorrelation could only be partially removed. It is important to point out that when higher order structures were tested, there were convergence problems in the fit of dow, wt and duw, in addition to non-significance in some parameters. In contrast, Rodríguez & Molina (2003) and Diéguez-Aranda et al. (2006) achieved a better autocorrelation correction in taper models for clones of Populus × euramericana and Pinus sylvestris, respectively.

Although the autocorrelation correction achieved in the present study was partial, this fact does not limit the practical applicability of the models, according to Hernández-Pérez et al. (2013), since it does not affect the prediction of the taper because the values of its parameters maintain logical consistency (Fig. 5). With regard to heteroscedasticity, an adequate correction was achieved by introducing weighting factors for dow, wt and duw, since a distribution pattern of the residuals was achieved that tends to be homoscedastic; however, in the diameter interval of 5 to 13 cm, G. aculeata had a wider dispersion in dow (Fig. 3b), which may be because the culms came from natural stands that have had minimal forestry management.

Although the diameter profile in culms has been little studied utilizing taper models, the satisfactory result obtained regarding the improvements in the statistical properties of model 1 (high significance of parameters, autocorrelation correction, and heteroscedasticity) confirms and ratifies the assumption that such models can be applied to bamboo taxa with low estimation errors. In Mexico, model 1 has also been successfully fitted in studies with coniferous and broadleaf tree species, such as that of Hernández-Ramos et al. (2017) for commercial plantations of Eucalyptus urophylla in which that model explained 96% of the observed variability, while Torres-Ávila et al. (2020) for hybrids between Pinus caribaea and Pinus elliotti found that the same model explained 97% of the observed variability. In this sense, in the present study, such a model was consistent because it explained at least 95% of the variability; together with the values obtained in the RMSE and bias statistics, it is inferred that model 1 can offer a reliable distribution of structural products of culms. Nevertheless, it was discovered that the basal part of the bamboo culms does not adhere to a neiloide-type dendrometric body, which translates into a partial deficiency of model 1. In this sense, Inoue et al. (2021) reported that the taper in bamboo culms can be determined by the so-called “Taper Index based on Form-Factor” that relates culm surface area with the form-factor corresponding to the apparent volume of the culm. Additionally, the difference calculated between the observed and estimated basal diameter is of the order of millimeters; therefore, the minimal differences presented did not affect the estimate of h. Besides, the most relevant aspect was to estimate the commercial height (h) as well as the commercial diameter over wall at the tip of the culm, whose mathematical expressions will make up the mathematical system of product distribution.

The prediction of dow, duw and dow_min is exemplified in Fig. 5, which was built with data from bamboo culms with similar Dow between taxa. In this graphical representation of the variables, the diameter of the culm at a height of 1.3 m (Dow) was used. The graphical analysis showed that the taper of B. oldhamii was smaller and different from that of the other two species (Fig. 5), which means that the decrease in diameter with respect to height was more gradual compared to the other taxa, whose decreases were more pronounced (abrupt). This aspect implies that this species, from the point of view of its taper, could have a greater potential for use for structural purposes. This dissimilarity may be due in part to the morphology of the clump and growth habit of the species, where B. oldhamii presents closed pachymorphic growth, which makes the culms emerge very close to each other, generating culms that are more homogeneous; in contrast, the genus Guadua presents open pachymorphic growth, where the shoots emerge with comparatively greater separation and therefore have more growing space for access to moisture, nutrients, and sunlight (Aguirre-Cadena et al., 2018).

Estimation of the wall thickness is key in the culms bamboo, which Trujillo (2016) states is the second most important dimensional property to take into account when the culms are to be used in structural design and construction. In this sense, B. oldhamii had the smallest thickness (Table 1), while G. angustifolia and G. aculeata had the greatest thickness (Figs. 3b and 3c), so the latter are promising sources because according to Hernández-Santiago & Torres-Hoyos (2020), the culms with dow_min ≥ 6 cm are products with structural quality that meet the technical specifications in the construction market. It was also determined that wall thickness in B. oldhamii represented 25% of dow and in G. angustifolia and G. aculeata 32%. Additionally, based on Sanquetta et al. (2015), from the dow and duw equations, further studies could be aimed at estimating the timber volume that corresponds exclusively to the volume of the culm wall, as an important factor in estimating biomass and fiber production.

Future studies on bamboo culm taper should be aimed at overcoming limitations, such as: (1) using a larger sample size, besides incorporating covariates and the mixed effects model technique, which will allow fitting a system of additive equations separately for each bamboo taxon; and (2) the technical aspect that the evaluated taper models do not adequately predict the basal part of the culms because it does not adhere to the neiloide-type dendrometric body. In this sense, the need to generate specific taper models for giant bamboo species that consistently estimate the basal part is highlighted. Based on McTague & Weiskittel (2021), for the basal part of the culms of the taxa studied, it is necessary to explore other types of functions that are capable of appropriately describing the observed shape, such as a segmented function of increasing type or through some spline function.

In summary, this study showed that taper models can be used to describe the diameter profile over and under wall of mature and commercial culms of B. oldhamii, G. aculeata and G. angustifolia. Based on the selection of a taper model with high predictive capacity and a wall thickness function, a system of additive equations was formed, and a mathematical system was generated to carry out the distribution by structural product of culms of the three species in natural stands and in commercial plantations in northeast Puebla, Mexico. Comparatively, G. angustifolia and G. aculeata are more tapered and have a greater wall thickness, and B. oldhamii provides the greatest amount of primary product. The system will provide support in decision-making to generate strategies aimed at optimizing the production and distribution of products with predefined diameter over wall dimensions and determining the lengths in linear meters of the culm sections to be used for structural purposes in construction. The fitting strategy that was applied using WNSUR with autocorrelation correction in combination with the Dummy Variables technique is robust, modern, and up-to-date. It also fulfills the additivity property, so it can be applied to other giant bamboo species in other regions of the world.