Mathematical structure of chromatographic optimization based on information theory: II. Analytical role of chromatographic variables

J

Mathematical Structure of Chromatographic Optimization Based on Information Theory: H. Analytical Role of Chromatographic Variables

Y. Hayashi* / R. Matsuda

National Institute of Hygienic Sciences, 1-18-1, Kamiyoga, Setagaya, Tokyo 158 (Japan)

Key Words Chromatography, general Optimization Information theory Kalman filter

Summary The influence of variables Z (mobile phase composition X, column length L, etc.) on chromatographic analysis is studied in terms of the two separate stages of the whole optimization process. In the first stage, peak separation is the central concern and the resolution R s can be related to

Shannon information 0j of peak i: ~i C ~R~ where C is a - ~ = az

coefficient (> 0). In the second stage, only chromatograms with no peak overlap are considered and R s is not con- nected with ~i" The analytical role of Z is evaluated by comparing the signs and magnitudes of the derivatives of the information, Oj , and its transmission rate, ~5' with respect to Z in both stages.

Introduction 1) In the preceding paper [1], the analytical role of the mobile phase composition, X, was studied in reversed-phase liquid chromatography in terms of the derivative of the mutual information ~... 0.' is known as FUMI (Function of Mutual

J J

Information) and denotes the precision for peak j. The derivative 0~j/OX was shown to take different mathematical expressions and analytical meanings in the different stages of the total optimization process. These stages are referred to as ~-stage and 9-stage [1].

The ~-stage involves chromatograms suffering overlap of analyte peaks (i.e., R s < 1.5). The O-stage comprises only overlap-free chromatograms (i.e., R~ ___ 1.5).

The most important utility of the derivatives is the derivation of the relationship between FUMI and R sin the q~-stage.

1) For a list of the symbols used see Table 1 of the preceding paper [1].

If one variable Z (column length L, etc.) is slightly changed by 0Z, the increase 01~in R s corresponds to the increase 0Oj in the information ~i ofpeakj except for coefficient C (> 0) (proof for this is given in the Appendix):

0~j C oRs O--Z -= OZ (1)

This relationship holds only in the O-stage, and not in the O-stage where greater separation (ORs/aZ > 0) causes u~" due peak broadening and spoils the precision (O0i/0Z < 0) as is well-known empirically [2].

In the present paper attention is also paid to the analytical efficiency Oj for peakj (see Table I). O: is the transmission rate of the information ~ j by means of c~romatography and represents the rapidity of analysis (see Appendix). The variables Z treated here are mobile phase composition, X, column length, L, mobile phase velocity, v, detection wavelength, L and plate number, N.

Other conclusions in the present paper are:

(i) peak separation is the most effective means in chromatography for obtaining the precision 0i. This is evident from the fact that 190j/0ZI in the q~-stage > [0r in the O-stage;

(ii) the direction of variables Z toward more informatio~ 0j is given by the signs of 0r

(iii) the influence of variables, Z, can be ranked by comparing the magnitudes of OCj/0Z. For example, if I%/a:f] > I a,/OL I, manipulation of mobile phase composition is more effective than that of columr~ length in improving r j;

(iv) the analytical role of lengthening the column without any change in the stationary phase and of the im- provement in N without any changes in L, etc. can be discriminated. Note that R s overlooks this difference (3Rs/0L = ORs/ON), although it can be recognized flora the disparate effects of L and N on analysis time and peak shape.

(v) The variables Z can be classified into three groups. For example, if da~j/d~j > 0 (Z = X, L), the control of Z can immediately improve precision $.. and rapidity ~J; if dt~i/dC0j < 0 (Z = v), any handlin~ of Z cannot improve ~j and 6j without the expense of the others,

374 Chromatographia VoI. 31, No. 7/8, April 1991 Originals

0009-5893/91/4 0374-07 $ 03.00/0 �9 1991 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH

Table !. FUMI and its related functions.*

~j =

8~ i =

O i =

0 =

~i - 8~j FUMI for a peak

l l o g ( A2 ] Intact information

-1--log ( "or (j)-xi +~:r (J) - z i / Information loss 2 -

q

Cj FUMI for q peaks j = t

t~i/1: q Flow of peak j

q

Oj (= r Information flow i = l

* For list of symbols see Table I of Part 1 [1]. The fundamental relation N = (zi/t~i) 2 used in this paper.

