Linear Multiple Regression Model of High Performance Liquid Chromatography

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1 Lnear Multple Regresson Model of Hgh Perforance Lqud Chroatography STANISLAVA LABÁTOVÁ Insttute of Inforatcs Departent of dscrete processes odelng and control Slovak Acadey of Scences Dúbravská 9, Bratslava SLOVAKIA utrrlabt@savba.sk, Abstract: - The ultple regresson analyss s proposed as ethod for presentaton results of hgh perforance lqud chroatography (HPLC), whch sultaneously depends on several paraeters. The coeffcents of lnear ultple regresson odel are assessed by the least square ethod. In ths way dependent varable retenton volue s expressed by ore than one explanatory varable (.e. by the olar ass, concentratons of eluent and teperature). The approach was to depct dependences of retenton volues on olar ass of polystyrene probes n tetrahydrofuran () and n xed eluents plus dethylforade eluents on the slca gel alkyl bonded colun packng. The dependences retenton volues on olar ass, concentratons of eluent and are shown. Key-Words: -Multple regresson analyss, odel specfcaton, hgh perforance lqud chroatography, calbraton curves Introducton Hgh perforance lqud chroatography (HPLC) belongs to the ost portant analytcal tools of odern chestry. HPLC separates coplex xtures of both low and hgh olecular speces to allow conclusons on saple coposton n ters of checal structure and archtecture of coponents and also n ters of ther olar ass averages and dstrbutons. HPLC s based on dfferences n retenton of separated substances wthn approprate statonary phase. The analyzed phase s ntroduced nto devce (usually a colun) flled wth statonary phase (eluent). The heart of all HPLC systes s the colun packng. The volue of oble phase needed to eluate the partcular substance fro the colun s easured. Ths s the retenton or eluton volue (V R ) and reflects the extent of retenton of analyze substance wthn the colun packng. Concentraton of analytes leavng the HPLC colun s ontored by eans of detectors, whch easure lght absorpton of the colun effluent. The dependence of the detector response on eluent volue s shown on a chroatogra. Many real polyers consst of dfferent checal acroolecular coponents. These are xed ntentonally to obtan the expected syste characterstcs or they arse durng the use of polyers (.e. lght degradaton). A de-forulaton ntentonally prepared coplex polyers xtures to specfy the role of ndvdual eleents of syste and to try to prove ther characterstcs n next. Analyss and olecular characterzaton of syste eleent obtaned durng constructon or degradaton poly-reactons allows better understandng and verfcaton. The a of degradaton reacton s exact knowledge of process echans that allows n the end to suggest the better systes. Drect analyss of checal structure of coplex polyers and especally of nor (<) coponents n ult-coponent polyer blends s ISSN: Issue 3, Volue 7, March

2 not possble by the classcal analytcal ethods. New unconventonal ethods of separaton of coplex ult-coponent polyer syste wth ephass on nor (<) coponents are needed. Selected ethods of hgh perforance lqud chroatography of polyers, (polyer HPLC) belong to the ost potental canddates. In polyer HPLC, the colun packngs are usually fored wth beds of porous partcles. Dependences of V R data on olecular characterstcs of polyer (s) under study are evaluated. The ost coon polyer HPLC ethod s sze excluson chroatography, SEC. SEC s based on the dependence of eluton rate of acroolecules on ther sze. Large acroolecules pereate only few pores of colun packng and ther progresson along the colun s fast. At the sae te, saller speces pereate any pores and ther eluton rate s low. As a result, acroolecules are separated accordng to ther sze. One speaks about excluson retenton echans. However, t s often necessary to separate acroolecules also accordng to ther checal coposton or physcal archtecture. The proble s that at least two olecular characterstcs