Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol168.Issue2.Dec2000:

Extensions to a procedure for generating locally identi®able

reparameterisations of unidenti®able systems

Neil D. Evans, Michael J. Chappell

*

School of Engineering, University of Warwick, Coventry CV4 7AL, UK

Received 8 September 1999; received in revised form 28 August 2000; accepted 30 August 2000

Abstract

In this paper extensions to an existing procedure for generating locally identi®able reparameterisations of unidenti®able systems are presented. These extensions further formalise the constructive nature of the methodology and lend themselves to application within symbolic manipulation packages. The extended reparameterisation procedure is described in detail and is illustrated with application to two known non-trivial examples of unidenti®able systems of practical relevance. Ó _{2000 Elsevier Science Inc. All rights} reserved.

Keywords:Unidenti®able systems; Indistinguishability; Reparameterisation procedure

1. Introduction

Given a postulated parameterised state space model, structural identi®ability is concerned with whether the unknown parameters within the model can be identi®ed uniquely from the (ideal, noise-free) experiment considered. Thus structural identi®ability analysis is an important step in the modelling process and is a necessary theoretical prerequisite to experiment design and system identi®cation or parameter estimation.

A problem that arises in structural identi®ability analysis is what to do with unidenti®able systems. This situation has been considered by various authors (see, for example, [1±7]) and in particular a methodology for generating locally identi®able reparameterisations of unidenti®able systems was introduced in [8,9]. A drawback of this methodology was that for certain steps in the process it was not possible to follow a speci®c calculation procedure. Under these circumstances www.elsevier.com/locate/mbs

*_{Corresponding author. Tel.: +44-1203 524 309; fax: +44-1203 418 922.}

E-mail address:mjc@eng.warwick.ac.uk (M.J. Chappell).

0025-5564/00/$ - see front matter Ó _{2000 Elsevier Science Inc. All rights reserved.}

these steps were performed either by inspection or using prior knowledge and experience. Nevertheless the methodology as introduced did permit the generation of locally identi®able reparameterised models.

There may of course be situations where a reparameterisation might not be wholly appropriate and where the model as originally postulated may be the one that needs to be studied. However, a reparameterisation may provide additional insight into the importance of certain (locally) iden-ti®able parameter combinations, particularly if they have physical signi®cance.

The aim of this paper is to extend the methodology introduced in [8,9] in order to provide less heuristic calculation procedures for those steps in the process requiring them. Consequently the steps in the procedure can be, to a much larger extent, performed automatically. This is greatly facilitated by recent advances in symbolic computation. To illustrate the application of symbolic computation to the reparameterisation procedure the M A T H E M A T I C AM A T H E M A T I C A source code used to per-form the analysis of Example 2 is provided in Appendix A. It should again be noted that the analysis can be carried out in most other commercially available systems.

The reparameterisation procedure, as presented here, will be based on the Taylor series ap-proach to structural identi®ability [10]. The extensions provided, however, are equally valid for the similarity transformation approach as discussed in [9]. The extensions to the procedure are illustrated with application to two known examples of unidenti®able systems of importance in biology and pharmacokinetics ([11,12,7], respectively). Much of the analysis in these examples was performed using the symbolic package M A T H E M A T I C AM A T H E M A T I C A [13] (see, for example, Appendix A), but could equally well have been performed within most other commercially available computer algebra software.

2. Taylor series approach to structural identi®ability

2.1. Structural identi®ability

Parameterised systems of the following form will be considered:

_

x…t;p† ˆf…x…t;p†;p† ‡u…t†g…x…t;p†;p†;

y…t;p† ˆh…x…t;p†;p†; …1†

x…0† ˆx0…p†;

wherex…t;p† 2Rn_{is the state variable, the input}

u…† 2U_{the set of all admissible controls, and the}

output is y…t;p† 2Rm. Let M…p† be a neighbourhood of x0…p† such that M…p† is a connected

manifold and has globally de®ned coordinates xˆ …x1;. . .;xn†T. It is assumed that f…;†, g…;†

andh…;†are real analytic onM…p†for allp2XRr, the set of all possible parameter values. It is also assumed that, for eachp, the system is complete (with respect toU_{); that is, for every control}

u2U _{and x0 2M…p† there exists a solution of the di€erential equation (1) satisfying x…0† ˆx0}

and x…t† 2M…p† for all t2R‡ [14]. Let Px0…p†

write pp ifPx0…p†

p …u† ˆ

Px0…p†

p …u† for all u2U. The following de®nition of structural identi®-ability from [15] will be used:

