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Mathematical Biosciences 168 (2000) 137±159
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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 nontrivial 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

*

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.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 4 7 - X

138

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

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) identi®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 A source code used to perform 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 approach 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 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:
_ p† ˆ f…x…t; p†; p† ‡ u…t† g…x…t; p†; p†;
x…t;
y…t; p† ˆ h…x…t; p†; p†;

…1†

x…0† ˆ x0 …p†;
where x…t; p† 2 Rn is the state variable, the input u…
† 2 U the set of all admissible controls, and the
output is y…t; p† 2 Rm . Let M…p† be a neighbourhood of x0 …p† such that M…p† is a connected
manifold and has globally de®ned coordinates x ˆ …x1 ; . . . ; xn †T . It is assumed that f…
;
†, g…
;
†
and h…
;
† are real analytic on M…p† for all p 2 X  Rr , the set of all possible parameter values. It is
also assumed that, for each p, the system is complete (with respect to U); that is, for every control
u 2 U and x0 2 M…p† there exists a solution of the di€erential equation (1) satisfying x…0† ˆ x0
and x…t†P
2 M…p† for all t 2 R‡ [14].
x0 …p†
Let

: u…
† ! y…
; p† denote the input-output map of (1) with initial condition
p
x…0† ˆ x0 …p†. Identi®ability of Eq. (1) is considered in the experiments …x0 …p†; U†. For p; p 2 X, we

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

Px0 …p†

Px0 …p†

write p  p if p …u† ˆ p
ability from [15] will be used:

139

…u† for all u 2 U. The following de®nition of structural identi®-

De®nition 2.1. Model (1) is said to be globally identifiable at p 2 X, if p  p, p 2 X, implies that
p ˆ p. It is locally identifiable at p 2 X, if there is some neighbourhood W  X of p such that
p  p, p 2 W , implies that p ˆ p. Otherwise the system is said to be unidentifiable.

The model is said to be globally (locally) structurally identifiable if it is globally (locally)
identi®able at almost all p 2 X.
2.2. Taylor series approach
The Taylor series approach is used for experiments with a single analytic or impulsive input,
that is, we take U 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 condition t ˆ 0‡ , say). These derivatives
are thus expressed solely in terms of the system parameters p. They can also be incorporated as the
successive coecients in the Taylor series expansion of y…t; p†, that is
ti
y…t; p† ˆ y…0‡ ; p† ‡ y…1† …0‡ ; p† t ‡


‡ y…i† 0 …0‡ ; p† ‡


;
i!

…2†

where
y…i† …0‡ ; p† ˆ lim‡
s!0

di y

…s; p†;
dti

i ˆ 1; 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
For a system of the form (1), it is assumed that a structural identi®ability analysis has been
performed using an appropriate technique (for example, the similarity transformation approach
[15], the Taylor series approach [10], or approaches based on di€erential algebra [19]; for a

comparison of methods see [20]) and that the system has been shown to be unidenti®able for the
experiment performed.

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N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

The following is a summary of a procedure that can be adopted for reparameterising unidenti®able systems to yield a system that is at least locally identi®able. This procedure is described 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
t ˆ 0‡ (namely Eq. (2)) from the system equations and initial conditions. This gives rise to the
sequence of coecients:
…0†

…1†

…i†

yj …0‡ ; p†; yj …0‡ ; p†; . . . ; yj …0‡ ; p†; . . .

for j ˆ 1; . . . ; 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 p 2 X, g…p† is then given by
3
2 …0† ‡
…0†
oy1 …0 ;p†
oy1 …0‡ ;p†
.
.
.
opr
7
6 op1
7

6
..
..
7
6
.
.
7
6 oy …0† …0‡ ;p†
…0†
oym …0‡ ;p† 7
6 m
...
7
6 op1
opr
7
6 oy …1† …0‡ ;p†
…1†
oy1 …0‡ ;p† 7

6
1
…3†
G…p† ˆ 6
...
7:
op1
opr
7
6
..
..
7
6
7
6
.
…1† ‡
6 oym…1† …0. ‡ ;p†
oym …0 ;p† 7

7
6 op
.
.
.
opr
5
4
1
..
..
.
.
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 nullspace NG of the Jacobian is
sought; suppose that
NG ˆ spanfn1 …p†; . . . ; nrÿq …p†g;
where ni …p† 2 Rr for each i.
3.1.3. Step 3: calculate the locally identi®able parameters
The new locally identi®able parameters for the reparameterisation, which are combinations of
T
the original parameters, denoted by / ˆ …/1 ; . . . ; /q † , must be solutions of the partial di€erential
equations given by