Information

~j = ~ - ~

= * . . . . . . . . . . . . . . . qbj = u ; j

8Z oz oz

O-stacLe

a~!

8z

a R s -- C . . . . . . . . ~**

dz

Figure 1 Mathematical properties of ~- and O-stages in whole optimization process.

Cdrrcsponding equation to ~), = - ~jin O-stage does not hold in (1)-stage cnoted by **); #** denote~s that 3f/aZ is not related to R,.

As ment ioned above, the analytical roles of variables Z can be comprehended f rom the effects of Z on ~i and analytical

as a means of gaining the maximum informat ion ~ or information flow t% The analytical effects of Z can be visually recognized f rom the ~ - O plots (see ref. [3]).

10 5

0 0

Figure 2

I . . f ~ "

z / "

/

20 40 L (c rn)

RB

Dependence of FUMI r and R~ on column length L. (- - -): FUMI r in cI,-stage. ( - - ) : FUMI #j in O-stage. (--.--): R s- (o): optimum peak separation ~s (maximum of ~). A~, At ~ 1 = 10000;

= 1; 1~ = 5.25; k i = 5; k i + i = 5.5; N = C..NL (C N = 5000110"cm'- 1);I: 0 = 67 s at L = 10; ~ = 1.53. Optimum separation ~ depends on peak re- solving powers of data processing [28].

Mathematical Properties o f D-Stage and O-Stage

Two peaks in a chromatogram are taken for comparison of r with R s. Figure 1 will help in unders tanding the different mathemat ical expressions of the 9- and ~-stages. FUMI ~j for peak j, informat ion loss St) i, intact information gj, etc. are listed in Table I.

Figure 2 shows the profiles of R s and FUMI r against co lumn length L. Peak j overlaps with peak j +J1. As L increases f rom zero, the information Cj first increases abruptly and R s also increases in the 4~-stage; however, in the a%stage, Cj decreases gently while R s cont inues to increase monotonously. The positive slope 0r in the ~ - stage is at tr ibutable to the chromatographm peak separa- t ion while the negative slope 35.j/3L comes f rom the peak broadening effect [3, 4].

Chromatographia Vol. 31, No. 7/8, April 1991 Originals 375

Let Rsbe the optimum peak separation. In the O-stage where every peak is free of overlap (R s > Rs) , the information loss ~5r is negligible (gr = 0). Therefore, r is equal to one constituent Vi of FUMI ej (= ~j - fir

* i = (2)

Thus, the derivative of FUMI is described as:

3Z 3Z (3)

In the O-stage, the area A: of peak j, standard deviation t~j and noise level g invotvedJin OV../aZ are the only dominat- ing factors of 3r We shodld note that the relationship between R s and FUMI (Eq. (1)) cannot be derived in this stage.

In the ~-stage, the magnitude f 3 (8r I of peak overlap overweighs the effect 13xgj/3Z ] of peak shape. Thus, the derivative 0r can be approximated:

aCj_: a (~r (4) 3Z 3Z

However, r ~ -~5(hi' because ~i takes appreciable values in both stages. This derivation can be confirmed by comparing the slopes 3r of the r line in Figure 2: the broken line represents - 3 (6r in the O-stage and the solid line represents 3~Itj/3Z in the O-stage.

The following statements are well-known:

(A) in the ~-stage, peak separation favors precision; (B) in the O-stage, further separation is not appropriate

for analysis.

Statements (A) and (B) will be contradictory without de- scription of the stages. This is because separation is used as the criterion. This paper demonstrates that with FUMI as an objective function, the above logical trade-off can be construed and replaced by a single statement: chromatographic conditions Z should be changed so that FUMI increases. For example, if 3r :/3L > 0, increasing the column length is recommended for the update state (State- ment (A)). If OC:/aL < 0, shortening of the column is pref-

J erable (Statement (B)). We should note that the above trade-off arises from the fact that always 3Rs/OL > 0.

Optimization in O-Stage The relationship (Eq. (1)) between R s and FUMI ej can be derived from the q~-stage (see Eq. (4) and Appendix). Eq. (1) offers the information-theoretical interpretation of R s.

