A novel fourth-order calibration method based on alternating quinquelinear decomposition algorithm for processing high performance liquid chromatography–diode array detection– kinetic-pH data of naptalam hydrolysis

Analytica Chimica Acta 861 (2015) 12–24

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A novel fourth-order calibration method based on alternating quinquelinear decomposition algorithm for processing high performance liquid chromatography–diode array detection– kinetic-pH data of naptalam hydrolysis Xiang-Dong Qing, Hai-Long Wu * , Xi-Hua Zhang, Yong Li, Hui-Wen Gu, Ru-Qin Yu State Key Laboratory of Chemo/Biosensing and Chemometrics, College of Chemistry and Chemical Engineering, Hunan University, Changsha 410082, China

H I G H L I G H T S

G R A P H I C A L A B S T R A C T


Five-way HPLC–DAD–kinetic-pH data were obtained for the first time.
A new algorithm, alternating quinquelinear decomposition (AQQLD), was developed.
Simulated data and real data were analyzed to explore the performance of AQQLD.
PARAFAC and AQQLD were applied to investigate the kinetics of naptalam.
It indicates the potential for the use of fourth-order data in complex systems.

A R T I C L E I N F O

A B S T R A C T

Article history: Available online 24 December 2014

Five-way high performance liquid chromatography–diode array detection (HPLC–DAD)–kinetic-pH data were obtained by recording the kinetic evolution of HPLC–DAD signals of samples at different pH values and a new fourth-order calibration method, alternating quinquelinear decomposition (AQQLD) based on pseudo-fully stretched matrix forms of the quinquelinear model, was developed. Simulated data were analyzed to investigate the performance of AQQLD in comparison with five-way parallel factor analysis (PARAFAC). The tested results demonstrated that AQQLD has the advantage of faster convergence rate and being insensitive to the excess component number adopted in the model. Then, they have been successfully applied to investigate quantitatively the kinetics of naptalam (NAP) hydrolysis in two practical systems. Additionally, the serious chromatographic peak shifts were accurately corrected by means of chromatographic peak alignment method based on abstract subspace difference. The good recoveries of NAP were obtained in these samples by selecting the time region of chromatogram. The elution time, spectral, kinetic time and pH profiles resolved by the chemometric techniques were in good agreement with experimental observations. It demonstrates the potential for the utilization of fourthorder data for some complex systems, opening up a new approach to fourth-order data generation and subsequent fourth-order calibration. ã 2014 Elsevier B.V. All rights reserved.

Keywords: Fourth-order calibration Parallel factor analysis Alternating quinquelinear decomposition High performance liquid chromatography Naptalam Kinetic

* Corresponding author. Tel.: +86 731 88821818; fax: +86 731 88821818. E-mail address: [email protected] (H.-L. Wu). http://dx.doi.org/10.1016/j.aca.2014.12.037 0003-2670/ ã 2014 Elsevier B.V. All rights reserved.

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

1. Introduction Calibration with the alternating multilinear decomposition (AMLD) algorithms implies building a useful model to process multilinear or approximately multilinear data systems. Wherein, alternating trilinear decomposition (ATLD) [1] and its variants (self-weighted alternating trilinear decomposition (SWATLD) [2] and alternating penalty trilinear decomposition (APTLD) [3]) have made significant progress in the analysis of second-order data. Extensions of ATLD and its variants have subsequently been developed for third-order data analysis, i.e., alternating quadrilinear decomposition (AQLD) [4], alternating penalty quadrilinear decomposition (APQLD) [5] as well as regularized self-weighted alternating quadrilinear decomposition (RSWAQLD) [6]. These algorithms hold the second-order advantage which is known as the ability to accurately achieve the concentrations of individual component of interest through separating the signals of target analyte(s) from those of uncalibrated background or interferences. They have been applied in many scientific fields, such as chemistry, medicine, food, environmental and single-cell science, as can be seen from an explosion in the volume of relevant literatures [7–11]. Multi-dimensional calibration methods can also be directly extended to higher-order calibration, i.e., fourth-order calibration. Up to now, there were only two types of algorithms which could be directly extended to fourth-order calibration for analysis of fiveway data arrays. The first type was based on computing the regression coefficients leading to prediction by combining data from calibration and test samples, including classical parallel factor analysis (PARAFAC) [12–15] and AMLD mentioned above. These algorithms can provide accurate curve resolution and concentration prediction of the compounds of interest. The second type was built on estimating loadings from calibration data only and then calculating regression coefficients after the test sample entered the scene, for example, the combination of residual multilinearization with unfolded partial least-squares (U-PLS/RML [16,17]) or N-way partial least-squares (N-PLS/RML [17,18]) or unfolded principal component analysis (U-PCA/RML [19]). The second algorithms do not yield pure constituent profiles, but they can furnish quantitative information of the anlytes of interest. Theoretically, the fourth-order calibration should exhibit the same advantage shown by three-way and four-way data with regard to the presence of uncalibrated components, i.e., the second-order

