Smoothing-and-differentiation-of-data-by-simplifie

Interferences.Methylenechlorideorchloroform-solublecarboxylicacids(acetylsalicylicacid,salicylicacid,etc.)whichinhibited4-,UPresponsewerereadilyextractedfromthesephasesusing0.ln’sodiumhydroxide.Span-typeexcipients,whichgaveaslightresponsetothe4-AAPreagent,wereeliminatedbyvirtueoftheirinsol-ubilityin10%aqueoussodiumchloride;whileinterferingTween-typeexcipientswereprecipitated(3)usingtheTweenreagent.Propyleneglycol,whenformu-latedatalevelof5.25%,contributedca.2to370totheobservedabsorbance.Thiswasminimizedbyamorefavorabledistributionoftheinterferencewithintheaqueousphasesusedinthepro-cedure.Placeboanalysesindicatedthatnointerferencewasobtainedwithsuchcommonexcipientsasstearicacid,stearylalcohol,cetylalcohol,petrola-tum,methylorpropyl-p-hydroxy-benzoates,sesameoil,orthimerosal.Lipotropicagents(betaineorcholine),allthecommonvitamins,andneomycinsulfatefailedtointerfere.ScopeofReaction.Inadditiontotheothertypesofsteroidsreportedtoreactwiththe4-AAPreagent,thefollowingsteroidsreactedatelevatedtemperature(boilingpointofmethanol)andatincreasedconcentration(2.0mg.per10.0ml.ofreagent):2a-hydroxy-methyl-17p-hydroxy-17a-methyl-5a-androst-%one(306mp);and2a-hydroxymethyl-17p-hydroxy-501-androst-3-one(306mp).Steroidswith-outaketogroupfailedtogiveanyresponse.Thequantitativeaspectsoftheaboveresponsestothe4-AAPreagentwerenotinvestigated.CorrelationsofChromophoreWave-lengthwithStructure.TheadditiveeffectofvariousringAandBsub-stituentsonthechromophoreoftheparentsaturated-3-ketosteroidispresentedinTable11.LITERATURECITED(1)Cohen,H.,Bates,R.W.,J.Am.Pharm.Assoc.,Sci.Ed.40,35(1951).(2)Hiittenrauch,R.,Z.Physiol.Chem.326,166(1961).(3)Pitter,P.,Chem.&Ind.42,1832(1962).RECEIVEDforreviewSeptember4,1963.’AcceptedMarch13,1964.SmoothingandDifferentiationofDatabySimplifiedLeastSquaresPro,ceduresABRAHAMSAVITZKYandMARCELJ.E.GOLAYThePerkin-ElmerCorp.,Norwalk,Conn.bInattemptingtoanalyze,ondigitaIcomputers,datafrombasicaIIycontinuousphysicalexperiments,numericalmethodsofperformingfa-miliaroperationsmustbedeveloped.Theoperationsofdifferentiationandfilteringareespeciallyimportantbothasanendinthemselves,andasapre-ludetofurthertreatmentofthedata.Numericalcounterpartsofanalogde-vicesthatperformtheseoperations,suchasRCfilters,areoftenconsidered.However,themethodofleastsquaresmaybeusedwithoutadditionalcom-putationalcomplexityandwithcon-siderableimprovementintheinformo-tionobtained.Theleastsquarescal-culationsmaybecarriedoutinthecomputerbyconvolutionofthedatapointswithproperlychosensetsofintegers.Thesesetsofintegersandtheirnormalizingfactorsaredescribedandtheiruseisillustratedinspectro-scopicapplications.Thecomputerprogramsrequiredarerelativelysim-ple.TwoexamplesarepresentedassubroutinesintheFORTRANlanguage.HEPRIMARYOUTPUTofanyexperi-Tmentinwhichquantitativeinformationistobeextractedisinfor-mationwhichmeasuresthephenomenonunderobservation.Superimposeduponandindistinguishablefromthisinforma-tionarerandomerrorswhich,regardlessoftheirsource,arecharacteristicallydescribedasnoise.Offundamentalimportancetotheesperimenteristheremovalofasmuchofthisnoiseaspossiblewithout,atthesametime,undulydegradingtheunderlyingin-formation.Inmuchexperimentalwork,theinfor-mationmaybeobtainedintheformofatwo-columntableofnumbers,Aus.B.Suchatableistypicallytheresultofdigitizingaspectrumordigitizingotherkindsofresultsobtainedduringthecourseofanexperiment.Ifplotted,thistableofnumberswouldgivethefamiliargraphsofTOTus.wavelength,pHus.volumeoftitrant,polarographiccurrentus.appliedvoltage,SMRorESRspectrum,orchromatographicelutioncurve,etc.Thispaperiscon-cernedwithcomputationalmethodsfortheremovaloftherandomnoisefromsuchinformation,andwiththesimpleevaluationofthefirstfewderivativesoftheinformationwithrespecttothegraphabscissa.Thebasesforthemethodstobedis-cussedhavebeenreportedpreviously,mostlyinthemathematicalliterature(4,6,8,9).Theobjectivehereistopresentspecificmethodsforhandlingcurrentproblemsintheprocessingofsuchtablesofanalyticaldata.Themethodsapplyaswelltothedeskcalculator,ortosimplepaperandpenciloperationsforsmallamountsofdata,astheydotothedigitalcomputerforlargeamountsofdata,sincetheirmajorutilityistosimplifyandspeeduptheprocessingofdata.Therearetwoimportantrestrictionsonthewayinwhichthepointsinthetablemaybeobtained.First,thepointsmustbeatafixed,uniformintervalinthechosenabscissa.Iftheindependentvariableistime,asinchromatographyorNMRspectrawithlineartimesweep,eachdatapointmustbeobtainedatthesametimeintervalfromeachprecedingpoint.Ifitisaspectrum,theintervalsmaybeeverydrumdivisionorevery0.1wavenumber,etc.Second,thecurvesformedbygraphingthepointsmustbecontinuousandmoreorlesssmooth-asinthevariousexampleslistedabove.ALTERNATIVEMETHODSOneofthesimplestwaystosmoothfluctuatingdataisbyamovingaverage.Inthisprocedureonetakesafisednumberofpoints,addstheirordinatestogether,anddividesbythenumberofpointstoobtaintheaverageordinateatthecenterabscissaofthegroup.Sest,thepointatoneendofthegroupisdropped,thenextpointattheotherendadded,andtheprocessisrepeated.Figure1illustrateshowthemovingaveragemightbeo