The derivative ORJaZ is equal to ar (= - ~ (8r except for coefficient C (= 2/(zv2f)) where f is a function ofR s (see Eqs. {A8-All)). Eq. (1) applies in the R s range [R s - (~a/2/4), Rs] where f increases from near 0 to 1 with increasing R s. A small value of R s leads to a large change 3Oj/3Z in FUMI.

Table II shows some examples of ORs/OZ and Or For the sake of simplicity, the mobile phase composttion X is described by the retention time x i for peak j, which is the only variable in the optimization of X. The different signs

of 3r a/3xl and O~b 2/3z2 reflect the relative positions of peaks 1 and 2 (*2 > ~1): the precision r increases with increasing z 2 and R s (x 1 is fixed). Statement A is confirmed in the O" stage. Increasing the column length improves r (3r > 0). The change dL affects xj and N, but 3@j/ON means im- provement in a stationary phase without any changes in L, etc.

Throughout this paper, all the variables Z are assumed to be shifted by the same ratio dZ/Z for comparing the relative magnitudes of 3O./0Z (or 30Y0Z). Increases in Z by dZ/Z leadto increases ~n q~ibY dej ~= Z(Or ) (dZ/Z)) and the coefficients Z (3~/3Z) are compared for evaluating the analytical effects of Z.

The amount of r in the *-stage is not sufficient due to peal~ overlap (6r > ~) and O i will not matter in an analytical sense. Yet the formal relfftionship between Oj and R s can be obtained if dXq/dZ = 0 (see Eq. (A14)).

For velocity, v, Eq. (1) will not hold. If the dependence of N on v (e.g., ref. [5]) is taken into FUMI, the assumption (13(8r > [/hgi/0Z I) will not be met because I dAi/dv I and [ daj/dv I i~a xgj cannot be neglected as corn 2

pared with [dN/dvl in 8r The power spectrum intensity a is kept constant, although the noise level Wc in a chromatogram varies according to the analysis time zq [6].

Optimization in O-Stage and Analytical Role of Z The partial derivatives 3r (= 3~Y3Z) are listed in

�9 J .I Table III. The increase in retention time xj, column length L, or velocity v adversely affects precision r in the O-stage, but increase in A. or N improves r Wavelength ~ is re-

J j" placed by peak area A i. Plate number N is assumed to be independent of v in the small range, dv.

The signs of 3r with respect to L and x 2 are opposite in the two stages. Statement (B) can be supported in the O- stage. Shortening a column is favorable for precision (3@./ 3L < 0). Unlike 31~2/0~ 2 ( > 0 ) in the *-stage, ej decreases in the O-stage with increasing xj (Or < 0) and R s (xj_ 1 is fixed) because of peak broadening effects. A superio~ stationary phase always improves ej (3r > 0 in both stages).

Table 11. Part ial der ivat ives of resolut ion Rs and m u t u a l information ej in O-stage.*

Z

�9 1:2

L

N

3R~ dZ 0Z

N v2 d ~

4 't 1

N Irz dzz

4 '1; 2

Rs dL

2 L

Rs dN

2 N

aej dZ 0Z

N v2 dz 1

2nit2 f x 1

N 1'2 dx z

2nl/2 f "c 2

Rs dL

l c 2 f L

R, dN

rt v2 f N

* L: xl = elL; x2= czL; N = c3 L. ciis cons tan t . For de r iva t ion of re la t ionship, see Append ix .

376 Chromatographia Vol. 31, No. 7/8, April 1991 Originals

Table 111. Partial derivatives of precision q5 i and analytical efficiency "3 i and dO/dr in O-stage.*

Z O*idZ aO~dZ ~O--t az 0Z dr

1 d~j _(Oq+ i_._i_" ] dxq 20q + 1 >0

l dL _ (.6j + 4_~_q )dL 4Oj+ 1 >0 L 4 L ~ Xq

(, v 2 v v "~q

Aj d A i 1 d A i 1_ > 0

Aj 'l;q A i "l'q

1 dN 1 dN 1 N - - - - - - >0 4 N 4Xq N "~q

The following relationships are used in derivation, q;j: j = q for O'0i/~'l: j for sake of simplicity (see Eq. (A14)); q means last peak which has strongest effect on aOj.

L: "q = cjL; N = CNL. v: x i = ci/v; A i = CAj/V. A: A = A (X).

The variables Z affect r in the O-stage through A. and o. i J J J a .~i" A sharp, large peak provides a large amount of ~j.