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159







o/i
o/ ÿ
;


; i n1 ; . . . ; nrÿq ˆ 0
op1
opr

for i ˆ 1; . . . ; q. In addition, these solutions must satisfy


o/i
rank
ˆ q:
opj qr

141

…4†

…5†

The partial di€erential equations given by (4) give rise to r ÿ q inner products that constitute an
orthogonality condition. The gradient vectors D/i must be orthogonal to the vector space
spanned by the vectors n1 ; . . . ; nrÿq , that is, the nullspace NG 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 of q locally identi®able parameterPcombinations, is constructed. This state-space
transformation preserves the input-output map, px0 …p† …
†, of the system.

Remark 3.1. Steps 2±4 in the procedure are non-trivial in the following sense:
Step 2. 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.
Step 3. 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.
Step 4. 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 performed 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 yields q independent 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. Let D be the distribution de®ned by
D ˆ spanfn1 ; . . . ; nrÿq g:

…6†

Thus D assigns to each vector w 2 W a vector space that is spanned by the vectors
n1 …w†; . . . ; nrÿq …w†. The open subset W is chosen such that the vectors n1 …p†; . . . ; nrÿq …p† are linearly independent for all p 2 W ; that is, the distribution D is non-singular. The Frobenius theorem
asserts that there exist q independent solutions of (4) (in which case D is said to be completely
integrable) if and only if the distribution is involutive and it is this condition that must be veri®ed.

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N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

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 on…p†=op the Jacobian matrix of n, that is,
1
0
on1 …p†
. . . onop1 …p†
op1
r C
B
.. C
..
on…p† B
. C
ˆB
B onr.…p†
C:
op
onr …p† A
@
...
op1

opr

Then testing whether (6) is involutive reduces to checking that, for all p 2 W and 1 6 i; j 6 r ÿ q,
ÿ


ÿ


rank n1 …p† . . . nrÿq …p† ˆ rank n1 …p† . . . nrÿq …p† ‰ni ; nj Š…p† ;

where

‰ni ; nj Š…p† ˆ

onj …p†
oni …p†
ni …p† ÿ
nj …p†
op
op

is the standard Lie Bracket [21].
Stage 2. Complementary vector ®elds nrÿq‡1 ; . . . ; nr , de®ned on W, are chosen such that the set
of vectors


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

is linearly independent for all p 2 W .
Stage 3. For each of the vector ®elds ni , i ˆ 1 to r, consider the ordinary di€erential equation
given by
z_ …t† ˆ ni …z…t††;

z…0† ˆ z0 :

…7†

wit …z†,

The flow, denoted
of ni is calculated; this is the smooth function of t and z such that the
solution of (7) is given by z…t† ˆ wit …z0 †.
Stage 4. Let  > 0 be some small real number, and de®ne U to be the set
fr 2 Rr : jri j <  for 1 6 i 6 rg. The function W : U ! Rr is constructed as
W…r† ˆ w1r1 


 wrrr …z0 †;
where  denotes composition with respect to the argument z. The initial state z0 is arbitrary,
provided W is well de®ned. For a suitable choice of , which might depend on z0 , the function W is
a di€eomorphism onto its range [21].
Stage 5. The inverse of W is determined. The last q rows of the inverse mapping give the independent solutions required of the partial di€erential equations (4). The new locally identi®able
parameter combinations can then be determined from these solutions.
Remark 3.2. The proof of the Frobenius theorem therefore provides a constructive method for
solving the partial di€erential equation (4) that reduces the problem to one of solving a system of
ordinary di€erential equations. This makes the procedure more amenable to symbolic computation within most commercially available systems.

