obtained in case B with any single reasonable parameter value set. But only one, at most, of the two sets of parameter values can be correct. Thus, one must proceed with extreme caution, and the small degree of discrimination possible from the case A fits and As is usually misleading. Further, any use of case A results outside of their fitting ranges is extremely dangerous. The most meaningful discrimination will be obtained from calculating Al's by the case B procedure, using the same most reasonable choice of parameter values in both equations. If the two equations under consideration seem to fit about equally well and no other parameter value information is available, reasonable values to use in the case B discrimination are the averages of the two sets of values found from the least squares fittings. Because of the wide use of least squares procedures, these matters will be further discussed in the next section.

The present case B results are closely related to some obtained by Barsch and Chang [3]. These authors compared, however, p-value predictions obtained from a certain lattice equation of state tailored for Csl with p values obtained from several other phenomenological equations of state. They found, for example, that using the same parameter values the BE2 approximated the lattice equation an order of magnitude better (in Ap) than did the KE. Although Barsch and Chang present VV, versus p curves for several of the equations of state considered herein, they do not give A curves and are not primarily concerned with establishing what accuracy in V is needed, for a given por z range to allow equation discrimination.

Even though Barsch and Chang assert the superiority of the BE2 over the other phenomenological equations they considered, as already mentioned the BE2 curve of the present figure 2 suggests that exceptionally accurate data or a very wide range will usually be required to allow meaningful discrimination to be made between the BE2 and the KE. Although Barsch and Chang's calculated | Ap values for the BE, and lattice equation were an order of magnitude smaller than those for the KE and lattice equation, the latter values themselves were still considerably smaller for the range 0 ≤ p ≤ 200 kbar than either the errors in Ap❘ calcu lated from the BE with experimental uncertainties in the parameters or those expected experimentally [3]. Thus, the actual discrimination between the BE and KE appears nugatory for this range. It seems doubtful that sufficiently accurate wide-range data yet exist to make adequate BE-KE discrimination possible unless the situation is very different for appreciably different parameter values than those used here and those used by Barsch and Chang, an unlikely possibility.

The curves of figures 1 to 5 are somewhat more general than they appear at first sight. First, since the normalized pressure variable z is used, the results are independent of the value of Ko. Second, since the Fo value used is quite close to unity, little specialization is introduced by the specific o value used. When Fo differs appreciably from unity, the present curves may still be used with the values reinterpreted as A values. For the UTE, ME, and BE, only the

additional parameter n enters. This quantity is usually found to be within the range 3 ≤ n ≤8; thus, the present value, near 6, is fairly typical. Further, changes in ʼn may be expected to change Al itself less than the I's entering A. On the other hand, the value used. near -7, is quite special since little is known thus far about the likely range of V for a variety of materials, and it probably can be positive as well as negative [1,5]. Nevertheless, we suggest that the present curves may be used, at a zero to first order level, for an initial estimate of discrimination possibilities between various equations for other materials besides CsI at 150 °C. Of course, the next order of assessment would employ an estimated parameter set (Vo, Ko, K¦, and K" values) for the material in question. This set could then be used, as herein, to generate Al curves for comparison with the estimated total errors of the experiment, all incorporated into the values.

As examples of such zero-order assessment, we may consider the data of Monfort and Swenson [6], Kell and Whalley [7], and Vedam and Holton [8]. Monfort and Swenson studied potassium metal up to z~0.4. Their volume data were given to four places, and they found a scatter of about 5 units in the last place. Although they primarily considered the BE,, the ME, was also introduced. The ME, curve of figure 5 shows a maximum || for these equations of about 7× 10-3. When the Monfort-Swenson data is normalized to a l。 near unity, allowing comparison of errors with present Al's, one may estimate that the data are accurate to perhaps 3 x 10-3 in normalized volume. Comparison suggests that one might then just be able to distinguish between the BE, and ME, for this range and accuracy. One of the present authors [1] has considered discrimination between the 3DGE and 3SE for the 0 °C water data of Kell and Whalley (max~0.05) and between the 3DGE and ME, for the 50 °C water data of Vedam and Holton (max~0.44). Similar zero-order comparison of probable errors in with the present A curves suggests that the 3DGE-3SE discrimination was near the borderline of possibility and was probably not very meaningful, while the 3DGE-ME: discrimination was somewhat more possible and certain.

