I
Edgar Heilbronner Physikalisch-Chemisches lnstitut der Universitat CH-4056 Basel, Switzerland
I
Position and Confidence Limits of an Extremum
I
The determination of the absorption maximum in wide bands
Two previous communications by De La Zerda et al. (I) and by Eriksson et al. (2) have dealt with methods for the precise determination of the position of the absorption maximum in wide hands. This is a particular example of a more general problem encountered in many physico-chemical investigations, namely the assessment of the position xeOof an extremum (minimum or maximum) of an unknown function y = y(x) defined within experimental error by N paired couples of measured data ( x , y,). Although the methods proposed in references ( I ) and (2) yield estimates x, for re0, the all-important question of how precise these "precise" determinations of x, really are remained open. In this contribution we shall concern ourselves with providing an objective answer in terms of confidence limits for x.0 and with developing a simple method for their numerical evaluation, suited for pocket calculators. Position and Confidence Limits of an Extremum In the interval A < x < B a r e given N experimental points (xu,y,), v = 1,2, . . .N , scattered around an unknown function y = y(x), which goes through a single extremum at position x,O within this interval. The values x u of the independent variable x are assumed to he known precisely, whereas the experimental values y, = y(x,) f c, of the dependent variable y are affected with random errors fc, of a priori unknown magnitude. It is always possible to choose the interval A, B in such a way that for all practical purposes the cubic polynomial v = n + i Z r + r ~ ~ + 6 ~ ~ (1) can he assumed to represent faithfully the unknown function y(x) in the interval A, A. Unbiased estimates a, b, c, d of the parameters a,0, y, 6 of eqn. (1)are then calculated according to standard least squares procedures by first solving the following set of three linear equations (3)
. ..
s11 Sl2
. .
s13
Sm Sm S n
Say
where the matrix elements are defined by
"-.
The estimate a of the constant ru of eqn. (1) is then obtained by inserting the results b, c, d of eqn. (2) into
-
o=F-bx-ex2-dx'
-
(4)
This yields the following least-squares approximation for the unknown function in eqn. (1): Y=o+bx+cz2+dx:'
(5)
The position x, of the single extremum (minimum or maximum) of eqn. (5) within the interval A, R, is ohtained by setting its derivative Y'(x) equal to zero, i.e., by solving the equation 0 = b + (2z)c
+ (3x2)d
240 1 Journal of Chemical Education
(6)
(The second root of eqn. (6), which lies outside the range A,
B, is an artifact and should be discarded). Apart from the inclusion of a cubic term in eqns. (1)and (5), the ahove procedure for the determination of x, is essentially the one advocated by Eriksson et al. (2). A standard analysis of variance yields the residual variance
..
where Y , = Y(xJ are the values of function (5) at positions I,. The denominator N - 4 is the number of degrees of freedom of the problem. The residual variance s2is a measure for the scatter of the N points x,, y, around the regression function (5). Finally the variances and covariances of the estimates b, c, d of eqn. (5) can he expressed in terms of s2 as follows (3)
The multipliers Cii are the matrix elements of the inverse of the left-hand side matrix of eqn. (2), i.e. C12
c22
(9)
which has to he evaluated when solving the eqns. (2) forb, c, and - - d. So far the ahove procedure follows the traditional curvilinear regression analysis outlined in all the textbooks on the topic, e.g., in ref. (3). I t is important to realize that this procedure yields the position x. of the extremum of the regression function Y(x) given in eqn. (5), whereas we are really interested in the nosition of the extremum xS0of the function Y(X) to be answered is: given in eqn: (1). Thus the crucial What are the lower ( x ~and ) upper (xu) confidence limits for the single extremum x,O of y(x), within the range A, B? The answer to this type of question has heen given by Hotelling (4) and for the case of a simple quadratic regression by Linder (5).Note that a similar problem is implied in the work of Box and Wilson (6). We shall now derive a simple procedure which yields X L and x u for our particular example, making use of Fieller's theorem (7). The right-hand side of eqn. ( 6 ) is a linear combination of the three estimates b, c, d of the parameters a, (3, y of eqn. (I), the variances and covariances of which are given in eqn. (8). According to the theory of error-propagation, the variance of the right-hand side of eqn. (6) is given by V(b
+ ( 2 x 1 ~+ (3x2)d) = V(b) + ( 2 ~ ) ~ V (+e )( 3 ~ ~ ) ~ V+( 2(2x)Cov(b, d) CI + 2(3x2)Cov(b,d ) + 2(2x)(3x2)Cov(c,d )
Inserting eqn. (8) into eqn. (10) yields
(10)
Dividing t h e right-hand side of eqn. (6) b y t h e square root of eqn. ( l l ) , i.e., b y i t s s t a n d a r d error, yields a ratio which i s distributed accordina to Student's t:
If we now replace t in eqn. (12) by a fixed value t ( P , @) of Student's t corresponding to a chosen significance level P and t o the n u m b e r 4 = N - 4 of degrees of freedom of t h e particular problem, we obtain, making use of eqn. (11)