De®nition 2.1. Model (1) is said to beglobally identifiable atp2X, ifpp, p2X, implies that pˆp. It is locally identifiable at p2X, if there is some neighbourhood W X of p such that pp, p2W, implies thatpˆp. Otherwise the system is said to beunidentifiable.

The model is said to be globally (locally) structurally identifiable if it is globally (locally) identi®able at almost allp2X.

2.2. Taylor series approach

The Taylor series approach is used for experiments with a single analytic or impulsive input, that is, we takeU_{to be the set containing the single input. The basis of the Taylor series approach}

([10,16]) is that the output or observation function y…t;p† and its successive time derivatives are evaluated at some known time point (usually an initial conditiontˆ0‡_{, say). These derivatives}

are thus expressed solely in terms of the system parametersp. They can also be incorporated as the successive coecients in the Taylor series expansion ofy…t;p†, that is

y…t;p† ˆy…0‡;p† ‡y…1†…0‡;p†t‡ ‡y…i†0…0‡;p†t i

i_!‡ ; …2†

where

y…i†…0‡;p† ˆ lim s!0‡

diy

dti…s;p†; iˆ1;2;. . .

Since the coecients in the Taylor series expansion are unique and, in principle, measurable, the identi®ability problem reduces to determining the number of solutions for the system parameters in a set of algebraic equations that are, in general, non-linear in the parameters.

For linear and bilinear systems, and systems of homogeneous polynomial form upper bounds are known for the number of derivatives that are required for a full identi®ability analysis ([17,18]). However, for more general forms of non-linear system, no upper bound on the number of derivatives is currently known and thus the technique may only yield sucient results for global identi®ability.

The Taylor series approach will form the basis of the reparameterisation procedure that will be presented in this paper. This will be an extension of that previously considered in [8,9].

3. The extended reparameterisation procedure

The following is a summary of a procedure that can be adopted for reparameterising un-identi®able systems to yield a system that is at least locally un-identi®able. This procedure is de-scribed in detail in [8,9].

3.1. Summary of the reparameterisation methodology

The case will be considered where the reparameterisation process is performed for a single input experiment. The process is performed sequentially using the following steps.

3.1.1. Step 1: calculate the Taylor series

The ®rst step involves calculating the Taylor series of the output, or observation, y…t;p† at

tˆ0‡ _{(namely Eq. (2)) from the system equations and initial conditions. This gives rise to the}

sequence of coecients:

y_j…0†…0‡_;p†;y…1†

j …0‡;p†;. . .;y …i†

j …0‡;p†;. . . for jˆ1;. . .;m:

3.1.2. Step 2: calculate the nullspace of the Jacobian matrix

The partial derivatives of the Taylor series coecients, with respect to the parameters, are calculated. The (possibly in®nite) Jacobian matrix, for some p2X, g…p† is then given by

G…p† ˆ

Since the original model is unidenti®able the rank of the Jacobian matrix, q, will be strictly less than r, the number of parameters (rank test of Pohjanpalo [16]). The value of q can often be established from the identi®ability analysis. Next a basis for the nullspaceNG of the Jacobian is sought; suppose that

NGˆspanfn1…p†;. . .;nrÿq…p†g;

whereni…p† 2Rr for each i.

3.1.3. Step 3: calculate the locally identi®able parameters

o_/i

o_p1; ; o_/i

o_pr

n1;. . .;nrÿq

ÿ

ˆ0 …4†

foriˆ1;. . .;q. In addition, these solutions must satisfy

rank o/i opj

qr

ˆq: …5†

The partial di€erential equations given by (4) give rise torÿq inner products that constitute an orthogonality condition. The gradient vectors D/i must be orthogonal to the vector space

spanned by the vectorsn1;. . .;nrÿq, that is, the nullspaceNG of the Jacobian matrix.