13

advantage [20,21], while also providing further benefits, such as richer analytical information, improved algorithmic resolution of serious collinear data and increased sensitivity and selectivity. Unfortunately, very little work has been conducted investigating all the potential advantages of fourth-order calibration to date. This may be due to the limited number of ways that methodologies for obtaining five-way data arrays can be applied or the considerable difficulties involved in the corresponding theoretical or experimental processes. The only work present in the literature, to date, is that of Maggio et al. who used a fast-scanning spectrofluorimeter to monitor the hydrolysis of carbaryl at different pH values to obtain the fourth-order excitation–emission–kinetic-pH fluorescence data. The data were then processed with two algorithms, U-PLS in combination with residual quadrilinearization (U-PLS/ RQL) and PARAFAC [22]. Furthermore, a new expression of the sensitivity estimation in fourth-order calibration with the U-PLS/ RQL model was also developed by Allegrini and Olivieri [17]. Naptalam (NAP) is a selective pre- and post-emergent herbicide and it may degrade to 1-naphthylamine (NAA), O-phthalic acid (PHT) and N-(1-naphthyl) phthalimide (NPI) in an aqueous environment. NAP is estimated to be of low toxicity but NAA is well known for its cytotoxic and genotoxic effects [23]. The two different pathways of degradation are presented in Fig. 1. It has been previously reported that phosphorescence [24], fourier transform infrared spectrometry (FTIR) [25] and chromatographic methods [26] can be applied to the determination of NAP and its metabolites in diverse matrices such as river water, drinking water, urine, etc. Over the last few decades, first-order calibration methods have been employed for the analysis of NAP and its metabolites [23,25]. For example, PLS has been applied to predict the concentrations of NAP, NAA and NPI in mixtures by FTIR spectrometry [25]. Unfortunately, the first-order calibration methods require a sufficiently large and representative calibration sample set and calibration samples containing all of the constituents present in future unknown samples, a requirement that makes analysis of real systems containing many unknown interferents very difficult. Owing to model development, secondorder or higher-order calibration methods successfully circumvent the problems associated with first-order methodologies, and provide us with an alternative method to easily study the kinetic processed of the analyte exploiting the second-order and extended advantages as previously described.

[(Fig._1)TD$IG]

Fig. 1. The degradation processes of naptalam in acid medium.

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In the work, a new fourth-order calibration method, alternating quinquelinear decomposition (AQQLD) based on pseudo-fully stretched matrix forms of the quinquelinear model, was developed for the processing of five-way HPLC–DAD–kinetic-pH data, which were obtained by recording the kinetic evolution of HPLC–DAD signals of samples at different pH values. A study comparing AQQLD with five-way PARAFAC by two simulated data sets was carried out to show the merits of the proposed algorithm. Then, these methodologies were successfully applied to investigate quantitatively the kinetics of naptalam (NAP) hydrolysis in two systems containing (1) indole-3-acetic acid (IAA) and 1-naphthaleneacetamide (NAD), (2) IAA, NAD and unexpected component (s), respectively, as interferents. Additionally, the serious chromatographic peak shifts which were observed at different pH were accurately corrected by means of chromatographic peak alignment method based on abstract subspace difference (ASSD) [27]. 2. Simulated and experimental data 2.1. Simulated data 2.1.1. Simulated excitation–emission–kinetic-pH fluorescence data An excitation–emission-matrix fluorescence (EEMF) data set was simulated for 16 samples with three components. The excitation profiles were reproduced by the following equations:

where s homo is the parameter of controlling the homoscedastic noise level; randn is a function which produces a matrix with the size of I  J; max(X..k) represents the element-wise product. Homoscedastic noise influence is investigated by testing different s homo values: 0.001, 0.005, 0.01, 0.02. 2.1.2. Simulated HPLC–DAD–kinetic-pH chromatography data A HPLC system with DAD on 16 samples containing three species is simulated. The chromatographic profiles were reproduced by the following equations: a1;i ¼ 100CHIð0:25; 9; iÞ;

a2;i ¼ 100CHIð0:25; 15; iÞ;

a:3;i ¼ 100CHIð0:25; 20; iÞ; where i = 1, 2, . . . , 40, CHI(a,b,i) = [a  i(b/2  1) e(i/2)]/{2(b/2) [(n/2  1)!]}. The spectral profiles were reproduced by the following equations: b1;j ¼ gsð0:4; 10; 12; jÞ;

b2;j ¼ gsð0:4; 7; 15; jÞ;

a1;i ¼ gsð0:5; 10:9; iÞ; b3;j ¼ gsð0:5; 4; 17; jÞ; a2;i ¼ gsð0:5; 8; 15; iÞ;

a3;i ¼ gsð0:5; 5; 20; iÞ; where i = 1, 2, . . . , 30; gs(x, a, b, i) refers to the value at x of a Gaussian function with standard deviation a and center b, i.e., gs(x, a,b,i) = xe((b  i)2/a2). The emission profiles were reproduced by the following equations: b1;j ¼ gsð1:0; 3; 9; jÞ;

b2;j ¼ gsð1:0; 6; 13; jÞ;

b3;j ¼ gsð1:0; 10; 20; jÞ; where j = 1, 2, . . . , 25; the kinetic profiles were reproduced by the following equations: c1;k ¼ eðaf kÞ ;

c2;k ¼ 1  eðaf kÞ ;

c3;k ¼ 0:5; where k = 0, 1, . . . , 24; the pH profile of each component was distributed in the range of 1.0–2.3. The concentration of each component was distributed in the range of 0–1.0 through orthogonal experimental design. Then, a five-way data array (30  25  25  5  16) was obtained. In addition, homoscedastic noise was added into the simulated data to approximate the real chemical instrumental measurement process. The type of noise was produced as follows: Homo  noise::k ¼ s homo
randnðI; JÞ  maxðmaxðX::k ÞÞ

where j = 1, 2, . . . , 35; the kinetic time, pH, concentration profiles and homoscedastic noise, respectively, were generated by the approach similar to that of the simulated fluorescence data. Then, a five-way chromatography data array with dimensions 40  35  30  11 16 was obtained.

