Review and comparison of methods to study the contribution of variables in artificial neural network models

Convinced by the predictive quality of artificial neural network (ANN) models in ecology, we have turned our interests to their explanatory capacities. Seven methods which can give the relative contribution and/or the contribution profile of the input factors were compared: (i) the ‘PaD’ (for Partial Derivatives) method consists in a calculation of the partial derivatives of the output according to the input variables; (ii) the ‘Weights’ method is a computation using the connection weights; (iii) the ‘Perturb’ method corresponds to a perturbation of the input variables; (iv) the ‘Profile’ method is a successive variation of one input variable while the others are kept constant at a fixed value; (v) the ‘classical stepwise’ method is an observation of the change in the error value when an adding (forward) or an elimination (backward) step of the input variables is operated; (vi) ‘Improved stepwise a’ uses the same principle as the classical stepwise, but the elimination of the input occurs when the network is trained, the connection weights corresponding to the input variable studied is also eliminated; (vii) ‘Improved stepwise b’ involves the network being trained and fixed step by step, one input variable at its mean value to note the consequences on the error. The data tested in this study concerns the prediction of the density of brown trout spawning redds using habitat characteristics. The PaD method was found to be the most useful as it gave the most complete results, followed by the Profile method that gave the contribution profile of the input variables. The Perturb method allowed a good classification of the input parameters as well as the Weights method that has been simplified but these two methods lack stability. Next came the two improved stepwise methods (a and b) that both gave exactly the same result but the contributions were not sufficiently expressed. Finally, the classical stepwise methods gave the poorest results.     

How sensitive are elasticities of long-run stochastic growth to how environmental variability is modelled?

A variable environment leaves a signature in a population's dynamics. Deriving statistical and mathematical models of how environmental variability affects population projections has - in the wake of reports of substantial climatic fluctuations - received much recent attention. If the model changes, then so too does the population projection. This is because a different model of environmental variability changes estimates of long-run stochastic growth, which is a function of demographic rates and their temporal sequence. Decomposing elasticities of long-run stochastic growth into constituent parts can assess the relative influence of different components. Here, we investigate the consequences of changing the environmental state definition, and therefore altering the shape of demographic rate distributions and their temporal sequence, by using age-structured matrix models to project vertebrate populations into the future under a range of environmental scenarios. The identity of the most influential demographic rate was consistent among all approaches that perturbed only the mean, but was not when only the variance was perturbed. Furthermore, the influence of each demographic rate fluctuated among projections by up to factors of six and two for changes to the variance and mean, respectively. These changes in influence depend in part upon how environmental variability - in particular, the color of environmental noise - is incorporated. In the light of predictions of increasing climatic variability in the future, these results suggest caution when drawing quantitative conclusions from stochastic population projections.     

Perturbation methods in optimal control problems

Iteration methods for control improvement in quadratic on state systems are proposed. These methods have been applied for solving the systems of improvement conditions. In order to ground the methods, elements of the theory of perturbations have been applied in this research.     

Uncertainty analysis of transient population dynamics

Two types of demographic analyses, perturbation analysis and uncertainty analysis, can be conducted to gain insights about matrix population models and guide population management. Perturbation analysis studies how the perturbation of demographic parameters (survival, growth, and reproduction parameters) may affect the population projection, while uncertainty analysis evaluates how much uncertainty there is in population dynamic predictions and where the uncertainty comes from. Previously, both perturbation analysis and uncertainty analysis were conducted on the long-term population growth rate. However, the population may not reach its equilibrium state, especially when there is management by harvesting or hunting. Recently, there has been an increased interest in short-term transient dynamics, which can differ from asymptotic long-term dynamics. There are currently techniques to conduct perturbation analyses of short-term transient dynamics, but no techniques have been proposed for uncertainty analysis of such dynamics. In this study, we introduced an uncertainty analysis technique, the general Fourier Amplitude Sensitivity Test (FAST), to study uncertainties in transient population dynamics. The general FAST is able to identify the amount of uncertainty in transient dynamics and contributions by different demographic parameters. We applied the general FAST to a mountain goat (Oreamnos americanus) matrix population model to give a clear illustration of how uncertainty analysis can be conducted for transient dynamics arising from matrix population models.