According to Fieller's theorem t h e two roots X L a n d x u of eqn. (13) in t h e interval A, B(A < X L < x u < B ) a r e t h e required confidence limits, meaning t h a t t h e position x." of the extrem u m of eqn. (1) lies with probability 1- P (or in other words with 100 ( 1 - P ) 9h security) between X L a n d x u . Note t h a t t h e numerical solution of eqn. (13) is a n easy task, because t h e software provided even with small desk computers includes a n iterative root-finder for user-defined functions. Example: The Determination of t h e Absorption Maximum of a Wide Band \\'r illulmre the pnrvdurr outlined in the prrvi) appl\.iug il I U the px;imple pruxdrd hy Frikwm rt nl. t?). The prunaq dm?. anvrlmnth. \ a d a h s d n n c v A arc shewn in Ficurr 1. Heeausk the h, hive bken chosen equidistsnt, it is possibleand more convenient to use the indices u as independent variables x,. The sample size is N = 18. Computing the matrix elements S, and S, according to eqn. (3) with*, = "and y . = A, 10",and inserting them mta eqn. (2)yields the follow in^ set of three linear equations forb, c, d
.k
""
5 0
10
5
IS
8
20
Figure 1. Example given by Eriksson el ai. (3(see Table l),with thirddegree polynomial fit. The shaded area indicates the range ( X L to xu) within which the true value xeo of the maximum lies with 99% security, i.e.. P = 0.01. For comparison, the 95% confidence limits (P = 0.05)and the 50% confidence limits (P = 0.5)are included.
h, = (270.8
+ (18 - x,)
0.4) nm = 274.86 nrn = 36380 cm-I
S, = 1/A,
(18)
The latter value differs by 40 cm-I from 6.- = 36420 cm-' found by Eriksson et al. (2) who have used a different regression function in their least-squares treatment. However, as we shall see presently this difference is not significant. To obtain confidence limits X L and xu for the positions xeoof the maximum of our band, we first insert the estimates a, b, c (eqn. (15)), s Z (eqn. (1711, and themultipliers (eqn. (18)) intoeqn. (13). Then we must choose a significance level for which we want to compute the confidence limits. There is a great tendency to opt for a 95% significance level ( P = 0.05), because this is the favorite of biologists, but for physical chemistry 99% is presumably a better choice, i.e., P = 0.01. However, to show how the confidence limits depend on P we compute them far P = 0.5,0.1,0.05, and 0.01. For g = 18 - 4 = 14 degrees of freedom the Student's t-values for P = 0.5,0.1,0.05,and0.01 are0.692,1.762,2.145,and 2.977,respect i d y . With those values inserted into eqn. (21) the confidence limits are
The solutions of eqn. (14) are b = 3.196093 X T T=
and with? = 125.3869 one ohtains from eqn. (4) and thus from eqn. (7)
The inverse of the left-hand side matrix of order 3 in eqn. (141, i.e., the matrix (9) of the multipliers C;, is (values rounded off to 10-9:
8.156
371= 36398 em-' G; = 36423 cm-' 3u = 36432 cm-I 6.u = 36453 em-'
x;, = 8.617 xu = 8.792 xu = 9.183
(19)
These confidence limits are displayed in Figure 1 and the 99% security range for rd' has been indicated by the shaded area. Thus we deduce from the experimental data quoted by Eriksson et al. (2) that the maximum GeOof the absorption band in question lies with 99% security between the limits 31. = 36333 cm-I and 6.u = 3fi453cm-', i.e., within a range of 120 em-'. Note that the confidence limits x~ and xu (and therefore GL and iru) do not lie symmetrically with respect to x, (or w e ) . This asymmetry will be ahserved even if the cubic t e r n in the regression function (eqn. (5)) vanishes or if a quadratic function (eqn. (1)) is assumed, as will be shown in the next section. Adaption of t h e Method t o Pocket Calculators
In real practice, the above results, i.e., those summarized in eqns. (Is), (16), and (171, are ohtained hy using the standard program for nonlinear (~olvnomial) regressions provided with all commercial mini. . or desk computers. We are now in oossession of all the oertinent information necessarv toanswer the question concerning the position of the hand maximum xen in a statistically meaningful fashion. Inserting the estimates (eqn. (16)) into eqn. (6) and solving for the root x , in the interval 1 < x < 18 yields (see Fig. 1) x, = 7.882
To a d a p t t h e above procedure t o (uon-programmable) pocket calculators, t h e following simplifications a r e introduced: 1) In planning the experiment we choose N equidistant values x,. for the independent variable .r
r. = XI
+
(U
- l)h
(20)
h being the step width between two successive values x., and x,.+~.As a consequence we are allowed to use the indices v = 1, 2 , . . . , N as independent variable in our calculation. 2) We assume that the function y(x) and thus the regression Volume 56. Number 4. April 1979 1 241
function ( e q n (5))are quadratic functions of x, rather than cubic ones. In most cases this will not entail a sirmifieant lws of information. 3) Fmally, we take advantage uf the properties uf orthogonal polym,miali 18, 9 ) fur which tabulated values are available in most slandard tablr~ofstatistics(9). I n this section we demonstrate t h e mechanics of t h e procedure. t h e derivation of which will be eiven i n the A o ~ e n d i x . T o t h i i e n d we choose two examples, &mmarized &+able 1 a n d Fienre 2. For both e x a m.~ i e (A:N s. = 6: B:N = 7) t h e values of ihe independent variahle v and the co;respondi& values of t h e deoendenr variable 3 . a r e listed in the resoective columns ( l i a n d (2) of 'l'able'l. These a r e t h e .'ex;erimental" points of Figure 2, to which a quadratic function is ro he fitted. Columns (31 and (4) contain the values and C2(u) of the linear and quadratic orth(tgnnal polynomials $1 and &, a hich have been copied from standard statistical tables (9).
