computer M e / . 110 Least-Squares Fitting of Multilinear Equations Richard T. O'Neilll Department of Chemistry Xavier University Cincinnati, OH 45207 David C. Flaspohler Department of Mathematics Xavier University Cincinnati, OH 45207
Recent articles in this Journal (1, 2) have stressed the importance of curve fitting of experimental data in the teaching and practice of chemistry. Optimally, the method used should (a) report the parameters fitted and their uncertainties, (b) weight data correctly, (c) be able to be implemented on a wide variety of computers and calculators, (d) be as general as possible and not restricted to the linear case, and (e) avoid iteration if this is at all possible. To satisfy criterion c, the algorithm employed should not use matrices as such, since calculators and microcomputers do not usually support MAT commands, and should complete the calculation in a single computing cycle. Iteration should be avoided if possible because the functions involved may not converge rapidly, good initial estimations may not be available, and the process may converge to only a local minimum, giving an answer that is incorrect, but not obviously incorrect. Methods have been described that solve the curve-fitting problem for the strictly linear case, with uncertainty in both dependent and independent variables (3),the general nonlinear case with the uncertainty in the dependent variable only (4), and the general case when there is uncertainty in both dependent and independent variables (5).However these methods all involve iteration. Author to whom correspondence should be directed. BITNET address: ONEILL@XAVIER. Table 1.
c = [ao1[11 General heat capacity C,T
For the process described in this paper, all uncertainty is assumed to reside in the dependent variable, more than one independent variable is allowed, and iteration is not required. Multilinear Equations
It has been stated that when the model equation cannot be arranged in linear form, the data analysis problem must be solved by iterative methods (5). Linear form in this sense refers to the parameters to be fitted and not to the variables. It is not restricted to bivariate (X, Y ) data and to the Y = mX b form. Thus the van der Waals equation, after series expansion and rearrangement:
+
is not linear in the independent variables (V, T ) but is linear in the parameters to be fitted (a, b, c = b2). We use the term multilinear to refer to an equation that describes a dependent variable or function of variables (independent and dependent) as the sum of constants multiplied by functions of independent variables: k
or, if restricted to k 5 3 as in this paper,
where Y is the dependent variable (error normally distributed); Xi, X2, . . . are the independent variables (negligible error); F, G, H, and K are functions given by the model equation; and a , 0, and 7 are the parameters to be determined. In the van der Waals equation given above, F = PV - RT, Gl=G=-l/V,G2=H=RT/V,G3=K=RT/V2,al=a=a, a 2 = 0 = b , w = 7 = b2.
+
+ [b2][RT/ v ]
+ [all [ TI
+ [a21 [ l21
-I-[Be][(-2XJ4 ~ R , + J ~P,J+I = [uoI[21 4- [ae][(-2XJ-k fundamental vibration-rotation bands of heteronuclear diatomic molecule ~ R .+J ~ P . J +JI ,
Journal of Chemical Education
Arizona State University, Tempe, AZ 85281
Examples of Multilinear Equations of Chemical Interest
PV-RT =[a][-1/V] [b] [RT/ V] van der Waals equation (after series expansion) P, V, T
40
edited by JAMESP. BIRK
+ 4J3 + 7J2+ 6J+ 2)]
Note that the parameters must be multipliers of the functions. An equation that cannot be rearranged to this form is not multilinear. Table 1 gives in multilinear form several examples of equations of chemical interest that are usually not considered to be linear. Data for these equations can be fit analytically, not just iteratively, to obtain the coefficients. The table is constructed so that the variables and parameters for each equation can be easily identified with the terms (F,G, H, K, a , 1 3 , ~ )in the multilinear form.
Table 2.
Algorithm for Estimation of Parameters
1. Enter N, C, M Comment: N is the number of observations. Cis the number of functions on the right-hand side of the model equation (the number of parameters to be estimated), M is the number of independent variables
The Procedure
In least-squares analysis for this type of function, the usual assumptions are that
where ei's are independent and normally distributed with mean zero and variance (SD - 1). The problem can be avoided by using the alternate method of calculating the variance indicated in Table 2, step 12, by asterisks. In any case, the highest precision available with the computer or calculator should be used. The algorithm has been tested with VAX, IBM-PC, Apple-11, and Commodore-64 computers using the same BASIC program.
+
Weighting
When all the error is assumed to be in the dependent variable, the weighting function for each point should be:
In the previous discussion it was assumed that a,, the uncertainty in Yi was the same for all Y. If it is not, then the global weight (1/(&F/5Y)2) should be multiplied by (l/ni2),all evaluated at the point in question, in order to calculate the weighting factor W;. The global part of the weight can be included explicitly for particular cases (6),but for reasons of generality, the procedure described here approximates the &F/&Y term as:
Conclusion
For situations where error exists in the independent variable(~) or where the equation cannot be arranged in a multilinear form, iterative techniques are needed. However, many equations of chemical interest that are not linear in the dependent variable(s) are nonetheless multilinear in form. If the error is essentially all in the dependent variable, these can conveniently be treated analytically, without iteration, by the technique described. 42
Journal of Chemical Education
John E. Douglas Eastern Washington University Cheney, WA 99004
Visualization of the electron orbital concept continues to challenge and intrigue chemical educators. The concept is crucial both to the nonscientist who is taking a liberal arts course in chemistry and should understand the basic structure of matter and to the serious chemistry student who is exploring the sophisticated nuances of atomic and molecular orbitals. The chemical educator deals with both of these groups. Twenty-five years ago Ogryzlo and Porter (1)pointed out that "The ideal model of an atomic or molecular orbital would be a cloud-like structure showing the probability of finding the electron at all points in space. . .". Admitting the difficulty of preparing such a model, they presented a method for computing contours and preparing solid models. Bordass and Linnett (2), Olcott (3),and Streitweiser and Owens (4) were among the first to use computer-generated threedimensional contour diagrams to represent atomic and molecular orbitals. The author (5)and Baughman (6)have used computer generated numerical grids as the basis of student exercises for plotting orbital contours. A recent issue of this Journal contained two articles on the subject, one by Brenneman (7) that presented contours for the angular functions of all orbitals through the g type, and one by Leibl (8) that presented a computer program for plotting two- and threedimensional contour diagrams. However, these traditional presentations of orbitals as distribution functions and contour surfaces are abstract and sometimes inaccurately simplified. The viewer has difficulty, visualizing the true nature of the electron cloud, especially just how diffuse or dense it actually is in different regions of the atom or molecule. Dot-density diagrams are a more realistic form of presentation but have been utilized only to a limited extent in some texts and by a few instructors. Frequently these are qualitative only and are not designed for interactive use by the student. Today, inexpensive microcomputers with powerful graphics capabilities enable the 25-year-old ideal of Ogryzlo and Porter to become a reality in the classroom with the easy production of accurate dot-density orbital diagrams either for demonstrations or as student exercises. One rudimentary computer program is available through a national exchange (9). This paper presents computer routines that can quickly produce accurate, graphic representations of electron clouds for both atomic and molecular orbitals, with all parameters easily adjustable. They are suitable for all levels of instruction. Simple plots may be produced in a minute or so, suitable for a live class demonstration. Longer times are re-
