Parameter Estimation - Part II

In the related literature, the estimation method illustrated in this and in the previous Unit for the IFM model is referred to as nonlinear regression (H94; Moilanen, 1999). It is beyond the aim of this course to discuss this topic. However, in the case of the basic IFM with rescue effect something similar to the usual linear regression can be done. As this has an interesting implication for the IFM parameterization, we will sketch out the logistic model and the relationship with Unit 10 and Section 11.1. In the usual linear regression model, parameters are estimated by minimizing the differences (or residuals) between the observed response and the model expected values, as summarized in Figure 11.1 (that is, the regression line is determined by the ordinary least square method).

Figure 11 - Scatter plot, regression line (blue) of Y on X and residuals (vertical red bars).

The estimated linear regression provides the expected value of the dependent variable Y, conditional to the dependent variable X=u, E(Y | X=u). For a 0-1 response variable, like presence/absence, this expected value E(Y | X = u) coincides with the (conditional) success probability P(Y = 1 | X = u). Let \(\theta\)(u) = P(Y = 1 | X = u). In the logistic model, it is assumed that

and therefore, the logit transformation

yields to express a transformation of E(Y | X = u) as linearly dependent from unknown parameters (from which the name of generalized linear model). Of course, k ≤ 1 dependent variables can be considered, yielding the last member of (11.3) to be in the form \(\beta_0+\sum_{j=1}^k\beta_ju_j\)

For the logistic model, the maximum likelihood method applied in Unit 10 and in Section 11.1 is the “equivalent” of the least square method. By proceeding as in Section 10.2 it is easy to prove that parameter estimation is indeed obtained by the minimization of

that is, by minimizing a comparison between the observed presence/absence data (\(y_i\)) and the expected presence/absence \(\theta(u_i)\).

The similarity between (11.4) and (10.2) is self-evident. Moreover, we know from Eq. (4.1) that for the basic IFM, the expected presence/absence \(\theta(u_i)\) is given by the incidence \(J_i\) of Eq. (10.3) and Eq. (10.4). Therefore, we only need to check if \(J_i\) can be expressed as in (11.2) or, alternatively, if \(ln (J_i / 1- J_i)\) is a linear combination of the unknown parameters and the dependent variable (u), as in Eq. (11.3).

We firstly consider Eq(10.4) and prove that Eq. (11.3) holds true.

\(1-J_i =\frac {A_0^xy^2/S_i^2A_i^x}{1+A_0^xy^2/S_i^2A_i^x}\)   and then   \(\frac{J_i}{1-J_i}=\frac{A_i^xS_i^2}{A_0^xy^2}\)  , so that    \(ln\frac{A_i^xS_i^2}{A_0^xy^2}=xln(\frac{A_i}{A_0})-2lny+2lnS_i\).

Therefore, by considering as independent variables the patch area and the patch connectivity, and naming \(\beta_0= -2 ln⁡y\), \(\beta_1=x\), and \(\beta_2=2\) we obtain

It can be clearly seen that the unknown parameters x and y can be estimated by regressing the presence/absence data (response) on the (scaled) patch area and the connectivity (independent variables) to estimate the unknown parameters \(\beta_0\) and \(\beta_1\). From the estimated values of \(\beta_0\) and \(\beta_1\) we can go back to x and y (\(y = exp(-\beta_0/2)\)). Note that as \(\beta_1=x\), then \(\beta_1\gt0\), while \(\beta_0\) can be any real number.

For the basic IFM without rescue effect, it is not possible to obtain an equation similar to Eq. (11.5). Indeed, from Eq. (10.3) we obtain

\(\frac{J_i}{1-J_i}=\frac{A_i^xS_i^2}{A_0^xy^2+A_0^xS_i^2}\)     so that    \(ln\frac{A_i^xS_i^2}{A_0^xy^2+A_0^xS_i^2}=xln\bigg(\frac{A_i}{A_0}\bigg) + ln\bigg(\frac{S_i^2}{y^2+S_i^2}\bigg)\)

where connectivity S_i and parameter y cannot be separated as in Eq. (11.5).