Finally, to the degree that the present A curves are reasonably general, it is worth mentioning that the sign changes for the BE-VUTE VBE-ME2 and BE-SE curves shown in figure 5 indicate that the BE, remains a closer approximation to the other three equations over a wider range than if such changes of sign were absent. This result is perhaps one reason why the BE, has been found to be of relatively general applicability in the past.

an yield valuable information about the systematic rrors arising from the use of the wrong model. Further, he use of exact data allows the usually mixed effects f random and systematic errors of this type to be ntirely separated. Since figures 1 to 3 and 5 show that he 3DGE is, in some sense, close in its predictions to everal of the other equations, it has been chosen here is the "correct" model for illustrative purposes. The xact data employed was thus generated by using in he 3DGE the 150 °C CsI parameter values already liscussed.

Table 3 shows the results of applying the least quares method in a few different situations of interest. lere and hereafter "linear" and "nonlinear" generally efer to linearity, or its absence, of the parameters ntering the model. Thus, by a "linear" equation we ill mean one linear in its parameters. The "linear" tuation cited is actually rendered nonlinear in the arameters by the weighting used [1]. Even though the odel is originally linear in the parameters, weighting the independent variable will lead to nonlinear arameter dependence except in the special simple ise (not considered here) of a linear relation between dependent and dependent variables. In a succeeding aper, we hope to investigate in some detail the imortance of and degree of bias frequently arising in the case of table 3 when random error is present. Here e continue to restrict attention to the exact data tuation.

The 3DGE is written in table 1 in a form involving e parameters nonlinearly. This form was required by e constraint of using Ko, n, and V as the basic paramers in each equation listed in the table. On the other and, the 3DGE may also be written in the linear form

Ko, n, and estimates to be calculated for comparison with their true values. Further, comparison of corresponding nonlinear and linear least square parameter estimates will allow bias arising from nonlinear least squares to be indentified and quantified.

The following definitions are useful in comparing least-squares parameter values with exact values. Let O be a specific parameter of the model; then denote the true value of 0 (here known) by 0, and the least-squares estimate as Ô. The relative error of the estimate is then 8 = (0-0)/00. When no random errors are present, d; will measure the systematic error in the ith parameter value. It is also of interest to compare the parameter error (0-0) with the standard deviation (sa)ø obtained for a given least squares estimate of 0. To do so, we define A= |(0-0)/(Sa) o|=| Ood/ (Sa). This measure will indicate possible systematic error in (Sa)o.

We have been discussing least squares results in the above as though they were exact solutions of the least squares equations. It is not widely appreciated that all the usual least squares computer routines may yield very inaccurate parameter values because of round-off errors [9]. For example, if Gaussian elimination with pivoting is used to solve the least squares equations, the number of accurate decimal digits in a 0, A, is A~ (C-n+1±1), where C is the number of (equivalent) decimal digits carried in the computer calculation and n is the number of free parameters. Clearly, if n> C, results of little value are likely to be obtained. Expression for A of this type were originally derived for linear least squares fitting of polynomials, but they seem to apply at least approximately to nonlinear equations as well. Recently, Wampler [10] has made a more detailed study of the matter for polynomials and discussed more complex routines which can yield very substantially higher solution accuracy.

The effects of roundoff are illustrated by the results of table 4. Elimination with pivoting was used to carry out least squares fitting of the 14-figure 3DGE data using the 3DGE equation in both its linear and nonlinear forms. Since c=14 and n=4, A~11±1. In Table 4, the d are calculated using 06=Vo, 01=Ko, 02=n, and Y. The quantity sa is the standard deviation. for the fit itself. The results show values of A between about 14 and slightly less than 11, in rough agreement

TABLE 4. Least squares results in the absence of systematic error: exact 3DGE data fitted by the 3DGE model

Parameter estimates

Parameter variance estimates

All nonlinear least squares fitting in the present paper has been carried out using the Deming iterative method of solution [1, 11]. Although this is an approximate method [12], the resulting errors in the estimated parameter values are generally negligible. O'Neill et al. [12] have presented a more accurate iterative solution of the least squares problem with weighting of both dependent and independent variable. applicable only for polynomial equations. We have recently generalized and improved this solution so that it applies to equations of any form and converges much more rapidly [13]. This method, applied to the situations of table 5, yields essentially the same 8, values as those in the table but A's some 25 to 40 percent larger than the tabular values. These increases thus mainly arise from smaller (sa)e's produced by the new least squares solution. Although the new method leads to an essentially exact (in the sense of iterative convergence) least squares solution when round-off errors are negligible, the results cited above and those in the table show the presence of large systematic errors in 8, and ▲, arising from wrong model choice. The rest of the present paper is primarily concerned with 's and p-weighting, where the differences between the Deming and improved estimates are negligible.