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

143

3.2.2. Extension of Step 4
The second extension of the methodology relates to the construction of an appropriate statespace 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 parameterised 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 transformation (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† ˆ d…t†) in [12]. The
equations governing the evolution of the system are given by
ls…t†x…t†
ÿ Kd x…t†;
Ks ‡ s…t†
ÿls…t†x…t†
;
s_ …t† ˆ
Y …Ks ‡ s…t††

…8†

x_ …t† ˆ

…9†

y…t† ˆ x…t†

…10†

with initial conditions
x…0‡ † ˆ b1 ;
s…0‡ † ˆ b2 ;

…11†
…12†

where x is the concentration of micro-organisms, s the concentration of growth-limiting substrate,
l the maximum velocity of the reaction, Ks the Michaelis±Menten constant, Y the yield coecient
and Kd is the decay rate coecient. The unknown parameter vector p is given by
T

p ˆ …l; Ks ; Y ; Kd ; b1 ; b2 † :

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N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

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 reparameterisation procedure has been extended in a non-heuristic fashion.
The analysis of [12] shows that b1 , l, Kd , b2 Y and b2 =Ks are globally identi®able parameter
combinations. For any two parameter values p; p 2 X, such that p  p, there exists a (linear)
di€eomorphism connecting the states

T
b2
T
…x; s† ˆ k…x; s† ˆ x; s ;
b2
T

is the state
where …x; s†
p ˆ …l; K s ; Y ; Kd ; b1 ; b2 †T .

vector

for

the

system

(8)±(10)

with

parameter

value

4.1.1. The nullspace of the Jacobian matrix
Using M A T H E M A T I C A it can be shown that the ®rst ®ve rows of the Jacobian matrix, after row
reduction, form the submatrix
1
0
1 0 0 0 0
0
B 0 1 0 0 0 ÿ Kb s C
B
2 C
Y C
G…p† ˆ B
B 0 0 1 0 0 b2 C:
@0 0 0 1 0
0 A
0 0 0 0 1
0

Since the model is unidenti®able the full Jacobian matrix has rank strictly less than 6 (the number
of parameters) and in this case q ˆ 5. Therefore the nullspace of the Jacobian can be calculated
from this submatrix. The vector

T
T 
Ks
Y
p2
p3
n1 …p† ˆ 0; ; ÿ ; 0; 0; 1 ˆ 0; ; ÿ ; 0; 0; 1
b2
b2
p6
p6
spans the nullspace NG .
4.1.2. Application of the Frobenius theorem
Let D be the distribution de®ned by D…p† ˆ span fn1 …p†g for all p 2 W ˆ fw 2 R6 : w6 6ˆ 0g.
This distribution is non-singular by the de®nition of the open set W and 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 for R6 , that is
T

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

T

are chosen as the complementary vector ®elds n2 …p†; . . . ; n6 …p†. For each of these vector ®elds the
¯ow wit …z0 †, that solves (7), is straightforward to calculate; for example
w2t …z† ˆ …z1 ‡ t; z2 ; z3 ; z4 ; z5 ; z6 †T :

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

145

The ¯ow of n1 is readily calculated (within M A T H E M A T I C A ) to be

T
z2 …t ‡ z6 † z3 z6
1
wt …z† ˆ z1 ;
; z4 ; z5 ; t ‡ z6 :
;
z6
t ‡ z6
Composing these ¯ows gives the function
1
0
r2 ‡ h1
B …r1 ‡ h6 †…r3 ‡ h2 †…h6 †ÿ1 C
C
B
B h6 …r4 ‡ h3 †…r1 ‡ h6 †ÿ1 C
C;
B
W…r1 ; . . . ; r6 † ˆ B
C
r5 ‡ h4
C
B
A
@
r6 ‡ h5
r1 ‡ h6

where …h1 ; . . . ; h6 † is any arbitrary initial state such that h6 6ˆ 0. Let U ˆ fr 2 R6 : jr1 j < jh6 jg.
Then W : U ! R6 is a di€eomorphism onto its range with inverse given by
1
0
b2 ÿ h6
C
B
l ÿ h1
C
B
ÿ1
C
B
h
K
…b
†
ÿ
h
6
s
2
2
ÿ1
C
W …p† ˆ B
B b2 Y …h6 †ÿ1 ÿ h3 C:
C
B
A
@
Kd ÿ h4
b1 ÿ h5

Since the hi are arbitrary scalars and independent of the unknown parameters, one can, for example set h6 ˆ 1 and hi ˆ 0 for all i 2 f1; . . . ; 5g. The new parameter combinations, that are at
least locally identi®able, are given by
Ks
/1 ˆ l; /2 ˆ ; /3 ˆ b2 Y ; /4 ˆ Kd ; /5 ˆ b1 ;
b2
which is in agreement with [9]. Note that the number of parameters has been reduced from six
…r ˆ 6† to ®ve …q ˆ 5†.
4.1.3. Coordinate transformation
All that remains to complete the reparameterisation process is to determine a coordinate
transformation n ˆ …n1 ; n2 †T that gives rise to a system parameterised by the new parameter
T
combinations /i . The state vector for the reparameterised model will be denoted by …x; s† .
In line with the observation for the original model, the output of the reparameterisation is given
by
y ˆ x ˆ n1 …x; s†:
Since the transformation preserves the input±output map this implies that
n1 …x; s† ˆ x