3.1.4. Step 4: derive a state-space transformation

A state-space transformation, which converts the original system into one that is parameterised by the new set ofq locally identi®able parameter combinations, is constructed. This state-space transformation preserves the input-output map,Px0…p†

p …†, of the system.

Remark 3.1.Steps 2±4 in the procedure are non-trivial in the following sense:

Step2. The rank de®ciency of the Jacobian matrix may not be easy to determine. However, this may be possible if some indication of the dependence between the unidenti®able parameters is provided by the previously performed identi®ability analysis.

Step3. It was previously thought [9] that there are no clear cut rules to apply to solve (4). The complexity of this calculation varies from example to example. In addition, there exists no unique solution to this family of equations.

Step4. Generation of the state-space transformation is typically performed by inspection and, to a certain extent, intuition.

3.2. Extending the procedure

3.2.1. Extension of Step 3

The ®rst extension relates to a more constructive method for performing the third step of the procedure. The calculation of the solutions of the partial di€erential equations (4) can be per-formed more readily by application of the constructive method of the proof of the Frobenius theorem (as provided in [21], Theorem 1.4.1). This method yieldsqindependent solutions of (4) as required and is performed in the following stages.

Stage 1. The nullspace vectors n1…p†;. . .;nrÿq…p† are considered as vector ®elds on some open

subset W, of _Rr_{, containing the parameter value}

p. LetD be the distribution de®ned by

Dˆspanfn1;. . .;nrÿqg: …6†

Thus D assigns to each vector w2W a vector space that is spanned by the vectors n1…w†;. . .;nrÿq…w†. The open subset W is chosen such that the vectorsn1…p†;. . .;nrÿq…p† are

In applications where the Jacobian matrix is rank de®cient by 1 this condition is automatically satis®ed since all one dimensional distributions are involutive [21]. More generally, the following provides a method for testing whether a non-singular distribution of the form (6) is involutive. Denote by o_n…p†=o_pthe Jacobian matrix of n, that is,

is the standard Lie Bracket [21].

Stage2. Complementary vector ®eldsnrÿq‡1;. . .;nr, de®ned onW, are chosen such that the set

of vectors

n1…p†;. . .;nrÿq…p†;nrÿq‡1…p†;. . .;nr…p†

is linearly independent for allp2W.

Stage3. For each of the vector ®eldsni, iˆ1 to r, consider the ordinary di€erential equation

given by

where denotes composition with respect to the argument z. The initial state z0 is arbitrary,

providedWis well de®ned. For a suitable choice of, which might depend onz0, the functionWis

a di€eomorphism onto its range [21].

Stage5. The inverse of W is determined. The lastq rows of the inverse mapping give the in-dependent solutions required of the partial di€erential equations (4). The new locally identi®able parameter combinations can then be determined from these solutions.

3.2.2. Extension of Step 4

The second extension of the methodology relates to the construction of an appropriate state-space transformation that yields the locally identi®able reparameterised version of the original postulated model. This extension relates to the generation of a reparameterised state-space model that is indistinguishable from the original in terms of the input-output behaviour.

For linear systems the concept of indistinguishability is well formulated and understood [22]. The techniques of such an analysis can be applied and appropriate indistinguishable models generated. This family of models can then be examined to eliminate those models not parame-terised by the locally identi®able combinations. The remaining models can then be used for the elementary construction of the state-space transformation. In particular for linear compartmental models this permits the incorporation of geometric rules for indistinguishability analysis as de®ned in [22] which makes the construction of the state-space transformation even more straightforward. This will be illustrated in the second of the examples presented in Section 4.

For non-linear systems indistinguishability analysis is not normally exhaustive and generally pairs of systems are compared in the analysis [23]. However, if a particular form for the trans-formation (i.e., ane as motivated by [24]) is chosen, then it is possible to search for all possible forms of model that incorporate the locally identi®able parameter combinations established in Step 3 of the procedure. This approach will be illustrated in the ®rst of the examples presented in Section 4.

