Introduction to the Least Squares Fit. Table of Contents. Least Squares Fitting with Excel.1

Transcription

1 Least Squares Fttng wth Ecel. Introducton to the Least Squares Ft Tale of Contents. Uses for a Least Squares Ft: Lnear Dependence. Methods of Fndng the Best Ft Lne: Estmatng, Usng Ecel, and Calculatng Analtcall 3. Calculatng a Least Squares Ft 4. Uncertant n the Dependent Varale, Slope, and Intercept 5. Calculatng the r Value 6. Sample Data and Settng up the Spreadsheet 6. Eample: Enterng a Sum 6. Eample: Enterng a Formula 6.3 Formulae for Ecel Usng Sample Data 6.4 Usng Ecel s LIEST Functon

2 Least Squares Fttng wth Ecel.. Uses for a Least Squares Ft: Lnear Dependence What s the reason for wantng to do a Least Squares Ft? Wh other wth fndng a est ft lne to a set of data n the frst place? Well, a graph s used to show whether there s a relaton etween the dependent varale the -as and the ndependent varale the -as. Usuall, one looks for a lnear relaton, that s to sa whether the data ponts fall roughl on a lne or not. A lnear relaton has the form a +, whch s useful for showng drect relatonshps such as Fma and VIR. A graph of the force of gravt vs. mass would eld a lne wth a slope equal to the acceleraton due to gravt. A graph of voltage vs. current would gve a value for resstance. Ths s good stuff! What aout equatons whch are non-lnear? How could calculatng a est ft lne usng the Least Squares Fttng method help wth that? Here are two eamples of equatons that ma appear nonlnear ut can e made lnear. The frst eample nvolves the magnetc force on electrons and the crcular moton the electrons m ev undergo n a unform magnetc feld. The equaton s eb, whch doesn t seem lke t r m could e graphed easl. However, one wants to graph the charge vs. the mass of the electron, as was done n 897 Sr J. J. Thompson. Therefore, that mess equaton can e rearranged as follows: ebr ev m m e B r ev eb r V V e m m m m B r Ths nvolves smple algera, however, one had to know ahead of tme what varales one wanted to graph. In order to graph charge vs. mass, the charge had to e alone on the left-hand sde, and the mass had to e on the rght-hand sde, to the frst power onl, and wth a prefactor of known varales V V, B, and r. Ths gves a lnear graph of the charge vs. the mass wth a slope of. B r Fgure : Lnearzaton of an equaton. A shows a graph of vs. of the quadratc functon a. B shows a graph of vs., a lnear relaton wth slope a. A B

3 Least Squares Fttng wth Ecel.3 The second eample nvolves the perod of a pendulum, gven T π l g. In la one measures 4π the perod and the length of the pendulum. What to do? Squarng the equaton gves T l. g Ths elds a lnear graph of the square of the perod vs. the length of the pendulum wth a slope of 4π. g. Methods of Fndng the Best Ft Lne: Estmatng, Usng Ecel, and Calculatng Analtcall A graph can e used to estmate the est ft lne as opposed to calculatng the est ft lne from the data ponts. The asc dea s to draw a lne through the data wth as man data ponts aove t as elow t wthn error. For full nstructons refer to the Makng Graphs secton n the la manual. Estmatng the est ft lne s a good dea when there are fewer than 5 data ponts. Pckng where the lne should go s a skll that mproves wth practce. The downsde of ths s that the est ft lne and the uncertantes reall are just estmates ased on ee-allng the graph. A step up from gettng a est ft lne hand s havng Ecel graph and ft a trendlne mentoned n the Graphng wth Ecel document. Ecel uses the Least Squares Ft method to calculate the est ft lne. Usng Ecel s a good dea for data sets larger than 5 ponts, for the program takes care of the whole process; ut, whle an Ecel graph wll gve the equaton of the est ft lne, t won t gve the uncertant n the slope. Also, t s alwas dangerous to use a result that s not full understood. Calculatng the est ft lne usng Least Squares Ft method s good for data sets wth more than 5 ponts and gves etter results for more data ponts. The analtc method s good for when error s small. The method wll gve numers for slope, ntercept, the uncertantes n the slope and ntercept, as well as the correlaton coeffcent an ndcaton of how good the data ft the lne. Ths document goes through the dervaton and the method of dong a least squares ft. At the end, there are nstructons for usng Ecel s LIEST functon, whch calculates all of the numers of the ft for ou. Wh other readng ths whole document when ou could just skp to the end? It s mportant to know where our numers come from rather than just acceptng them from a program. Fgure : Comparson of A data where t s eas to estmate the est ft lne and B data where t s est to use a Least Squares Ft. The ke to calculatng a Least Squares Ft s a well-organzed data sheet tred of hearng that?. Equatons wll e used to calculate the values for the slope, the ntercept, and the uncertant n those values. Throughout the document, the equatons wll refer to the ndependent varale as just and the dependent varale as just. When applng Least Squares Fttng to a data set, keep n mnd