…13†

and in particular x ˆ x. The initial state for the reparameterised model is given by
T

T

…x…0†; s…0†† ˆ n…b1 ; b2 † ˆ …b1 ; n2 …b1 ; b2 †† :

…14†

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N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

Finally, note that the derivative with respect to time of the state vector for the reparameterised
model is given by
d
d
n…x; s† ˆ Dn …x; s†;
dt
dt
where
! 
on1
on1
1
Dn ˆ onox2 onos2 ˆ on2
ox

…15†

0
on2
os

ox

os



:

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 determine n. Thus
lxs
…16†
ÿ Kd x
Ks ‡ s
and




on2 lxs
on2
lxs
ÿ
ÿ Kd x ‡
Y …Ks ‡ s†
ox Ks ‡ s
os

…17†

must be functions of x; s; /1 ; . . . ; /5 . To simplify the analysis, an ane map (as motivated by [24])
given by
T

n…x; s† ˆ …x; a1 x ‡ a2 s ‡ a3 † ;
with a2 6ˆ 0, will be considered as a candidate for n. Expression (16), therefore, becomes
/1 x…s ÿ a1 x ÿ a3 †
ÿ /4 x;
…a2 Ks ‡ …s ÿ a1 x ÿ a3 ††

…18†

while (17) becomes


/1 x…s ÿ a1 x ÿ a3 †
/1 x…s ÿ a1 x ÿ a3 †
a1
ÿ /4 x ÿ a2
:
a2 Ks ‡ …s ÿ a1 x ÿ a3 †
Y …a2 Ks ‡ …s ÿ a1 x ÿ a3 ††

…19†

Rearranging expression (18) gives
a 2 Ks
;
…/1 ÿ /4 †x ÿ /1 x
s ÿ a1 x ‡ …a2 Ks ÿ a3 †
which implies that a2 Ks , a1 and a3 are functions of the new parameters. From expression (19), it is
seen that
a2

/1 x…s ÿ a1 x ÿ a3 †
Y …a2 Ks ‡ …s ÿ a1 x ÿ a3 ††

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 a2 b2 must also be a
function of the new parameters. Hence
a1 ˆ F1 …/†;

a2 ˆ

F2 …/†
F4 …/†
ˆ F3 …/†Y ˆ
;
Ks
b2

a3 ˆ F5 …/†;

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

147

where Fi is a function of the new parameters for i ˆ 1; . . . ; 5. The particular choices, given by
F1 …/†  F5 …/†  0;

F2 …/† ˆ /2 ;

F3 …/† ˆ

1
;
/3

F4 …/† ˆ 1;

give rise to the linear state space transformation
n…x; s† ˆ



1
x; s
b2

T

:

The corresponding reparameterised state space model is given by
/1 s…t†x…t†
ÿ /4 x…t†;
/2 ‡ s…t†
_ ˆ ÿ/1 s…t†x…t† ;
s…t†
/3 …/2 ‡ s…t††
x_ …t† ˆ

…20†
…21†

y…t† ˆ x…t†

…22†

with initial conditions
x…0‡ † ˆ /5 ;
s…0‡ † ˆ 1;
where
/1 ˆ l;

/2 ˆ

Ks
;
b2

/3 ˆ b2 Y ;

/4 ˆ Kd ;

/5 ˆ b1 :

Remark 4.1. The form of the original model (8) and (9) has been preserved. In the reparameterisation 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 compartmental model of Fig. 1, used for describing the dynamic behaviour of bromosulphthalein, was shown to be unidenti®able in [7] using the similarity transformation approach. There is also discussion in [7] concerning the addition of constraints on the parameters to
make the model structurally globally identi®able. This provides another method by which the
model of Fig. 1 can be reparameterised.

148

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

Fig. 1. Linear four compartment model from [7].

