Chemometrics. Derivatives in Spectroscopy. Part I The Behavior of the Derivative. Howard Mark and Jerome Workman Jr.

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Chmomtrics Drivativs in Spctroscopy Part I Th Bhavior of th Drivativ Howar Mark an Jrom Workman Jr. Jrom Workman Jr. srvs on th Eitorial Avisory Boar of Spctroscopy an is vic-prsint of rsarch for Argos (Waltham, MA).

First rivativs an scon rivativs ( T/ or A/ ) ar both commonly us in morn spctroscopy, particularly in nar-infrar (NIR) spctroscopy.

Th mathmatics an bhavior of th rivativ is inpnnt of th particular spctroscopic tchniqu to which it is appli, howvr.

Stuis of th application of rivativs to spctroscopy go back at last as far as 953 ( 3).

NIR spctroscopy as wll as custom harwar an softwar sign an vlopmnt. H can b rach via -mail at hlmark@proigy.nt. contains a goo bibliography of th work bfor its apparanc (4).

1 Chmomtrics Drivativs in Spctroscopy Part I Th Bhavior of th Drivativ Howar Mark an Jrom Workman Jr. Jrom Workman Jr. srvs on th Eitorial Avisory Boar of Spctroscopy an is vic-prsint of rsarch for Argos (Waltham, MA). H may b rach by -mail at jworkman@argos.com. Drivativs of spctra (T/ or A/, an thir wavnumbr quivalnts in Fourir transform infrar spctroscopy) hav bn known an us in spctroscopy for a long tim. First rivativs an scon rivativs ( T/ or A/ ) ar both commonly us in morn spctroscopy, particularly in nar-infrar (NIR) spctroscopy. Thy also njoy wispra us in som nonoptical spctroscopic tchniqus, such as nuclar magntic rsonanc an lctron spin rsonanc spctroscopis. Th mathmatics an bhavior of th rivativ is inpnnt of th particular spctroscopic tchniqu to which it is appli, howvr. But bcaus our own backgrouns ar in optical spctroscopy, w will iscuss it, whr prtinnt, in trms of th spctroscopy w know bst. Stuis of th application of rivativs to spctroscopy go back at last as far as 953 ( 3). A mor rcnt papr Howar Mark srvs on th Eitorial Avisory Boar of Spctroscopy an runs a consulting srvic, Mark Elctronics, Trrac Avnu, Suffrn, NY 090, that provis assistanc, training, an consultation in NIR spctroscopy as wll as custom harwar an softwar sign an vlopmnt. H can b rach via -mail at hlmark@proigy.nt. contains a goo bibliography of th work bfor its apparanc (4). Sinc th avnt of NIR spctroscopy as a popular analytical tchniqu, th routin us of rivativ spctra has burgon, along with th application to this mtho of spctroscopic analysis. Along with th incras applicability, intrst has grown in th backgroun an bhavior of rivativs. Dav Hopkins spcially has l th way in unrstaning th bhavior of first an scon rivativs, particularly thir computation using Savitzky Golay convolution functions (5, 6). W o not plan to al with that aspct too xtnsivly at this tim, howvr. Th application of rivativs is not without problms, howvr, spcially whn th concrn is to accuratly rprsnt th rivativ of a givn ata spctrum. Thrfor, unrstaning th natur of th problms ncountr so that th propr cisions can b ma rgaring how th rivativ shoul b calculat is crucial to obtaining optimum rsults. Figur illustrats som of th problms of rivativs. This figur picts som of th basic bhaviors unrlying th us of th rivativs for spctroscopic analysis. Th top curv in Figur rprsnts a synthtic spctrum with two Gaussian (normal) bans, on of 0-nm banwith an on of 60-nm banwith. Spctroscopic ban shaps ar convntionally consir to b ithr Gaussian or Lorntzian; in this column w will concntrat on Gaussian ban shaps; thrfor all our figurs ar bas on Gaussian-shap bans. W will, howvr, trat Lorntzian bans at appropriat points. In Figur w prsnt normal bans with spacing btwn wavlngth points of nm, a numbr that will bcom important latr on. Th mil curv rprsnts th first rivativ an th bottom curv th scon rivativ of th absorbanc ban. W ar putting th trm rivativ in quots, bcaus thy ar, in fact, not tru rivativs. Th finition of a rivativ inclus th stp of taking a limit as iffrncs approach zro. In th ral worl, with ral ata, w can nvr calculat a tru rivativ bcaus w must comput th iffrncs btwn finit ata points, an ths must b takn ovr finit intrvals, so that comput rivativs ar approximations of th actual rivativ. 3 Spctroscopy 8(4) April 003

