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This matrix can be evaluated directly without the need of creating the A and A r matrices, both of which are much larger. The number of elements in the ATA matrix is equal to the number of parameters squared. In general, if there are enough independent data points, and the parameters being estimated are not perfectly correlated, then Eq. (16) can be solved for e. Equation (16) is the classic linear algebra form for the problem of M linear equations in M unknowns. Consequently, there are numerous methods to solve it.
Thirty synthetic data points were generated 23 j. Beechem, this volume . [l] P A R A M E T E R ESTIMATION BY LEAST-SQUARES 23 to represent a single exponential decay of the type defined by Eq. (1). 5. 0. 1 standard deviation was added to these synthetic data. 3. The baseline was held constant so the graph would contain only two dimensions. The general method will in principle work with any number of dimensions. Figure 3 presents a series of contours for which the least-squares norm is constant; each contour represents an increase in the value of s 2 with respect to the optimal value.
The confidence region for a given parameter is defined by the extreme values found for that parameter, after searching in "all" directions, which correspond to an s 2 less than the critical value defined by the F-statistic. In practice we search the variance space in carefully selected specific directions from the point of minimum s 2 instead of in all possible directions. Because only specific directions are searched, the amount of computer time required is proportional to the number of directions searched.