2.2. Experimental data 2.2.1. Reagents and chemicals Naptalam (NAP) and 1-naphthaleneacetamide (NAD) were purchased from Aladdin Chemical Co., Ltd., (Shanghai, China). Indole-3-acetic acid (IAA) was obtained from Sigma Chemical Co., Ltd., (St. Louis, MO, USA). The concentrations of working standard solutions of NAP, NAD, and IAA were 401.6, 207.2, and 53.4 mg mL1, respectively. 2.2.2. Instrumentation and chromatographic conditions Analysis was performed using an LC-20AT liquid chromatographic system (Shimadzu Corporation, Japan), which consisted of a degasser, a pump, a manual injector provided with a 20 mL loop, a column oven and a deuterium lamp–diode array detector (DAD). The column for HPLC separation was a HypersilODS analytical column (Shimadzu, Japan) with a length of 150 mm, an inner diameter of 4.6 mm and a particle size of 5.0 mm. The LC solution software was used for controlling the instrument and data acquisition. The mobile phase was 5% acetic acid–acetonitrile (40:60), pumped at a flow rate of 1.0 mL min1 with sample injection volume of 20 mL. Photometric detection was performed in the range 190–450 nm, with a spectral resolution of 1.2 nm. Data were obtained over an integration period of 640 ms per spectrum. In the MATLAB environment, home-made programs were written and used for data analysis. All calculations were performed on a microcomputer utilizing Microsoft Windows 7 operating system.

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2.2.3. Prediction sample sets 2.2.3.1. Set number 1. Samples containing IAA and NAD as interferents. For NAP, a set of 4 calibration solutions containing analyte of concentration 8.03, 16.06, 24.10, and 32.13 mg mL1, respectively, was prepared by adding 3.0 mL phosphate buffer (pH 1.5, 1.7, or 1.9) to the appropriate working standard solutions of the analyte, and diluting to the mark with ultrapure water in 25.0 mL volumetric flasks. A set of 6 test spiked samples was prepared in 25.0 mL volumetric flasks by appropriate dilution of a mixture of the corresponding working standard solutions and the interfering agrochemicals IAA and NAD with ultrapure water before 3.0 mL phosphate buffer (pH 1.5, 1.7, or 1.9) were added. The contents of interference in each test sample were 0.5 mL IAA of 207.2 mg mL1 and 1.0 mL NAD of 103.0 mg mL1. The kinetic evolutions of HPLC– DAD signals of these solutions at different pH values were then recorded, as described below. For each sample, the reaction mixture was continuously heated in a water bath with a constant temperature of 70  C after its preparation, and at a series of specific moments during the reaction (0, 5, 10, 15, 20, and 25 min) about 1.0 mL of the reaction solution was withdrawn to a centrifuge tube (10 mL round bottom) by a Pasteur pipette, then the centrifuge tube containing the reaction solution was immediately placed into ice-water to cool for 30 s before the HPLC–DAD analysis. The degradation reaction was carried out in three different acid mediums (pH 1.5, 1.7, and 1.9 phosphate buffers, respectively). An HPLC–DAD– kinetic–pH data array with dimensions 291 76  6  3 (retention time  wavelength  reaction time  pH) was obtained. All ten samples were measured during the reaction at the series of specific moments previously specified and the first five-way data array was constructed with dimensions 291 76  6  3  10. Then, the obtained data were subjected to five-way analysis. The retention time and spectral ranges used were selected after suitable consideration of the time and spectral regions corresponding to maximum signals for the analyte and minimizing background signals.

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2.2.3.2. Set number 2. Samples containing IAA, NAD and unexpected component(s) as interferents. The building of set no. 2 was conducted by following the same procedure described for the building of set no. 1, with the exception that the reaction solution was withdrawn using a syringe connected silicone tube rather than Pasteur pipette in the process of transferring reaction solutions. Here, (an) unexpected interferent(s), which may come from silicone tube, was (were) carried into the reaction system. The sample set became more complex than set no. 1. So, it was also constructed as a prediction set to investigate the performance of fourth-order calibration based on AQQLD. Similarly, the second five-way data array with dimensions 291 76  6  3  10 was obtained and also subjected to five-way analysis. 3. Theory 3.1. Quinquelinear model for fourth-order calibration Suppose that fourth-order data are collected in a five-way data array Xqu of size I  J  K  L  M. Each element xijklm of the five-way data array can be written in the following way: xijklm ¼

N X ain bjn ckn dln emn þ eijklm n¼1

i ¼ 1; 2; :::; I; j ¼ 1; 2; :::; J; k ¼ 1; 2; :::; K; l ¼ 1; 2; :::; L; m ¼ 1; 2; :::; M: (1) where ain,bjn, ckn, dln and emn correspond to five underlying profile matrices A, B, C, D and E of Xqu with sizes of I  N, J  N, K  N, L  N and M  N, respectively; The eijklm term is the element of the fiveway residue data array Equ of size I  J  K  L  M. N denotes the number of factors, which is really the total number of detectable physically-meaningful components of interest as well as the interferents and the background. Fig. 2 shows the graphical representation of the quinquelinear model of the five-way data array Xqu. Where A, B, C, D and E are the five underlying profile matrices of Xqu with I  N, J  N, K  N, L  N and M  N, respectively; Iqu is the

[(Fig._2)TD$IG]

Fig. 2. The quinquelinear model for fourth-order calibration.

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five-way diagonal core array of size N  N  N  N  N with ones on the superdiagonal and zeros elsewhere; Equ is the five-way residue data array of size I  J  K  L  M. Quinquelinear model can also be expressed in the following forms:

cyclic symmetry for quinquelinear decomposition can be extended from four-way cyclic symmetry [28]. The visualization of the property for quinquelinear model is shown in Fig. 3.

