Assume that the population of the target regions is accurately characterized by the regression model \(\mbox{Pop}_t = X_t\beta + e_t\), that the error has mean zero, variance \(\mbox{var}(e_t)=\sigma^2 \mbox{E}[\mbox{Pop}_t]\), and that the errors are mutually uncorrelated. Then the dasymetric map, in which the population estimates are given by \(\widetilde{\mbox{Pop}}_t = X_t\hat{\beta}\), and \(\beta^{\tiny \mbox{GLS}}\) is the Generalized Least Squares regression estimate, is in fact the area-to-area regression kriging (best linear unbiased) predictor of the target population. Furthermore, the prediction error variance is and the covariance is .
Denote the covariance between two region \(t\),\(t'\) by \(C_{tt'}\). By assumption, the following formulas for the covariances hold: \(C_{tt}=k \mu_t\), \(C_{tt'}=0\), and the covariance between target and source is \(C_{ts}=k \mu_t\). Note: \(\mu_t=\mbox{Pop}_t = X_t\beta\)
The kriging predictor is defined as the linear predictor \(\widehat{\mbox{Pop}}_t=\sum_s w_{ts} \mbox{Pop}_s\), where \(w_{ts}\) are kriging weights to be solved for.
The best linear unbiased predictor is then defined as: \[\min E(\mbox{Pop}_t-\sum_s w_{ts}\mbox{Pop}_s)^2\]
subject to the adding up constraints \(\sum_s w_{ts}X_s=X_t\)
Solving this via the lagrange method, we get: \[\min E(\mbox{Pop}_t\mbox{Pop}_t - 2 w'_{t}\mbox{Pop}_s\mbox{Pop}_t + w'_{t}\mbox{Pop}_s\mbox{Pop}_{s'}w_{t'}) - 2\lambda (w'_{t}X_{s}-X_{t})\]
Setting the derivative to zero produces the equations: \[\frac{\partial}{w_{t}} = -2 C_{st} + 2 w'_{t}C_{ss} - 2\lambda' X_=0\] \[\frac{\partial}{\lambda} = w'_{t}X_{s}-X_{t}=0\] Writing in matrix notation we get: \[\begin{bmatrix}C_{ss} & X_s\\ X'_s& 0\end{bmatrix}\begin{bmatrix}w_t\\ -\lambda\end{bmatrix} = \begin{bmatrix} c_{ts}\\ X_t \end{bmatrix}\] and after rearranging, we get:
\[\begin{bmatrix}w_t\\-\lambda \end{bmatrix}=\begin{bmatrix}C_{ss} & X_s\\ X'_s& 0\end{bmatrix}^{-1}\begin{bmatrix} c_{ts}\\ X_t \end{bmatrix}\]
Using the formula for the inverse of a partitioned matrix (see for example: http://en.wikipedia.org/wiki/Block_matrix_pseudoinverse ), we can derive the following formulas for w and \(\lambda\): \[w_t = C^{-1}_{ss}(I - X_s(X'_sC^{-1}_{ss}X_s)^{-1}X'_sC^{-1}_{ss}) c_{ts} + C^{-1}_{ss} X_s (X'_sC^{-1}_{ss}X_s)^{-1} X_t\]
\[ \lambda_t = (X'_sC^{-1}_{ss}X_s)^{-1}X'_sC^{-1}_{ss}c_{st} - (X'_sC^{-1}_{ss}X_s)^{-1}X_t\]
After collecting terms, we get \[\lambda_t = (X'_sC^{-1}_{ss}X_s)^{-1}(X'_sC^{-1}_{ss}c_{st}-X_t)\] and after substution for \(C_{ss}\) we get: \[\lambda_t = (X'_s\frac{1}{X_s\beta}X_s)^{-1}(X'_s \frac{1}{X_s\beta}X_t\beta-X'_t)\]
Substituing the expression for w into the linear predictor, we get: \[w'_t\mbox{Pop}_s = X_t(X'_sC^{-1}_{ss}X_s)^{-1}X'_sC^{-1}_{ss}P_s + c_{ts}C^{-1}_{ss}(P_s - X_s (X'_sC^{-1}_{ss}X_s)^{-1} P_s )\] and after collecting terms, we get \[w'_t\mbox{Pop}_s = X_t \hat{\beta}^{\tiny \mbox{GLS}} + c_{ts}C^{-1}_{ss}(P_s-X_s\hat{\beta})\] where \(\hat{\beta}^{\tiny \mbox{GLS}}=(X'_sC^{-1}_{ss}X_s)^{-1}(X'_sC^{-1}_{ss}\mbox{Pop}_s)\).
Since \(C_{ss}\) is a diagonal matrix with terms \(X_s\beta\) and \(c_{st}\) is the scalar \(X_t\beta\), we have that the kriging estimator is the RWDM estimator: \[X_t\beta + \frac{X_t\beta}{X_s\beta}(\mbox{Pop}_s-X_s\beta)\] or more simply: \(\frac{X_t\beta}{X_s\beta}\mbox{Pop}_s\) - the dasymetric map estimate.
In practice, we will have the Feasible GLS estimator, where the covariance is estimated, not known. We leave it to an advanced econometrics text, such as Davidson and MacKinnon (1993: p. 300) to show in what ways Feasible GLS and GLS are equivalent.
The kriging prediction variance is \[C_{tt} - 2w'_tc_{ts} + w'_tC_{ss}w_t\] \[C_{tt} - 2w'_{t}c_{ts} + w'_t(c_{ts}+X_s\lambda)\] \[C_{tt} - w'_{t}c_{ts} + \lambda' X_t\]
\[C_{t,t'} - w'_tc_{st'} - w'_{t'}c_{st} + w'_tC_{ss}w_{t'}\] \[X_t\beta \delta(t=t') - w'_t (C_{ss}w_t-X_s\lambda_t) - w_t (C_{ss}w_{t'}-X_s\lambda_{t'})+w'_sC_{ss}w_s\] \[\lambda_{t'}+\lambda_t=var(\hat{\beta})(X'_s \frac{X_s\beta}{X_s\beta} - X'_s) = 0\] \[X_t\beta \delta(t=t') -w'_sC_{ss}w_s\] \[X_t\beta \delta(t=t') - \frac{x_t\beta}{x_s\beta}X_s\beta \frac{x_{t'}\beta}{x_s\beta}\]
Thus, we have the desired result, the variance is \(\sigma^2 X_t\beta(1-\frac{X_t\beta}{X_s\beta})\) and the covariance is \(\sigma^2(-\frac{X_t\beta X_{t'}\beta}{X_s\beta})\)