with the formula. There appears to be no significant tendency for the linear results to be better than the nonlinear ones, and one can scarcely conclude that much of the bias of table 3 is showing up here. In fact, bias is only important when random errors are present; in the A cases of table 3, bias approaches zero as the random error goes to zero. Incidentally, since do is zero in the linear case when Vo is taken fixed at its exact value, Ao is given in its place; since the true value of A is zero, this is an absolute, not relative error. The results of table 4 were calculated with N=37 points, covering the range 0 ≤ z < 0.476. Let the maximum value of z be denoted zr. In earlier work [1], weighting of both the dependent and independent variable data values was discussed. The related standard deviations were denoted σ for the V variable and σ, for the p variable. The p-weighting of table 4 takes σ, 1 and σ = 0 (weighting of dependent variable only), while the V-weighting uses σ,, = 0, σ = 1. In the linear equation case, the V-weighting chosen leads to somewhat different weighting of the actual independent variable x used [1]. Table 4 indicates slightly improved results for the V-weighting over the p-weighting, and no bias arising from V-weighting in the nonlinear situation is apparent. The differences between the sa's for p and weighting arise because the p-weighting sa is a measure of the least-square fit residuals when they are all in pressure and is here in kilobars, while for V-weighting the residuals are all forced to be in reduced volume and sa is then dimensionless. The ratio between the sa's is roughly Ko.

Although we shall use the usual inaccurate Gaussianelimination-with-pivoting solution of the least squares equations in the following, all inaccuracies introduced. thereby are four or more orders of magnitude smaller than the systematic errors we consider. Such systematic errors in parameters and standard deviations are illustrated in table 5, where the 3DGE data are fitted with p-weighting using the incorrect BE2 model. Results for 8, and A, are first given for two different ranges of z, from zero to ~ 0.048 and ~0.48. Note the strong increases in these error measures both with range and with the index i. Also included in the table are fitting results obtained for the complement range, all points contained in the second range but not in the first. As might be expected, the parameter estimates are somewhat worse for this coverage than for the largest span shown, even though sa itself is somewhat better.

429-270 - 71 - 4

Figures 6 to 12 show how the systematic errors, represented by the 8,'s and by sa, change as the fitting range is extended. As usual, dashed lines indicate negative values. All 8; results shown were obtained with p-weighting: σ =0, σ, 1. Results obtained with V-weighting were closely similar. Unlike the di`s, which are relative, the sa's are absolute and, with p-weighting, measure the overall goodness of fit in pressure units, as already mentioned. Thus, for example, figure 6 indicates that sa for the BE, fitting over the range 0≤ z <2 is nearly 0.1 kbar. All sa curves were obtained with p-weighting except the one marked (sa), on figure 9. Here we compare the sa's obtained from p and I weighting. The (sa) values are somewhat more than Ko times smaller than (sa)p values here. Note that, as expected, (s) and do. the relative error in Io, are quite close together over much of the range.

2

For weighting, sa is an overall measure of the residuals in I. Its absolute value in figure 9 at z = 0.143 of about 3 × 10 7 (the maximum magnitude of a volume residual was 6.5 × 107) is about two orders of magnitude smaller than the ABE-FOGE of - 5 × 10 5 shown in figure 3 for the same z. But this last figure is that applying when the correct parameter values are used in both equations. As expected, the least squares parameter adjustment in the BE, fitting of the exact 3DGE data makes it difficult to conclude (without independent knowledge of parameter values) that the BE is the wrong model, as it is here. With some random errors in the 3DGE model data, least squares fitting using the BE, and KE, for example, would again generally lead to results which wouldn't allow one to

identify either the BE2 or KE as an incorrect model, even though they both would be.

The results of figures 6 to 12 show that when the range is extended, relative errors in all the parameters increase when wrong models are used. Further, the higher-order parameters are more inaccurate than the lower-order ones for all the ranges shown. Not much added accuracy in the higher-order parameters cat be obtained by reducing the range and, in practica cases where random error is present, generally n added accuracy will be achieved by such reduction. I

Figure 10 stops with a z, of 0.476 because the volum predicted by the ME is negative for z 1.85, preclud ing a meaningful fit with z, 2.00. Note that & for the KE and 3SE is so large that its values must be divided by 10 and 100, respectively, to allow plotting on the present scale. For the 3SE, even 8 must be divided by 10 as well. These results illustrate an important gen eral point. The figures show that the BE and KE ar the best least-squares simulators of the 3DGE mode as far as sa is concerned. Yet even for the relativel low z value of 0.143 (p= 15 kbar), V is about 10 per cent high for the BE and is of even the wrong sig for the KE. The average residuals arising from syste matic error would, when all in volume, be mostly less than 106 in magnitude here. Even for the best data currently available such small residuals would be ob scured by random error. Thus we see that it is possible that two different equations, both wrong (as here) of