On the Equivalence Between a Commonly Used Correlation Coefficient and a Least Squares Function

Transcription

1 On the Eqivalence Between a Commonly Used Correlation Coefficient and a Least Sqares Fnction Diane C. Jamrog George N. Phillips, Jr. Yin Zhang Janary 2003 (revised October 2003) Abstract Many objective fnctions have been proposed in X-ray crystallography to solve the moleclar replacement (MR) problem and other optimization problems. In this paper, we establish the eqivalence between optimizing two target fnctions: a commonly sed correlation coefficient and a least sqares fnction. This eqivalence may be in neighborhoods abot the global optima or the entire MR variable space depending on whether the average vales of the observed and calclated data are sbtracted from observed and calclated data. In addition, we also present an argment that the correlation coefficient between strctre factor magnitdes is likely to perform better than the correlation coefficient between intensities. This was confirmed by the MR program SOMoRe, especially when low-resoltion data were sed. 1 Introdction A major goal in X-ray crystallography is to qantitatively compare the observed and calclated diffraction patterns for a moleclar strctre being solved. This comparison Department of Comptational and Applied Mathematics, Rice University, Hoston, Texas, USA. Department of Biochemistry and Department of Compter Sciences, University of Wisconsin-Madison, Madison, Wisconsin, USA. 1

2 is sefl in the evalation of trial protein models, refinements of strctres, and error estimation. The measre of closeness between observed and calclated intensities (or strctre factor magnitdes) is determined by a target fnction. The choice of target fnction has been debated, and mch effort has been pt into developing new target fnctions. In particlar, solving the moleclar replacement (MR) problem is often a critical step in determining a detailed moleclar strctre. The MR problem is an optimization problem to determine the orientation and position of a model protein that prodce calclated intensities closest to those observed from a crystal with nknown atomic strctre. Varios target fnctions for the MR problem have been discssed. See [1, 2, 4, 6, 8, 11, 12, 14], for example. In this article, we establish that maximzing a correlation coefficient, which is a commonly sed objective fnction for the MR problem, is eqivalent to minimizing a least sqares fnction when the calclated intensities are properly scaled and some mild assmptions are met. In other words, the two respective optimization problems have the same set of global optimizers. 2 Objective fnctions We introdce a correlation coefficient and a least sqares fnction, and then we prove that the set of global optimizers of the respective optimization problems are identical nder some mild assmptions. 2.1 Correlation coefficient The Pearson correlation coefficient is often sed to solve the MR problem both becase it can be interpreted in terms of Patterson fnctions [3, 6] and becase it is scale invariant. In the X-ray crystallography literatre, this correlation coefficient is typically written as: C(I c (), I o h ( Ih c ) = () Ic () )( Ih o Io ) [ h ( Ih c() Ic () ) 2 ] 1/2 [ h ( Ih o Io ) 2 ] 1/2, (1) where I o h and I c h() are the observed and calclated intensities occrring at the lattice point h, R n specifies the orientation and translation of the model being positioned 2

3 by MR, h is the smmation over all h (or intensities) in the resoltion range, and I o and I c are the average vales of the observed and calclated intensities, respectively. Of corse, strctre factor magnitdes ( Fh c and Fh ) o can be sed in place of intensities; Fh c 2 = Ih c and F h o 2 = Ih o. The correlation coefficient can be written in more general terms as C(w(), w o ) = w() T w o w o w() = cos w(), wo, (2) where cos w(), w o is the cosine of the angle between the two vectors w() R m and w o R m. The vectors w() and w o are typically defined to be F c () k F c () k and F o k F o k, respectively. If k = 1, then strctre factor magnitdes are sed, and if k = 2, then intensities are sed. Ths, the correlation coefficient is scale invariant, becase the correlation coefficient is eqal to the cosine of an angle, and scaling either vector does not change the cosine of the angle between the two vectors. Obviosly, C(w(), w o ) [ 1, 1]. However, if the average vales are not sbtracted from the observed and calclated intensities, then C(w(), w o ) [0, 1] becase both w() and w o will be non-negative. Finally, the MR problem can be posed as 2.2 Least Sqares fnction min (1 C(w(), w o )). (3) A natral target fnction to measre the disagreement between the observed and calclated intensities is the following least sqares fnction: L(w(), α) = α w() w o 2, (4) where α R is a scale factor, w() R m is the vector of calclated intensities, and w o R m is the vector of observed intensities. In general, w() and w o can be either F c () k and F o k, respectively, or F c () k F c () k and F o k F o k, respectively, for k = 1 or 2. If the least sqares fnction is sed, then either the calclated or the observed intensities shold be scaled becase the observed intensities are measred on a relative scale dring the X-ray crystallography experiment. We choose to scale the calclated intensities, bt the the same effect can be achieved by scaling the observed 3