2 Chmomtrics Th absorbanc spctrum in Figur is ma from synthtic ata, but mimics th bhavior of ral ata in that both ar rprsnt by ata points collct at iscrt an (usually) uniform intrvals. Thrfor th calculation of a rivativ from actual ata is rally th computation of finit iffrncs, usually btwn ajacnt ata points. W will now rmov th quotation marks from aroun th trm, an simply call all th finit-iffrnc approximations a rivativ. As w shall s, howvr, oftn ata points that ar mor wily spra ar us. If th ata points ar sufficintly clos togthr, thn th approximation to th tru rivativ can b quit goo. Nvrthlss, a tru rivativ can nvr b masur whn ral ata is involv. Figur, howvr, still shows a numbr of charactristics that rval th bhavior of rivativs. First of all, w not that th first rivativ crosss th x-axis at th wavlngth whr th absorbanc pak has a maximum, an has maximum valus (both positiv an ngativ) at th point of maximum slop of th absorbanc bans. Ths charactristics, of cours, rflct th finition of th rivativ as a masur of th slop of th unrlying curv. For Gaussian bans, th maxima of th first rivativs also corrspon to th stanar viation of th unrlying spctral curv. Th scon rivativ, in contrast, has its maximum valu at th sam wavlngth as th unrlying pak, although in th ngativ irction. Th scon rivativ crosss th x- axis at th point of maximum slop of th unrlying (first rivativ) curv, an bcaus of that, prsnts a much sharpr ban than th unrlying absorbanc ban os. Th problm ariss, howvr, that this sharpning ffct is accompani by th cration of two artifact paks, th two positiv-going paks that flank th ngativ-going portion of th scon rivativ. In complicat spctra, thrfor, it can somtims b ifficult to istinguish tru spctral faturs from th artifacts crat by th scon rivativ calculation. Finally, w not that th magnitus of both th first an scon rivativs of th narrow absorbanc ban ar consirably gratr than corrsponing magnitus for th wir absorbanc ban. This charactristic is a rflction of th fact that th slop of th narrowr ban truly is gratr than that of th broar ban of th sam hight, as can b sn in th xpan viws of th two absorbanc bans in Figur. For th sam X, th narrow absorbanc ban has a much largr valu of Y than th broa absorbanc ban os, thrfor Y/X (th rivativ) is largr for that ban. A similar situation is tru for th scon rivativ as wll. Thr is a furthr consiration as wll; th mathmatical finition of a normal curv inclus a prmultiplying factor of /(() / ), which maks th ara unr th normal curv qual to unity. Thrfor, th wir th banwith, th smallr th maximum valu of th curv will b, furthr rucing th slop as compar to a narrowr ban. It is intrsting an usful to consir this quantitativly. Th xprssion for th normal istribution is (7).5.0 [a] Th corrsponing xprssion for th Lorntzian istribution is (8, s p. ) [b] whr is th masur of banwith (an quals th stanar viation for th normal curv) an is th wavlngth corrsponing to th pak cntr. W not parnthtically hr that quation a inclus th prmultiplying factor for constant ara. Th xprssion for a normal curv of constant maximum hight (of unity) will b simply Th first rivativ of th normal istribution, from th xprssion in quation a, thn, is x y Wavlngth Y () / Y (X) Y X X y x Figur. Two Gaussian absorbanc bans an thir rspctiv first an scon rivativs (finit iffrncs). Th top spctrum rprsnts a synthtic Gaussian absorbanc spctrum, th mil a first rivativ an th bottom a scon rivativ. Not that th orinat of th first rivativ has bn xpan by a factor of 0 an th scon rivativ by anothr factor of 0. Th wavlngth spacing btwn ata points is nm. Th narrow ban has a banwith (full with at half hight) of 0 nm, th broa on is 60 nm. X X () / [] [3] April 003 8(4) Spctroscopy 33

