V March 4 2015 Contents 1 Introduction 2 Polynomial Baselines 21 Syntax parameters 2 22 General Use 2 23 Fitting polynomial baselines using least squares ID: 60313
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FittingBaselinestoSpectraClaudiaBeleites Claudia.Beleites@chemometrix.gmbh&#x-34.;妉 DIARamanSpectroscopyGroup,UniversityofTrieste/Italy(2005{2008)SpectroscopyImaging,IPHT,Jena/Germany(2008{2017)OPV,JKI,Berlin/Germany(2017{2019)ArbeitskreisLebensmittelmikrobiologieundBiotechnologie,HamburgUniversity,Hamburg/GermanChemometricConsultingandChemometrixGmbH,Wolfersheim/Germany(since2016)May27,2020Contents 1Introduction 1 2PolynomialBaselines 1 2.1Syntax¶meters .......................................2 2.2GeneralUse .............................................2 2.3Fittingpolynomialbaselinesusingleastsquares .......................2 2.4Themechanismofautomatically ttingthebaselineinspc. t.poly.below ........3 2.5Specifyingthespectralrange ...................................4 2.6Fittingpolynomialsofdi erentorders .............................5 2.7Thenoiselevel ...........................................5 3RubberbandMethod 1IntroductionThisdocumentdiscussesbaselinecorrectionmethodsthatcanbeusedwithhyperSpec.hyperSpecprovidestwo ttingfunctionsforpolynomialbaselines,spc.fit.polyandspc.fit.poly.below.Anotherpossibilityisspc.rubberband,a\rubberband"methodthatdeterminessupportpointsby ndingtheconvexhullofeachspectrum.Thebaselinesarethenpiecewiselinearor(smoothing)splinesthroughthesupportpoints.Pleasenotethataspecializedpackageforbaseline tting,baseline[ 1 ],existsthatprovidesmanymoremethodsto tbaselinestospectroscopicdata.UsingbaselinewithhyperSpecobjectsisdemonstratedinvignette("hyperspec").2PolynomialBaselinesIncontrasttomanyotherprogramsthatprovidebaselinecorrectionmethods,hyperSpec'spolynomialbaselinefunctionsdoleastsquares ts.However,thebaselinescanbeforcedthroughparticular points,ifthisbehaviourisneeded.Themaindi erencebetweenthetwofunctionsisthatspc.fit.polyreturnsaleastsquares tthroughthecompletespectrumthatisgivenin t.towhereasspc.fit.poly.belowtriesto ndappropriatespectralregionsto tthebaselineto.2.1Syntax¶metersspc.fit.poly(fit.to,apply.to=fit.to,poly.order=1spc.fit.poly.below(fit.to,apply.to=fit.to,poly.order=1,npts.min=NULL,noise=0)fit.to:hyperSpecobjectwiththespectrawhosebaselinesaretobe tted.apply.to:hyperSpecobjectgivingthespectralrange,onwhichthebaselinesshouldbeeval-uated.IfapplyisNULL,ahyperSpecobjectwiththepolynomialcoecientsisreturnedinsteadofevaluatedbaselines.poly.order:polynomialorderofthebaselinesnpts.min:minimalnumberofdatapointsperspectrumtobeusedforthe t.npts.mindefaultstothelargerof3times(poly.order+1)or thofthenumberofdatapointsperspectrum.Ifnpts.minpoly.order,awarningisissuedandnpts.min-poly.order+1isused.noise:avectorgivingtheamountofnoise,seebelow.2.2GeneralUseBothfunctions tthepolynomialtothespectralrangegiveninhyperSpecobject t.to.Ifapply.toisnotNULL,thepolynomialsareevaluatedonthespectralrangeofapply.to.Otherwise,thepolynomialcoecientsarereturned.Subtractingthebaselineisuptotheuser,itiseasilydoneashyperSpecprovidesthe(minus)operator.2.3FittingpolynomialbaselinesusingleastsquaresCommonly,baselinesare tusing(single)supportpointsthatarespeci edbytheuser.Also,usuallysupportpointisusedforapolynomialoforder.Thisapproachisappropriateforspectrawithhighsignaltonoiseratio.Suchabaselinecanbeobtainedbyrestrictingthespectrain t.tototherespectivepoints(see gure 1 ):�bl-spc.fit.poly(chondro[c(1,101),,c(633,1788)],chondro[c(1,101)])�.99;冘plot(chondro[c(1,101)],plot.args=list(ylim=c(200,600)),col=1:2)�.99;冘plot(chondro[c(1,101),,c(633,1788)],add=TRUE,col=1:2,+lines.args=list(type="p",pch=20))�.99;冘plot(bl,add=TRUE,col=1:2) 600700800900100011001200130014001500160017001800 200300400500600Dn