2D Standardization

8/8/2019 2D Standardization

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By: ParvanehBy: ParvanehEbrahimiEbrahimi



A Brief Introduction to PDS



- ( )Piece Wise Direct Standardization PDS- ( )Piece Wise Direct Standardization PDS

M a steaster

S la v ela v e

i

S u b s eu b s e

22S u b s eu b s e

11

- +i u i v

r r j

==

When shift occurs in columns dimension When shift occurs in columns dimension

= (F diag r= (F diag r jj))

…0 0 …0 0 0 0…0

0

=BandWidth 3



- ( )Piece Wise Direct Standardization PDS- ( )Piece Wise Direct Standardization PDS

When shift occurs in rows dimension When shift occurs in rows dimension

Masteraster Slavelave

j

-

+

j

r

j

s

q q j

==

= (F diag q= (F diag q jj))=BandWidth 3

0 0… 0

…0 0



:In this paper

A method is presented to standardize two-dimensional responses (e.g., GC/MS, LC/UV)measured on multiple instruments or on a

single instrument under different operationalconditions.



INTRODUCTION

The second-order data collected from liquidchromatography with ultraviolet spectroscopicdetection (LC/UV) can have a shift in retention time,due to temperature and pressure fluctuations and

column aging, as well as a wavelength shift in the UVspectrometer and a spectroscopic intensity variation,due to the misalignment of the monochromator andlight source intensity changes.

To make things worse, when such instrumentalvariations occur in second-order data, it is difficult todecide whether the variations come from one order orthe other or both.



Theory

first-order piecewise direct standardization (PDS):

R1=R2 F The banded diagonal matrix F is arranged in such a way that theresponse at every specific channel on the first instrument can be

represented as a linear combination of the responses in a smallwindow near this specific channel on the second instrument.second-order piecewise direct standardization (PDS):

N1= A N2 B T h e le ft tra n sfo rm a tio n w ill co rre ct fo r sh ift an d in te n sity

d iffe re n ce s b e tw e e n th e ro w s o f 1 a n d 2 ( . .,e g standardizing

) :th e LC o rd e r w ith th e fo llo w in g b a n d e d d ia g o n a l m a trix

( = , ,… , )i i 1 2 p

The right transformation will correct for shift and intensity differences between the columns of 1 and 2 ( . .,e g standardizing the UV

) :order with the following banded diagonal form

( = , ,… , )j j 1 2 q

,1 N2 (dimensioned m x ) n

,1 N2 (dimensioned m x ) n

(dimensioned m x ) p

≤p m

B(dimensioned n x ) q

≤q n



Numerical solution

It is insightful to write out the expression for a typical element in N1,N1,ij,as a function of a corresponding local submatrix of the elements

in N2 by the use of the two right and left transformation matrices:

N2 is a (u + v + 1) * (r + s + 1) submatrix of N2

ai appears in the expressions for all elements on the ith row of N1and bj appears in the expressions for all elements on the jth columnof N1



A nonlinear least-squares method is proposed here to solvefor all nonzero parameters in A and B simultaneously. The

numerical procedure used is the Gauss-Newton method:

Thus the valid ranges for i and j in eq.are:

u + 1 ≤ i ≤ m- v and r + 1 ≤ j≤ n – sAs a result, p and q are given by:

p = m - u - v and q = n- r -s



Uniqueness of solution

there are p x q usable elements in N1, the number of independent elements in N1, is limited by its rank and is givenas:

rank (N1) x max(m,n)

necessary and sufficient condition for the uniqueness of asolution is:

This equation implies that second-order standardization canbe accomplished with only one sample, preferably acomplicated sample with high rank.Compared to first-order standardization, where multiplestandard samples are required, this may be considered as one

additional second order advantage.



/omputer Simulation of LC UV Data

The chromatograms and spectra for nine components weregenerated using Gaussian peaks. Seven out of these ninecomponents were used to generate a bilinear response matrixN:

The remaining two components were combined with two of the components included in N to form another bilinearresponse M:

To simulate a deviation from the Lambert-Beer law:



:true responses N1 and M1 on the firstinstrument



The responses on the second instrument were simulatedthrough a nonlinear shift of both the chromatographic andspectral axes. This nonlinear shift was generated using a

quadratic form a + bx + cx 2

N and M were subjected to this shift to generate the bilinearresponses on the second instrument, N2,M2.

In the simulation, N1 and N2 were used as the standard

sample responses while M1 and M2 were used as the testsample responses.

h e d a ta o n th e s e c o n d in stru m e n t w a s

:im u la te d b y th e fo llo w in g p ro c e d u re



Responses from two instruments



To measure the stability of the proposed parameter estimation

method, a standard deviation was also calculated for A and B,which is the mean standard deviation of all elements in theestimated A or B for Six noise levels:0.00%, 0.05%, 0.10%,0.50%, 1.00%,and 5.00%(as compared to the A and B calculated at 0.00% noise level).

( ) & ( )T D A S T D B



Modeling ErrorIn each simulation, A and B were estimated from the standardsample responses on both instruments and a residual matrixwas calculated for equation which indicates thegoodness of the model.From this individual matrix, a modeling error was calculatedas the mean of the standard deviations of all elements.

N1= N2 B



/C UV experiment

One standard sample(3 dyes) and one test sample(3 dyes)were made using4 different dyes. Two chromatographic runs were performedwith a one week interval.“The responses of the two samples measured during the first

run”



When these two samples were measured during thesecond run, the responses became significantly

different:



Residual matrix of the samplesafter standardization



Conclusion

v The computer simulation has indicated the capacity of second-order standardization in modeling both thenonlinear shifts and nonlinear intensity changes, with arelatively small bandwidth for the right and lefttransformations.

vv In both the computer simulation and the LC/UV

experimental data study, the extreme importance of thedesign of the standard sample is observed.

v

v

v On the other hand, the selection of bands in thetransformation matrices can be flexible within a certainrange with relatively stable standardization

performance.



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2D Standardization

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