4. Examples

4.1. Batch reactor

The following mathematical model for a batch reactor was introduced by Holmberg in [11] and was shown to be unidenti®able for an impulsive input experiment (u…t† ˆd…t†) in [12]. The equations governing the evolution of the system are given by

_

x…t† ˆls…t†x…t†

Ks‡s…t†

ÿKdx…t†; …8†

_

s…t† ˆ ÿls…t†x…t†

Y…Ks‡s…t††

; …9†

y…t† ˆx…t† …10†

with initial conditions

x…0‡† ˆb1; …11†

s…0‡_{† ˆb}

2; …12†

wherexis the concentration of micro-organisms,sthe concentration of growth-limiting substrate, lthe maximum velocity of the reaction,Ksthe Michaelis±Menten constant,Ythe yield coecient

and Kd is the decay rate coecient. The unknown parameter vector pis given by

pˆ …l; Ks; Y; Kd; b1; b2† T

A globally identi®able reparameterisation of this system was presented in [8,9] using the basic methodology described in Section 3.1. We now revisit this example to illustrate how the repa-rameterisation procedure has been extended in a non-heuristic fashion.

The analysis of [12] shows that b1, l, Kd, b2Y and b2=Ks are globally identi®able parameter

combinations. For any two parameter values p;p2X, such that pp, there exists a (linear) di€eomorphism connecting the states

where …x;s†T is the state vector for the system (8)±(10) with parameter value pˆ …l; Ks; Y; Kd; b1; b2†

T

.

4.1.1. The nullspace of the Jacobian matrix

UsingM A T H E M A T I C AM A T H E M A T I C Ait can be shown that the ®rst ®ve rows of the Jacobian matrix, after row

Since the model is unidenti®able the full Jacobian matrix has rank strictly less than 6 (the number of parameters) and in this case qˆ5. Therefore the nullspace of the Jacobian can be calculated from this submatrix. The vector

n1…p† ˆ 0;

spans the nullspaceNG.

4.1.2. Application of the Frobenius theorem

Let D be the distribution de®ned by D…p† ˆspanfn1…p†g for all p2W ˆ fw2R6 :w6 6ˆ0g.

This distribution is non-singular by the de®nition of the open setWand has dimension 1. Since all one-dimensional distributions are automatically involutive it is completely integrable by the Frobenius theorem. The proof of this provides the constructive method that will be used in the following.

The ®rst ®ve elements of the canonical basis forR6, that is

…1; 0; 0; 0; 0; 0†T;. . .;…0; 0; 0; 0; 1; 0†T

are chosen as the complementary vector ®eldsn2…p†;. . .;n6…p†. For each of these vector ®elds the

¯ow wi

t…z0†, that solves (7), is straightforward to calculate; for example

w2_t…z† ˆ …z1‡t; z2; z3; z4; z5; z6† T

The ¯ow ofn1 is readily calculated (withinM A T H E M A T I C A) to beM A T H E M A T I C A

Composing these ¯ows gives the function

W…r1;. . .;r6† ˆ

Then W_:U !R6 is a di€eomorphism onto its range with inverse given by

Wÿ1…p† ˆ

Since the hi are arbitrary scalars and independent of the unknown parameters, one can, for

ex-ample set h6 ˆ1 and hi ˆ0 for all i2 f1;. . .;5g. The new parameter combinations, that are at

least locally identi®able, are given by

/₁ ˆl; /₂ ˆKs

b2

; /₃ ˆb2Y; /4ˆKd; /5ˆb1;

which is in agreement with [9]. Note that the number of parameters has been reduced from six

…rˆ6† to ®ve…qˆ5†.

4.1.3. Coordinate transformation

All that remains to complete the reparameterisation process is to determine a coordinate transformation nˆ …n1;n2†

T

that gives rise to a system parameterised by the new parameter combinations/_i. The state vector for the reparameterised model will be denoted by…x;s†T.

