Some Remarks on the Method of Least Squares, with Application to Direct and Inverse Location Problems
Yu. V. Linnik · Theory of Probability and Its Applications · 1957
We consider the problem of a least squares estimation of the matrix of “elements” $A = \left\| {\begin{array}{*{20}c} {a_1 } \\ \vdots \\ {a_n } \\ \end{array} } \right\|$ by means of the equations \[Y = X^{(0)} + XA,\] where $X^{(0)} = X_{N1^{(0)} } = \left\| {\begin{array}{*{20}c} {x_{01} } \\ {x_{02} } \\ \vdots \\ {x_{0N} } \\ \end{array} } \right\|;\, X = X_{Nn} = \left\| {x_{rj} } \right\|$ are known matrices; $Y = \left\| {\begin{array}{*{20}c} {y_1 } \\ \vdots \\ {y_n } \\ \end{array} } \right\|$ an unknown matrix; $L = L_{N1} = Y + \Delta $ the matrix of observations, $\Delta = \Delta _{N1} = \left\| {\begin{array}{*{20}c} {\Delta _1 } \\ \vdots \\ {\Delta _N } \\ \end{array} } \right\|$ the error matrix, which is a normal unbiased vector with the correlation matrix $B_\Delta = \sigma ^2 P^{ - 1} .$. $P = \left\| {\begin{array}{*{20}c} {p_1 0 \cdots 0} \\ {0p_2 \cdots 0} \\ { \cdot \cdot \cdot \cdot \cdot \cdot } \\ {0 \cdots \cdots p_N } \\ \end{array} } \right\|p_i > 0$, is the known weight matrix, $\sigma ^2 $ an unknown parameter. The following theorem on confidence regions is proved: Theorem.Let$G = G_{mn} $be a known${m \times n}$matrix with rangem; $H = \left\| {\begin{array}{*{20}c} {h_1 } \\ \vdots \\ {h_m } \\ \end{array} } \right\| = GA$; $Z - \left\| {\begin{array}{*{20}c} {z_1 } \\ \vdots \\ {z_m } \\ \end{array} } \right\|$the general coordinate vector; $C = X^T PX$; $K = GC^{ - 1} G^T $; $\tilde A = \left\| {\begin{array}{*{20}c} {\tilde a_1 } \\ \vdots \\ {\tilde a_n } \\ \end{array} } \right\|$the matrix of least square estimates ofA: \[ \tilde V = X^{(0)} - X\tilde A - L;\, \tilde H - G\tilde A.\]ThenKis non-singular, and the confidence ellipsoid$E_{\gamma _0 } $\[ (Z - \tilde H)^T \cdot K^{ - 1} \cdot (Z - \tilde H) = \gamma _0 [p\tilde v\tilde v] \]covers the point$Z = H$with the probability$p_0 $, where$F_{m,N - n} (\gamma _0 ) = p_0 ,F_{m,N - n} (x)$being Fisher’s distribution withmand${N - n}$degrees of freedom. The extreme cases are : $G = E;\, H = A;\, K = C^{ - 1} ;\, K = C$ and : $G = \| {00, \cdots ,1,0, \cdots ,0} \|$ (1 on i-th place); $K = GC^{ - 1} G^T = \{ C^{ - 1} \} _{ii} ,\, E_{\gamma _0 } $ of the type ${{(z - \tilde a_i )^2 } / {\{ C^{ - 1} \} _{ii} = \gamma _0 [p\tilde v\tilde v]}}$ which is tantamount to the construction of the well known separate confidence intervals for $a_i $’s involving Student’s distribution. An application to geodetic location problems on a plane is given. In this case $n = 2$, and the equation $F_{2,N - 2} (\gamma _0 ) = p_0 $ is solvable in elementary functions.