4 Least Squares Fttng wth Ecel.4 whch varale s the ndependent one the varale one changes n la and whch one s the dependent one the varale one checks n la to see what effects changes n the ndependent varale produce. 3. How to Calculate a Least Squares Ft Consder the dstances from each pont to the est ft lne, as shown n Fgure 3, also called the devatons. If a lne s a reall good ft, those devatons wll e as small as possle. The Least Squares Fttng method attempts to mnmze the square of the devatons. Squarng the devatons makes the math far more manageale, where workng wth the asolute values of the dstances would lead to dscontnuous dervatves. The sum of all of the squares of the devatons s called the resdual, χ χ s pronounced k to rhme wth gu, gven equaton, where are the data ponts and true s the -value from the est ft lne. true χ If the data follows a lnear relaton, then true can e epressed n the general form m true +, makng the resdual + m m χ ote that ths formula stll uses the general form of the equaton for the lne: and m have not een specfed! To fnd the lne that est fts the data, the resdual should e as small as possle. Varales and m must e chosen so that the mnmze χ. Ths s where the method gets ts name: makng the sum of the squares of the devatons the least t can e. An statstcal ook wll gve the detals of ths mnmzaton; the are also avalale on the weste under Useful Documents and Westes. The end result s that the values of and m are gven the followng equatons: 3 m 4 Fgure 3: Eample of devatons from the est ft lne lue.

5 Least Squares Fttng wth Ecel.5 4. Uncertant n the Dependent Varale, Slope, and Intercept The formula for the standard devaton s. That formula apples to fndng the standard devaton of a numer of measurements of the same true value, whch are randoml dstruted around that true value assumng that sstematc errors have een reduced. An eample of that would e fndng the dstance to a lock sttng on the tale: varous measurements of the dstance mght e slghtl off the true dstance to the lock, ut that true dstance s the same for all the measurements ecause the lock s sttng stll. The true value for the dstance s appromated the average of the dstance measurements. Consder the measurement of the dstance to a glder as t moves along a track. Tme s the ndependent varale and the dstance to the glder s the dependent varale. In ths eample these data are separate measurements n themselves, each one randoml dstruted around the true value for the dstance at a partcular nstant. If lots of measurements of the dstance to the glder were taken at one nstant lots of rangers nstead of just one, those measurements would e randoml dstruted around a true value for the dstance to the glder at one nstant n tme. The true value for the dstance to the glder at that nstant could e appromated the average value of the measurements from all those rangers --- ut that s just one nstant! The mean of the dstance s a good appromaton to the true value of the dstance f the glder s not movng. If the glder was gven a push and s movng along a track, measurements of dstance versus tme would follow a lnear pattern. As seen n Secton 3, the est ft lne true m + s a good appromaton to the true value of the dstance at an tme f the glder s movng. The standard devaton of the dependent varale,, s calculated a lttle dfferentl than. When the dependent varale s changng wth the ndependent varale n a lnear fashon, the standard devaton of the dependent varale can e found the followng: 5 true m In the formula for the standard devaton,, the factor efore the sum s, ut n equaton 5 for t s. Wh the dfference? Consder a data set wth onl two ponts.e.. Two ponts defne a lne, so of course! the lne through those two ponts s a perfect ft. For a data set consstng of less than three ponts, the uncertant s undefned. The uncertant n, the ntercept, and m, the slope, can e found propagaton of the uncertant n the dependent varale,, n the formula for and m. In dong ths, t s assumed that most of the error comes from the dependent varale. Ths propagaton ma e found n an statstcs ook and s also avalale on the weste under Useful Documents and Westes.

6 Least Squares Fttng wth Ecel.6 The uncertantes are gven the followng equatons: m 5. Calculatng the r Value 6 7 Thanks to the efforts of everone from Mr. Baage to Mr. Gates, t s no longer necessar to watch the est ears of ou lfe slp past whle ou hand-process large data sets. In fact, at the push of a utton we can even see how well an TWO data sets relate to each other lnear regresson. But as ou mght suspect, the qualt of all ths cer-math must e reported wth somethng more ojectve than the words good or ad. So when our lnear regresson program spts out a slope and a - ntercept, t proal also gves ou a numer laeled ether r or correlaton coeffcent. What s that numer? The frst thng ou need to know aout t s that t OLY tells ou how well the two data sets match a STRAIGHT LIE. It does not tell ou how well the data matches an other mathematcal functon. The second mportant thng to know s that the correlaton coeffcent has a range from 0 to. If ever matched par of data values s a pont that fts eactl on a sngle lne, r wll equal ; a perfect ft. If the ponts are close to the lne ut not perfect, r wll e less than a small amount. Fnall, f the ponts are all over the place, and not even close to favorng the chosen lne, r wll equal 0. Ths means there s no LIEAR relaton etween the two data sets. A thrd thng to know s that we ve een usng the coeffcent s nckname. Its full name s the Pearson product-moment correlaton coeffcent. Knowng ths ma get ou past a tough queston on Jeopard some da. ow let s go ack to that second thng. How do ou get a calculaton to e when the ponts match the lne and 0 when the mss? Start calculatng the average of all the values. On a graph, ths average wll e a horzontal lne that runs through all the data ponts around the md-heght pont, as shown the dotted lne n Fgure 4. Wh start wth the average? The horzontal average lne s no etter Average of ponts 7.3 Fgure 4: Average of dependent varale values.