All the results of the O-stage can be explained by the above Principle. For example, spending more time on the obser- vation process makes the analysis more precise, if the Velocity v is decreased because of the increased Aj (~)~)i/ .~v < 0), and not if x; or L is increased due to the peak broadening effects (~r < 0, O~i/OL < 0) [3, 4].

Tables II and III show that for every Z, I ~ i / b Z I in the ~-Stage > I bO~:/3ZI in the O-stage (e.g., if R s = 1 and Rs = 1.22, then J f = 0.5 (see Eq. (A9)). Thus, Result (i) is Proven. The above inequality also supports the assumption for the O-stage ( 13 (8~j)lbZ [ > [ ~agj/0Z I).

From Table III, the order of effects [ 3#i/3Z [ is evident: Ai > ~: = v > L = N The selection of detection wavelength .k [s of]primary importance in improving #j for optimization tn the O-stage. However, this is not true for the analytical efficiency Oj (see below).

l~ the analytical efficiency O i, Table III shows that decrease in xj or L and increase in v, Aj or N make Oj high. b'or this examination of the signs of 30i/3Z, the experi- ~entalfactsthatr >> 1 areusedandOi - [1/(2Xq)] >> 0(e.g., ~1 = 10, x_ = 100 s~ [3, 4]. The order o~ contribution to Oj is: ~j ~ L > v ~ >> A. > N O. depends on the analysis time Xq rn J " J ore than changes in peak shape O~/i/OZ). Although L has the least effect on ~;, 00,/0L is the second strongest factor. The selection of Z. (~i) also ranks differently with Cj and O~.

The mobile phase composition X (here ~j) is the most commonly manipulated variable in liquid chromatography. The utility of this methodology is supported by the above Conclusion that the effects of X on analysis, a#i/0X and ~6j/3X, always rank high or highest. Optimization of X is a POWerful means in chromatography of improving both Precision and analytical efficiency. In general, the direction of Z towards more information can be given by a vector expression: grad (*) = (3r 30jilL, O~j/bv, ...).

Variables Z can be classified into three groups according to the signs of dO;/d~;. It is well-known that dO~/d~i = (bOi/bZ)/ (a,j/0Z). If dO~j/dq~j > 0 (X, L, X, N) (see Ta'ble'III), bbth ,j and Oj can be increased by the procedure of Z [3]. That is, the precision and rapidity of analysis can be obtained at once by manipulation, especially of X or L. If dOj/dr < 0 (v), either r or 0 i can be improved at the expense of tfae other [3]. The injection interval of overlapped chromatography is a variable of constant #i (dOj/d#j = o~), unless peaks overlap strongly [7]. It is difficult to imagine variables of constant O i (dOi/d~ i = 0). The above situation can be visualized by the O-i9 plots (see ref. [3]).

( la~-~dZ ) and The re la t ive d i f fe ren t ia l s 3#j/~jl_= =-r

0Oj/O i - ~ dZ are introduced for fair comparison

between ~r and ~Oi/~Z of different scales. Table IV shows that for every Z, /~0/Oi [ __. [~r162 ] in the O-stage. The analytical efficiency O i can be expected to be more improved by the optimizatton procedures dZ than is the precision #j [3]. This is the reason why rapid assays (maximum O) of many samples such as in industrial product control can be accomplished by optimization in the O-stage; the conditions to provide overlap-free chromatograms are easy to find, and the most rapid analysis is found in the O- stage [3, 4].

In the O-stage, the information flow 6 is more conspicuous than O, whereas the mutual information �9 is more important in the q~-stage as mentioned above. Two stages �9 and O were named after this fact.

The effects of N and L are the same in the O-stage (3q~i/0L = ~q~/~N). That is, the chromatographic roles of N, increase by lengthening the column and by using a new superior stationary phase, cannot be discerned in this stage (bRs/bL = 0Rs/0N ). It is in the O-stage that clear difference appears: L is linearly related to the "column efficiency" N as is well-known (N = %L), but increase in L spoils at once


Table IV. Relative differentials OOil0j and O'~i/Oi in O-stage.

