When M parameters are to be determined from a single experiment containing N events, the error formulas of the preceding section are applicable only in the rare case in which the errors are uncorrelated.. Errors are uncorrelated only for = 0 for all cases with i j. For the general case we Taylor-expand w() about (*):
The second term of the expansion vanishes because ðw / ð = 0 are the equations for *
Neglecting the higher-order terms, we have
(an M-dimensional Gaussian surface). As before, our error formulas depend on the approximation that () is Gaussian-like in the region i i*. As mentioned in Section 4, if the statistics are so poor that this is a poor approximation, then one should merely present a plot of (). (see Appendix IV).
According to Eq. (9), H is a symmetric matrix. Let U be the unitary matrix that diagonalizes H:
Let and . The element of probability in the -space is
Since |U| = 1 is the Jacobian relating the volume elements dM and dM, we have
Now that the general M-dimensional Gaussian surface has been put in the form of the product of independent one-dimensional Gaussians we have
According to Eq. (10), H = U-1 . h . U, so that the final result is
M parameters (11)
(A rule for calculating the inverse matrix H-1 is
If we use the alternate notation V for the error matrix H-1, then whenever H appears, it must be replaced with V-1; i.e., the likelihood function is
Assume that the ranges of monoenergetic particles are Gaussian-distributed with mean range 1 and straggling coefficient 2 (the standard deviation). N particles having ranges x1,..., xN are observed. Find 1*, 2*, and their errors . Then
The maximum-likelihood solution is obtained by setting the above two equations equal to zero.
The reader may remember a standard-deviation formula in which N is replaced by (N - 1):
This is because in this case the most probable value, 2*, and the mean, 2 , do not occur at the same place. Mean values of such quantities are studied in Section 16. The matrix H is obtained by evaluating the following quantities at 1* and 2*:
According to Eq. (11), the errors on 1 and 2 are the square roots of the diagonal elements of the error matrix, H-1:
|(this is sometimes
called the |
error of the error).
We note that the error of the mean is 1/sqrt[N] where = 2 is the standard deviation. The error on the determination of is /sqrt[2N].
The matrix Vij is defined as the error matrix (also called the covariance matrix of ). In Eq. 11 we have shown that V = H-1 where Hij = - ð2 w / (ði ðj). The diagonal elements of V are the variances of the 's. If all the off-diagonal elements are zero, the errors in are uncorrelated as in Example 2. In this case contours of constant w plotted in (1, 2) space would be ellipses as shown in Fig. 2a. The errors in 1 and 2 would be the semi-major axes of the contour ellipse where w has dropped by ½ unit from its maximum-likelihood value. Only in the case of uncorrelated errors is the rms error j = (Hjj)-½ and then there is no need to perform a matrix inversion.
Figure 2. Contours of constant w as
a function of 1
and 2. Maximum
likelihood solution is at w = w*. Errors in
2 are obtained
from ellipse where w = (w* - ½).
In the more common situation there will be one or more off-diagonal elements to H and the errors are correlated (V has off-diagonal elements). In this case (Fig. 2b) the contour ellipses are inclined to the 1, 2 axes. The rms spread of 1 is still 1 = sqrt[V11], but it is the extreme limit of the ellipse projected on the 1-axis. (The ellipse "halfwidth" axis is (H11)-½ which is smaller.) In cases where Eq. 11 cannot be evaluated analytically, the *'s can be found numerically and the errors in can be found by Plotting the ellipsoid where w is 1/2 unit less than w * . The extremums of this ellipsoid are the rms error in the 's. One should allow all the j to change freely and search for the maximum change in i which makes w = (w * - ½). This maximum change in i, is the error in i and is sqrt[V11].