7 Least Squares Fttng wth Ecel.7 than an other guess as a model for the lne that fts the data. It gves a worst case scenaro to use as a ass of comparson. To see how good or ad ths ft s, ou could take the dfference etween, and add them up... each actual value and the average Ooops! That s alwas gong to e zero defnton. The average s the value that s rght n the mddle of a data set. Half the values are larger and half are smaller, so the sum of postve dfferences wll eactl match the negatve dfferences. To remed ths, after fndng each dfference, square t to make all values postve and then add them all up snce we are onl usng ths numer for comparson, we ll just rememer to square all other values that wll e compared to t: 8 Ths sum of the square of the dfferences s a good wa of determnng how good the ft s. A small value for ths sum ndcates all the numers are all prett close to the average. A large value ndcates the numers are wdel scattered. Ths sum wll e used as a comparson for an addtonal attempts to fnd a lne that fts the data. You then calculate a calc value for each value usng the equaton of the lne ou wsh to tr to ft to the data: calc m + Fgure 5: Straght lne ft lack to data wth the equaton of the ft and correlaton coeffcent Average of -values red s 7.3. ote the smaller devatons for the est ft lne. To see how well ths lne matches the data, a smlar sum of squared dfferences s calculated. calc 9 It mght seem that a smple comparson wth equaton 8 mght e the wa to evaluate how good ths ft s, ut we can do much etter. Consder sutractng equaton 9 from equaton 8 to get equaton 0. calc 0

8 Least Squares Fttng wth Ecel.8 If the chosen lne calc m + fts no etter than the horzontal average lne of the numers, then equaton 9 wll e just as large as equaton 8. If that s the case, the result of 0 wll e zero. calc 0 If the chosen lne calc m + the correspondng data value s an eact ft, then each calculated value s eactl equal to, and equaton 9 wll e zero. If that s the case, the result of equaton 0 wll e equaton 8. 0 So equaton 0 s equal to zero for a lne that completel msses the data and equal to equaton 8 for an eact match to the data. If we dvde equaton 0 equaton 8, then t wll equal zero for a lne that completel msses and equal for an eact match. Ths process of dvdng equaton 0 equaton 8 s called normalzaton. When normalzed, the correlaton coeffcent, r, s adjusted to have a range from 0 to. calc r 3 You ma see other formulae for r that nvolve standard devatons. The are alternate methods to calculate the same numer. In the past, calculators had onl standard devaton uttons so these other formulae ma have een easer to use. Toda t s common for the correlaton coeffcent to e ult nto the software, so the dffcultes of calculaton are no longer an ssue. We have developed the equaton usng averages ecause t s much easer to see wh t takes the form t does. Fgure 6A s almost a perfect ft, for the ponts fall ver close to the est ft lne. Fgure 6B s a good ft, for the ponts fall farl close to the est ft lne. Fgure 6C s a ad ft, for the ponts do not fall close to the est ft lne and f ou look closel ou ll note that the seem to fall n a A C Fgure 6: Graphs wth dfferent r values. A s a perfect ft. B s a good ft. C s a ad ft, not lnearl related. D s a ad ft, zero relatonshp etween the varales. calc B D

9 Least Squares Fttng wth Ecel.9 paraola and not a lne!. Fgure 6D s a ad ft for the ponts do not fall along the chosen est ft lne. The data n Fgure 6D falls on a horzontal lne, whch ndcates that the varales are unrelated, for changes n the ndependent varale produce no changes n the dependent varale. 6. Sample Data and Settng up the Spreadsheet Here s a run-through of the method of Least Squares Fttng usng the sample data shown n Fgure 7 accordng to the tales n Secton 6.3. otce that the frst row s oldfaced, as well as rows 3 through 3. Ttles of columns should alwas e oldfaced, and t helps the results and ther ttles stand out from the data. Fgure 7: Screen capture of the worksheet constructed followng the steps n ths document. ote that the means have een formatted to show onl one decmal place, the est ft lne calculatons onl two decmal places, and the remanng numers no decmal places. Ths was done to make the worksheet easer to read. 6. Eample: Enterng a Sum Cell D3 should have the sum of A:A. An tme a formula s entered n Ecel, precede t an equal sgn; otherwse, Ecel thnks the entr s just tet. Clck on D3, tpe an equal sgn n the cell, go to the ame Bo, and select the SUM functon. In the Functon Arguments wndow that appears, use the umer feld to select all of the ndependent varale data. Do ths clckng the utton net to the feld. The Functon Arguments wndow wll collapse, leavng onl the umer feld showng. Use the mouse ponter to select all of the ndependent varale data,