3 Chmomtrics () / X (X) [4] Y (X) X [9b] () / Equation 6a is riv from th constant-ara xprssion for th normal curv; from th constant-hight xprssion w obtain [5] [6a] [6b] Th origin of th faturs sn qualitativly in Figur can b obsrv in ithr of quations 6a or 6b. Whn X, thn th rivativ is zro, an th sign of th rivativ changs from positiv whn X < to ngativ whn X >. Th prsnc of th ngativ xponntial trm nsurs that th rivativ will asymptotically approach zro as X approachs infinity in both irctions. Similarly, from quation 6a w can riv th xprssion for th scon rivativ of th normal istribution Y Y (X) () / X (X) () / X (X) X () / ( X ) X Y (X) 3 () / () / (X) () / X X (X) () / X X (X) X [7] [8] [9a] For th Lorntzian istribution, from quation b th first rivativ is [0] [] An thn th scon rivativ of th Lorntzian istribution is Y (8(X)) (( 4(X) ) ) ( 4(X) ) 4 Y (X) 8(X) ( 4(X) ) ( (8(X)) 4(X) ) ( 4(X) ) 4 8( 4(X) ) ( 4(X) ) 4 8(X)( 4() ) ( 4(X) ) 4 Y 8( 4(X) ) ( 4(X) ) 4 6( ( 4(X) ) (X)4(X) 3 ) ( 4(X) ) 4 Y 8( 4(X) ) ( 4(X) ) 4 6( (X)4(X) 3 )(8(X)) ( 4(X) ) 4 (X) ( 4(X) ) [] [3] [4] [5] An from quation 6b w obtain 34 Spctroscopy 8(4) April 003

4 Chmomtrics Y 8( 4 8(X) 64(X) 4 ) ( 4(X) ) 4 [6] Going back to quations 6 an, how o th magnitus of th rivativs chang with? Sinc th maximum first rivativ occurs whn X, lt us substitut for X in quation 6a an for th normal istribution w gt an in quation for th Lorntzian istribution [7] [8] For th normal istribution, th xponntial trm has bcom a constant, an w s that th maximum magnitu of th rivativ is invrsly proportional to (rgarlss of whthr w start with th constant-ara xprssion or th constant-hight xprssion). This confirms our obsrvation from Figur. For th Lorntzian istribution, w s that th rivativ crass with th fourth powr of th banwith. Similarly, th maximum scon rivativ occurs whn X, so insrting this quality into quation 9a for th normal istribution givs us An substituting X 0 into quation 6 givs us th corrsponing valu for th Lorntzian istribution Th ngativ sign in quations 9 an 0 tlls us that th maximum scon rivativ is a ngativ valu, which agrs with Figur, an it also tlls us that th magnitu of th scon rivativ crass invrsly as th squar of (for th normal ban shap) an invrsly as th fifth powr of (for th Lorntzian ban shap), that is, as th banwith of th absorbanc ban, incrass. This xplains why th riva Y 8(0) 8(4 64(0) 4 )8( (0) 4(0) ) 4 ( 4(0) ) 4 8( (X) 4(X) 4 ) MAX 8() 6 ( 4() ) 5 5 Y MAX 0 = () / () / (8 4 ) 8 ( 4(X) ) 4 () / () / [9] [0] tivs of th broa absorbanc ban cras with rspct to th narrow absorbanc ban as w s in Figur, an mor so as th rivativ orr incrass. Th Bhavior of Comput Drivativs Now, quations 6 an 9 ar mathmatically xact. But as w obsrv whn iscussing Figur, a rprsntation of a rivativ bas on finit iffrncs is only an approximation. How goo is this approximation, an how quickly os it gt ba? That pns somwhat on how th rivativ is calculat. W ma a point of noting that th rivativ in Figur was calculat from synthtic ata, with abscissa (wavlngth) spacing of nm. This valu of spacing was chosn so that th two mthos of calculation woul fault to th sam rsult. W not abov that th finition of a rivativ inclus th opration of ivision by X (or by, in th mathmatically xact cas). Som computr programs that purport to calculat rivativs o not inclu th stp of prforming that ivision, whil othrs o. Th rsults will vary consirably in th two cass. W will bgin our iscussion by consiring th simplr cas, whr w o not ivi by X. This provis th numrator trm for th rivativ finition, an also for th approximation; this allows us to xamin th bhavior of that trm in isolation. In som cass, this is all that is us or n: it provis a qualitativ obsrvation of th ovrall shap of th spctrum that is of intrst, for xampl. Somtims it is on this way whn th ata ar us for quantitativ or qualitativ analysis, an th spctral ata from th unknown sampls, th sampls which ar to b analyz on a routin basis, ar trat th sam way as th calibration ata. In, sinc th numrator trm iffrs from th corrct rivativ approximation only by a scaling factor, it can b ifficult to tll just from looking at th rivativ curv whthr it is a corrctly calculat rivativ or not, spcially if th scal is not prsnt. On th othr han, computing only th numrator trm is not rcommn whn rsults ar to b compar btwn iffrnt instrumnts or laboratoris. It is also not rcommn whn prforming thortical stuis is of intrst, or whn th rsults of xprimnts ar to b compar to thortical xpctations, sinc it os not, in gnral, rflct th actual valu of th tru rivativ. Givn th minor computational burn, howvr, th propr computation of incluing th ivision shoul always b on. Hr w start with th xamination of th numrator trm alon for its pagogical valu. Th qustion ariss: Bcaus th finition of th rivativ spcifis taking a limit as iffrncs approach zro, wouln t th bst rsults b obtain from using th smallst possibl iffrncs? Th answr is Ys, but.... Th but rflcts th fact that whil synthtic ata ar nois-fr, ral ata contain nois. In this column w consir only th nois-fr synthtic ata w crat, but it is clar that with ral ata, containing ral an irrucibl nois, computing smallr an smallr iffrncs will vntually bring us to th point whr th iffrncs qual an thn bcom lss than April 003 8(4) Spctroscopy 35

