SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA


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1 SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands Emal: We present sx ways to ft a straght lne to measurement data. The computatonal methods dffer n the way weghts are attrbuted to the data and measurement uncertantes are taken nto account. A numercal example shows the results of the dfferent methods. We conclude that even the best method s ncomplete and moreover s not mplemented n well known commercal programs. KEYWORDS: regresson, straght lne model, least squares, weghts, uncertantes Introducton A common case n the teachng laboratory s that students measure a quantty Y dependent on a quantty X. Examples are verfcatons of Ohm s and Hooke s laws. The result s a lst of measurement data (x,y ), =1...n. In measurement theory we assume that these data are realsatons of ndependent stochastc varables (X, Y ), whch are charactersed by probablty dstrbutons lke the normal dstrbuton. If the measurement s performed correctly, not only (x,y ) have been obtaned, but also the correspondng standard uncertantes (u(x ),u(y )). These standard uncertantes are estmates of the standard devatons σ, σ ) of the ( X Y measurement varables. Often we ask our students to ft a model to the measurement data. The model s a functon f whch descrbes the expected dependency: y = f ( x; α1... α m ) The α j are parameters, whch we want to estmate from the data, wth ther uncertantes u(α j ). Before the students can make a ft, four choces have to be made: the model, the fttng method, how weghts are attrbuted to the data, how the measurement uncertantes are taken nto account. The last two choces may be related, because often weghts are derved from uncertantes. Note: uncertantes may be dfferent for dfferent values of, both measurement varables have uncertantes. We should make clear to our students from the start that these choces are relevant and we should oblge them to make ratonal choces and to publsh these, together wth the results. For the smple case of a straght lne model, we wll now show how these choces may nfluence the outcomes of the fttng procedure. (1)
2 The straght lne model Often we lnearze the data by transformng them, for example logarthmcally. We do ths because a straght lne model can be nspected graphcally by eye and algorthms for the straght lne case are generally avalable, even on hand calculators. The general straght lne model predcts that the relaton between the dependent varable (y) and the ndependent varable (x) s: y = α + β x () Our task s to get estmates a and b of the parameters α and β and uncertantes u(a) and u(b) from the measurement data. In many cases not the general straght lne, but the straght lne through the orgn s approprate: y = β x (3) An example s the determnaton of the gravtatonal feld strength g from pendulum measurements. Ths example demonstrates the need for transformed data: to get a straght lne model we need perod squared aganst pendulum length or perod aganst the square root of pendulum length. One way to choose between these two models Eq.() or Eq.(3) s to nspect the estmate a of α after a general straght lne ft has been made. If a does not dffer sgnfcantly from zero, the drect proportonalty model may be the rght choce. Therefore the uncertanty u(a) should be known. Smple hand calculators do not provde ths, so we need more advanced calculators, lke a PC, and an approprate applcaton lke Orgn [1] or SgmaPlot [] or programs wth approprate algorthms, lke those descrbed by Press et al [3]. Another way to choose between models s to use a measure of goodness of ft. Expermentalsts generally use the chbyeye method, as t s called by Press et al. [4]: they judge f the ft looks good. For example, by usng a ruler we judge f a straght lne can be drawn through all uncertanty regons defned by the uncertanty (or error) bars. Statstcans want an objectve measure, for example the χ test. Here χ s the sum of (weghted) squares of the dfferences between measured and calculated yvalues, usng the estmates. See for example Press et al. [4]. The method of least squares The method of least squares s generally used to get estmates of the model parameters. The least squares crteron states that χ should be mnmzed. In the straght lne case: χ = N = 1 w ( y a bx ) w s the weght attrbuted tot the th datapont. Least squares estmators often have good propertes n terms of bas and mean square error and have some computatonal advantages. However the method of least squares s only one of a number of estmaton methods. Some programs offer socalled robust methods. See for example Press et al. [3]. (4)
3 Attrbutng weghts and usng uncertantes The estmates wll depend on the way the weghts w are attrbuted to the data and the way the measurement uncertantes (u(x ),u(y )) are used. We consder three possbltes: Uncertantes n x and y are neglected. They may even be absent. Nonetheless we can calculate values of the uncertantes u(a) and u(b) of the parameter estmates! Ths s done by usng the devatons between measured and ftted yvalues to estmate σ Y. There are some hdden assumptons n ths proces, lke the assumpton that the model s correct. See for example Press et al. [4]. Although t s aganst the rules of measurement, neglectng uncertantes s qute common practce, gven the fact that applcatons lke Orgn [1] and SgmaPlot [] use t as the default method. Clearly, ths wll not stmulate our students to take the trouble of payng attenton to measurement uncertantes! Uncertantes n y are accounted for, but those n x are not. Ths s a standard case n the lterature. See for example Press et al. [4]. There are two solutons: one wth uncertanty propagaton and one wthout. In the latter case the uncertantes are only used n weghng the data, almost always by puttng w =1/u(y ). Uncertantes n x and y are both accounted for. Ths s the real problem, whch seems hard to solve. See for example the dscusson n Press and Teucholsky [4]. Exstng solutons are not mplemented n wellknown applcatons. As stressed by Press et al. [3], the estmators a and b are not statstcally ndependent. In general, ther covarance u(a,b) dffers from zero. Therefore, f more than one parameter estmate s needed n a calculaton, we have to account for the covarance n uncertanty propagaton. See Taylor [5] for a smple example. Some commercal programs do not provde covarances or make t rather dffcult to obtan them. An example: stressng a sprng Table 1 shows faked measurement data on the stress of a sprng. Table 1. Faked data of the stress r of a sprng by an appled force F r (cm) u(r) (cm) F u(f) We have added tenson to the sprng before t s stressed, to press the cols together n the state of rest. Therefore, the stressforce curve does not pass through the orgn. Fgure 1 shows a graph of the data.