Z

L

Aj

N

04)i _ 1 8r d Z 4)i ~i OZ

1 dxi 2~j % i

1 dL 44) i L

1 dv 20 i v

1 dAi

r A1

1 dN

0~i _ 1 ~oi dZ 15 i ~j 3Z

L, 241qJ Xq

-6+ ' ) dL I. 40j) L

1 dAj

0i A1

i dN 40j N

the precision 4): and analytical efficiency t~ i in the )5-stage (d~j/dL < 0 anal d)sj/dL < 0). On the other hand, an im- provement in a stationary phase is always acceptable (8~i/dN > 0 in both stages). These different analytical roles of N and L cannot be recognized by R s (always ~Rd0N = ~Rs/OL ). Our earlier paper showed that v is the most prominent factor in O over the wide range of the (I)-)5 space (O > cp )5 >_ c2, etc.; c i is a constant) [3]. The different conclusion of the present paper will hold for the small range (dO, d)5).

F U M I as Information Theory in Analytical Chemistry

This section considers the similarities and differences between FUMI and the orthodox information theory of Eck- schlager and Stepanek [8].

FUMI seemingly lacks a priori probability and refers only to aposterioriprobability (the precision of measurements). The former is indispensable for formulating some measures in information theory [8]. However, FUMI meets the prerequisite. The original form of FUMI for peak j is given [9] as follows:

The mutual information can be explained as the amount d information which the measurement process can provide for analysts and the more the mutual information, the better the estimation [11]. The optimum measureme~ t process can be designed so that the maximum of the mutual: information or the minimum of the filtering error Pk is achieved [11].

The above statements are cited from the engineering lit-! erature in the 1970's [11, 12], but nearly a decade later, the: information theory of Kalman filter appeared in the field of analytical chemistry [10, 13-18]. Thijssen et al. derived fla- formation measures called information yield and total i~" formation yield [10]. The latter is just the mutual informa" tion shown in Eq. (5) and corresponds to FUMI except that FUMI gives the clear relationship between the informatio~ and signals (peak shape and overlap) [9]. Without the relationship, this series of studies on the mathematic~ l structure of optimization could not be established.

Even if the apriori probability is left in the FUMI formula, the results and discussion of the present and preceding papers will not be altered at all. The a priori probability corresponding to P0 is independent of any chromatograph" ic variables Z treated here. In other words, the variance of the analyte concentration (Po) is never controllable by any chromatographic manipulation, but is an environmental variable [1]. That is, 0log (det P0)/0Z = 0. Furthermore, the effects of the covariance P0 on the filtering error Pk iS known to decrease as the Kalman filtering proceeds witla time [11].

FUMI does not concern all the error sources in chromatog" raphy such as the injection error e which is actually one of the most important disturbances. However, the optimization of e with respect to Z will make no sense in theory and practice because 0e/3Z = 0. Similar suggestions have been given by Cela et al. [19].

We should note that the optimization theory presented here concerns the a posteriori probability and the "quanti- tative" aspects of chromatography. The importance of a priori probability is beyond question for "qualitative" anal- yses.

I [log (det Po) - log (det Pk)] (5) i [1,k] =

where Pk denotes the error covariance matrix of a Kalman filter; P0 is diagonal and denotes the initial errors (the initial entropy [10]). In FUMI, all the diagonal elements of Po are assumed to be the same (unity). This assumption has also been adopted by some approaches of orthodox information theory where the apriori probabilities go 0 (i) of all the identities i are equal (ga 0 (i) = 1/n 0 for i = 1, 2, ..., %) [8].

FUMI coincides with an information measure described by Eckschlager and Stepanek [8]:

I = H [P (X)] - Z (6)

where H [P (X)] denotes the initial entropy; E is the mean of the entropy for conditional probabilities and is called equivocation. This measure, I, is known in engineering as the mutual information [11].

C o n c l u s i o n s

This paper has tried to elucidate the mathematical structure of optimization by clarifying the unique influences of variables Z on precision ~J and analytical efficiency)5, in the �9 N and )5-stages. It is concluded that the theory of lbUMI entails the concepts of traditional R~-methods for optimization as a special case in the r (see Eq. (1)). In the ~3-stage, FUMI also includes the concepts of Rs-minimum method [20] and the sensitivity function s i [1].

A c k n o w l e d g e m e n t

The authors would thank Dr. K. Okudai ra and Dr. A. Nakamura for their useful suggestions.