5 Chmomtrics (a) First iffrnc (b) First iffrnc (c) First iffrnc Spacing 5 nm Spacing nm 3 45 Spacing 40 nm 4 Spacing 5 nm 67 6 Wavlngth Figur. First iffrncs calculat using iffrnt spacings btwn th ata points us to calculat th finit iffrnc for th numrator trm only, as an approximation to th rivativ. Th unrlying curv is th 0-nm banwith absorbanc ban in Figur, with ata points vry nm. (a) iffrnc spacings 5 nm, (b) spacings 5 40 nm, (c) spacings nm Wavlngth Spacing 40 nm Wavlngth Spacing 90 nm Absorbanc Wavlngth Figur 3. Drivativ comput ovr a vry larg spacing. With a larg spacing, on point us for th iffrnc is on th baslin an th othr tracs ovr th shap of th curv. th nois lvl. Drivativ calculations ar in known to b fraught with nois problms. In th intrst of xamining th bhavior of th rivativ, howvr, w ar going to ignor th ffct of th nois in this column, although w will vntually rturn to that qustion. Th way to minimiz nois ffcts is to xaggrat th iffrncs, by computing finit iffrncs at largr an largr wavlngth intrvals, an this is oftn on in practic. In Figur w prsnt th rsults of computing finit iffrnc approximations to a rivativ (for th normal cas), using iffrnt spacings (that is, th wavlngth iffrnc btwn th ata points w comput th finit iffrnc btwn; w will somtims call this X an frly intrmix th two trms). For th rivativs in Figur, th unrlying absorbanc curv is th narrowr on from Figur, having a 0-nm banwith. W s from Figur that, in contrast to th mathmatically ial bhavior of a tru rivativ, th bhavior of a finit iffrnc pns on how it is calculat. As Figur a shows, at small spacings, th shap of th comput iffrnc curv closly mimics th tru rivativ, an has a magnitu that is proportional to th spacing. Figur b shows that as th spacing incrass, svral changs occur: Th rlationship btwn th iffrnc spacing an th magnitu of th rivativ parts from th gr of proportionality w obsrv at smallr spacings. As th spacing incrass, th maximum valu of th comput iffrnc asymptotically approachs th valu of unity. Thr is a shift in th wavlngth corrsponing to th maximum valu of th rivativ. Clos xamination of Figur b will rval a cras in th slop of th iffrnc curv at th point it crosss th x-axis, vn though w ar not using th nominator trm of th rivativ calculation. Figur c shows that at sufficintly larg spacing valus, th concpt of this bing a rivativ braks own ntirly. Th rivativ curv has sparat into two faturs, ach of thm apparing to b a normal curv, although on of thm is ngativ. As th spacing continus to incras, th two faturs mov farthr an farthr apart. Figur 3 shows how this occurs. Whn th spacing is vry wi that is, wir than th brath of th absorbanc ban nar th baslin on of th points us to comput th iffrnc is always on th baslin, whil th othr point ris ovr th pak an tracs its shap. As th point of th rivativ slis along th x-axis, vntually th two points xchang rols, an th othr fatur is trac out, but with th opposit sign. Now w look at th scon rivativ similarly. Som of this has bn prsnt prviously in th litratur (9), although in lss tail than w o hr. Figurs 4a c prsnt scon rivativs calculat using th sam spacings as for th iffrncs in Figur. In Figur 4 w s that th scon 3 36 Spctroscopy 8(4) April 003