(1) Five fully slicewise matrices,

Five-way PARAFAC, which is a direct extension of its trilinear model [11,12,29], is commonly carried out through an alternating least-squares minimization scheme. According to the Eqs. (7)–(11), one can obtain five equations as follows,

      Xjkl ¼ Ediag dðlÞ diag bðjÞ diag cðkÞ AT þ Ejkl

Xklm

     ¼ Adiag eðmÞ diag cðkÞ diag dðlÞ BT þ Eklm 

(2)

3.2. Five-way PARAFAC

A ¼ XIJKLM ½ðE  D  C  BÞT þ

(17)

B ¼ XJKLMI ½ðA  E  D  CÞT þ

(18)

(3)

      Xi::lm ¼ Bdiag aðiÞ diag dðlÞ diag eðmÞ CT þ Ei::lm

(4)

  Xijm ¼ CdiagðbðjÞ ÞdiagðeðmÞ Þdiag aðiÞ DT þ Eijm

(5)

C ¼ XKLMIJ ½ðB  A  E  DÞT þ

(19)

      Xijk ¼ Ddiag cðkÞ diag aðiÞ diag bðjÞ ET þ Eijk

(6)

D ¼ XLMIJK ½ðC  B  A  EÞT þ

(20)

E ¼ XMIJKL ½ðD  C  B  AÞT þ

(21)


Five fully stretched matrices,

XIJKLM ¼ AðE  D  C  BÞT þ EIJKLM

(7)

XJKLMI ¼ BðA  E  D  CÞT þ EJKLMI

(8)

XKLMIJ ¼ CðB  A  E  DÞT þ EKLMIJ

(9)

XLMIJK ¼ DðC  B  A  EÞT þ ELMIJK

(10)

XMIJKL ¼ EðD  C  B  AÞT þ EMIJKL

(11)

3.3. AQQLD The ATLD algorithm was proposed by Wu et al. in 1998 as an improvement of the traditional PARAFAC algorithm without any constraints. It has the features of fast convergence and being insensitive to excess factors. Moreover, it has the second-order advantage, which allows for accurate quantification of the analytes of interest even in the presence of uncalibrated interference and background. An extension of ATLD has since been developed for third-order data analysis, i.e., alternating quadrilinear decomposition (AQLD). AQLD not only retains the positive features of ATLD, but can also improve resolution and quantitative analysis

[(Fig._3)TD$IG]


Or five pseudo-fully stretched matrices,

  X::::m ¼ Adiag eðmÞ ðD  C  BÞT þ E::::m

(12)

  Xi:::: ¼ Bdiag aðiÞ ðE  D  CÞT þ Ei::::

(13)

  X:j::: ¼ Cdiag bðjÞ ðA  E  DÞT þ E:j:::

(14)

  X::k:: ¼ Ddiag cðkÞ ðB  A  EÞT þ E::k::

(15)

  X:::l: ¼ Ediag dðlÞ ðC  B  AÞT þ E:::l:

(16)

where  symbolizes the Khatri–Rao product; + symbolizes the Moore–Penrose generalized inverse and diag denotes the generation of a diagonal matrices with diagonal elements equal to the elements of bracket. Due to the inner cyclic symmetry of the quinquelinear model, the five expressions of each form are mathematically equivalent. The property of inner mathematical

Fig. 3. Visualization of the property for quinquelinear model named as five-way cycle symmetry.

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

capability and provide more information. More details of these valuable properties were presented in original reference [4]; so, it was not described here. According to Eqs. (12)–(16), five objective functions can be obtained as:

s ðEÞ ¼

M X

jjX::::m  AdiagðeðmÞ ÞðD  C  BÞT jj2F

(22)

m¼1

s ðAÞ ¼

I X jjXi::::  BdiagðaðiÞ ÞðE  D  CÞT jj2F

(23)

i¼1

s ðBÞ ¼

J X jjX:j:::  CdiagðbðjÞ ÞðA  E  DÞT jj2F

(24)

j¼1

cTðkÞ ¼ diagmðD X::k:: ððB  A  EÞ Þ Þ

(30)

dðlÞ ¼ diagmðEþ X:::l: ððC  B  AÞT Þþ Þ

(31)

T

The steps of the AQQLD algorithm have been described as follows: (1) choose the number of factors, N; (2) randomly initialize matrices A, B, C and D; (3) compute E using Eq. (27); (4) compute A using Eq. (28) and scale A to be column-wise normalized; (5) compute B using Eq. (29) and scale B to be column-wise normalized; (6) compute C using Eq. (30) and scale C to be column-wise normalized; (7) compute D using Eq. (31) and scale D to be column-wise normalized; (8) compute E using Eq. (27); (9) update A, B, C, D and E according to steps 4–8, until the following stop criterion is reached: SSRðmÞ  SSRðm1Þ j j
K X s ðCÞ ¼ jjX::k::  DdiagðcðkÞ ÞðB  A  EÞT jj2F

(25)

SSRðmÞ

SSR

is the residual sum of squares, 0 1 J X I X K X L X M X ¼ kE qu k2F ¼ @ eijklm2 A, m is the current i

k¼1

s ðDÞ ¼

L X jjX:::l:  EdiagðdðlÞ ÞðC  B  AÞT jj2F

(26)

l¼1

Based on the above five objective functions, a new algorithm called alternating quinquelinear decomposition (AQQLD) has been developed. AQQLD minimizes the five objective functions in an alternating manner, i.e., E for fixed A, B, C and D; A for fixed B, C, D and E; B for fixed A, C, D and E; C for fixed A, B, D and E; and D for fixed A, B, C and D. According to the least-squares principle, A, B, C, D and E can be updated by using the Eqs. (27)–(31). Therefore, the least-square solutions of AQQLD are described as follows: eTðmÞ

17 T þ

þ

þ

T þ

¼ diagmðA X ::::m ððD  C  BÞ Þ Þ

(27)

aTðiÞ ¼ diagmðBþ Xi:::: ððE  D  CÞT Þþ Þ

(28)

bðjÞ ¼ diagmðCþ Xj ððA  E  DÞT Þþ Þ

(29)