4 F r (cm) Fgure 1. Graph of the data of table 1. Lnes are drawn by hand, ftted to the error regons Table shows the results of the calculatons. All calculatons have been done n Maple. The formulas and the programs used can be obtaned from the author. Table. Results of fttng a straght lne model weght functons and dfferent ways to calculate uncertantes y = α x + β to the data of table 1, usng dfferent weght uncert. prop. a u(a) b (N/cm) u(b) (N/cm) u(a,b) (N /cm) 1 no /u(y) no /u(y) yes, y 1 yes, n.a. x,y 1/(u(y) +b u(x) ) yes n.a. x,y by eye (lnes n fg.1) by eye n.a. We have obtaned the sxth soluton by takng our ruler, drawng two lnes whch just pass through the uncertanty regons, one wth maxmum and one wth mnmum slope, and calculatng the slope b as the mean slope and the ntersecton a as the mean of ntersectons of the yaxs. See fgure 1. We have determned the uncertantes as half the dfferences of ntersectons and slopes. Statstcans wll quckly pont out that ths method depends on a personal nterpretaton of extremes, does not gve the covarance u(a,b) or a measure of goodness of ft and does not work on other than straght lne models. Nonetheless, manly because t accounts for uncertantes n x, we recommend t as second best. An addtonal advantage may be that t convnces our students that the uncertanty bars, whch we oblge them to draw, do any good.
5 The best method s the ffth one n table, because t does not contan an element of personal taste as the paper and pencl method does and correctly takes the uncertantes n account. Drawbacks of ths method are that s ncomplete, because u(a,b) s lackng, and that t s avalable for the general straght lne model only. Worse, to our knowledge t s not mplemented n any commercal program. Takng a closer look at table, we see that a and b are mostly effected by weghng. Ths s because delberately we have chosen the ffth data pont somewhat outlyng, although ts large uncertanty does not make t a real outler from the straght lne pont of vew. We also see that no value of a or b s sgnfcantly dfferent from another value, whch may ndcate that the straght lne model s a good one n ths case. Note that the uncertantes u(a) and u(b) are larger f uncertantes n x are taken nto account, as t should be. The lower value of u(a) and u(b) f no uncertanty propagaton s used may be an ndcaton that the uncertantes are overestmated. The dfference n the values of u(a,b) rases the queston what happens f we use both a and b n a calculaton. We have calculated the quotent q=a/b usng values from the frst and the second row of table wth proper error propagaton: q q q q u( q) = u( a) + u( b) + u( a, b) (5) a b a b The results are q=0.49 cm wth u(q)=0. cm (11% contrbuted by the covarance term) and q=0.69 cm wth u(q)=0.15 cm (16% contrbuted by the covarance term). These results are not sgnfcantly dfferent. Concluson The algorthm descrbed by Press and Teucholsky [5] solves our straght lne ft problem correctly. However t does not estmate the covarance and therefore cannot be used for uncertanty calculaton f the parameters of the lne both are needed n a calculaton. If t s not avalable a smple paper and pencl method may be as good f uncertantes n both x and y are mportant. The message of ths paper s that we should make aware our students of the ptfalls of parameter estmaton, especally those concernng weghng and uncertanty analyss, before we let them solve fttng problems usng a computer. As always, they should learn to thnk frst before pushng the buttons. Lterature 1. Orgn s a trademark of Mcrocal Software. We have used verson SgmaPlot s a regstered trademark by SPSS Inc. We have used verson W.H. Press, B.P. Flannery, S.A. Teucholsky and W.T. Vetterlng, Numercal Recpes n Pascal, Cambrdge Unversty Press, New York (1989) 4. W.H. Press and S.A. Teucholsky, Fttng Straght Lne Data wth Errors n Both Coordnates, Comp. n Phys., 6, pp (199) 5. J.R. Taylor, Smple examples of correlaton n error propagaton, Am. J. Phys., 53, pp (1985)
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