cm-1I / a.u. Figure1Fittingalinearbaselinethroughtwopoints.Ifthesignaltonoiseratioisnotideal,wavelengthsthatwork neforonespectrum(black)maynotbeappropriateforanother(red).However,ifthesignaltonoiseratioisnotideal,apolynomialwithsupportingpoints(i.e.withzerodegreesoffreedom)issubjecttoaconsiderableamountofnoise.Ifontheotherhand,moredatapointsconsistingofbaselineonlyareavailable,theuncertaintyonthepolynomialcanbereducedbyaleastsquares t.Bothspc.fit.polyandspc.fit.poly.belowthereforeprovideleastsquares ttingforthepoly-nomial.spc.fit.poly tstothewholespectralregionof t.to.Thus,forbaseline ttingthespectraneedtobecuttoappropriatewavelengthrangesthatdonotcontainanysignal.Inordertospeedupcalculations,theleastsquares tisdonebyusingtheVandermondematrixandsolvingtheequationsystembyqr.solve.This tisnotweighted.Aspectralregionwithmanydatapointsthereforehasgreaterin\ruenceontheresultingbaselinethanaregionwithjustafewdatapoints.Itisuptotheusertodecidewhetherthisshouldbecorrectedforbyselectingappropriatenumbersofdatapoints(e.g.byusingreplicatesoftheshorterspectralregion).2.4Themechanismofautomatically ttingthebaselineinspc.fit.poly.belowspc.fit.poly.belowtriestoautomatically ndappropriatespectralregionsforbaseline tting.Thisisdonebyexcludingspectralregionsthatcontainsignalsfromthebaseline tting.Theideaisthatalldatapointsthatlieabovea ttedpolynomial(initiallythroughthewholespectrum,thenthroughtheremainingpartsofthespectrum)willbetreatedassignalandthusbeexcludedfromthebaseline tting.Thesupportingpointsforthebaselinepolynomialsarecalculatediteratively:1.Apolynomialoftherequestedorderis ttotheconsideredspectralrange,initiallytothewholespectrumgivenin t.to 600700800900100011001200130014001500160017001800 50010001500Dn cm-1I / a.u.spectrum 1 Figure2Iterative ttingofthebaseline.Thedotsgivethesupportingpointsforthenextiteration'sbaseline,color:iterations. 17001705171017151720172517301735174017451750 200300400500Dn cm-1I / a.u. Figure3In\ruenceoffit.toonthebaselinepolynomial.Theblackbaselineis ttothespectralrange17001750cm,theblueto17201750cmonly.2.Onlythepartsofthespectrumthatliebelowthispolynomialplusthenoiseareretainedassupportingpointsforthenextiteration.Thesetwostepsarerepeateduntileithernofurtherpointsareexcluded,orthenextpolynomialwouldhavelessthannpts.minsupportingpoints.Thebaselinesandrespectivesupportingpointsforeachiterationofspc.fit.poly.below(chondro[1],poly.order=1)areshownin gure 2 .2.5SpecifyingthespectralrangeItispossibletoexcludespectralregionsthatdonotcontributetothebaselinefromthe tting,whilethebaselineisusedforthewholespectrum.Thisselectionofappropriatespectralregionsisessentialforspc.fit.poly.Butalsospc.fit.poly.belowcanbene tfromnarrowerspectralranges:the ttinggainsspeed.Thedefaultvaluefornpts.mindependsonthenumberofdatapointsperspectrum.Thusoneshouldalsoconsiderrequiringmoresupportpointsthanthedefaultvaluesuggests. 60080010001200140016001800 50010001500Dn cm-1I / a.u.spectrum 1 60080010001200140016001800 50010001500Dn cm-1I / a.u.spectrum 1 60080010001200140016001800 50010001500Dn cm-1I / a.u.spectrum 1 Figure4Baselinepolynomial ttothe rstspectrumofthechondrodatasetoforder0{2(lefttoright).Thedotsindicatethepointsusedforthe ttingofthepolynomial.