T

j

k

m

l

iteration number and e = 106. A maximal number of 500 iterations are adopted to avoid possible unduly slow convergence. Because the five-way array decomposition only provides relative values, the analyte concentrations in the prediction samples are obtained after a calibration step is performed by regression of the relative concentrations of each component of interest against its standard concentrations, which is a fourthorder calibration. The MATLAB code used in this paper can be obtained upon request via email from the corresponding author. 4. Results and discussion 4.1. Simulated data arrays 4.1.1. The convergence property and accelerating capacity of AQQLD For each data array, 10 runs of both AQQLD and PARAFAC with random initialization were carried out. All of the runs of AQQLD have converged to satisfactory results within many fewer iterations than PARAFAC. The last three columns of Tables 1 and 2 listed the performances of AQQLD and PARAFAC for the simulated data arrays with N = 3. At low noise level anoise = 0.001 for the two simulated data, PARAFAC required 42 and 82 iterations to converge, respectively. The increase of the noise level from 0.001 to 0.02, the average

Table 1 The influence of N on the final results obtained by PARAFAC and AQQLD for two simulated three-component data sets with noise level anoise = 0.01. Simulated data

N

Algorithm

Recovery (%)a

RMSEP (104)

Fitnessb

EEMF

3

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD

100.0c 100.0 100.0 100.0 100.2 100.0

4.41 8.50 7.53 7.57 123.58 8.04

0.9932 0.9911 0.9931 0.9930 0.9931 0.9929

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD

100.0 100.0 100.8 100.0 99.9 100.0

2.67 3.78 67.91 3.75 41.06 4.03

Iterative number/time(s) Min

4 5

HPLC

a b

3

0.9919 0.9915 4 0.9857 0.9848 5 0.9920 0.9929 h P 1 The root-mean-square error of prediction (RMSEP) can be calculated in terms of the formula as RMSEP ¼ I1 cact Fitness used to assess the fitness between the fitted and real data array is calculated as follows:

Max

Average

23/4.6 6/0.7 28/7.4 15/1.6 66/20.6 28/3.8

36/7.4 9/1.0 147/36.5 91/9.2 196/62.1 117/15.0

26/5.9 6/0.8 72/18.0 36/3.7 107/34.0 64/8.0

53/42.4 5/3.1 32/31.5 14/7.0 61/75.8 16/9.5

101/81.2 8/4.5 114/110.4 47/22.4 161/200.6 62/34.9

68/55.8 6/3.7 78/75.5 26/12.5 87/108.4 36/20.8

 cpred

2 i1=2

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Table 2 The final results obtained by PARAFAC and AQQLD for two simulated three-component data sets with three different noise levels. Simulated data

l

Algorithm

Recovery (%)

RMSEP (104)

Fitness

EEMF

0.001

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD

100.0a 100.0 100.0 100.0 100.0 100.0

0.46 0.90 1.97 4.73 8.30 16.22

0.9999 0.9999 0.9982 0.9978 0.9733 0.9707

31/6.7 5/0.7 28/5.7 4/0.5 22/4.7 6/0.8

48/10.3 9/1.1 39/8.1 9/1.1 32/6.8 8/1.0

42/9.0 6/0.9 34/7.1 6/0.8 28/5.9 7/0.9

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD

100.0 100.0 100.0 100.0 100.0 100.0

0.24 0.36 1.85 1.98 4.97 6.98

0.9999 0.9999 0.9980 0.9979 0.9691 0.9683

52/42.4 5/3.1 57/46.0 5/3.0 45/37.6 5/3.0

117/94.8 11/6.1 99/79.4 10/5.4 141/116.7 8/4.4

82/66.4 7/4.0 74/60.0 7/4.0 60/49.9 6/3.5

Iterative number/time(s) Min

0.005 0.02

HPLC

0.001 0.005 0.02

a

Max

Average

The values listed are averaged over 10 trials with random initialization.

convergence rate of PARAFAC has a corresponding increase tendency, which is consistent with the observations reported in literature [30]. However, AQQLD can converge much faster. Even at low noise level anoise = 0.001 for EEM data, AQQLD required only an average number of 6 iterations (i.e., 0.9 s) to find the correct results, which is at least 7 times in terms of iterations and 10 times in terms of computation time faster than PARAFAC. For larger HPLC data, AQQLD still required only an average number of 7 iterations (i.e., 4.0 s), which is at least 12 times in terms of iterations and 17 times in terms of computation time faster than PARAFAC. Even when the noise anoise was as high as 0.02, AQQLD still converge much faster than PARAFAC. But the influence of the noise level on the convergence rate of AQQLD was not so striking compared with PARAFAC. In conclusion, either in terms of iterations or computation time, AQQLD has a faster convergence rate than PARAFAC. 4.1.2. The influence of factors on the results’ qualities obtained by AQQLD The simulated three-component fluorescence data with noise level anoise = 0.01 was firstly investigated. From Table 1, it can be seen that the increase of N from 3 to 5 greatly influenced the result qualities of PARAFAC but had hardly influenced on the results of AQQLD. When N = 3, PARAFAC and AQQLD provided good results. The values of fitness and RMSEP for PARAFAC were 0.9932 and 4.41 104, respectively, better than the results of AQQLD (0.9911 and 8.50  104, respectively). The correlation coefficients obtained by AQQLD were slightly inferior to those of PARAFAC (the third and fourth columns of Table 3). These results indicated PARAFAC performed somewhat better than AQQLD. When N = 4, AQQLD provided satisfactory results as good as PARAFAC. However, when the component number of 5 was chosen, PARAFAC did not improve the model fit, leading to worse results (RMSEP increased from 7.53 to 123.58 (104)). Moreover, out of the 10 trials, PARAFAC provided wrong results of curve resolution for 2 times (not shown). In contrast, AQQLD still found the true solutions rather than their linear combinations. The value of RMSEP was only 8.04  104 and its fitness was 0.9929. The provided correlation coefficients were better than the ones of PARAFAC. It meant that AQQLD did not require an accurate estimation of the number of factor in mixtures. Then, a chromatography data with noise level anoise = 0.01 was simulated to investigate the performance of AQQLD in comparison with PARAFAC. The final results for both algorithms with N = 3, 4 and 5 were summarized in the last six rows of Tables 1 and 3. For N = 3, AQQLD provided the results, including recovery, RMSEP, fitness and correlation coefficient, similar to the ones gave by

