1. Curve fitting methods for complex and rank deficient chemical data
14th Iranian Workshop on Chemometrics
Curve fitting methods for complex and rank deficient chemical data
A. Naseri
Faculty of Chemistry, University of Tabriz, Tabriz, IRAN

2. Model-based non-linear fitting A = C E + R based on beer-lambert law
The task of model-based data fitting for a given matrix A, is to determine the best parameters defining matrix C, as well as the best pure responses collected in matrix E.
A = C E + R based on beer-lambert law A C E R = +

3. The quality of the fit is represented by the matrix of residuals
The quality of the fit is represented by the matrix of residuals. Assuming white noise, the sum of the squares, ssq, of all elements ri,j is statistically the best measure to be minimized
ssq
ssq = ΣΣ r2 I,j

4. Classical non-linear least squares fitting (traditional analysis)
The residuals are a matrix R which is a function of all parameters, represented by the vector k:
Target transform fitting

5. The difference between the two methods lies in the fact that TTF treats the residual vectors rE individually and thus is able to fit the parameters defining the rEs individually, independently of all the others.

6. Any optimization method can be used .
fminsearch uses the Nelder-Mead simplex (direct search) method.
fminsearch Multidimensional unconstrained nonlinear minimization (Nelder-Mead).
X = fminsearch(FUN,X0) starts at X0 and attempts to find a local minimizer X of the function FUN. FUN is a function handle. FUN accepts input X and returns a scalar function value F evaluated at X. X0 can be a scalar, vector or matrix.

7. Rank deficiency in concentration profiles
Rank deficiency and fitting
Second order kinetics k
A + B C
Rank deficiency in concentration profiles
[A] + [C] = [A]0
[B] + [C] = [B]0
a [A] - [B] + (a - 1) [C] = 0
[B]0 = a [A]0
Linear dependency
[B] + [C] = a [A] + a [C]

8. A = C E + R
[A]0 = 1
[B]0 = 1.5
k = 0.3
E = C \ A

9. Calculated pure spectra according to E = C \ A

10. Reconstructed data
Measured data
Residuals

13. Dimerization Equilibrium
Dimerization equilibria can be seen in organic dyes solutions.
Organic dyes are used in lasers, fluorescence spectroscopy , protein labeling, … .

14. Different geometrical coupling are seen in dimerization equilibria such as H-coupling and J coupling.
Dimerization equilibria can be seen in concentration ranges M.

15. M + M D
K D = [D] [M] 2
Mass balance
roots function of MATLAB can be used to find roots of this polynomial equation.
Negative and imaginary roots must be eleminated.

16. A function to calculate the concentration of monomer and dimer

17. Detecting the presence of dimerization
Plotting Normalized spectra
PCA analysis of data

18. Log(KD)=4
Singular values

19. Grid search for finding KD
Noisy data
Singular values

20. Resolved and real profiles

21. Dimerization of protonated form at constant pH
2 HM D
M- + H+
K D = [D] [HM] 2
K a = M [H] [HM]
Mass balance

22. Log(KD)=5, pKa=5, pH=5.4
Singular values
Three component system with rank=2 ; rank deficient system

23. Conditional dimerization constant can be obtained by fitting on dimerization polynomial equation:
Log(KD’)=3.90
𝐾 𝐷 = 𝐾 𝐷 ′ (1+ 𝐾 𝑎 [𝐻] ) 2

26. Dimerization of two forms, protonated and deprotonated, at constant pH
2 HM HD
2H+ + 2M D
K HD = [HD] [HM] 2
K D = [D] [M] 2
K a = M [H] [𝐻𝑀]
Mass balance
2 𝐾 𝐷𝐻 𝐻 𝐾 𝐷 𝐾 𝑎 2 [𝐻𝑀] 𝐻 2 + 𝐾 𝑎 𝐻𝑀 − 𝐻 2 𝐶 0 =0

27. Singular values
Four components system with rank=2 ; rank deficient system
𝐻𝐷 =[𝐷]× 𝐾 𝐷𝐻 𝐾 𝐷 𝐾 𝑎 2 [𝐻] 2

28. Running experiment in other pH
Singular values

29. Augmentation of two data sets
pH=5.5
pH=9
Singular values

30. Fitting

31. Dimerization of protonated form at non-constant pH

32. Dimerization in the presence of Inert
Impurity in solvent
Impurity in solute
Singular values
Singular values

33. Dimerization and Trimerization Equilibrium
M + M D
M + M +M T
K D = [D] [M] 2
K D = [T] [M] 3
Mass balance
roots function of MATLAB can be used to find roots of this polynomial equation.
Negative and imaginary roots must be eliminated.

