Tipificación del feminicidio en el Perú

The selection of th e model param eters which give th e b e st approximation to actual d ata points th en becomes an optim isation problem. I t is im portant however to remember th a t th is p ro cess m ust fit into th e time scales available within a chrom atographic separation. Since th e optimisation p ro cess re q u ire s a complete chromatogram to analyse, on-line control of th e separation using th e c u rr e n t sep ara tio n ’s d ata is not possible.

Two possible a ltern a tiv e s however a re possible, e ith e r u se data from previous sep aratio n s to control th e position of th e p ro d u ct fractio n in th e c u rre n t sep aratio n or

use th e an aly sis to decide which collected fractio n s should be pooled to g eth er to form th e p ro d u ct fraction.

With e ith e r of th ese a ltern a tiv e s th e re is a sig n ifican t period of time available to c a rry out th e optim isation process. This time la sts from when th e elution of material from th e column is completed u n til th e loading of th e next sep aratio n is complete.

However not all of th is time is available fo r deconvolution as o th er processes m ust be carried out such as identification (see c h a p te r 3) and control decisions (see c h ap ter 4) based on th e information gained from th e deconvolution. A more detailed discussion of th e time re q u ire d for all th ese p ro cesses is given in c h a p te r 5.

The Box-Complex algorithm (Box, 1965) is used in th is th e s is and is described below.

2.2.4.1 Box-complex optimisation

The Box-complex tech n iq u e (Box, 1965) is a directed se arc h optim isation method sim ilar to th e Simplex method (Nelder and Mead, 1965) b u t with a co n strain ed feasible space.

Initially a num ber, n, of c o n stra in ts (equal to th e num ber of param eters) a re set. For th e ith param eter:

^ X. k h. (2.8)

These c o n stra in ts define th e feasible space. Initially n+1 (w here n is th e num ber of p aram eters to be optimised) v e rtic e s a re random ly selected, eg. for th e ith vertex:

X, = gf + - g,) (2.9)

The lowest point in th e complex is th e n found and th is is replaced by a point a > 1 from th e centroid of th e rem aining points, th e new point being co-linear with th e old point and th e centroid. If th e new point is still th e lowest it is replaced with a point half way betw een th e old point and th e centroid of th e remaining points. If an explicit c o n stra in t is violated th en th e offending point is moved j u s t inside th e feasib le space and th en th e point is modified in th e same manner as fo r a rep eatin g lowest point.

This pro ced u re is rep eated until e ith e r an a rb itr a r y maximum num ber of iteratio n s has been exceeded or th e o bjective function is below a certain th resh o ld (see fig u re 2.7).

One advantage of th is method is th a t it does not re q u ire d e riv a tiv e s of th e objective function. This makes th e mathematics of th e problem much sim pler b u t makes th e ra te of convergence slower, since th e ra te of descent of th e objective function tow ards zero is not used.

The Box-Complex optim isation and indeed any o th er optim isation method may fail to obtain an optimum solution to a given problem. This may be due to one of a num ber of reaso n s which a re listed in tab le 2.1.

Table 2.1 Box-Complex Algorithm Failure Modes

Description of problem Consequences Possible solutions 1 A minimum does not exist within

the feasible space drfined by the constraints.

Algorithm attempts to move the complex vertex outside the feasible space. The parameters are forced against the constraints

Modify the constraints

2 The randomly chosen complex does not converge onto a global minimum.

A local minimum is found. Restart the algorithm and check that the same solution is reached. If it is then it can be assumed that the global minimum has been found since the starting point is randomly chosen.

3 The lowest point in the complex repeats after modification as the lowest point This will occur if the objective function is concave, ie. the objective function decreases towards the centroid.

The complex will collapse onto the centroid at a non-zero (or low) value rf the objective function

Use an alternative objective function or peak model.

2.2.4.2 Deconvolution failure

T here a re a num ber of situ atio n s w here deconvolution methods may fail. All methods of deconvolution have a limit on th e degree of peak sep aratio n th e y can accommodate before th ey begin to fail. With c u rv e fittin g th is o ccu rs when th e overlapping peaks could each comprise a num ber of d ifferen t combinations of peaks. This is described mathematically as ill conditioning, ie. th e re is no unique solution. The d eg ree of sep aratio n below which th e method will fail is dependent on a num ber of fa c to rs including th e relativ e h eig h ts of th e overlapping peaks and th e d eg ree of skew ie. th e exact peak shape. For example, a small peak and a relativ ely larg e peak sep arated by th e same differences in rete n tio n time of elution volume, as two larg e peaks will exhibit ill conditioning a t la rg e r d ifferen ces in rete n tio n time th an th e larg e pair. The c u rv e fittin g method of deconvolution does not g u aran tee to find th e global optimum, for in stan ce th e method may select a se t of param eters which is not th e se t which gives th e lowest possible difference between th e two chrom atograms. Also th e re is no g u aran tee th a t th e method will, even if it does actually find th e global minimum, produce peak functions which co rresp o n d to th e actual elution profiles of th e individual com ponents. Such a situ atio n may occur even if th e problem is not ill conditioned. This is because th e actual elution profiles may not co rresp o n d to th e peak models used in th e deconvolution process.

I t is th e re fo re n e ce ssa ry to determ ine u n d er what conditions th e p a rtic u la r deconvolution method will fail to produce th e c o rre c t r e s u lt and how a c c u ra te th e re s u lt will be when th e model converges to th e desired o b jectiv e function value. The accu racy will be determ ined by th e d eg ree of peak overlap and will also be affected by th e size of th e difference betw een th e two chrom atograms. The e ffects of th e se two facto rs will also need to be determ ined, as well as th e most suitable and stab le objective function. The following sections explore th e effects of th e se fac to rs on th e co n v erg en ce of deconvolution methods.