S t e p IV. Using the abbreviations
BI = S d S n = S2VIS22 (24) where the S I , are those defined in eqn. (21) and S I T Sz2 , are taken from standard tables (9) we compute the parameters B2
wberes2istaken fromeqn. (23),t(P, 4 ) isstudent's t for (1 - P) 100% security and for the degrees of freedom 4 = N - 3, and the coefficients Q, R, U are listed in Table 2 for the ao~rooriatenumber N. Step V. Solving the quadratic equ&on'
Step I. Compute the sums
.-. S t e p II. The position u. of the extremum (maximum) of the leastsquare adjusted regression function is obtained from
where S1, and Szy are the sums (eqn. (21))and where the factor FN is taken from the list given in Table 2 for the appropriate value of N. If the corresponding values x, are nredrd, they are ohtsined by inserring the result rcqn. t221~m u eqn. 1201. S t w 111 Cumnut. the realdual wriance s'nrrordine to
IY
Example :
N =odd = 7
The values SII and Sg2arr taken from the rahlwof orthogonol pol"nomials ( 9 , whew they nre listed at the hottom o f ~ n r h m l o m n ~ t ' t h e
Table 1. Two Examples for the Method Adapted to Pocket Calculators Example A: N = even = 6 (1) u
(2) Y,
(3) Ed")
(4) E2(4
1 2
1 10 15 16 .. 10
-5 -3 -1 1
-5 -1 -4 -4
3 4
6
5
5
(5) Y&(v)
(6) Y&(YI
-5 -30 -15 18 . 50 Sly = 53
5 -10 -60 -84
-
Step I: Step 11: u. = 3.9989 Step Ill: sZ = 0.3262 For 99% security: t(O.OO1; 3) = 5.841 Step IV: p = 0.6150: q = -1.3760: r = 0.5275 Step V: UL = 3.764: uu = 4.373 (99% security)
.
-
-
Step I: S,, = -12 Step 11: u. = 3.8696 Step Ill: sZ = 2.3929 For 99% security: t(O.OO1; 4) = 4.604 Step IV: p = 8.3606; q = 2.8163; r = -1.5794 S t e ~V: a = 3.303: nl = 4.304 199% SecuriNI 242 1 Journal of Chemical Education
.. 50 -
Figure 2. Examples tar Uw treatment adapted to pocket calculators.The shaded area indicates lhe range (qto Y,,) wilhin Which the true value ve0of the maximum
lies within 99% security. The values Er are those of the (linear)first anhogonal polynomial for the values of u given above (see Table 2).
S, = -93
Table 2.
.-
S2).= -138
Factors for the Pocket Calculator Adanted Procedure
yields the roots CU and EL which are the required cnnfidence Limits for Ee0 with (1 - P ) 100%security on the scale, i.e., with reference to the linear orthogonal polynomial h. The corresponding values UL and by substitution into vu are obtained from EL and
The mefficients FN = (S2A2)/(2SnA2)have been tabulated in Tahle 2. With respect to the residual variance s2, calculated according to eqn. (23) the variances of BI and Bz are given by
V(B3 = s2/Su V(B3 = s2/Szz where Al = 2 if N = even and A, = 1if N = add,
Because of the orbhogonality of the polynomials £1 and $2the covariance of B, and B2 is zero. Consequently the variance of the left-hand expression of eqn. (35) is
Appendlx Derivation of the Formulae For the independent variahle u the linear and quadratic orthogonal polynomials are 11 =
(0
("
- 5) = 3
y)
--
(38)
r
In practice it is convenient to use the polynomials
