Error estimates

After the iteration has converged, one can determine error estimates for the obtained parameters $x_i$. The method used here follows the standard multivariate regression analysis [18,19] Assume that we take the scaled experimental observables and perturb them with a random noise that has zero mean value. The true experimental energies can now be thought of as being random variables but only one sample that has the values $\sqrt{W_j}e_j^{{\text{exp}}}$ is known. The deviation of each model parameter $x_i$ from its mean can then be calculated from Eq. (12) as

$$x_i-\langle x_i\rangle = \sum_j \left(\left( J^T J\right)^{-1} J^T\right)_{{i}{j}} \left(y_{j}-\langle y_{j}\rangle\right)\,.$$ (15)

Then, the correlation matrix of parameters $x_i$ and $x_{i'}$ becomes

$$\langle\left(x_i-\langle x_i\rangle\right)\left(x_{i'}-\langle x_... ...e\right)\rangle \!=\! \delta^2_{\text{rms}} \left( J^T J\right)^{-1}_{i{i'}}\,,$$ (16)

where

$$\delta_{\text{rms}} = t_{\alpha/2,m-n}\Delta_{\text{rms}}$$ (17)

and $t_{\alpha/2,m-n}$ is Student's t-distribution [20] for $m-n$ degrees of freedom, necessary here because of the small sample size. In Eq. (16) we have assumed that $y_j$ and $y_{j'}$ are independent random variables whose expectation values vanish when $j\neq j'$ and all have the same standard deviation, i.e.,

$$\langle\left(y_j-\langle y_j\rangle\right)\left(y_{j'}-\langle y_{j'}\rangle\right)\rangle = \delta_{j{j'}} \delta^2_{\text{rms}}\,.$$ (18)

The average values of parameters $\langle x_i\rangle$ are determined by the least square fitting procedure, $\langle x_i\rangle=x_{0,i}$. It is also assumed that the least square fitting gives an accurate estimate of the standard deviation of the observables $e_j$. With these assumptions from Eq. (16) we get the following formula for the confidence interval of $x_i$ with $(1-\alpha)$ probability:

$$\Delta x_i \equiv \sqrt{\langle \left(x_i-\langle x_i\rangle\right)^2\rangle} = \delta_{\text{rms}} \sqrt{\left( J^T J\right)^{-1}_{ii}}\,.$$ (19)

It is now clear that small SVD values that appear in the inverse matrix of Eq. (11) spoil confidence intervals of all parameters, and have to be removed, as in Eq. (13). One should observe that Eq. (19) does implicitly depend on the weights through the definitions of Eqs. (5),  (6), and (8).

We have to stress at this point that the error estimates of Eq. (19) have quite different meaning for the exact and inaccurate models discussed at the beginning of this section. In the first case, errors of the parameters result solely from the statistical noise in the measured observables--their variances are supposed to be known and define the weights in Eq. (2) as $W_j=\left(\Delta e_j\right)^{-2}$. Therefore, within the exact model, the assumption of equal variances, Eq. (18), is well justified. Such a model then gives the minimum value of $\Delta_{\text{rms}}^2$ near 1, i.e. the $\chi^2$ test.

For a inaccurate model, the error estimates of Eq. (19) only give information on the sensitivity of the model parameters to the values of the observables. They correspond to the situation where the experimental values are artificially varied far beyond their experimental uncertainties, so as to induce tangible variations in the values of the parameters. Eq. (18) then means that the range of this variation is inversely proportional to $\sqrt{W_j}$, i.e. it is commensurate with the importance attributed to a given observable. Here, the error estimates may depend on the weights, and are thus affected by their choices, and similarly so are the values of the parameters.

We are now in a position to discuss the mass predictions and error propagation. Suppose that we apply the model of Eq. (1) not only to the measured masses but also to the masses of unknown nuclei,

$${\tilde e}_j = f_j({\vec x})\,,$$ (20)

where the tilde means that the set of observables ${\tilde e}_j$ includes not only those used for the fit, $j=1,\ldots,m$, but also observables for, $j=m+1,\ldots,M$.

The error estimates of Eq. (19) allow us to estimate uncertainties of the predicted observables. With the same assumptions as before, but now with the parameters $x_i$ from the least square fit for both observables inside and outside the fitted set, we get

$$\left({\tilde e}_j-\langle {\tilde e}_j\rangle\right)^2 = \sum_{i... ...ft(x_i -\langle x_i \rangle\right) \left(x_{i'}-\langle x_{i'}\rangle\right)\,,$$ (21)

where ${\tilde I}_{ji}$ are the regression coefficients, Eq. (7), of observables ${\tilde e}_j$ with respect to the model parameters $x_i$. Then, using Eq. (16) the confidence intervals of predicted observables become

$$\Delta {\tilde e}_j = \sqrt{\langle \left({\tilde e}_j-\langle {\... ...rms}}\sqrt{ \left({\tilde I}\left(J^T J\right)^{-1}{\tilde I}^T\right)_{jj}}\,.$$ (22)

Equations (19) and (22) form the basis of the error analysis of our mass fits. The calculated error bars (19) of parameters $x_i$ must then be further scrutinized to analyze which parameters are necessary and which should be removed from the model. The confidence intervals (22) constitute estimates of predictivity of the model. Note that they should also be calculated for the observables that have actually been used in the fit. It is these intervals, and not the residuals $y_j/\sqrt{W_j}$, which have to be analyzed when discussing the quality of the model. It is obvious that the residuals can be arbitrarily small for some observables, or for some types of observables (e.g., masses of semimagic spherical nuclei), while the model can still be quite uncertain in describing these same observables.