The experiment under consideration is a unit impulsive input to compartment 1, that is
u…t† ˆ d…t†. The compartmental model presented in Fig. 1, with zero initial conditions, can be
formulated as the state space model given by
dx1 …t†
ˆ ÿa31 x1 …t† ‡ a13 x3 …t†;
dt
dx2 …t†
ˆ ÿa42 x2 …t† ‡ a24 x4 …t†;
dt
dx3 …t†
ˆ a31 x1 …t† ÿ …a03 ‡ a13 ‡ a43 †x3 …t†;
dt
dx4 …t†
ˆ a42 x2 …t† ‡ a43 x3 …t† ÿ …a04 ‡ a24 †x4 …t†
dt

…23†
…24†
…25†
…26†

with initial conditions
x1 …0‡ † ˆ 1;

x2 …0‡ † ˆ x3 …0‡ † ˆ x4 …0‡ † ˆ 0

…27†

and output
y1 …t† ˆ x1 …t†;

…28†

y2 …t† ˆ x2 …t†:

…29†

The vector of parameters will be denoted by p, where
p ˆ …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

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

1
B0
B
B
B0
B
B0
B
G…p† ˆ B
B0
B
B0
B
B0
@
0

0
1
0
0
0
0
0

0
0
1
0
0
0
0

0
0
0
1
0
0
0
..
.

0
0
0
0
1
0
0

0
0
0
0
0
1
0

1

149

1

a04 a24
C
a43 …a42 ÿa04 † C

0

C
C
C
a24
C
a43
C
C:
0
C
C
a24 a42
C
a43 …a04 ÿa42 † C
C
0
A

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

T
a04 a24
a24
a24 a42
n1 …p† ˆ ÿ 1;
; 0; ÿ
; 0;
;1
a43 …a04 ÿ a42 †
a43
a43 …a42 ÿ a04 †

T
p2 p4
p4
p4 p6
ˆ ÿ 1;
; 0; ÿ ; 0;
;1 :
p7 …p2 ÿ p6 †
p7
p7 …p6 ÿ p2 †
4.2.2. Application of the Frobenius theorem
Let D be the distribution de®ned by D…p† ˆ spanfn1 …p†g for all p 2 W ˆ fw 2 R7 : w2 6ˆ w6 ;
w7 6ˆ 0g. This distribution is non-singular on the open set W and 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 ®elds n2 …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 ®eld n1 the ¯ow can be readily calculated within most symbolic computation packages and
is found to be given by
T

p1 …t; z†
z4 z7
2z2 z6 …t ‡ z7 †
1
; z3 ;
; t ‡ z7 ;
wt …z† ˆ z1 ÿ t;
; z5 ;
2…t ‡ z7 †
p1 …t; z†
t ‡ z7
where

p1 …t; z† ˆ t…z2 ‡ z4 ‡ z6 † ‡ z2 z7 ‡ z6 z7 ÿ

q
2
2
…t…z2 ‡ z4 ‡ z6 † ‡ z7 …z2 ‡ z6 †† ÿ 4z2 z6 …t ‡ z7 † :

Composing these ¯ows together gives the function W de®ned by
1
0
ÿr1 ‡ r2 ‡ h1
1
C
B
p …r; h†…r1 ‡ h7 †ÿ1
2 2
C
B
C
B
r
‡
h
4
3
C
B
ÿ1
C;
B
h7 …r5 ‡ h4 †…r1 ‡ h7 †
W…r1 ; . . . ; r7 † ˆ B
C
C
B
r6 ‡ h5
C
B
@ 2…r1 ‡ h7 †…r3 ‡ h2 †…r7 ‡ h6 †p2 …r; h†ÿ1 A
r1 ‡ h7

150

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

where
h
2
p2 …r; h† ˆ h7 …h2 ‡ r3 † ‡ h7 …h6 ‡ r7 † ‡ r1 …h2 ‡ h4 ‡ h6 ‡ r3 ‡ r5 ‡ r7 † ÿ ÿ 4…h7 ‡ r1 †

…h2 : ‡ r3 †…h6 ‡ r7 † ‡ …h7 …h2 ‡ h6 ‡ r3 ‡ r7 † ‡ r1 …h2 ‡ h4 ‡ h6 ‡ r3 ‡ r5 ‡ r7 ††2

i1=2

and h ˆ …h1 ; . . . ; h7 † is an arbitrary initial condition. For simplicity let hi ˆ 0 for all 1 6 i 6 6 and
h7 ˆ 1. Then the inverse of W can be seen to be
1
0
a43 ÿ 1
C
B
a03 ‡ a43 ÿ 1
C
B1
B …a04 ‡ a24 ‡ a42 ÿ a24 a43 ‡ p3 …p†† C
C
B2
C;
a13
Wÿ1 …p† ˆ B
C
B
C
B
a24 a43
C
B
A
@
a31
1
…a04 ‡ a24 ‡ a42 ÿ a24 a43 ÿ p3 …p††
2
where

p3 …p† ˆ

q
…a04 ‡ a24 ‡ a42 ÿ a24 a43 †2 ÿ 4a04 a42 :