34. Singular values

36. Obtaining of other thermodynamic parameters
T=298K
T=325K
T=350K
After analyzing KD can be obtained:
Log(KD)=5.71
Log(KD)=5.24
Log(KD)=4.86

37. Van’t hoff Equation

38. Studying of dimerization equilibria by temperature changes
Spectra of a solution in different temperatures
Singular values

41. Grid search was used to find ΔH and ΔS.

42. Different views

44. Ssq surfaces for data in the absence of noise

45. Ssq surfaces for data in the presence of noise

48. Fminsearch function was used for fitting.

49. (fminsearch function)
ΔH (kJ/mol)
ΔS (J/mol.K)
Real
60.0 90
Grid Search
-59.89
-89.5
Nelder-Mead
(fminsearch function)
59.95-
89.71-
Residuals
Concentration profiles; resolved and simulated
Spectral profiles; resolved and simulated

50. Methylene Blue dye
غلظت 5-10×2/5 مولار از متیلن بلو تهیه شد و طیف آن در دوازده دمای 9، 12، 16، 5/19، 23، 26، 8/30، 5/36، 41، 8/44، 49 و 56 درجه ثبت شد. نقطه ایزوبستیک مشاهده شد. به منظور جلوگیری از تداخل تعادل اسید باز pH محلول در2/00 تثبیت شد.
طبف جذبی محلول 5-10×2/5 مولار متیلن بلو در دوازده دمای متفاوت
طیف نرمالیزه متیلن بلو 5-10×5/2 مولار در دو دمای (a 9 و (b 56 درجه سانتی گراد

51. آنالیز رنگ نیل بلو و نوترال رد
نمایش پنج مقدار منفرد اول برای سه رنگ متیلن بلو، نیل بلو و نوترال رد
طیف جذبی محلول 5-10×2/75 مولار نیل بلو در یازده دمای متفاوت
طیف نرمالیزه نیل بلو 5-10×2/75 مولار در دو دمای (a 15 و (b 65 درجه سانتی گراد
طبف جذبی محلول 5-10×4/5 مولار نوترال رد در ده دمای متفاوت
طیف نرمالیزه نوترال رد 5-10×4/5 مولار در دو دمای (a 7 و (b 52 درجه سانتی گراد

52. نمودار حاصل از جست و جوی شبکه ای برای سه رنگ متیلن بلو، نیل بلو و نوترال رد
5-10×2/5 مولار
نوترال رد
5-10×4/5 مولار
نیل بلو
5-10×2/75 مولار

53. نمودار log(ssq) بر حسب آنتالپی و آنتروپی های مینیمم شده برای سه رنگ متیلن بلو، نیل بلو و نوترال رد
متیلن بلو 5-10×2/5 مولار
نیل بلو5-10×2/75 مولار
نوترال رد5-10×4/5 مولار

54. آنتالپی و آنتروپی بدست آمده به کمک دو روش بهینه سازی برای سه غلظت متفاوت از رنگ متیلن بلو، نیل بلو و نوترال رد
متیلن بلو
غلظت
5-10×2/75
5-10×3/5
5-10×4/5
نیلدر-مید
33/54∆H=-
42/38-∆S=
37/18∆H=-
57/76-∆S=
27/64∆H=-
21/00-∆S=
جست وجوی شبکه ای
33/60∆H=-
42/60-∆S=
37/20∆H=-
57/89-∆S=
27/80∆H=-
21/60-∆S=
نیل بلو
نوترال رد
غلظت
5-10×2/75
5-10×3/5
5-10×4/5
نیلدر-مید
35/62∆H=-
43/00-∆S=
39/54∆H=-
54/45-∆S=
33/11∆H=-
28/28-∆S=
جست وجوی شبکه ای
35/70∆H=-
43/50-∆S=
39/50∆H=-
54/39-∆S=
32/80∆H=-
27/19-∆S=
غلظت
5-10×2/5
5-10×3/5
5-10×4/5
نیلدر-مید
59/55∆H=-
122/00-∆S=
97/57∆H=-
247/00-∆S=
69/53∆H=-
158/81-∆S=
جست وجوی شبکه ای
59/50∆H=-
122/10-∆S=
97/60∆H=-
247/30-∆S=
69/50∆H=-
158/90-∆S=

55. پرو فایل غلظتی مربوط به سه رنگ متیلن بلو، نیل بلو و نوترال رد حاصل از برازش (a مونومر (b دیمر
متیلن بلو 5-10×2/5 مولار
نیل بلو5-10×2/75 مولار
نوترال رد5-10×4/5 مولار

56. پرو فایل غلظتی مربوط به سه رنگ متیلن بلو، نیل بلو و نوترال رد حاصل از برازش (a مونومر (b دیمر
متیلن بلو 5-10×2/5 مولار
نیل بلو5-10×2/75 مولار
نوترال رد5-10×4/5 مولار

57. Dimerization in the presence of Inert
Singular values
Three component system with rank=2 ; rank deficient system

58. Results of fitting to Dimerization model without inert
Real:*
Resolved:__

59. Simulating oscillating chemical reactions using Microsoft Excel Macros
Tutorial review: Simulating oscillating chemical reactions using Microsoft Excel Macros
Abdolhossein Naseri*, Hossein Khalilzadeh and Saheleh Sheykhizadeh
Analytical and Bioanalytical Chemistry Research, Accepted to publish

60. Oscillating reactions are one of the most interesting topics in chemistry and analytical chemistry. Fluctuations in concentrations of one the chemical reaction species (usually a reaction intermediate) create an oscillating chemical reaction.
In oscillating systems, the reaction is far from thermodynamic equilibrium.
In these systems at least one autocatalytic step is required.
Developing an instinctive feeling for how oscillating reactions work will be invaluable to future generations of chemists.