4 intensities by 1/α. As a reslt, the MR problem can be posed as the minimization of the disagreement between observed and calclated intensities over all possible linear scale factors and all possible orientations and translations of the model protein: min L(w(), α). (5), α The least sqares fnction is generally not sed as an objective fnction for the MR problem bt has been sed as an objective fnction for rigid body refinement, a comptational process that is often sed to refine a MR soltion. Most likely, the MR problem is not posed as (5) becase at face vale (5) may appear to be a more difficlt optimization problem than (3) de to the higher dimension of the variable space. However, the first two lemmas of the next section sggest the appropriate scale factor α. If this scale factor is sed, then optimizing the above least sqares fnction does not involve any more variables than optimizing the correlation coefficient. 3 Proof of Eqivalence In this section, we present for lemmas and a theorem establishing the eqivalence between minimizing L(w(), α) and 1 C(w(), w o ). In other words, (, α ) is a global minimizer of L(w(), α) if and only if is also a global minimizer of 1 C(w(), w o ). Two optimization problems will be referred to as eqivalent if the two sets of global optimizers are identical. This eqivalence will be symbolically denoted as. The seqence of theoretical reslts begins with a lemma that ses the reslt that if w() 0, then the optimal scale factor for the least sqares fnction is β() = w() T w o /(w() T w()). The second lemma establishes that minimizing L(w(), α) is eqivalent to minimizing L(w(), β()) when the above optimal scale factor β() is sed. The third lemma shows that 1 C 2 (w(), w o ) = L(w(), β())/ w o 2 provided w o 0 and w() 0. Finally, the theorem ties all these reslts together to show the eqivalence between minimizing L(w(), α) and 1 C(w(), w o ) nder mild assmptions. The role the assmptions play with respect to the regions of eqivalence are discssed following the theorem. 4

5 3.1 Theoretical reslts Lemma 1 For, v R m and 0, min α α R v 2 = T v 2 T v = v ( 2 1 cos 2, v ). (6) Proof: For fixed and v, the optimal scale factor is α = T v/( T ) or the soltion to the normal eqations for the minimization problem above. Now, sing this optimal scale factor, ( T v ) T ( T v ) T v T v = ( T ) 2 v T 2 T v T T T v + v T v, = v T v (T v) 2 T, ( = v T v 1 (T v) 2 ). v T v T Finally, sing the definition, cos, v = T v/( v ), v T v ( 1 (T v) 2 ) v T v T = v 2 ( 1 cos 2, v ). (7) Lemma 2 Let L(w(), α) be the least sqares fnction as defined in (4), where w o R m and w : R n R m and α R. Assme there exists R n sch that w() T w o > 0. (8) Then where min,α L(w(), α) min L(w(), β()), (9) β() = w()t w o w() 2. (10) Proof: Let f(, α) = α w() w o 2 and g(v) = β(v)w(v) w o 2. (11) 5