In line with the observation for the original model, the output of the reparameterisation is given by

y ˆxˆn1…x;s†:

Since the transformation preserves the input±output map this implies that

n1…x;s† ˆx …13†

and in particularxˆx. The initial state for the reparameterised model is given by

…x…0†;s…0††Tˆn…b1;b2† ˆ …b1;n2…b1;b2†† T

Finally, note that the derivative with respect to time of the state vector for the reparameterised

The right-hand side of (15) must be a function of the state vector…x;s†T and the new parameters only and these facts will be used to determinen. Thus

lxs

witha26ˆ0, will be considered as a candidate for n. Expression (16), therefore, becomes

/1x…sÿa1xÿa3†

which implies thata2Ks,a1 anda3are functions of the new parameters. From expression (19), it is

seen that

a2

/₁x…sÿa1xÿa3†

Y a… 2Ks‡ …sÿa1xÿa3††

must have coecients that are functions of the new parameters. Therefore a2=Y must also be a

function of the new parameters. Considering the initial state (14) shows that a2b2 must also be a

function of the new parameters. Hence

whereF_{i is a function of the new parameters for iˆ1;. . .;5. The particular choices, given by}

F_1…/† F_{5…/† 0;} F_{2…/† ˆ/}

2; F3…/† ˆ

1

/₃; F4…/† ˆ1;

give rise to the linear state space transformation

n…x;s† ˆ x; 1

b2

s

T

:

The corresponding reparameterised state space model is given by

_

x…t† ˆ/1s…t†x…t†

/₂‡s…t† ÿ/4x…t†; …20†

_

s…t† ˆ ÿ/1s…t†x…t†

/₃…/₂‡s…t††; …21†

y…t† ˆx…t† …22†

with initial conditions

x…0‡† ˆ/₅;

s…0‡† ˆ1;

where

/₁ ˆl; /₂ ˆKs

b2

; /₃ ˆb2Y; /4ˆKd; /5ˆb1:

Remark 4.1.The form of the original model (8) and (9) has been preserved. In the reparamete-risation one of the unknown parameter values …b2† has been included in some of the other

parameter values (/₂ and/₃) and hence the number of parameters has decreased by 1.

Although the theory only guarantees that the new parameter combinations are at least locally identi®able, in this example they are actually globally identi®able.

4.2. A four compartment linear model

The experiment under consideration is a unit impulsive input to compartment 1, that is

u…t† ˆd…t†. The compartmental model presented in Fig. 1, with zero initial conditions, can be formulated as the state space model given by

dx1…t†

dt ˆ ÿa31x1…t† ‡a13x3…t†; …23†

dx2…t†

dt ˆ ÿa42x2…t† ‡a24x4…t†; …24†

dx3…t†

dt ˆa31x1…t† ÿ …a03‡a13‡a43†x3…t†; …25†

dx4…t†

dt ˆa42x2…t† ‡a43x3…t† ÿ …a04‡a24†x4…t† …26†

with initial conditions

x1…0‡† ˆ1; x2…0‡† ˆx3…0‡† ˆx4…0‡† ˆ0 …27†

and output

y1…t† ˆx1…t†; …28†

y2…t† ˆx2…t†: …29†

The vector of parameters will be denoted by p, where

pˆ…a03; a04; a13; a24; a31; a42; a43†T:

4.2.1. The nullspace of the Jacobian matrix

After row reduction, the Jacobian matrix with respect to the parameters of the Taylor series coecients is given by

G…p† ˆ

The rank of this matrix is 6 and so the Jacobian is rank de®cient by 1. The nullspace of this matrix is spanned by the vector

n1…p† ˆ

4.2.2. Application of the Frobenius theorem

Let D be the distribution de®ned by D…p† ˆspanfn1…p†g for all p2W ˆ fw2R7 :w2 6ˆw6;

w76ˆ0g. This distribution is non-singular on the open set Wand since it is one dimensional it is

involutive. Hence by the Frobenius theorem this distribution is completely integrable. The ®rst six vectors of the standard basis for _R7, namely