sultaneously affect the retenton volue of a coplex polyer, naely sze (olar ass) and checal coposton or archtecture of acroolecules. It s necessary to dscrnate these two effects n order to ndependently deterne two or even three olecular characterstcs of a polyer saple. To do so, two or even several retenton echanss are cobned wthn one sngle colun or wthn a seres of dfferent coluns. The ost potental retenton echanss alternatve to sze excluson are based on dfferences n the adsorpton of acroolecules on surface of colun packng or n the enthalpc partton of acroolecules between the volues of statonary phase and oble phase. Correspondngly, one speaks about adsorpton and enthalpc partton retenton echans of separaton. The pore excluson s, however always present as an addtonal retenton echans. Consequently, one deals wth the coupled retenton echanss adsorpton plus excluson or enthalpc partton plus excluson. The a of couplng of retenton echanss n polyer HPLC s to ether enhance or to suppress separaton of acroolecules accordng to one olecular characterstc so that the effect of other characterstc(s) can be neglected. The results obtaned wth coupled ethods of polyer HPLC are coplex and t s dffcult to dscrnate effect of partcular retenton echanss on the resultng retenton volues. Experental results are atheatcally processed n order to obtan proved descrpton and generalzed elucdaton of retenton echans. The a s to descrbe a syste wth optal nuber of paraeters, whch to would allow generalzed conclusons and predctng of the behavor of polyer systes. Novel approaches to lqud chroatography separaton of synthetc polyers were elaborated, whch cobne two or several separaton echanss. These approaches are called coupled ethod of polyers hgh perforance lqud chroatography. The a s to ether enhance or suppress separaton of acroolecules accordng to one olecular characterstc so that the effect of other characterstc(s) can be neglected. Unconventonal lqud chroatography ethods of separaton of coplex ult-coponent polyer systes [3] are atheatcally processed n order to obtan odel for HPLC process and generalzed elucdaton of the retenton echanss. The reander of the paper s organzed as follows: Secton deals wth calbraton curves. Secton 3 explans odel specfcaton usng ult regresson analyss. Secton 4 contans exaples of lnear ult regresson odel and n Secton 5 are done conclusons. Optal Desgn of Calbraton Curves for HPLC by Polynoal Regresson One portant step n the data treatent of HPLC experent s odelng of calbraton curves (CC). The dependence retenton volue on olar ass s called calbraton curves. The CC s are generally not lnear. Fro atheatcal pont of vew for soluton above proble as a dependence retenton volue on olar ass (CC) we can wrte: Y = f(x ) + ε ; =,,...,n, () where x s value of ndependent varable of HPLC experent,.e. olar ass Y s easured response of HPLC experent retenton volue f s unknown functon ε s rando error n s a nuber of easureents ISSN: Issue 3, Volue 7, March

3 We would be wllng to assue that functon f s suffcently sooth over the range of nterest, so we represent the functon f adequately by a polynoal. If we assue that functon f(x ) =P (x ) = a + a x + a x a x, =,,...,n,, () s a polynoal of degree, where a are unknown, we wll call the setup odel P. Let Y, a, ε denote colun vectors, (transpose lne vectors Y = (Y, Y,...,Y n ), a = (a, a,...,a ), ε = (ε, ε,..., ε n )) and X denote a atrx of type (n)/ (+) gven by: x x X = Μ Μ xn x x x Μ n Λ Λ Λ Λ x x Μ xn (3) Usng soe level of portance t s possble to defne coprose optal crteron, whch detects nforaton about exactness of paraeters and share of odel, respectvely. Dfferent polynoal odels have been evaluated and copared, not only for odel ft but also for ther predctve propertes. log M Table Kolóna Kroasl C-, A, 5x7.8 Mw DMF , , , , , , , , then the odel P can be wrtten as : Y = Xa + ε, (4) where x s take at least + dstnct values and n +. Then unque least squares estator of a s gven by a * = (X X) - XY = (a *,a *,...,a * ). (5) 6 5 Kroasl C-, A, 5x7.8 thf df 5 thf thf 8 thf 7.5 thf 7 thf 5 thf Queston s, whch crteron can be consdered as coproses between the ncopatble goals of nference about regresson functon under an assued odel and checkng odel adequacy. For evaluaton estated regresson functon (n our case polynoal order) s possble to use several ways [4]:. Coeffcent of ultple deternaton (R ). Test based on analyss varance 3. The root ean square error 4. An analyss of resduals. Crteron that we used s applcaton root ean square error (RMSE) ethod gven by E= n = ( y y n ' ). (6) log M 4 3,5 3, 3,5 4, 4,5 5, Retenton volue [L] Fg. Dependences olar ass on retenton volue In HPLC, agntude of paraeter whch represents olar ass, s too large so fro practcal and hstorcal arguents n next there s consdered wth ther logarth. For creatng odel descrbng calbraton dependences n HPLC n frst we suppose that the eprcal forula has the for: ISSN: Issue 3, Volue 7, March

4 and P ( x) M = Ae, (7) log M = log A + P (x). (8) An eprcal functon we wanted to have nal (for establshed polynoal coeffcents by usng least square ethod) and have the for: RMSE: ε 5 =.84. Graf of estated polynoal functons ( nd, 3 rd, 5 th orders) for experental data fro Tab. are shown on Fg.3. Φ( a n = so [ a, a,..., a + a x,log ) = a x log M ] n (9) = Φ( a ( a, a,..., a + a x a,log M log M ) = ) =. () Solvng ths syste of lnear equatons we obtan polynoal coeffcents a, a,...,a n. Dependence olar ass on retenton volue at concentraton of eluent whch s equal, for experental data fro Tab., estated by polynoal of nd order, s gven by: Y =.x.33x+8.8 +ε. () The root ean square error: ε = See Fg. Dependence olar ass on retenton volue where the concentraton of eluent s equal of, for experental data fro Tab., estated by polynoal of 3 rd order s gven by: Y=.4x 3.3x -.7x ε 3. () RMS error: ε 3 =.56 See Fg. For the polynoal odel of 5-th order for experental data fro Tab. we obtan next coeffcents: Y= -.4x x 4 5.6x x x ε 5, (3) Fg. Calbraton curves for experental data fro Table., where... represents polynoal nd order, represents polynoal 3 rd order, represents polynoal 5 th order Evdently, for every value of eluent concentraton we obtan dfferent calbraton curve, whch s very ense and for presentaton results of HPLC experent non avalable. Syste for evaluaton of calbraton curves n lqud chroatography s constructed as Mult Agent Syste (MAS). Syste conssts fro decson agent and agent for odelng and processng dfferent cases of CC. In ths work the agent for processng CC s P 3 ost reflected, as ths agent s very sutable for our a. The shape of cubc parabola depends on value D = 3a a a. If D< then polynoal P 3 has one axu and one nu, whch characterstc s used for analyss for exaple for obtanng nforaton about tendences n behavor of acroolecules n HPLC. For presentaton results HPLC s portant to observe not only CC, that represents only dependences two varables (olar ass and retenton volue), but also nfluence of next varables that evdently to nfluence on process HPLC. For observng ultple dependences that have nfluence on process HPLC a ultple regresson s a sutable tool. ISSN: Issue 3, Volue 7, March

5 3 Model specfcatons usng ultple regresson Multple regresson (MR) analyss s a statstcal tool for understandng the relatonshp between two or ore varables n a syste. In other way, MR s a regresson gvng condtonal expectaton value of gven varable n ters of two or ore varables. Generally, MR nvolves a varable to be explaned called the dependent varable and addtonal explanatory varables that are thought to produce or be assocated wth changes n the dependent varable. The portant varables that systeatcally ght nfluence the dependent varable, and for whch data can be obtaned (easured), typcally should be ncluded n the odel. All reanng nfluences, whch should be sall ndvdually, but can be substantal n the aggregate, are ncluded n an addtonal rando error ter. MR s a procedure that separates the systeatc effects (assocated wth the error ter) fro the rando effects and also offers a ethod of assessng the success of the process. MR analyss s soetes well suted for analyss of data about copetng theores. There are several possble explanatons for the relatonshp aong a nuber of explanatory varables to assess the statstcal data pertnent to these theores. MR regresson also ay be useful n:. Deternng whether or not a partcular effect s present. Measurng the agntude of partcular effect 3. Forecastng what a partcular effect would be, but for an ntervenng event. Moreover, n nterpretng the results of MR analyss, t s portant to dstngush between correlaton and causalty. Two varables are correlated when the events assocated wth varables occur ore frequently together than one would expect by chance. Correlaton between two varables does not ply that one event causes the second to occur. Therefore, n akng causal nferences, t s portant to avod spurous correlaton. MR allows the expert to choose aong alternatve theores or hypothess and asssts the expert n sortng out correlatons between varables that are planly spurous fro those that reflect vald relatonshps. Model specfcaton nvolved several steps, each of whch s fundaental to the success of the research effort. Models are often characterzed n ters of the paraeters nuercal characterstcs of the odel. MR uses a saple to obtan estate of the values of paraeters of the odel an estate assocated wth a partcular explanatory varable s a regresson coeffcent. Multple regresson analyss goes beyond the calculaton of correlatons; t s a ethod n whch a regresson lne s used to relate the average of the varable the dependent varable to the values of other explanatory varables. As a result, regresson analyss can be used to predct the values of one varable usng the values of others. Steps for odel specfcaton:. Choosng the dependent varable the varable to be explaned should be approprate for analyzng the queston n the ssue.. Choosng the explanatory varable that s relevant to the ssues n the case. The explanatory varable that allows the evaluaton of alternatve hypotheses ust be chosen approprately. 3. Choosng the addtonal explanatory varables an attept should be ade to dentfy the addtonal known or hypotheszed explanatory varables, soe of whch are easurable and ay support alternatve substantve hypotheses that can be accounted for by the regresson analyss. 4. Choosng the functonal for of the MR odel the expert ust also choose the proper for of the regresson odel, ost frequently lnear for, or rare nonlnear for. Regresson results can be nterpreted n purely statstcal ters, through the use of sgnfcance tests, or they can be nterpreted n a ore practcal, non-statstcal anner []. In our case, we use the latter. ISSN: Issue 3, Volue 7, March

6 4 Lnear ult regresson odel of HPLC In general, lnear ult regresson odel takes the followng for: Y = k + k X + k X k n X n + ε, (4) where Y represents the dependent varable (.e. the varable to be explaned), X, X,,..., X n represent the explanatory varables and ε error, whch represents the collectve unobservable nfluence of any otted varables. Coeffcents k, k,... represent ultple regresson coeffcents whch are estated by fttng the equaton to the data by soe approprate ethod. Usng the unconventonal HPLC ethod we obtan experental data set {V R, log M, } =,,...n, (5) where V R s retenton volue logm s log olar ass s concentraton of eluent For creatng a odel for HPLC n frst we suppose that eprcal forula has the for: log M agntude of the effect of the change n explanatory varable on the dependent varable but also an estate of the relablty of the paraeter estates and easure of the overall goodness-offt of a regresson odel. Experental data for polystyrene n neat and xed eluent + DMF are shown n Table : Kroasl C-8, A, x5x7.8 Mw 9 Table experental data 8 7 DMF V R = k *logm + k * +k. (6) For deternaton of coeffcents k, k, k fro (3) we nze functon Φ gven by dscrete set of ponts () (least square