PARAFAC. Both algorithms can fit model well and yield satisfactory quantitative results. For N = 4, the predicted recovery and RMSEP of AQQLD were 100.0% and 3.75  104, respectively, better than the ones of PARAFAC (100.8% and 67.91 104, respectively). The lowest correlation coefficients provided by AQQLD and PARAFAC were equal to 0.9998 and 0.9995, respectively. For N = 5, AQQLD still gave the results better than the ones of PARAFAC. Moreover, PARAFAC provided wrong results of curve resolution for one time (not shown). However, AQQLD is always able to provide satisfactory and robust results. These results indicated that AQQLD has the property of being insensitive to the excess factors used for calculation in fourth-order calibration.

Table 3 The correlation coefficients obtained by PARAFAC and AQQLD for the analysis of two simulated data sets with different component number (anoise = 0.01). Data

Mode

N=3

N=4

N=5

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD EEMF A

B

C

D

E

HPLC

A

B

C

D

E

a

a1 a2 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3 e1 e2 e3

1.0000a 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 0.9996 0.9997 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9888 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 0.9999 0.9994 0.9995 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 0.9996 0.9998 1.0000 0.9998 0.9996 1.0000 0.9995 1.0000 1.0000 0.9997 0.9995 1.0000 0.9999 0.9999

1.0000 1.0000 1.0000 0.9999 0.9996 0.9996 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

a1 a2 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3 e1 e2 e3

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 0.9998 1.0000 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.9995 1.0000 1.0000 0.9988

1.0000 1.0000 1.0000 1.0000 0.9998 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.9999 1.0000 0.9997 0.9998 1.0000 0.9999 1.0000 1.0000 1.0000 0.9998 1.0000 0.9998 0.9996 1.0000 0.9999

1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

The values listed are averaged over 10 trials with random initialization.

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24 Table 4 The correlation coefficients obtained by PARAFAC and AQQLD for the analysis of two simulated three-component data sets with three different noise levels. Data

Mode

0.001

0.005

0.02

PARAFAC AQQLD PARAFAC AQQLD PARAFAC AQQLD EEMF A

B

C

D

E

HPLC

A

B

C

D

E

a

a1 a2 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3 e1 e2 e3

1.0000a 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 0.9997 0.9984 0.9982 0.9999 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 1.0000

a1 a2 a3 b1 b2 b3 c1 c2 c3 d1 d2 d3 e1 e2 e3

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 0.9999 0.9995 0.9997 1.0000 0.9999 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

The values listed are averaged over 10 trials with random initialization.

4.1.3. The influence of noise level on the results’ qualities obtained by AQQLD Since noise always presented in chemical data arrays, the influence of the noise level on the results’ qualities should, therefore, be carefully investigated. A comparison between the results obtained by AQQLD and PARAFAC analyzing two simulated

[(Fig._4)TD$IG]

19

three-component data arrays with three different noise levels, respectively, is shown in Tables 2 and 4. It can be seen that when at relatively low noise level anoise = 0.001, the results of AQQLD and PARAFAC were in perfect consistence with the real data arrays. Their fitness between the fitted and real data arrays were no less than 0.9999. The correlation coefficients obtained by the both algorithms were equal to 1.0000. When anoise increased to 0.005, the results’ qualities of the two algorithms were still very satisfactory with their fitness no less than 0.99, correlation coefficient no less than 0.9998 and the recoveries of 100%. Further increase of the noise level seems have some influences on the results’ qualities of the both algorithms. However, even when anoise was as high as 0.02, the results of AQQLD were still acceptable, though they were slightly inferior to those of PARAFAC. In practice, noise level seldom exceeded 2% of the maximal absorbance of the data sets, so AQQLD was suitable for practical use in chemometrics. 4.2. Real data sets 4.2.1. Kinetic evolution of naptalam Naptalam (NAP) is rather stable in alkaline solutions at room temperature but undergoes a fast hydrolytic process when the pH is lower than 2.0 and temperature is higher than 50  C. In the experiment, the hydrolysis of NAP was performed in the presence of phosphate buffer at pH 1.5, 1.7, or 1.9, respectively, and at a constant water bath temperature of 70  C. Under hydrolytic conditions, NAP rapidly degraded to 1-naphthylamine (NAA), phthalic acid (PHT) and N-1-naphthylphthalimide (NPI), which is displayed in Fig. 4(c). It was reported that the rate constants (k1) for hydrolysis of NAP at pH 1.49, 1.77, and 1.88 at a temperature of 40  C were 1.75  104 s1, 1.70  104 s1, and 1.63  104 s1, respectively [31]. Very clearly, the differences between the three k1 values were very small and could therefore be neglected under the heating temperature of 70  C. Fig. 4(a) showed the observed degradation profiles of NAP in the presence of the phosphate buffer at pH 1.5, 1.7, and 1.9, respectively, at 70  C, which implied that the pseudo first-order kinetics of the NAP hydrolysis was pH-independent in the pH range from 1.5 to 1.9. However, an effect of H+ concentration on the

Fig. 4. The observed degradation profiles in different pH phosphate buffer. (a) The degradation profiles of NAP at 286.0 nm in calibration sample 1 at pH 1.5, 1.7 and 1.9, respectively; (b) HPLC–DAD chromatograms at 276.5 nm for calibration sample 1 at the degradation time of 10 min at pH 1.5, 1.7 and 1.9, respectively; (c) HPLC–DAD chromatograms at 276.5 nm for calibration sample 1 at a series of specific moments in phosphate buffer of pH 1.9.

