Simultaneous equations, error-in-variable models, and model integration in systems ecology

Abstract

Numerous dynamic ecological models of varied time and spatial scales exist in systems ecology. In general, small-scale models are more accurate, more capable of reflecting tiny local variations in eco-processes, and more sensitive to the outside disturbances than large-scale models. On the other hand, large-scale models are more comprehensive, and usually describe the ecosystem's average properties. There has been increased interest in how to integrate accurate small-scale models with comprehensive large-scale models. The two-stage or three-stage least squares regression is the classic parameter estimation method for such purposes. In this study, a two-stage error-in-variable method is introduced to estimate the parameters for model integration. It is proved theoretically that when the restriction is exactly identifiable, the two-stage least squares regression and the two-stage error-in-variable model produce the same estimates. If the restriction is over identifiable, both methods have solutions, but the estimates are not necessarily identical. For under identifiable systems, the estimate from the error-in-variable model still exists, but the estimate from the two-stage least squares regression is not valid any more. An example is provided to demonstrate how to use the two-stage error-in-variable model in a step-by-step fashion.

Introduction

To study how an ecological system functions, it is a common practice to first break down the system to sub-systems. The outputs from some sub-systems are treated as the inputs to others, and all sub-systems are coupled by the outputs and inputs to become a complete system (see the simplest example as in Fig. 1). In forestry, this practice is often referred to as scaling (Jarvis, 1995), linking (Somers and Nepal, 1994), aggregation (O'Neill and Rust, 1979), disaggregation (Zhang et al., 1993), or ‘integration’ (Daniels and Burkhart, 1988, Tang, 1991). The similar problems analogous to the Fig. 1 example also occurred in systems ecology, which is the problem of variable aggregation in ecological simulation models (Luckyanov et al., 1983). Luckyanov (1983/1984) had studied the problem of linear aggregation and separability in linear models of ecological systems and proved a theorem of the existence of a general solution. An attempt was also made on the problem of aggregation in nonlinear ecological system models (Logofet and Svirezhev, 1986, Iwasa et al., 1987). These studies are theoretically oriented, and the authors assumed the parameters in the systems are either given or known, and made efforts on finding the conditions such that a system can be aggregated. The problems in forestry are usually opposite, i.e. it is known that the variables in the system can be aggregated, however, the parameters in the system are generally unknown and have to be estimated from sampling data. In this study, we will concentrate our efforts on the parameters estimation method for integrated ecological models.

The system in the figure can be represented by equations dx₁/dt=f₁(t, x₁, u₁, a₁), dx₂/dt=f₂(t, x₂, u₂, a₂) and dx₃/dt=f₃(t, x₁, x₂, x₃, a₃), where t is time, x_i are the states of the system, u_i are the inputs to the system, and a_i are the parameters to be estimated. As a typical simultaneous estimation problem, the solution to the equations can be expressed as x₁=F₁(t, u₁, a₁), x₂=F₂(t, u₂, a₂), and x₃=F₃(t, x₁, x₂, a₃). There are at least two approaches available to estimate the parameters in the system. First, the three sub-systems could be fitted separately from u₁ and u₂ and X₁, X₂, and X₃ to obtain the estimated parameters $\text{a}\text{̂}{}_1\text{,}\text{a}\text{̂}{}_2\text{,}\text{and}\text{a}\text{̂}{}_3\text{;}$ hence, the estimated states should satisfy equations:

$$\text{X}{}_1\text{=F}{}_1\text{(t,}\text{u}{}_1\text{,}\text{a}\text{̂}{}_1\text{),}$$

$$\text{X}{}_2\text{=F}{}_2\text{(t,}\text{u}{}_2\text{,}\text{a}\text{̂}{}_2\text{),}$$

$$\text{X}{}_3\text{=F}{}_3\text{(t,}\text{X}{}_1\text{,}\text{X}{}_2\text{,}\text{a}\text{̂}{}_3\text{).}$$

However, the parameters can also be estimated by integrating the three sub-systems as a single model such as:

$$\text{x}{}_3\text{=F}{}_3\text{(t,}\text{F}{}_1\text{(t,}\text{u}{}_1\text{,}\text{a}{}_1\text{),}\text{F}{}_2\text{(t,}\text{u}{}_2\text{,}\text{a}{}_2\text{),}\text{a}{}_3\text{).}$$

The parameter estimates $\text{a}\text{̄}{}_1\text{,}\text{a}\text{̄}{}_2\text{,}\text{and}\text{a}\text{̄}{}_3$ through Eq. (4) are usually different from $\text{a}\text{̂}{}_1\text{,}\text{a}\text{̂}{}_2\text{,}\text{and}\text{a}\text{̂}{}_3\text{.}$ In terms of prediction, the result from Eq. (4) with u₁ and u₂ as input should be more accurate than the prediction through Eq. (3) with X₁ and X₂ as input, which are outputs from , where u₁ and u₂ are input (George et al., 1982). If the correlation between the variables in Eq. (1) or Eq. (2) is low and the chain of serial linking is long, the estimation errors would be large, and such propagated errors as a result of indirect prediction are not necessarily random. In fact, if the output from the previous stage sub model is used as the input to the model in the next stage, then the variables are endogenous, and the estimates are biased. If the model chain is very long, the accumulation of such biases will make the final estimate problematic (see Chapter 14, George et al., 1982).

If , , are linear, then econometricians have defined the system to be linear simultaneous equations with X₁, X₂, and X₃ as endogenous variables. Two-stage or three-stage least square regression (George et al., 1982) is the classic parameter estimation method for such equations. This method is capable of eliminating the error propagation and the parameters estimated are asymptotically unbiased. In forestry, Borders (1989) discussed the applicability of linear simultaneous equations in forest growth-and-yield modeling. Estimation procedures are well documented. However, the procedures work only on the identifiable equations without the restrictions between the equations. The fact is that, in systems ecology, unidentifiable simultaneous equations are frequently found, and very often these equations are nonlinear. The objective of this study is to introduce a two-stage regression method based on the error-in-variable model to solve such problems.