6 To prove the lemma, we show (, α ) U = { (ũ, α) sch that f(ũ, α) f(, α) (, α) R n R } (12) if and only if and V = { ṽ sch that g(ṽ) g(v) v R n } (13) α = β( ). (14) Let (, α ) U. Assmption (8) implies w( ) 0. Hence, as shown in Lemma 1, the niqe soltion to min γ γ w( ) w o 2 (15) is well defined as γ = w( ) T w o / w( ) 2 = β( ). Therefore, g( ) = β( ) w( ) w o 2 α w( ) w o 2 = f(, α ) β(v) w(v) w o 2, (16) that is, g( ) g(v) for arbitrary v. Ths, V. Moreover, f(, α ) = α w( ) w o 2 β( ) w( ) w o 2 = g( ), (17) becase (, α ) is a global minimizer of f(, α). Ths, α = β( ), since α w( ) w o 2 = β( ) w( ) w o 2 and β( ) is the niqe minimizer of (15). In addition, g( ) = f(, α ). Now, let v V, and sppose f(v, β(v )) > f(, α ). This ineqality implies g(v ) > g( ), a contradiction. Therefore, (v, β(v )) U. Lemma 3 Let C(w(), w o ) be the correlation fnction as defined in (2), where w o R m and w : R n R m. Let L(w(), α) be the least sqares fnction as defined in (4), and β() be the scale factor as defined in (10). Assme w() 0 and w o 0. Then 1 C 2 (w(), w o ) = L(w(), β()) w o 2. (18) Proof: Since w() 0, by Lemma 1, L(w(), β()) = w o 2 ( 1 cos 2 w(), w o ) = w o 2 ( 1 C 2 (w(), w o ) ). (19) 6

7 Therefore, becase w o 0, 1 C 2 (w(), w o ) = L(w(), β()) w o 2. (20) Lemma 4 Let C(w(), w o ) be the correlation fnction as defined in (2), where w o R m and w : R n R m is continos fnction on a compact set D R n. Assme γ 1 = min cos w(), w o, γ 2 = max cos w(), w o, and γ 1 < γ 2, (21) where the minimm and maximm are taken over the set D. Then over D Proof: Clearly, Similarly, min min 1 C 2 (w(), w o ) min 1 C(w(), w o ). (22) 1 C 2 (w(), w o ) max min 1 C(w(), w o ) max C 2 (w(), w o ) max cos 2 w(), w o. (23) cos w(), w o. (24) Now, given assmption (21), is a global maximizer of cos w(), w o if and only if cos w( ), w o = γ 2. Similarly, is a global maximizer of cos 2 w(), w o if and only if cos w( ), w o = γ 2 since γ 2 2 > γ2 1. Theorem 1 Let C(w(), w o ) be the correlation fnction as defined in (2), where w o R m and w : R n R m is a continos fnction on a compact set D R n. Let L(w(), α) be the least sqares fnction as defined in (4), and β() be the scale factor as defined in (10). Assme there exists R n sch that and w() T w o > 0, (25) γ 1 = min cos w(), w o, γ 2 = max cos w(), w o, and γ 1 < γ 2, (26) where the minimm and maximm are taken over the set D. Then over the set D min,α L(w(), α) min 1 C(w(), w o ). (27) 7

8 Proof: Given (25), w( ) 0, and by Lemma 2, Given (25), by Lemma 3, min,α L(w(), α) min L(w(), β()). (28) min L(w(), β()) min 1 C 2 (w(), w o ). (29) Finally, given (26), by Lemma 4, min 1 C 2 (w(), w o ) min 1 C(w(), w o ). (30) 3.2 Regions of eqivalence The assmptions of the lemmas and theorem are satisfied for the observed and calclated intensities, either in neighborhoods abot a global minimizer or for all in the MR variable space. First for the MR problem, assmption (25) shold always be satisfied becase w() = I c () 0 and w o = I o 0. The calclated and observed intensities shold not all be eqal to zero. Similarly, if w() = I c () I c () and w o = I o I o, then w() 0 becase the calclated intensities become less bright at a fairly rapid rate as their distance from the origin in reciprocal grows [13, p. 165]. For the same reason, w o 0. Second, whether assmption (26) holds for any in the optimization variable space D depends on the definition of w() and w o. (For example, in MR may be eqal to (θ 1, θ 2, θ 3, x, y, z) and D = [0, 2π] 3 [0, 1] 3.) Assmption (26) implies that γ 1 cos w(), w o γ 2, (31) where γ 1 < γ 2. If w() = I c () I c and w o = I o I o, then (26) may be satisfied only in a neighborhood of a global minimizer rather than for all. If the average vales are sbtracted, then the cosine of the angle between the two vectors, w() and w o may be large and violate assmption (26). However, if the model protein is accrate enogh, then in a neighborhood of the global minimizer, the initial 8