20

[(Fig._5)TD$IG]

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

Fig. 5. Contour plots of the HPLC–DAD data; figures (a)–(c) are for calibration sample 1, test sample 1 containing two plant hormones and test sample 1 containing two plant hormones and unexpected component(s), respectively, at the degradation time of 10 min and pH 1.9.

retention time for the hydrolytic product of NAA was observed and shown in Fig. 4(b). As can be seen, an increase on the pH value from 1.5 to 1.9 caused a significant increase on the retention time of NAA. However, the retention time shifts of NAA among different samples were not observed at the same pH value (see Fig. 4(c)). Fig. 5 showed the contour plots of the HPLC–DAD data for calibration sample 1 and test sample 1 in set nos. 1 and 2, respectively, at a degradation time of 10 min and pH 1.9. It was clear that there was chromatographic coelution problem in these samples, due to the overlap of the analyte and interferent signals (Fig. 5(b) and (c)). It illustrated the difficulty in directly studying the hydrolysis process of NAP in complex mixtures. 4.2.2. Set no. 1 The set of 10 samples only containing IAA and NAD as interferents was firstly investigated with the aid of PARAFAC and AQQLD, respectively. Fig. 5(a) showed three distinct time domains from the raw data. Because of peak shifts originating from NAA, the time domain of 2.01–2.99 min was split into two domains from 2.57 min. So, we can study the hydrolysis process of NAP according to the four elution domains, i.e., in the range 1.47–2.00

(I), 2.01–2.57 (II), 2.58–2.99 (III) and 3.98–4.56 (IV) min for PHT, NAP, NAA and NPI, respectively. As mentioned above, the serious chromatographic peak shifts of NAA at different pH (Fig. 6(a)) can be clearly observed in the time domain of III. So, the abstract subspace difference method (ASSD) on basis of the abstract chromatographic peak was applied for peak alignment. Firstly, the calibration sample with maximum concentration of NAA was chosen to be the reference sample, and the number of significant components for the reference and test samples was roughly chosen as 2. The elution chromatographic segment of NAA was chosen to be the chromatographic domain between 2.58 and 2.99 min. Then, the peak shift points for all samples were estimated and aligned by ASSD. The aligned chromatogram was shown in Fig. 6(b). It can be seen that the problem of chromatographic peak shifts was successfully resolved by ASSD. Then, PARAFAC was used to process the obtained five-way data array based on four different elution chromatographic segments. The analysis based on the core consistency diagnostic test (CORCONDIA) [32] indicated that four factors for the model II were necessary. The analysis of models I, III and IV were done by following a similar procedure to that described for the analysis of II.

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

[(Fig._6)TD$IG]

21

Fig. 6. Chromatograms for 1-naphthylamine at all samples. (a) Raw chromatographic profiles; (b) aligned by ASSD.

[(Fig._7)TD$IG]

Fig. 7(a1–e1) showed the normalized elution time, spectral, kinetic, pH and relative concentration profiles of NAP and its hydrolysis products, which were resolved from the HPLC–DAD data of model II by PARAFAC. It was found that the obtained elution time

and spectral profiles of NAP for the four-component model were in good agreement with the actual ones. The average recovery (AR), standard deviation (SD), root mean square error of prediction (RMSEP) and t-test were presented in the

Fig. 7. Normalized elution time (a1), spectral (b1), kinetic (c1), pH (d1) and relative concentration (e1) profiles for NAP and its hydrolysis products, which were resolved from HPLC–DAD of set no. 1 by PARAFAC. Dark green solid, pink long-dash, dark red short-long, blue dash-dot and green dash-dot-dot lines designate the resolved profiles of NAP, NAA, IAA and NAD, PHT and NPI, respectively. The green dotted line represents the actual profile of NAP. The other lines denote the interferences and the background in samples of set no. 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

Table 5 Quantitative prediction results and statistical validation parameters in the presence of interferences between naptalam and two plant hormones or unexpected components by using fourth-order calibration methods based on PARAFAC and AQQLD algorithms for five-way data, respectively. Sample no.

Predicted concentration (mg mL1)

Actual value (mg mL1)

Set no. 1

I II III IV V VI AR S.D. (%)b T-testc RMSEP (mg mL1) Iterative number Time(s)

8.03 12.85 17.67 22.49 27.31 32.13

[Recovery (%)] Set no. 2

PARAFAC

AQQLD

PARAFAC

AQQLD

7.92 [98.5]a 12.76 [99.3] 17.60 [99.6] 22.76 [101.2] 27.23 [99.7] 32.28 [100.5] 99.8 0.7 0.48 0.010 265 32.9

8.01[99.7] 12.75 [99.2] 17.64 [99.8] 22.70 [100.9] 27.25 [99.8] 32.03 [99.7] 99.9 0.4 0.56 0.0072 47 3.0

8.06[100.4] 13.25 [103.1] 17.84 [100.9] 22.17 [98.6] 27.13 [99.3] 30.72 [95.6] 99.7 1.8 0.29 0.042 151 15.4

8.18 [101.9] 13.34 [103.8] 18.03 [102.0] 22.22 [98.8] 27.06 [99.1] 30.79 [95.8] 100.2 2.3 0.19 0.043 76 3.4

a

The values listed are averaged over 5 trials with random initialization. AR, average recovery; S.D., standard deviation. pffiffiffi c T ¼ ðX  m0 Þ=ðS= nÞ, here X is the average recovery, m0 is 100%, n is the degree of freedom (where n + 1 is the number of evaluated levels), and confidence level is 95%, 5 T 0:025 ¼ 2:57. b