61. 𝐴→𝑋 (1.a)
𝐴+𝑋 →2𝑋 (1.b)
𝑋+𝑌 →2𝑌 (1.c)
𝑌 →𝐵 (1.d)

62. Using kinetic theory, the concentration-time equations of the species in a reaction mechanism can be defined by a system of ordinary differential equations (ODEs).
Euler's method is a routine numerical integration method. This method uses a form of the Taylor series expansion truncated to the first derivative. Starting with the concentration at time 𝑡 𝑖−1 , the concentration at 𝑡 𝑖 is estimated using the derivative at t (Eq. 2).
𝑐 𝑡 𝑖 =𝑐 𝑡 𝑖−1 + 𝑑𝑐(𝑡) 𝑑𝑡 ∆𝑡

63. 𝑋+𝑌 𝑘 𝑍 (3.a)
𝑑 𝑐 𝑋 𝑑𝑡 =−𝐾 𝑐 𝑋 𝑐 𝑌 (3.b)
𝑐 𝑋 𝑡 𝑖 = 𝑐 𝑥 𝑡 𝑖−1 −𝑘 𝑐 𝑋 𝑐 𝑌 ∆𝑡 (3.c)

64. Mechanism of oscillating reaction
Lotka–Volterra model
𝐴+𝑋 𝐾 𝑋 (6.a)
𝑋+𝑌 𝐾 𝑌 (6.b)
𝑌 𝐾 3 𝐵 (6.c)

65. 𝑑 𝑐 𝑋 𝑑𝑡 = 𝐾 1 𝑐 𝐴 𝑐 𝑋 − 𝐾 2 𝑐 𝑋 𝑐 𝑌 (7.a)
𝑑 𝑐 𝑌 𝑑𝑡 = 𝐾 2 𝑐 𝑋 𝑐 𝑌 − 𝐾 3 𝑐 𝑌 (7.b)
𝑑 𝑐 𝐵 𝑑𝑡 = 𝐾 3 𝑐 𝑌 (7.c)
𝑑 𝑐 𝐴 𝑑𝑡 =− 𝑘 1 𝑐 𝐴 𝑐 𝑋 (7.d)
𝑐 𝑋, 𝑡 𝑖 = 𝑐 𝑋, 𝑡 𝑖−1 + 𝑑 𝑐 𝑋 𝑑𝑡 ∆𝑡= 𝑐 𝑋, 𝑡 𝑖−1 + 𝐾 1 𝑐 𝐴 𝑐 𝑋, 𝑡 𝑖−1 − 𝐾 2 𝑐 𝑋, 𝑡 𝑖−1 𝑐 𝑌, 𝑡 𝑖−1 ∆𝑡 (8.a)
𝑐 𝑌, 𝑡 𝑖 = 𝑐 𝑌, 𝑡 𝑖−1 + 𝑑 𝑐 𝑌 𝑑𝑡 ∆𝑡= 𝑐 𝑌, 𝑡 𝑖−1 + 𝐾 2 𝑐 𝑋, 𝑡 𝑖−1 𝑐 𝑌, 𝑡 𝑖−1 − 𝐾 3 𝑐 𝑌, 𝑡 𝑖−1 ∆𝑡 (8.b)
𝑐 𝐵, 𝑡 𝑖 = 𝑐 𝐵, 𝑡 𝑖−1 + 𝑑 𝑐 𝐵 𝑑𝑡 ∆𝑡= 𝑐 𝐵, 𝑡 𝑖−1 + 𝐾 3 𝑐 𝑌, 𝑡 𝑖−1 ∆𝑡 (8.c)

66. One of the spreadsheet applications developed by Microsoft is Microsoft Excel. It has a grid of cells arranged in letter-named columns and numbered rows for organizing data manipulations. It allows sectioning of data to view its dependencies on various factors from different perspectives.
Microsoft Excel has a programming ability, Visual Basic for Applications (VBA). This ability allows the user to employ a wide variety of numerical methods, e.g., solving differential equations of mathematical physics. The Macro Recorder is a common and easy way to generate VBA code. It records actions of the user and generates VBA code in the form of a macro. Running the macro lets users to repeat those actions automatically.

69. Thank you for your attention