It is presented a new method of cubic piecewise smooth interpolation applied to experimental data obtained by glycemic profile for diabetics. This method is applied to create a soft useful in clinical diabetology. The method give an alternative to the Akima's procedure of the derivatives computation on the knots from [Akima, J. Assoc. Comput. Mach., 1970] and have an optimal property

Alexandru Mihai Bica

Emanuel Paul

Luiza Demian

Marian Degeratu

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Surveys in Mathematics and its ApplicationsISSN 1842-6298Volume 1 (2006), 41 49OPTIMAL ALTERNATIVE TO THE AKIMASMETHOD OF SMOOTH INTERPOLATIONAPPLIED IN DIABETOLOGYAlexandru Mihai Bica, Marian Degeratu,Luiza Demian and Emanuel PaulAbstract. It is presented a new method of cubic piecewise smooth interpolation applied toexperimental data obtained by glycemic prole for diabetics. This method is applied to create asoft useful in clinical diabetology. The method give an alternative to the Akimas procedure of thederivatives computation on the knots from [1] and have an optimal property.1 IntroductionWe construct here a method of cubic piecewise smooth interpolation which combinethe calculus of the derivatives on the interior knots, given by the Akimas procedure(see [1]), with the cubic piecewise smooth interpolation formula from [6] and a newprocedure to compute the derivatives on the rst two and last three knots. Suchnew procedure is proposed here. The derivatives on the rst two knots and on thelast three knots are calculated such that to be minimized the quadratic oscillationin average (dened below) of the smooth interpolation function. The quadraticoscillation in average (dened in [2]) was introduced to measure the geometricaldistance between the graphs of the interpolation function and of the polygonal linejoining the interpolation points. In the interior knots where we can use the Akimasmethod (see [1]), we prefer this method because give a natural computation of thederivatives.This numerical method is implemented here to obtain a proper soft which is testedand used on diabetology measurements. For patients which present blood-glucose2000 Mathematics Subject Classication: 57R12, 65D05.Keywords: piecewise smooth interpolation, smooth approximations, quadratic oscillation inaverage, least squares method.The research of the rst author on this paper is supported by the grant 2Cex-06-11-96 of theRomanian Government******************************************************************************http://www.utgjiu.ro/math/sma42 A.M. Bica, M. Degeratu, L. Demian and E. Paulhomeostasis, the numerical method can be combined with the mathematical modelobtained in [3] the aim to determine the critical hypoglycemia characteristic for suchpatients.2 The cubic piecewise smooth interpolation.In the plane tOx consider the points (ti; xi) ; i = 0; n where ti, represent momentsof time and xi represent measured values of a parameter (which varies in time). Letthe vector x = (x0; :::; xn) ; hi = ti   ti 1; i = 1; n and the slopes,mi =xi+1   xiti+1   ti; i = 0; n  1: (1)We use these slopes to compute x0i; ; i = 2; n  3, applying the Akimas procedurefrom [1],x0i =jmi+2  mi+1j mi 1 + jmi 1  mi 2j mi+1jmi+2  mi+1j+ jmi 1  mi 2j; i = 2; n  3 (2)By (2) we see that mi; i = 0; n  1 is not enough to