third column of Table 5. All predicted results were seen to good, suggesting that the fourth-order calibration method based on PARAFAC is capable of quantitatively analyzing NAP hydrolysis in the presence of IAA and NAD as uncalibrated interferents. Useful information about the reaction could be obtained from the analyte kinetic profiles. The obtained kinetic parameters were displayed in the second row of Table 6. The k1 was 0.1537 min1 and the half-life (t1/2) was 4.51 min. The results revealed that the degradation reaction of NAP followed pseudo first-order kinetic model. Furthermore, the quinquelinear decomposition also provided an additional pH profile of NAP. In Fig. 7(d1) a horizontal linear relationship between UV absorption intensity of NAP and pH was exhibited. It indicated that the pseudo first-order kinetics of the NAP hydrolysis was pH-independent under experimental conditions, which was in good agreement with experimental observations. The profiles of NAA and NAD were fitted into the same straight line as that of NAP by the algorithm. It implied that there was a serious collinearity of pH profiles among the two component factors in the data system. However, the fourth-order calibration method based on PARAFAC successfully circumvented the problem and provided satisfactory results. AQQLD was then applied to the same set. With similar to PARAFAC analysis process, all of the resolved profiles (a2–e2) are showed in Fig. 8. It can be observed that the obtained spectral profiles (a2 and b2) of NAP were almost the same as the actual ones. The quantitative and statistical results of AQQLD were summarized in the fourth column of Table 5. It can be seen that the prediction ability of AQQLD appeared to be reasonably nice even better than the one of

Table 6 Regression equation, correlation coefficient, rate constant and half life of degradation reaction of naptalam. Data

Algorithm Regression equation

Set 1

PARAFAC AQQLD

Set 2

PARAFAC AQQLD

Correlation coefficient

Rate constant (min1)

Half life (min)

y = 0.1537x + 0.0597 0.9989 y = 0.1537x + 0.0525 0.9986

0.1537 0.1537

4.51 4.51

y = 0.1546x + 0.0575 0.9984 y = 0.1533x + 0.0531 0.9981

0.1546 0.1533

4.48 4.52

PARAFAC. In comparison to PARAFAC, AQQLD provided better recoveries, lower RMSEP, less iterations and computation time. Fig. 8(c2) also showed the resolved kinetic profile by AQQLD. The obtained kinetic information was shown in the third row of Table 6. One can see that AQQLD gave the same results as the ones of PARAFAC. Additionally, the pH profiles of NAP, NAA and NAD provided by AQQLD were consistent with the ones gave by PARAFAC. In conclusion, these results from set no. 1 provided strong evidence that fourth-order calibration based on PARAFAC and AQQLD improved resolution and quantitative analysis capability and provided more information. Moreover, AQQLD converged obviously much faster than PARAFAC in the case. 4.2.3. Set no. 2 The proposed method was then applied to study more complex samples, which contained IAA, NAD and (an) unexpected component(s). The presence of unexpected interfering species made the study of the hydrolysis of NAP more difficult. However, the fourthorder calibration method based on PARAFAC and AQQLD provided an alternative to directly quantifying NAP even in the presence of uncalibrated interferents by exploiting the second-order advantage. Using the same procedure as for the analysis of sample set no. 1, good results were obtained in term of predicting the elution time, spectral, kinetic, pH, and concentration profiles of NAP from sample set no. 2. Only the elution time profiles obtained by the both algorithms are shown in Fig. 9(a3–a4), the others are not displayed here because the results were similar to those of sample set no. 1. It is important to note that IAA and NAD, as two interferents, were fitted into one profile due to constant ratio of their concentrations in all of the predicted samples (see Fig. 9). These coeluted components produced significant interference for the direct analysis of the hydrolysis process of NAP. However, the elution time profiles of NAP found by PARAFAC and AQQLD were almost identical to the actual ones. The obtained quantitative and statistical study of the results (the last two columns of Table 5) indicated that neither the quantitative results nor RMSEP were affected by the unexpected component(s). For more complex samples, AQQLD gave the results similar to or even better than the ones found by PARAFAC. The t values were obtained for n degrees of freedom at a 95% significance. Herein T = 0.78 < T50.025=2.57 suggested that the results were also accurate

X.-D. Qing et al. / Analytica Chimica Acta 861 (2015) 12–24

[(Fig._8)TD$IG]

23

Fig. 8. Normalized elution time (a2), spectral (b2), kinetic (c2), pH (d2) and relative concentration (e2) profiles for NAP and its hydrolysis products, which were resolved from HPLC–DAD of set no. 1 by AQQLD. For interpretation of the references to color and symbols in this figure legend, the reader is referred to Fig. 7.

[(Fig._9)TD$IG]

Fig. 9. Normalized elution time profiles (a3 and a4) for NAP and its hydrolysis products, which were resolved from HPLC–DAD of set no. 2 by PARAFAC and AQQLD, respectively. For interpretation of the references to color and symbols in this figure legend, the reader is referred to Fig. 7.

and reliable. In terms of either computation time or iterations, AQQLD still had a faster convergence rate than PARAFAC. On the other hand, the k1 and t1/2 provided by AQQLD were 0.1533 min1 and 4.52 min, respectively, better than the ones of PARAFAC (0.1546 min1 and 4.48 min, respectively). 5. Conclusions In this paper, we have developed a new alternating quinquelinear decomposition (AQQLD) algorithm for the processing of fiveway data, which was obtained by recording the HPLC–DAD signals of samples as a function of the reaction time and pH values. Simulations were carried out to investigate the performance of AQQLD in comparison with the classical PARAFAC. Then, these

methodologies have been successfully applied to investigate quantitatively the kinetics of naptalam (NAP) hydrolysis in two practical sample systems even in the presence of uncalibrated interferents. Satisfactory results were obtained for the determination of NAP in these samples by the both algorithms. Moreover, the newly introduced algorithm, AQQLD, held the advantages of fast convergence and being insensitive to the excess factors used in calculation, indicating the potential for the utilization of fourth-order data for some complex systems. However, in the near future, more related works on the development of more new theories or the production of more novel applications on the basis of fourth-order calibration methods are worth continued investigation to elucidate further the advantages of fourth-order calibration.

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