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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 img156 = 0 for all cases with i neq j. For the general case we Taylor-expand w(alpha) about (alpha*):





Equation 9     (9)

The second term of the expansion vanishes because ðw / ðalpha = 0 are the equations for alpha*


Neglecting the higher-order terms, we have


(an M-dimensional Gaussian surface). As before, our error formulas depend on the approximation that curlyL(alpha) is Gaussian-like in the region alphai approx alphai*. 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 curlyL(alpha). (see Appendix IV).

According to Eq. (9), H is a symmetric matrix. Let U be the unitary matrix that diagonalizes H:

Equation 10     (10)

Let img710 and img711. The element of probability in the beta-space is


Since |U| = 1 is the Jacobian relating the volume elements dMbeta and dMgamma, 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

Equation 11 Maximum
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

Equation 11a     (11a)

Example 2

Assume that the ranges of monoenergetic particles are Gaussian-distributed with mean range alpha1 and straggling coefficient alpha2 (the standard deviation). N particles having ranges x1,..., xN are observed. Find alpha1*, alpha2*, 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, alpha2*, and the mean, baralpha2 , 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 alpha1* and alpha2*:


According to Eq. (11), the errors on alpha1 and alpha2 are the square roots of the diagonal elements of the error matrix, H-1:

Equation (this is sometimes called the
error of the error).

We note that the error of the mean is 1/sqrt[N] sigma where sigma = alpha2 is the standard deviation. The error on the determination of sigma is sigma/sqrt[2N].

Correlated Errors

The matrix Vij ident img156 is defined as the error matrix (also called the covariance matrix of alpha). In Eq. 11 we have shown that V = H-1 where Hij = - ð2 w / (ðalphai ðalphaj). The diagonal elements of V are the variances of the alpha's. If all the off-diagonal elements are zero, the errors in alpha are uncorrelated as in Example 2. In this case contours of constant w plotted in (alpha1, alpha2) space would be ellipses as shown in Fig. 2a. The errors in alpha1 and alpha2 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 Deltaalphaj = (Hjj) and then there is no need to perform a matrix inversion.

Figure 2

Figure 2. Contours of constant w as a function of alpha1 and alpha2. Maximum likelihood solution is at w = w*. Errors in alpha1 and alpha2 are obtained from ellipse where w = (w* - ½).
(a) Uncorrelated errors.
(b) Correlated errors. In either case Deltaalpha12 = V11 = (H-1)11 and Deltaalpha22 = V22 = (H-1)22. Note that it would be a serious mistake to use the ellipse "halfwidth" rather than the extremum for Deltaalpha.

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 alpha1, alpha2 axes. The rms spread of alpha1 is still Deltaalpha1 = sqrt[V11], but it is the extreme limit of the ellipse projected on the alpha1-axis. (The ellipse "halfwidth" axis is (H11) which is smaller.) In cases where Eq. 11 cannot be evaluated analytically, the alpha*'s can be found numerically and the errors in alpha 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 alpha's. One should allow all the alphaj to change freely and search for the maximum change in alphai which makes w = (w * - ½). This maximum change in alphai, is the error in alphai and is sqrt[V11].

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