6 Chmomtrics (a) Scon iffrnc (b) Scon iffrnc (c) Scon iffrnc Spacing nm Spacing 5 nm Spacing 5 nm Spacing 40 nm 0 0 Wavlngth 0 0 Wavlngth Spacing 40 nm 0 Wavlngth Spacing 90 nm Figur 4. Scon iffrncs calculat using iffrnt spacings btwn th ata points us to calculat th finit iffrnc for th numrator trm only, as an approximation to th rivativ. Th unrlying curv is th 0-nm banwith absorbanc ban in Figur, with ata points vry nm. (a) iffrnc spacings 5 nm, (b) spacings 5 40 nm, (c) spacings nm. rivativ is subjct to som of th sam ffcts as th first rivativ: Linar (proportional) chang in amplitu at small spacings Nonlinar chang in amplitu at larg spacings. On th othr han, thr is no shift in th wavlngth of th cntral maximum, although Figurs 4b an 4c show that th artifact paks o chang thir wavlngth. Rplacing th shift in wavlngth, howvr, is a broaning of th cntral pak. W not abov that on charactristic of th scon rivativ is th narrowing of this pak compar with th unrlying absorbanc ban. As th spacing ovr which th rivativ is comput incrass, howvr, this rsolution nhancmnt ffct crass an vntually isappars. Th 0 Drivativ valu Spacing Scon rivativ First rivativ Figur 5. Maximum comput rivativ magnitu trmin by th spacing of th points us in th computation. Not that th sign of th scon rivativ has bn rvrs to simplify comparison with th first rivativ bhavior. rason is similar to that for th first rivativ, as shown in Figur 3: at vry larg spacings th points us to comput th rivativ vntually win up simply tracing ovr th unrlying absorbanc ban, with th rsult that, sinc scon rivativs ar ssntially comput from thr points, thr copis of th unrlying absorbanc ban ar prouc, albit with iffrnt signs. In Figur 5 w show th variation of th comput rivativs as trmin by th spacing us in th computation. Anothr fatur that can b sn in Figur 5, which is also obsrvabl in Figur 4 albit with som ifficulty, is that at small spacings th maximum rivativ valu is not simply proportional to th spacing but changs fastr than proportionatly to th spacing; th ovrall curv of calculat rivativ valu vrsus spacing is sigmoial. W continu in our nxt column by xamining th bhavior of th rivativ calculation whn th ivision of th Y trm is ivi by th X trm, to form an approximation to th tru rivativ. Rfrncs. F. Singlton an G.L. Collir, British patnt no. 760,79, appli for Dcmbr 6, F. Singlton an G.L. Collir, Chm. In. (Lonon), 59 (955). 3. A.T. Gis an C.S. Frnch, Appl. Spctrosc. 9, 78 (955). 4. M.J.D. Low an H. Mark, Appl. Spctrosc. 4, 9 30 (970). 5. D. Hopkins, NIR Nws (3), 3 5 (00). 6. D. Hopkins, Nar Infrar Analysis, 3 (00). 7. H. Mark an J. Workman, Spctroscopy (9), (987). 8. J.D. Ingl an S.R. Crouch, Spctrochmical Analysis (Prntic- Hall, Uppr Sal Rivr, NJ, 988). 9. G.E. Ritchi an H. Mark, NIR Nws 3(), 4 6 (00). April 003 8(4) Spctroscopy 37

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