Table 3 Actual, refitted, and recovered covariance matrices of regression coefficients in weighted least squares equations in Table 1
From: Model errors in tree biomass estimates computed with an approximation to a missing covariance matrix
Eq. # | Actual | Refitted | Recovered | Recovered (robust) | P 1 ×100 | P 2 ×100 | P 3 ×100 |
|---|---|---|---|---|---|---|---|
3 | \(\left( {\begin{array}{*{20}c} {2.62 \times 10^{ - 3} } & { - 0.132} \\ { - 0.132} & {8.67} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {2.15 \times 10^{ - 3} } & { - 0.249} \\ { - 0.249} & {4.11} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {2.66 \times 10^{ - 3} } & { - 0.0995} \\ { - 0.0995} & {5.00} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {3.54 \times 10^{ - 3} } & { - 0.175} \\ { - 0.175} & {10.4} \\ \end{array} } \right)\) | 0.1 | 0.2 | 0.0 |
4 | \(\left( {\begin{array}{*{20}c} {0.268} & { - 0.747} \\ { - 0.747} & {2.25} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.203} & { - 1.60} \\ { - 1.60} & {1.26} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.300} & { - 0.723} \\ { - 0.723} & {1.90} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.493} & { - 1.28} \\ { - 1.28} & {3.51} \\ \end{array} } \right)\) | 13 | 8 | 0.0 |
7 | \(\left( {\begin{array}{*{20}c} {0.816 \times 10^{ - 3} } & { - 0.0336} \\ { - 0.0336} & {1.66} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {3.78 \times 10^{ - 3} } & { - 0.115} \\ { - 0.115} & {4.37} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {1.36 \times 10^{ - 3} } & { - 0.0417} \\ { - 0.0417} & {1.61} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.578 \times 10^{ - 3} } & { - 0.0251} \\ { - 0.0251} & {1.27} \\ \end{array} } \right)\) | 1 | 0.8 | 29 |
8 | \(\left( {\begin{array}{*{20}c} {0.139} & { - 0.328} \\ { - 0.328} & {0.813} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.521} & { - 1.01} \\ { - 1.01} & {2.11} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.192} & { - 0.381} \\ { - 0.381} & {0.82} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.297} & { - 0.630} \\ { - 0.630} & {1.41} \\ \end{array} } \right)\) | 0.0 | 7 | 4 |
11 | \(\left( {\begin{array}{*{20}c} {5.53 \times 10^{ - 4} } & { - 0.0322} \\ { - 0.0322} & {2.19} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {2.13 \times 10^{ - 3} } & { - 0.0778} \\ { - 0.0778} & {3.46} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {5.78 \times 10^{ - 4} } & { - 0.0213} \\ { - 0.0213} & {0.959} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {1.40 \times 10^{ - 3} } & { - 0.0549} \\ { - 0.0549} & {2.30} \\ \end{array} } \right)\) | 0.0 | 16 | 0.5 |
12 | \(\left( {\begin{array}{*{20}c} {7.84 \times 10^{ - 2} } & { - 0.227} \\ { - 0.227} & {0.698} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.153} & { - 0.342} \\ { - 0.342} & {0.813} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {8.96 \times 10^{ - 2} } & { - 0.202} \\ { - 0.202} & {0.485} \\ \end{array} } \right)\) | \(\left( {\begin{array}{*{20}c} {0.202} & { - 0.457} \\ { - 0.457} & {1.06} \\ \end{array} } \right)\) | 0.0 | 38 | 4 |