…1; 0; 0; 0; 0; 0; 0†T;. . .;…0; 0; 0; 0; 0; 1; 0†T;

are chosen as the complementary vector ®eldsn2…p†;. . .;n7…p†. As before the ¯ows corresponding

to these vector ®elds, which are the solutions of (7), are straightforward to calculate. For the vector ®eldn1 the ¯ow can be readily calculated within most symbolic computation packages and

is found to be given by

w1_t…z† ˆ z1

Composing these ¯ows together gives the functionWde®ned by

where

Therefore the new locally identi®able parameter combinations are

/₁ˆa03‡a43; /2 ˆa13; /3 ˆa24a43; /4 ˆa31;

/₅ˆa04‡a24‡a42; /6ˆa04a42:

4.2.3. Coordinate transformation

Note that, in this example, the model is compartmental and, as can be readily veri®ed, minimal. Therefore, to simplify the construction of an appropriate coordinate transformation and corre-sponding reparameterised state space model, the geometric rules of [22] are applied (see Appendix B for details). These rules are applied to a general four compartment (linear) system to obtain the candidate model for indistinguishability given in Fig. 2.

LetAdenote the state matrix for the reparameterised model and xthe state vector. The input and output matrices for the reparameterised model remain those for the original system, namely

Bˆ…1 0 0 0†T and Cˆ 1 0 0 0

0 1 0 0

;

respectively. Since the reparameterised model, characterised by the triple …A;B;C†, is indistin-guishable from the original there exists a non-singular transformation Tsuch that

BˆTB; …30†

CˆCTÿ1; …31†

AˆTATÿ1; …32†

Aˆ

Using knowledge gained from the reparameterisation, and application of the geometric rules,Ais given by

Therefore Eq. (32) implies thatT is of the form

Tˆ

a42ˆ a04

Eqs. (35) and (37) imply that

t6 ˆ

Substituting the ®rst expression for t6 into the remaining equations and rearranging gives

t4 ˆ

Rearranging the last two equalities gives the following relationships between the new parameters

a02, a04, a24 and a42:

For the reparameterised model eachaijmust be a scalar or a function of the new parameters. All

that remains is to choose suitable elements t4 and t6 such that this is the case.

Suppose that a reparameterisation is sought with no elimination from the third compartment, in which case it is necessary to seta43ˆ/1. Hence, from Eq. (39),t6 ˆ/1=a43anda24ˆ/3=/1. In

addition, if we assume that a02ˆ0 then Eqs. (41) and (42) (considered as a quadratic in …a04‡a24†) give

Combining these equations it is seen that

a42ˆ

Aˆ

ÿ/₄ 0 /₂ 0

0 ÿa42 0 /3=/1

/₄ 0 ÿ…/₁‡/₂† 0

0 a42 /1 ÿ…a04‡ …/3=/1†† 0

B B @

1

C C A ;

wherea04 and a42 are given by Eqs. (44) and (43) respectively, and

/₁ ˆa03‡a43; /2 ˆa13; /3ˆa24a43; /4ˆa31;

/₅ ˆa04‡a24‡a42; /6ˆa04a42:

The corresponding state space transformation T is given by (33) where t4 is given by (45) and

t6 ˆa24. For certain ranges of parameter values aij the reparameterised model is also

compart-mental in structure. In this particular example it can be readily shown that the reparameterised state space model is structurally locally identi®able.

5. Conclusions

In this paper the problem of what to do with unidenti®able parameterised systems has been considered. The problem has been addressed by presenting a means for generating reparamete-risations of such systems that are, at least locally, identi®able. The procedure described is an extension of that introduced in [8,9]. This extension essentially makes the process less heuristic, particularly within symbolic manipulation packages. The examples presented illustrate this point and show that, particularly for the second example, symbolic computation is an invaluable tool for such analysis [25].

The ®rst of the two extensions to the procedure that have been introduced relates to a more constructive means for the calculation of identi®able parameter groupings (via the proof of the Frobenius theorem [21]). It would appear that the choice of complementary vector ®elds made in Step 3(ii) plays an important role in the form of the solutions to the di€erential equation (7) that emerge. Hence the identi®able parameter combinations obtained may be a€ected by this choice. In the examples presented, a `natural' choice for the complementary vector ®elds was made.

