Methods of regression analysis

In this section we briefly review the methods used in the standard linear regression method [14]. Along with presenting the necessary definitions and main results, we will also discuss several aspects that are specific to our particular problem of nuclear mass fits.

Let us assume that we have a model describing $j=1,\ldots,m$ observables $e_j$ in terms $i=1,\ldots,n$ parameters $x_i$, i.e.,

$$e_j = f_j({\vec x})\,.$$ (1)

To find an optimal set of parameters, a fitting procedure has to be used, whereupon the rms deviation (including in regression analysis, a $1/(m-n)$ normalization)

$$\Delta_{\text{rms}}^2 = \frac{1}{m-n} \sum_{j=1}^m W_j \left( f_j({\vec x}) - e_j^{\text{exp}} \right)^2$$ (2)

between experimental values of observables, $e_j^{\text{exp}}$, and the observables given by model is minimized by adjusting the model parameters. This is called the least square fitting procedure. As is usually the case, the number of observables is larger than the number of parameters, $m > n$.

Each term in the sum of Eq. (2) is multiplied by a weight factor $W_j>0$. In this respect we can single out two limiting situations of an exact and an inaccurate model:

• The model of Eq. (1) is exact and deviations in Eq. (2) result solely from imprecisely measured experimental values. In this case, one takes the weights $W_j=\left(\Delta e_j\right)^{-2}$, where $\Delta e_j$ are experimental variances of observables $e_j$.
• The model of Eq. (1) is a poor approximation of reality and deviations in Eq. (2) are much larger than the experimental variances of observables. In this case, the choice of weights is quite arbitrary and can only be based on intuition. By using different weights one can, in fact, differentiate between the importance of various observables in determining the model parameters. It is clear that the result of adjustment may crucially depend on the choice of weights.

In the nuclear mass fits discussed in the present paper, we obviously have the case of an inaccurate model, by which typical experimental errors are of the order of a few tens of keV [15], but can also be as low as about 100eV [16], while average deviations of mass models do not go below about 0.6MeV [1]. In the case of several different kinds of observables included in the fit, dependence of the results on weights is obvious, see e.g. the recent comprehensive analysis in Ref. [5]. However, even if only nuclear masses are fitted, the 'natural' choice of weights, $W_j=1$, is only a choice, and many other choices are possible, i.e. depending on whether one wants to put more weight into the measured values of light or heavy, or stable or exotic nuclei. We will illustrate this point in Sec. 3 below.