9 angle between the observed and calclated data shold be small enogh so that sbtracting the average vales will not increase the angle so mch as to violate (26). We now give a concrete example that shows that if the means are sbtracted, then there may be regions for which the fnction 1 C(w(), w o ) and the least sqares fnction are not eqivalent. Sppose the means are sbtracted and C(w(), w o ) has a local minimm at sch that C(w( ), w o ) < 0. Then, 1 C(w(), w o ) will have a local maximm at, bt w o 2 (1 C 2 (w(), w o )) = L(w(), β()) will have a local minimm at. Ths, optimization of the two fnctions will not be eqivalent near. In contrast, if the means are not sbtracted, then the cosine will always be non-negative, and assmption (26) and Lemma 4 will hold for all ; that is, eqivalence between the two fnctions will hold for the entire optimization variable space D. (Of corse, the above argments are the same if strctre factor magnitdes are sed in place of intensities.) Finally, we note that for the least sqares fnction there does not seem to be a jstification for sbtracting I o and I c from the observed and calclated intensities. On the other hand, when the correlation coefficient is sed as a traditional rotation fnction, Brnger notes that the nmerator of the correlation coefficient is eqal to the real space rotation fnction given some assmptions [3], and for the real space rotation fnction, the very large sprios origin peak is damped by sbtracting the average vales of the intensities; see [10, 13], for example. 4 Intensities verses strctre factor magnitdes Dring or development of the MR program SOMoRe [9], we sed sets of low-resoltion intensities and strctre factor magnitdes to compte srrogate fnctions, that is, fnctions that cold be sampled relatively qickly to identify regions of the MR variable space were soltions might exist. Based on or nmerical experimentation with SOMoRe, we feel that C( F o, F c ) is likely to be more accrate that C(I o, I c ) especially when lowresoltion data is sed. Similarly, Glykos and Kokkindis also advocate the general se of strctre factor magnitdes over intensities [7]. For example, dring some of SOMoRe s deterministic searches of the srrogate fnction, good starting points for local optimization cold not be fond when C(I o, I c ) was sed bt cold be fond when C( F o, F c ) was sed. (By good starting points, we mean starting 9

10 points that were sfficiently close to a global minimizer sch that the local optimization method BFGS cold converge to the soltions of the MR problems.) Besides the nmerical evidence presented in [9], we believe that C( F o, F c ) is more accrate becase this fnction essentially encorporates a weighting of the intensities according to the error in their measrements. The observation of the diffraction intensities dring an X-ray crystallography experiment is a stochastic process with nderlying Poisson statistics. Ths, the error in the measrements is proportional to the sqare root of the intensities. A proper point-wise weighting scheme of w h I h wold have a mltiplier of w h = 1/ I h, and this weighting effectively prodces C( F o, F c ) from C(I o, I c ). 10

11 Acknowledgments Diane Jamrog was spported in part by a training fellowship from the Keck Center for Comptational Biology (NSF GRT Grant BIR , NSF RTG BIR , and NLM 5 T15 LM07093) and NSF Grant DMS George N. Phillips, Jr. was spported by the National Institte of Health GM Yin Zhang was spported in part by DOE Grant DE-FG03-97ER25331, DOE/LANL Contract and NSF Grant DMS

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