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Question 6.6.5: In geography, local models of terrain are constructed from d...

In geography, local models of terrain are constructed from data \left(u_{1}, v_{1}, y_{1}\right), \ldots,\left(u_{n}, v_{n}, y_{n}\right), \text { where } u_{j}, v_{j}, \text { and } y_{j} are latitude, longitude, and altitude, respectively. Describe the linear model based on (4) that gives a least-squares fit to such data. The solution is called the least-squares plane. See Figure 7.

6.7
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We expect the data to satisfy the following equations:

\begin{aligned}&y_{1}=\beta_{0}+\beta_{1} u_{1}+\beta_{2} v_{1}+\epsilon_{1}\\&y_{2}=\beta_{0}+\beta_{1} u_{2}+\beta_{2} v_{2}+\epsilon_{2}\\&\vdots \vdots\\&y_{n}=\beta_{0}+\beta_{1} u_{n}+\beta_{2} v_{n}+\epsilon_{n}\end{aligned}.

\text { This system has the matrix form } y=X \beta+\epsilon \text {, where }

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