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Trend Models and Estimation

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Random Fields for Spatial Data Modeling

Part of the book series: Advances in Geographic Information Science ((AGIS))

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Abstract

In the preceding chapter we defined the trend as the component of a random field that represents the large-scale variations. In this chapter we will discuss different approaches for estimating the trend. Trend estimation is often the first step in the formulation of a spatial model.

There is no subject so old that something new cannot be said about it.

Fyodor Dostoevsky

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Notes

  1. 1.

    In some cases a nonlinear transformation of the data is performed before estimating the trend.

  2. 2.

    In spatial data analysis, the residuals obtained by removing the trend from the data often are not uncorrelated, in contrast with the commonly used assumption in classical regression analysis.

  3. 3.

    The residuals υ implicitly depend on the values of the regression coefficients β.

  4. 4.

    The common convention in ridge and LASSO regression problems is that the predictor variables are standardized (i.e., their mean is equal to zero and their variance to one), and the response variable is centered, i.e., its mean is zero.

  5. 5.

    Lignite is a form a low-quality coal used for electricity generation in various countries.

  6. 6.

    Ash water free refers to the ash that remains after moisture is removed by burning. A low count of remaining ash is preferable.

  7. 7.

    The normalization of the sampling coordinates is not necessary in this example. It can improve numerical stability, however, if the coordinates involve very large numbers. The transformation does not alter the relative distances between pairs of sampling points, and thus it maintains existing correlations.

  8. 8.

    We explain this in more detail in Chap. 6 which focuses on Gaussian random fields.

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Hristopulos, D.T. (2020). Trend Models and Estimation. In: Random Fields for Spatial Data Modeling. Advances in Geographic Information Science. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1918-4_2

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