A parsimonious approach for modeling uncertainty within complex nonlinear relationships

Abstract

Citation: Strimbu, B. M., A. Amarioarei, and M. Paun. 2017. A parsimonious approach for modeling uncertainty within complex nonlinear relationships. Ecosphere 8(9):e01945. 10. 1002/ecs2.1945 Abstract. 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.

Nonlinear parsimonious forest modeling assuming normal distribution of residuals

“…4. To assess the impact of truncation on the transformations, we focused on the functions that contains a Taylor series expansion, as computed by Strimbu et al (2017). However, the (4)…”

“…In this study, we have also included formulas for the variance Var(Y) = Y 2 − 2 [Y] (Grimmett and Stirzaker 2002), which requires the estimation of the mean, [Y] , and of the second moment, Y 2 . The formulas for the expectation of Y were proven by (Strimbu et al 2017); therefore, in this study we will include results for the Y 2 . The final formulations for a series of common transformations are (proof in "Appendix"):…”

“…The reduced number of nonlinear functions for which unbiased estimation exists (Nelder and Wedderburn 1972), the difficult to implement framework proposed by Neyman and Scott (1960) for bias correction when the dependent variable is transformed, and the lack of accuracy associated with heuristics prompted the development of unbiased estimates for 10 transformations of the predicted variable that are commonly encountered in forest modeling (Strimbu et al 2017). For the 10 functions, complex nonlinear models can be developed exactly, as parameters are estimated without using heuristics.…”

“…The approach proposed by Strimbu et al (2017), which avoids heuristic estimations, provides unbiased results when transforming the predicted variable, Y, with a differentiable function f. The method for correcting the bias induced by the change of the dependent variable is based on the assumption that between f(Y) and a set of predictor variables, X, there is a linear relationship where b is the vector of coefficients for the independent variables X, ε are the residuals, which are normally distributed with mean 0 and variance σ 2 , ∼ N 0, 2 .…”

“…The simulated data used by the author to guide the selection of the number of terms present in the Taylor series expansion is likely to be challenged by real problems, which are more complex than simulated data. Also, the method presented by Strimbu et al (2017) does not provide unbiased estimates for the variance of the back-transformed Y. Estimation of variance of the predicted values is mandatory for computing the confidence intervals. Therefore, the objective of the present study is twofold: first, to present the unbiased estimates of the variance of the back-transformed variable, and second, to estimate a computational efficient Taylor series expansion that would provide unbiased results for the transformations involving Taylor series expansion.…”