Appendix 1. Terminology of Analytical Efficiency

The lexicographic definition of efficiency is "a quality of Producing a satisfactory result without wasting time or energy,, [21]. An analysis of high efficiency may be regard- ed as one that can rapidly treat many samples with a satisfactory precision. A large value of the analytical effi- Ciency O (see Table I) represents the same state.

The definition of ~ is different from the well-known concepts of efficiency in science such as the thermal efficiency (the ratio of output joule to input joule). Information efficiency has been defined in an orthodox way [22], but Corresponds to the time-information performance [8, 23]. In chromatography, the plate number N is often called efficiency [24], possibly because N can be related to the number of peaks which can be precisely quantified in fixed time according to the peak capacity [25]. However, lengthening the column produces inconveniences in this expression as stated in the text.

The analytical efficiency 0 was adopted in the previous Papers for system evaluation rather than analysis time Xq [3, 4]. Even if Xq of the least-time analysis with a sufficient amOunt of precxsion �9 is less than "c_ of the most efficient analysis (the maximum ~), more i~formation O can be Collected from the latter in a fixed time period.

2. Derivation of a(R,)/az, a(aC~)/az and OOj/OZ

For two peaks, i~i of peak j is described by the following equation [26, 27]:

8r (~:f (j)) 1 (1 ~:~ (J) -_~J.) = - ~- log T* ~a~z. ~ ) (A1)

The first peak (j = 1) overlaps at the right-hand side of the peak and the filter-off point ~:f (1) can be given as [9]: ~:f (1) = ~2 - 13~ The filter-off point nf (2) of the SeCond peak which has left-hand side overlap is that ~f (2) = 2 h - (/;1 + [301) [27]. Eq. (A1) is denoted by a function fi:

8r i = _ 1 log (fJ) (A2)

Noting nf (j), we obtain:

1 x2- x l - [~~ (A3) fl = ~- 4- ~1/2.1:51

1 '~2- f2 = �88 + (A4)

/1:1/2. (l 2 L

The functions fi take the following forms by the use of cr i = ,~i/NV2:

1 / ;2- "C1 x,1/2 ~1:2 (A5) fl = ~" *" ~ • gl/2. I;1

1 /;2--Xl x,1/2 I]'gl (A6) f2 = ~" + ~ l-~ gl/2./;2

Using the approximat ions that 2% 2z2--/;2+/;i and x21xl ~. 1 because of the small R s in the ~-stage, the same equation can be obtained from Eqs. (A5) and (A6):

1 4Rs 13 (A7) f = - - t 2 /~1/2 •1/2

If f = 1, then ~ i = 0 and the optimum peak separation R s is given as [27]:

Substituting Eq. (A8) into Eq. (A7), we obtain:

f : 1 - - ~ (R~s- Rs) (A9)

Since I >_ f > 0 ~ e e Eq. (A4)) [26], f applies in the Rsrange, [Rs- (X1~2/4) ' Rs]"

From Eqs. (A9) and (A1) (f = fi)' we obtain:

d (8(~j) _ 2 (A10) dR s nl/2f

Since d ( 8 ~ j ) / d R s = (a(a,p/az)/(aRs/az), (ae/az) = - (d (fiCj)/dR s) (3RJOZ) (see Eq. (4)). If C is written as:

C = ~ (All) /t1/2 f

then Eq. (1) can be obtained. From Eqs. (A2) and (A9), the derivative of 80j is given as:

~(Sd~j)_ 1 ~fj (A12) 0Z 2f 3Z

Using Eq. (A5), we get ~fa/3xl:

Of a q:2 ,Nl/2 + 'C2 (A13)

Since N v2 >> [~ (e.g., [~ = 3.25, N = 10000) and xl -- x2, equation b4h/3za in Table II can be obtained. 3~z/Sx2 can also be derived in the same way. ~. /SN, ~i/BL and OR~/OZ can also be derived (see the legend of Table II).

The partial derivative of 0 i can be described as follows:

O0i_ 1 O~ i Cj dXq (A14) t)Z ~:q t~Z ,~q2 dZ

If d/;q/dZ = 0, R s can simply be related to Oj.


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Received: Nov. 26,1990 Revised manuscript received: Jan. 14,1991 Accepted: Jan. 25, 1991 B


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Mathematical structure of chromatographic optimization based on information theory: II. Analytical role of chromatographic variables