> A Prediction Approach to Agricultural Land Use Estimation Using a
> High-Resolution Spatial Model of Land Use Dynamics
>
> Ambrosio L.
>
> Marín C.
>
> Iglesias L.
>
> Universidad Politécnica de Madrid. Spain

Abstract

The prediction approach for finite populations using (super population)
linear models is now well - established. Nevertheless, for count and
binary data non linear models are more adequate. In this study a
generalized linear mixed model (multinomial) of land use decisions is
proposed to estimate agricultural land uses, in a prediction approach
framework. The spatiotemporal correlation structure is assessed using
semivariograms models and a panel data set on land uses and on auxiliary
variables observed in a spatial sample. The model is fitted using a
generalized estimating equation approach. Parameter estimators and
predictors, as well as their standard error estimators, are consistent,
even if the working spatiotemporal correlation structure is not the true
one. They are biased in small samples and the degree of bias increases
substantially if spatiotemporal correlation is ignored.

**The model**

Let ![](media/image1.wmf)be the acreages on ![](media/image2.wmf) land
use categories within the sampling unit ![](media/image3.wmf), where
$s_{i}$ is a vector of coordinates which locates ![](media/image4.wmf)
within the studied region. The working model is ![](media/image5.wmf)
where ![](media/image6.wmf) and ![](media/image7.wmf) is a row vector of
![](media/image8.wmf) explanatory variables of known values,
![](media/image9.wmf) is a column vector of ![](media/image10.wmf)
unknown parameters, ![](media/image11.wmf)is the ![](media/image12.wmf)
vector of the past observations ![](media/image13.wmf)until time
![](media/image14.wmf) and ![](media/image15.wmf) is a column vector of
![](media/image16.wmf) unknown parameters. The covariance matrix is
![](media/image17.wmf) where ![](media/image18.wmf)and
![](media/image19.wmf)are the variance functions.
![](media/image20.wmf)is the correlation matrix, where
![](media/image21.wmf) is the parameter's vector of the spatiotemporal
correlation structure: the process is assumed to be second order
stationary and isotropic and a product-sum spatiotemporal function is
considered as variogram \[Iaco *et al.* (2001)\]. The model specified
for the whole ![](media/image22.wmf)sampling data is
![](media/image23.wmf) and the covariance matrix is
![](media/image24.wmf). The estimator ![](media/image25.wmf) of
![](media/image26.wmf) is obtained by iteratively reweighted least
squares (McCullagh and Nelder, 1989) using an estimating equations
approach ( Zeger and Liang, 1986; Liang and Zeger, 1986).

**The predictor**

The predictor of ![](media/image27.wmf)where ![](media/image28.wmf) is
the ![](media/image29.wmf)land use acreage observed in the sampling unit
![](media/image30.wmf) not included in the sample is \[Valliant (1985),
Gotway and Stroup, 1997)\] ![](media/image31.wmf) where elements in
![](media/image32.wmf) are as ![](media/image33.wmf)

where ![](media/image34.wmf) is the correlogram function. The
performance of ![](media/image35.wmf) and of ![](media/image36.wmf)was
evaluated by Monte Carlo. Results for one hundred repetitions of a panel
data set with ![](media/image37.wmf)and ![](media/image38.wmf) are shown
in the following table.

Predictor performance

+-------------+-------------+-------------+-------------+-------------+
| Quantiles   | Predictor   | Standard    |             | Coefficient |
|             | Bias        | error bias  |             | of          |
|             |             | (%)         |             | Variation   |
|             | (%)         |             |             | (%)         |
+-------------+-------------+-------------+-------------+-------------+
|             |             | Taking into | Ignoring    |             |
|             |             | account     | spa         |             |
|             |             | spa         | tiotemporal |             |
|             |             | tiotemporal | correlation |             |
|             |             | correlation |             |             |
+-------------+-------------+-------------+-------------+-------------+
| ![](media/i | -5.06       | -24.87      | -47.34      | 4.01        |
| mage39.wmf) |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| ![](media/i | -1.49       | -14.92      | -33.79      | 10.51       |
| mage40.wmf) |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+
| ![](media/i | 1.95        | -4.67       | -23.66      | 16.13       |
| mage41.wmf) |             |             |             |             |
+-------------+-------------+-------------+-------------+-------------+

REFERENCES

De Iaco, S., Myers, D.E. and Posa, D. (2001). Space-time analysis using
a general product-sum model. *Statistics & Probability Letters* 52:
21-28.

Gotway, C.A. and Stroup, W.W. (1997). A generalized linear model
approach to spatial data analysis and prediction. *Journal of
Agricultural, Biological and Environmental Statistics* 2: 157-178.

Liang, K.Y. and Zeger, S. (1986). Longitudinal data analysis using
generalized linear models. *Biometrika.* 73: 13-22.

McCullagh, P. and Nelder, J.A. (1989). *Generalized linear models*.
Chapman and Hall: London.

Valliant, R. (1985). Nonlinear prediction theory and the estimation of a
proportion in a finite population. *Journal of the American Statistical
Association* 80: 631-641.

Zeger, S.L. and Liang, K.Y. (1986). Longitudinal data analysis for
discrete and continuous outcomes. *Biometrics* 42: 121-130.