10 Least Squares Fttng wth Ecel.0 meanng cells A through A. A dashed o ndcates what data s selected. If a mstake s made, clck on the umer feld, delete the nformaton, and tr selectng agan. When done selectng data, clck the utton net to the umer feld, whch wll epand the Functon Arguments wndow. Clck the Ok utton. The short method of dong ths s just to tpe SUMA:A. 6. Eample: Enterng a Formula Fndng the sum of the square of,, requres frst settng up a column C whch calculates the squares. In cell C the formula for A s A^. To fll column C from C all the wa to C wth the squares of the respectve cells A:A, drag the fll handle mentoned n the Introducton to Ecel document, Secton.4 Fllng Data. Gve the column the ttle of ^ n cell C. 6.3 Formulae for Ecel Usng Sample Data Set up the columns A through K as descred n Fgure 8. These columns wll e used to calculate the necessar sums. Fgure 8: Formulae for the columns n Ecel usng sample data from Fgure 7. Column Rows Contan Sample Ecel Formula A n/a B n/a C A^ D B^ E A*B F G A-$G$4 B-$G$5 H F^ I G^ J F*G K B-$G$7-$G$8*A^ m

11 Least Squares Fttng wth Ecel. Calculate the sums n the cells usng Fgure 9 usng the sample data from Fgure 7. These sums wll e used to calculate the slope, the ntercept, the uncertantes, and the correlaton coeffcent. Calculate the values for the Least Squares Ft accordng to the formulae n Fgure 0. Fgure 9: Formulae for the cells whch calculate the sums. Cell Formulae D3 SUMA:A D4 SUMB:B D5 SUMC:C D6 SUMD:D D7 SUME:E D8 SUMF:F D9 SUMG:G D0 SUMH:H D SUMI:I D SUMJ:J D3 m SUMK:K G3 n/a G4 D3/G3 G5 D4/G3 Fgure 0: Formulae for the cells whch calculate the m +,,, m, and r value. Cell Formulae G7 D5*D4-D3*D7/G3*D5-D3*D3 G8 m G3*D7-D3*D4/G3*D5-D3*D3 G9 m SQRT/G3-*D3 G0 SQRTG9^*D5/G3*D5-D3*D3 G m SQRTG3*G9^/G3*D5-D3^ G r calc D-D3/D

12 Least Squares Fttng wth Ecel. 6.4 Usng Ecel s LIEST Functon Congratulatons for makng t through ths eercse n Least Squares Fttng! Ths eercse set up qute a spreadsheet. There s, n fact, a much shorter wa of gettng the values of the Least Squares Ft, the uncertantes, and the correlaton coeffcent, nvolvng an Ecel functon. Wh make the spreadsheet f there s a functon that alread does the ft? Gong through the work of settng up the spreadsheet creates a famlart wth the method of Least Squares Fttng. As was mentoned efore, t s dangerous to use a result wthout understandng t. The LIEST functon can return a sngle value for the slope of the lne or t can return all of the values of the Least Squares Ft. It takes as arguments { n }the dependent varale data, { n } the ndependent varale data, an ndcator whch can set the ntercept of the equaton to zero, and an ndcator whch can dspla the values of the ft n addton to the slope. Before usng the functon, enter the ndependent and dependent varale data n columns A and B, just lke n Fgure 7. ow, select s lank cells n a lock, two columns wde and three rows hgh. Make sure the upper left-most cell s the frst cell clcked when makng the selecton, as shown n Fgure. Fgure : Selectng the cells for the LIEST functon. Red arrows ponts to the cell whch was clcked frst and where the formula wll e entered. Clck F and enter the formula LIESTB:B,A:A,,TRUE. When done enterng the formula, press the CTRL ke, SHIFT ke, and ETER ke all at once; ths wll tell Ecel to wrte the results of the LIEST functon n the selected cells wth the pattern shown n Fgure. ote that Ecel wll onl dspla the numercal results; keep n mnd whch cell refers to whch quantt! A B Fgure : A Pattern of results from LIEST functon, B what Ecel wll dspla. JB 5/3/007 least squares fttng wth Ecel.doc