�system.time(spc.fit.poly.below(chondro,NULL,npts.min=20))usersystemelapsed0.3550.0230.378�system.time(spc.fit.poly.below(chondro[,,c(min~700,1700~max)],NULL,npts.min=20))usersystemelapsed0.1910.0000.190Thechoiceofthespectralrangeinfit.toin\ruencestheresultingbaselinestoacertainextent,asbecomesclearfrom gure 3 .2.6Fittingpolynomialsofdi erentordersFigure 4 showstheresultingbaselinepolynomialofspc.fit.poly.below(chondro[1],poly.order=order)withorder0to3forthe rstspectrumofthechondrodataset.2.7ThenoiselevelBesidesde ningaminimalnumberofsupportingpoints,a\noiselevel"maybegiven.Consideraspectralrangeconsistingonlyofnoise.Theupperpartof gure 5 illustratestheproblem.Asthebaseline ttingalgorithmcannotdistinguishbetweennoiseandrealbandsappearingabovethe ttedpolynomial,theresultingbaseline(black)istoolowifthenoiseparameterisnotgiven.Settingthenoiselevelto4(2standarddeviations),the ttingconvergesimmediatelywithamuchbetterresult.Theresultingbaselinesforspc.fit.poly.below(chondro[1],poly.order=1,noise=12)ofthewholespectrumareshowninthemiddleandlowerpartof gure 5 noisemaybeasinglevalueforallspectra,oravectorwiththenoiselevelforeachofthespectraseparately.3RubberbandMethodParticularlyRamanspectraoftenshowincreasingbackgroundtowards.Inthiscase,polyno-mialbaselinesofteneitherneedhighorderorresidualbackgroundisleftinthespectra.Inthatcase,smoothingsplines ttedthroughthesupportingpointsareagoodalternative.Fortheparacetamolspectrum( g. 6 ),anoiselevelof300countsandtheequivalentof20degreesoffreedomworkwell. 0102030405060708090100 9698100102104 600700800900100011001200130014001500160017001800 50010001500Dn cm-1I / a.u.spectrum 1 Figure5Iterative ttingofthebaselinewithnoiselevel.Upperpart:e ectsofthenoiseparameteronthebaselineofaspectrumconsistingonlyofnoiseando set:withoutgivingnoisetheresultingbaseline(black)isclearlytoolow.Anoiselevelof4resultsinthebluebaseline.Thelowerpartshowthebaseline ttingwithnoiselevelonthecompletespectrum.Colour:iterations,dots:supportingpointsfortherespectivelynextbaseline.Dashed:baselineplusnoise.Allpointsabovethislineareexcludedfromthenextiteration.�bl-spc.rubberband(paracetamol[,,175~1800],noise=300,df=20)However,thereispossiblysomebackgroundleftbetween1200and1750cm-1wheretheoriginalspectrumisslightlyconcave.Thiscanbecorrectedbybendingthespectrumbeforeapplyingtherubberbandcorrection( g. 7 ):�bend-5e4*wl.eval(paracetamol[,,175~1800],function(x)x^2,normalize.wl=normalize01)�.99;冘bl-spc.rubberband(paracetamol[,,175~1800]+bend)-bendReferences[1]KristianHovdeLiland,TrygveAlm y,andBj rn-HelgeMevik.Optimalchoiceofbaselinecor-rectionformultivariatecalibrationofspectra.AppliedSpectroscopy,64:1007{1016,2010.SessionInfoRversion3.6.3(2020-02-29)Platform:x86_64-pc-linux-gnu(64-bit)Runningunder:Ubuntu18.04.4LTSMatrixproducts:defaultBLAS:/usr/lib/x86_64-linux-gnu/openblas/libblas.so.3LAPACK:/usr/lib/x86_64-linux-gnu/libopenblasp-r0.2.20.so 20040060080010001200140016001800 1000020000300004000050000Dn cm-1I / a.u. (a)paracetamolwiththerubberband ttedbaseline. 20040060080010001200140016001800 010000200003000040000Dn cm-1I / a.u. (b)Correctedspectrum.Figure6Rubberbandbaselinesfortheparacetamolspectrum. 20040060080010001200140016001800 10000300005000070000Dn cm-1I / a.u. (a)Bentparacetamolspectrumandrubberbandbaseline. 20040060080010001200140016001800 010000200003000040000Dn cm-1I / a.u. (b)Correctedspectrum.Figure7Rubberbandbaselinesfortheparacetamolspectrumafterbending.locale:[1]LC_CTYPE=de_DE.UTF-8LC_NUMERIC=CLC_TIME=de_DE.UTF-8[4]LC_COLLATE=CLC_MONETARY=de_DE.UTF-8LC_MESSAGES=de_DE.UTF-8[7]LC_PAPER=de_DE.UTF-8LC_NAME=CLC_ADDRESS=C[10]LC_TELEPHONE=CLC_MEASUREMENT=de_DE.UTF-8LC_IDENTIFICATION=Cattachedbasepackages:[1]gridstatsgraphicsgrDevicesutilsdatasetsmethodsbaseotherattachedpackages:[1]hyperSpec_0.99-20200527xml2_1.3.2ggplot2_3.3.0lattice_0.20-41loadedviaanamespace(andnotattached):[1]Rcpp_1.0.4.6magrittr_1.5tidyselect_1.0.0munsell_0.5.0[5]colorspace_1.4-1R.cache_0.14.0R6_2.4.1jpeg_0.1-8.1[9]rlang_0.4.6dplyr_0.8.5tools_3.6.3gtable_0.3.0[13]png_0.1-7R.oo_1.23.0latticeExtra_0.6-29withr_2.2.0[17]ellipsis_0.3.0lazyeval_0.2.2digest_0.6.25assertthat_0.2.1[21]tibble_3.0.1lifecycle_0.2.0crayon_1.3.4R.rsp_0.43.2[25]RColorBrewer_1.1-2purrr_0.3.4vctrs_0.2.4R.utils_2.9.2[29]testthat_2.3.2glue_1.4.0compiler_3.6.3pillar_1.4.4[33]scales_1.1.0R.methodsS3_1.8.0pkgconfig_2.0.3