A plethora of forest models were developed by transforming the dependent variable, which introduces bias if appropriate corrections are not applied when back-transformed. Many recognized models are still biased and the original data sets are no longer available, which suggests ad hoc bias corrections. The present research presents a procedure for bias correction in the absence of needed information from summary statistics. Additionally, we developed a realistic correction of the square root transformation based on a truncated normal distribution. The transformations considered in this study are the logarithm, the square root and arcsine square root. Using simulated data we found that uncorrected back-transformation created biases by as much as 100 percent. The generated data revealed that depending on available information, that bias can still be present after correction. In addition to generated data we corrected the site index of Douglas-fir and ponderosa pine in Oregon USA, tree volume of 27 species from Romania, stand merchantable volume for longleaf pine in Louisiana and East Texas USA, and canopy fuel weight in Washington USA. Using only the available information, the unbiased back-transformed estimates can change from ≤1 percent (i.e. the site index and canopy fuel weight) to ≥⅓ (tree and stand volume).

Advancements in information technology led environmental scientists to the illusion that efforts should be mainly focused on developing models that reduce uncertainty rather than on models adjusted to the existing uncertainty. As a result, environmental relationships are represented by non‐parsimonious and suboptimal models, which in many instances could be even wrong. The objective of this research was to provide scientists focused on modeling ecosystem processes with a procedure that supplies parsimonious correct results. The procedure transforms the response variable to achieve a linear model and the normality of the residuals. After the parameters of the transformed model are estimated, the bias induced by back‐transforming is corrected. We have computed the bias corrections for 11 of the most popular functions from the power, trigonometric, and hyperbolic families by considering the truncated normal distribution, when necessary. Using generated data, we have shown that the proposed procedure supplies unbiased results. We have identified a sample size artifact of data generation such that when the variance increases the truncation of distribution starts altering the corrections of predicted values, sometimes by more than 50% from the actual values. Our results indicate that uncertainty, measured by variance, impacts the analysis in a non‐intuitive way when the defining domain of the response variable is restricted. The subtle way of influencing the development of complex nonlinear models by uncertainty advocates the usage of parsimonious linear models, which are less sensitive to the method of processing data. Finally, ecosystem processes should be modeled with strategies that consider not only processes and computation aspects, but also uncertainty, in particularly reducing variance to levels with no significant impact on the results.

The parameters of nonlinear forest models are commonly estimated with heuristic techniques, which can supply erroneous values. The use of heuristic algorithms is partially rooted in the avoidance of transformation of the dependent variable, which introduces bias when back-transformed to original units. Efforts were placed in computing the unbiased estimates for some of the power, trigonometric, and hyperbolic functions since only few transformations of the predicted variable have the corrections for bias estimated. The approach that supplies unbiased results when the dependent variable is transformed without heuristic algorithms, but based on a Taylor series expansion requires implementation details. Therefore, the objective of our study is to investigate the efficient expansion of the Taylor series that should be included in applications, such that numerical bias is not present. We found that five functions require more than five terms, whereas the arcsine, arccosine, and arctangent did not. Furthermore, the Taylor series expansion depends on the variance. We illustrated the results on two forest modeling problems, one at the stand level, namely site productivity, and one at individual tree level, namely taper. The models that are presented in the paper are unbiased, more parsimonious, and they have a RMSE comparable with existing less parsimonious models.

To avoid the transformation of the dependent variable, which introduces bias when back-transformed, complex nonlinear forest models have the parameters estimated with heuristic techniques, which can supply erroneous values. The solution for accurate nonlinear models provided by Strimbu et al. (Ecosphere 8:e01945, 2017) for 11 functions (i.e., power, trigonometric, and hyperbolic) is not based on heuristics but could contain a Taylor series expansion. Therefore, the objectives of the present study are to present the unbiased estimates for variance following the transformation of the predicted variable and to identify an expansion of the Taylor series that does not induce numerical bias for mean and variance. We proved that the Taylor series expansion present in the unbiased expectation of mean and variance depends on the variance. We illustrated the new modeling approach on two problems, one at the ecosystem level, namely site productivity, and one at individual tree level, namely stem taper. The two models are unbiased, more parsimonious, and more precise than the existing less parsimonious models. This study focuses on research methods, which could be applied in similar studies of other species, ecosystem, as well as in behavioral sciences and econometrics.