computex00; x01; x0n 2; x0n 1; x0ntoo. Therefore H. Akima propose in [1] an articial computation ofm 2; m 1; mm; mn+1; mn+2and then the treatment of the end points is a weakness of the method (as it ismentioned in [1] and [5]). Moreover in [1] is not given the error estimation.To interpolate the data (ti; xi) ; i = 1; n; we dene F : [t0; tn]  ! R by hisrestrictions Fi; i = 1; n; to the intervals [ti 1; ti] ; i = 1; n. The functions Fi arecubic polynomials and have expression as in [6] and [5]:Fi (t) =(ti   t)2 (t  ti 1)h2i x0i 1  (t  ti 1)2 (ti   t)h2i x0i++(ti   t)2 [2 (t  ti 1) + hi]h3ixi 1+(t  ti 1)2 [2 (ti   t) + hi]h3ixi; t 2 [ti 1; ti] ; (3)that is, with other notations,Fi (t) = Ai(t)  x0i 1 +Bi(t)  x0i + Ci(t)  xi 1 + Ei(t)  xi: (4)******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/smaOptimal alternative to smooth interpolation applied in diabetology 43Denition 1. ([2]) Corresponding to the points (ti; xi) ; i = 0; n; let the polygonalline D (x) : [t0; tn]  ! R having the restrictions to the intervals [ti 1; ti] ; i = 1; n;Di given byDi (t) = xi 1 +xi   xi 1ti   ti 1 (t  ti 1):The quadratic oscillation in average of F is (F; x) =vuuut nX :i=1tiZti 1[Fi (t) Di (t)]2dt:We will obtain x00; x01; x0n 2; x0n 1; and x0n such that the quadratic oscillationin average of F to be minimal. In this aim we will consider the residualR x00; x01; x0n 2; x0n 1; x0n=nXi=1tiZti 1[Fi(t) Di(t)]2dt: (5)3 Optimal property and the error estimationTheorem 2. There exist an unique pointx00; x01; x0n 2; x0n 1; x0n2 R5 which min-imize the quadratic oscillation in average  (F; x) :Proof. We will minimize the residualR x00; x01; x0n 2; x0n 1; x0n=nXi=1tiZti 1[Fi(t) Di(t)]2dt:applying the least squares method. Therefore, we solve the systems given by theconditions@R@x00= 0;@R@x01= 0 (6)@R@x0n 2= 0;@R@x0n 1= 0;@R@x0n 1= 0: (7)******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/sma44 A.M. Bica, M. Degeratu, L. Demian and E. PaulThe system (6), according to (4), have the form :8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:0@ t1Zt0A21 (t) dt1A  x00 +0@ t1Zt0A1 (t) B1 (t) dt1A  x01 = t1Zt0A1 (t) D1 (t) dt  0@ t1Zt0A1 (t)  C1 (t) dt1A  x0  0@ t1Zt0A1 (t)  E1 (t) dt1A  x10@ t1Zt0A1 (t) B1 (t) dt1A  x00 +0@ t1Zt0B21 (t) dt+t2Zt1A22 (t) dt1A  x01 = t1Zt0B1 (t) D1 (t) dt  0@ t2Zt1A2 (t) B2 (t) dt1A  x02  0@ t1Zt0B1 (t)  C1 (t) dt1A  x0  0@ t1Zt0B1 (t)  E1 (t) dt1A  x1++t2Zt1A2 (t) D2 (t) dt 0@ t2Zt1A2 (t)  C2 (t) dt1A  x1  0@ t2Zt1A2 (t)  E2 (t) dt1A  x2:(8)Since@2R@x020= 20@ t1Zt0A21 (t) dt1A > 0and the determinant of the Hesse matrix @2R@x020@2R@x00@x01@2R@x00@x01@2R@x021!is1 = 40@ t1Zt0A21 (t) dt1A 0@ t1Zt0B21 (t) dt1A  40@ t1Zt0A1 (t) B1 (t) dt1A2++40@ t1Zt0A21 (t) dt1A 0@ t2Zt1A22 (t) dt1A > 0we infer that the system (8) have unique solutionx00; x01: The system (7) according******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/smaOptimal alternative to smooth interpolation applied in diabetology 45to (4), is :8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:0B@ tn 2Ztn 3B2n 2 (t) dt+tn 1Ztn 2A2n 1 (t) dt1CA  x0n 2 +0B@ tn 1Ztn 2An 1 (t) Bn 1 (t) dt1CA  x0n 1=  tn 2Ztn 3Bn 2 (t) [An 2 (t)  x0n 3 + Cn 2 (t)  xn 3 + En 2 (t)  xn 2  Dn 2 (t)]dt tn 1Ztn 2An 1 (t) [Cn 1 (t)  xn 2 + En 1 (t)  xn 1  Dn 1 (t)]dt0B@ tn 1Ztn 2An 1 (t) Bn 1 (t) dt1CA  x0n 2 +0B@ tn 1Ztn 2B2n 1 (t) dt+tnZtn 1A2n (t) dt1CA  x0n 1++0B@ tnZtn 1An (t)Bn (t) dt1CA  x0n =  tn 1Ztn 2Bn 1 (t) [Cn 1 (t)  xn 2 + En 1 (t)  xn 1  Dn 1 (t)]dt tnZtn 1An (t)  [Cn (t)  xn 1 + En (t)  xn  Dn (t)]dt0B@ tnZtn 1An (t) Bn (t) dt1CA  x0n 1 +0B@ tnZtn 1B2n (t) dt1CA  x0n ==  tnZtn 1Bn (t)  [Cn (t)  xn 1 + En (t)  xn  Dn (t)]dt(9)Since@2R@x02n 2= 20B@ tn 2Ztn 3B2n 2 (t) dt+tn 1Ztn 2A2n 1 (t) dt1CA > 0and the determinant of the Hesse matrix0@ @2R@x02n 2 @2R@x0n 2@x0n 1@2R@x0n 1@x0n 2@2R@x02n 11Ais2 = 40B@ tn 1Ztn 2A2n 1 (t) dt1CA0B@ tn 1Ztn 2B2n 1 (t) dt1CA  40B@ tn 1Ztn 2An 1 (t) Bn 1 (t) dt1CA2+******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/sma46 A.M. Bica, M. Degeratu, L. Demian and E. Paul+40B@ tn 2Ztn 3B2n 2 (t) dt1CA 0B@ tn 1Ztn 2B2n 1 (t) dt+tnZtn 1A2n (t) dt1CA++40B@ tn 1Ztn 2A2n 1 (t) dt1CA 0B@ tnZtn 1A2n (t) dt1CA > 0;and in addition for the determinant of the Hesse matrix0BB@@2R@x02n 2@2R@x0n 2@x0n 1@2R@x0n 2@x0n@2R@x0n 1@x0n 2@2R@x02n 1@2R@x0n 1@x0n@2R@x0n 2@x0n@2R@x0n 1@x0n@2R@x02n1CCAwe have = 80B@ tn 1Ztn 2A2n 1 (t) dt+tn 2Ztn 3B2n 2 (t) dt1CA  [0B@ tnZtn 1A2n (t) dt1CA 0B@ tnZtn 1B2n (t) dt1CA  0B@ tnZtn 1An (t)Bn (t) dt1CA2] + 80B@ tnZtn 1B2n (t) dt1CA  [0B@ tn 1Ztn 2A2n 1 (t) dt1CA 0B@ tn 1Ztn 2B2n 1 (t) dt1CA 0B@ tn 1Ztn 2An 1 (t) Bn 1 (t) dt1CA2]++80B@ tnZtn 1B2n (t) dt1CA 0B@ tn 1Ztn 2B2n 1 (t) dt1CA 0B@ tn 2Ztn 3B2n 2 (t) dt1CA > 0;according to the Cauchy-Buniakovski-Schwarzs inequality, follows that the system(9) have unique solutionx0n 2; x0n 1; x0n: Consequently, there exist an unique pointx00; x01; x0n 2; x0n 1; x0n2 R5 for which the residualRx00; x01; x0n 2; x0n 1; x0nis minimal. Because(F; x) =qR x00; x01; x0n 2; x0n 1; x0n;we infer that for this point, the quadratic oscillation in average, (F; x) is minimal.******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/smaOptimal alternative to smooth interpolation applied in diabetology 47From (3) follows that F 2 C1 [t0; tn] ; F =2 C2 [t0; tn], but F 00 is piecewise contin-uous. Then, F 00 is bounded.Let Fx00; x01; x0n 2; x0n 1; x0nthe smooth interpolation function dened by hisrestrictions in (3) and obtained in the above theorem. For this function we have theerror estimation obtained below.Theorem 3. If xi; i = 0; n are values of a function f : [t0; tn]  ! R, that isf (ti) = xi; 8i = 0; n and if f 2 C1 [t0; tn] with Lipschitzian rst derivative, then thefunction F 2 C1 [t0; tn] given in (3) realize the piecewise smooth interpolation of thefunction f on the knots ti; i = 0; n and the following error estimation holds :kf   Fkc 6 L0 +Mmaxfh2i : i = 1; ng; (10)where L0 is the Lipschitz constant of f 0 andM = maxfmax(F 00i (ti 1) ; F 00i (ti)) : i = 1; ng:Proof. Consider ' = f   F: Since f (ti) = F (ti) = xi; 8i = 0; n; we infer that' (ti) = 0; 8i = 0; n: Therefore, for any i = 1; n there exist i 2 (ti 1; ti) such that'0 (i) = 0:For any t 2 [t0; tn] there exist j 2 f1; :::; ng such that t 2 [tj 1; tj ]: We have,jf (t)  F (t)j =Z ttj 1[f 0 (s)  F 0 (s)]ds Z ttj 1(f 0 (s)  f 0  j+ f 0  j  F 0  j+ F 0  j  F 0 (s))ds Z ttj 1(L0 s  j+ F 00j C  s  j)ds  (L0 + F 00j C)  h2j :Since F 00i is rst order polynomial 8i = 1; n we getF 00i C = max(F 00i (ti 1) ; F 00i (ti));and obtain the estimation (10). We conclude that similar error estimation holds forthe smooth interpolation function Fx00; x01; x0n 2; x0n 1; x0n:4 