ethod): 6 5 Kroasl C-8, A, 5 u, x5x7.8 Φ(k,k, k ) = [ k kogm +k + k - V R ] n. (7) =,,...,n So we obtan syste of lnear equatons: log M 4 3 DMF thf 9 thf 8 thf 7 thf Φ( k Φ( k, k, k k, k, k k ) Φ( k, k, k ) = ; = ; k ) =. (8) Soluton of ths syste of lnear equatons deternes coeffcents k,k,k. The least squares regresson provdes not only paraeter estates that ndcate the drecton and Retenton volue [L] Fg.3 Dependences of retenton volue on olar asses calbraton curves for dfferent concentraton of eluent The retenton volue V R s the subject of our nterest of explanaton, and thus t has been chosen as dependent varable. Explanatory varables are choosng next varables: the percentage of ISSN: Issue 3, Volue 7, March

7 concentraton of eluent and olar asses. We ntend to estate the dependent varable V R n the for where exst two explanatory varables (logm, ): V R = k + k logm + k + ε. (9) Ths suggested equaton s now ftted by approprate ethod for estate regresson coeffcents. Usng the least square ethod for estate regresson coeffcent we obtan the results as shown n Table : M T V NV e =.658 k = k = k = Table 3 Results of ultple regresson estaton for experental data fro Table where NV - s an estate retenton volue k, k, k - are estate regresson coeffcents, e - s addtonal rando error, M - represents a log of olar ass, - s eluent concentraton V R - s retenton volue. Usng the results fro Table we can express the V R as follows: V R = M ε, () where ε =.658 (addtonal rando error) s connected wth varables whch are not ncluded n eprcal for (9) and whch ndvdually have sall nfluence on dependent varable (retenton volue), but can be substantal n aggregaton. The results of lnear ultple-regresson for experental data fro Tab. expressed by () are shown n Fg. 4. V R log M Fg. 4 Dependences of retenton volue on olar asses and concentratons eluent The rando errors of regresson coeffcents tell us how uch the paraeter estate s lkely vary fro saple to saple. ISSN: Issue 3, Volue 7, March

8 V R T log M Kroasl C-8, A; 5x7.8 (teperature nfluence) Mw 5 o C ,43 4,48 5, 4,93 4,89 5, , 4,6 5,9 5,7 5,5 5, ,68 4,3 5, 5,43 5,59 6, ,63 4,5 5,5 5,49 5,8 6, ,58 3,55 4,68 5, 6,6 7,7 5,58,69 3,33 4,4 7, ,56,58,79 3, ,5,55,7,87 6.8,5,53,69,66 4 o C 35 o C 3 o C 3 o C Table 4 HPLC experental data for nfluence teperature on dependency olar ass on retenton volue Lnear ultple regresson odel for explanng dependence olar ass and teperature on retenton volue can to use eprcal for: logm Fg.5 Lnear ultple regresson odel for dependences teperature and olar ass on retenton volue V R = k + k logm +k T +ε () where logm represents olar ass T represents teperature V R represents retenton volue k, k, k represents regresson coeffcents ε represents addtonal rando error Usng experental data fro Tab.4 for teperature nfluence, HPLC process we can express by equaton: V R = M.6T +.55+ε, ( ) ε =.854 log M Kroasl C-8 Polystyrenes DMF: (83:7 wt.) 5 o C 4 o C 35 o C 3 o C 3 o C 3 o C V R [L] The addtonal rando error ε s connected wth events that exst n the process HPLC, but they are not taken n eprcal for (as they agntudes are not relevant, or for whch does not exst experental data). Fg. 6 Dependency retenton volue on olar ass for dfferent teperatures (calbraton curves for dfferent teperatures) The ore coplcated odel, where we ntent to suppose that lnear odel contan three explanatory varables (olar ass, concentraton of eluent, teperature) can have the for: ISSN: Issue 3, Volue 7, March