The second of the two extensions relates to the construction of an appropriate state space transformation that gives rise to a reparameterised version of the original system. For non-linear systems this is heavily dependent upon the particular mathematical forms of the model considered and transformation chosen. However for linear systems general rules of indistinguishability analysis can be applied [22]. These make the construction of the transformation a more formal process as illustrated in the second example.

tailor the choice of reparameterisation with respect to the new identi®able parameter combina-tions and their signi®cance relative to the physical system and the experiments performed.

While in many examples the reparameterisation may be intuitive or arise from close inspection of the model, the second example demonstrates that this is not always the case. For such systems this algorithm provides a constructive method whereby a reparameterisation, which is at least locally identi®able, can be generated.

Acknowledgements

This work was supported by EPSRC Grant GR/M11943 `New Approaches to Identi®ability Analysis and their Application to Electronic Nose Experiments'. We are most grateful to Pro-fessor Keith Godfrey (School of Engineering, University of Warwick), and Dr Michael Chapman (School of MIS-Mathematics, Coventry University) for their helpful comments and discussion during the development of this paper.

Appendix A. Analysis of Example 2 using M A T H E M A T I C AM A T H E M A T I C A

Within M A T H E M A T I C A, the model characterised by the tripleM A T H E M A T I C A …A;B;Cmat†, and unknown parameter vector p are input as follows:

The following function (calcrow) calculates the next coecient in the Taylor series (for a given outputy) and appends the corresponding row of the Jacobian to a submatrixJ:

Since the rank of this matrix is the maximum it can be (i.e., 6) the single nullspace vector can be calculated from it:

We now consider the nullspace vectorn…p† as a vector ®eld f…†:

The corresponding ordinary di€erential equation given byz_{_}…t† ˆf…z…t††,z…0† ˆ …h1;. . .;h7† T

, is solved:

The corresponding ¯ows for the complementary vector ®elds are given by

By composing these ¯ows together the di€eomorphism (on a suitable domain of de®nition) W…r† is obtained:

The initial statehˆ …h1;. . .;h7†T is arbitrary, provided that W is well-de®ned. To obtain the

Since thehi, iˆ1;. . .;7, are scalars, to simplify this mapping we sethiˆ0,i6ˆ7, andh7ˆ1.

The new parameter combinations are obtained from the last 6 rows:

Hence the new parameter combinations are given by

/₁ ˆa03‡a43; /2ˆa13; /3ˆa24a43; /4ˆa31;

/₅ ˆa04‡a24‡a42; /6 ˆa04a42:

Appendix B. Application of geometric rules [22] to Example 2

The ®rst compartment is perturbed and observed while the second compartment is also ob-served. The shortest path from the ®rst to the third compartments must have length 3. Therefore

a21ˆ0 and either

a31ˆ0; a416ˆ0; a24ˆ0; a346ˆ0; a236ˆ0

or a41ˆ0; a316ˆ0; a23ˆ0; a436ˆ0; a246ˆ0:

We do not wish to consider models which only di€er in the labelling of compartments 3 and 4, thus a suitable choice is made. Since in the original model a31 is globally identi®able it is

ap-propriate to label the shortest path as 1!3!4!2. This rules out the case above where

a31ˆ0.

Rule 2.The number of compartments with a path to a given observed compartment is preserved (including paths of length zero).

There must be two compartments (including the compartment itself) with a path to compart-ment 1. This can only occur if

a136ˆ0; a12ˆa14ˆa32ˆa34ˆ0:

Applying this rule to the other observed compartment (compartment 2) yields no further infor-mation.

Rule 3. The number of compartments that can be reached from a perturbed compartment is preserved (including the perturbed one itself).

No new information is gained by the application of this rule.

Rule 4. The number of traps is preserved.

A set of compartments, in which there is a path from any given compartment to any other in the set, but no path exists to any compartment outside of it (including the environment) is called a trap.

There are no traps in the original model so it is not possible fora02ˆa42ˆ0, ora02ˆa04ˆ0.

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