ApplicationFrom the above section follows that the presented method optimally improves on therst two knots and on the last three knots, the Akimas method from [1]. Moreover,gives the error estimation of order O h2: This numerical method is used to obtain******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/sma48 A.M. Bica, M. Degeratu, L. Demian and E. Paula soft applicable in diabetology at the tting of glycemic prole experimental data.The soft was created in C# and was tested on data harvested in October 2006for ve patients.As example, for the patient no. 4 blood- glucose levels (in mg/dl) was measuredat the hours 7:00,9:30,12:30,15:00,18:00,20:30,0:00,3:00 and the values obtained were120,92,114,135,110,130,105,86 (in mg/dl). Variation trends on those moments were :-11.20, 7.33, -0.99, 8.37, -7.20, 11.60, -14.53, 0.93. The graphic result of this patientis:Graphic for patient no. 4The presented method is su¢ ciently accurate in the aim to approximate the dailyevolution of the blood glucose levels (continuously given by the Holder program)and cheaper than the Holder method which use a consumable enzyme. Therefore,represent a more economic method.References[1] H. Akima, A new method for interpolation and smooth curve tting based onlocal procedure, J. Assoc. Comput. Mach. 17 (1970), 589-602. Zbl 0209.46805.[2] A. M. Bica, Mathematical models in biology governed by di¤erential equations,PhD Thesis Babeş-Bolyai University Cluj-Napoca 2004.[3] A. Bica, Lyapunov function for a bidimensional system model in the blood glucosehomeostasis and clinical interpretation, Int. J. Evol. Equ. 1, no. 1 (2005), 6979.MR2144218 (2005m:92010). Zbl 1094.34029.******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/smaOptimal alternative to smooth interpolation applied in diabetology 49[4] A. M. Bica, M. Curil¼a and S. Curil¼a, Optimal piecewise smooth interpolationof experimental data, Proceedings of ICCCC2006 (International Conference onComputing, Comunications, Control, 2006), Oradea, 74-79.[5] C. Iacob(ed), Classical and modern mathematics, vol.4, Ed. Tehnica, Bucharest1983 (in Romanian).[6] K.Ichida, F.Yoshimoto and T. Kiyono, Curve tting by a Piecewise CubicPolynomial, Computing 16 no. 4 (1976), 329338. MR0403168(53 #6981). Zbl0321.65010.Alexandru Mihai Bica Marian DegeratuUniversity of Oradea, University of Oradea,Department of Mathematics and Informatics, Department of Mathematics and Informatics,Str. Universitatii no.1, 410087, Oradea, Str. Universitatii no.1, 410087, Oradea,Romania. Romania.email: smbica@yahoo.com, abica@uoradea.ro email: mariand@uoradea.roLuiza Demian Emanuel PaulUniversity of Oradea, University of Oradea,Faculty of Medicine and Pharmacy, Department of Mathematics and Informatics,Str. Universitatii no.1, 410087, Oradea. Str. Universitatii no.1, 410087, Oradea.Romania. Romania.email: paul_emanuel21@yahoo.com******************************************************************************Surveys in Mathematics and its Applications 1 (2006), 41 49http://www.utgjiu.ro/math/sma

Optimal Alternative to the Akima's Method of Smooth Interpolation Applied in Diabetology

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Optimal Alternative to the Akima's Method of Smooth Interpolation Applied in Diabetology

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