9 V R = k +k logm +k +k 3 T +ε (3) Agan, the error ε (addtonal rando error) s connected wth other events that are not expressed n (). In future we ntend to nclude further addtonal explanng varables that express V R and that have the nfluence (we suppose that sall) on nterpretaton of the experent. In [3] V R s expressed only as polynoal dependence on logm. It s evdent that the process of HPLC s the functon of several other addtonal varables. On the other hand, the dependency of retenton value on olar asses s n general nonlnear, so we ntend to use a ore precse as prevous explaned lnear odel a nonlnear MR odel for HPLC n our future work. 5 Concluson Hgh Perforance Lqud Chroatography separates coplex xtures of olecular speces to allow conclusons n ters of ther checal structure, archtecture and olar ass dstrbutons. As the drect analyss of checal structure of coplex ult-coponents polyers and especally of nor (<) coponents s not possble, new unconventonal ethods of separaton are needed. Selected ethods of HPLC belong to ths class of ethods. Unconventonal lqud chroatography ethods of separaton of coplex ult-coponent polyer were atheatcally processed by experental data for elucdaton of the retenton echanss. Multple regresson analyss s a tool for understandng the relatonshp n syste to be observed. Multple regresson analyss s well suted to the analyss of data copetng alternatve theores or hypothess n whch exst several possble explanatons for the relatonshp aong a dependent varable and nuber of explanatory varables. Proble s, on the other hand, how to obtan experental data to be useful for our assuptons. For understandng process of HPLC at frst there were studed the relaton between olar ass and retenton volue whch s represented by calbraton curves. Calbraton curves were expressed by polynoals of dfferent orders for dfferent values of concentratons of eluent. In the past, dependences retenton volue had been studed on one varable only the olar ass. It was realzed by polynoal regresson of the nd and 3 rd or 5th order. By approxatons calbraton curve does not be solved proble of HPLC process calbraton curves are only one part of HPLC that explan process of HPLC. Polynoal approxatons of calbraton curves are daetrcal dfferent for ndvdual concentratons, so we cannot detect how they change by nfluence of concentraton of eluent. For our goal to descrbe process of HPLC t s nsuffcent. Ths queston s partly solved by lnear ultple regresson odel. Usng ultple regresson to establsh regresson coeffcents for expressng eprcal functonal for for retenton volue n HPLC represents a next step of explanaton dependent varable as the dependence on ore than one explanatory varables. In HPLC any dfferent events s present that have soe nfluence on ths process and that have not been consdered n the prevous case. In (9) n the eprcal functonal for explanatory varables are ncluded: olar ass and eluent concentraton. It s evdent that n process HPLC are present ore than two explanatory varables. Ths realty s connected wth error, explaned n (). Ths error can be decreased when the eprcal functonal for for expresson process HPLC wll be expressed wth ore explanatory varables (olar ass, concentratons of eluent, teperature,...) that ay have a sall nfluence ndvdually but n aggregaton ay be portant. Soetes s a proble to obtan experental data for next explanatory varables. Indeed, calbraton curves that are basc n HPLC process are not generally lnear, so an eprcal functon for ultple regresson wll be nterestng to suggest n non-lnear for. Experences wth polynoal regresson approxatons of calbratons curves there wll be used. Soluton of non-lnear ultple regresson odel s not trval proble. In the future we wll deal wth next events and addtonal varables that ay have soe portance n proble HPLC by a non-lnear ultple regresson odel. Acknowledgeents: Fnancal support fro the APVV Slovak Grant Agency (project 59 7). The author s thankful to Dr. D. Berek for hs advce and to Mrs. J. Tarbajovska for her excellent techncal assstance. ISSN: Issue 3, Volue 7, March

10 References [] I. Rustanov, T.Farcas, F.Ahed, F.Chan, R LO Brutto, H.M.Mc Nar, Y.V. Kazakevch, J.Chroatogr. A,93, 49 [] Reference anual on Scentfc Evdence [3] D.Berek, S. Labatova, WSEAS Proc. Studes n Sulaton and Modelng, Vancouver, 7, pp.9 96 [4] D.L.Massart, B.G.M. Vandegnste, L.M.C.Buydens, S.De Jong, P.J.Lew, J.Seyers- Verbeke, Handbook of Cheoetrcs and Qualetrcs: Part A, Elsever, Asterda, 997 ISSN: Issue 3, Volue 7, March