Therefore the new locally identi®able parameter combinations are
/1 ˆ a03 ‡ a43 ;

/2 ˆ a13 ;

/5 ˆ a04 ‡ a24 ‡ a42 ;

/3 ˆ a24 a43 ;

/4 ˆ a31 ;

/6 ˆ a04 a42 :

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 corresponding 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.
Let A denote the state matrix for the reparameterised model and x the state vector. The input
and output matrices for the reparameterised model remain those for the original system, namely


1 0 0 0
T
;
B ˆ …1 0 0 0†
and
Cˆ
0 1 0 0
respectively. Since the reparameterised model, characterised by the triple …A; B; C†, is indistinguishable from the original there exists a non-singular transformation T such that
B ˆ TB;

…30†

C ˆ CTÿ1 ;
ÿ1

A ˆ TAT ;
where

…31†
…32†

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

151

Fig. 2. Summary of the indistinguishability analysis after the application of the geometric rules from [22]. All non-zero
rate constants are in bold type. In addition, either a02 6ˆ 0, or both a04 ; a42 6ˆ 0.

0

ÿa31
B 0
AˆB
@ a31
0

0
ÿa42
0
a42

a13
0
ÿ…a03 ‡ a13 ‡ a43 †
a43

1
0
C
a24
C:
A
0
ÿ…a04 ‡ a24 †

Eqs. (30) and (31) imply that T is given by
1
0
1 0 0 0
B0 1 0 0 C
C
TˆB
@ 0 t1 t2 t3 A:
0 t4 t5 t6
Using knowledge gained from the reparameterisation, and application of the geometric rules, A is
given by
1
0
ÿ…a01 ‡ a31 †
0
a13
0
C
B
0
a24
0
ÿ…a02 ‡ a42 †
C:
AˆB
A
@
a31
0
ÿ…a03 ‡ a13 ‡ a43 †
0
a42
a43
ÿ…a04 ‡ a24 †
0
Therefore Eq. (32) implies that T is of the form
0
1
1 0 0 0
B0 1 0 0 C
C
TˆB
…33†
@0 0 1 0 A
0 t4 0 t6

with a01 ˆ 0, and

a02 ‡ a42 ˆ a42 ‡
a24 ˆ

a24
;
t6

a24 t4
;
t6

…34†
…35†

152

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

a42 ˆ




a24 t4
a04 ‡ a24 ÿ
t4 ÿ a42 t4 ‡ a42 t6 ;
t6

a43 ˆ a43 t6 ;

a04 ‡ a24 ˆ a04 ‡ a24 ÿ

a24 t4
:
t6

…36†
…37†
…38†

Eqs. (35) and (37) imply that
t6 ˆ

a24 a43
ˆ
:
a24 a43

…39†

Substituting the ®rst expression for t6 into the remaining equations and rearranging gives
t4 ˆ

a02 ‡ a42 ÿ a42 a04 ‡ a24 ÿ a04 ÿ a24
a24 a42 ÿ a24 a42
ˆ
ˆ
:
a24
a24
a24 …a04 ‡ a24 ÿ a42 †

…40†

Rearranging the last two equalities gives the following relationships between the new parameters
a02 , a04 , a24 and a42 :
a02 ‡ a04 ‡ a24 ‡ a42 ÿ /5 ˆ 0;
2

…a04 ‡ a24 † ÿ /5 …a04 ‡ a24 † ‡ …/6 ‡ a24 a42 † ˆ 0:

…41†
…42†

For the reparameterised model each aij must 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 set a43 ˆ /1 . Hence, from Eq. (39), t6 ˆ /1 =a43 and a24 ˆ /3 =/1 . In
addition, if we assume that a02 ˆ 0 then Eqs. (41) and (42) (considered as a quadratic in
…a04 ‡ a24 †) give
a04 ˆ ÿa42 ‡ /5 ÿ

/3
;
/1


1=2 !

1
/3 a42
/
2
ÿ 3:
/5  /5 ÿ 4 /6 ‡
a04 ˆ
2
/1
/1
Combining these equations it is seen that

q
1
2
a42 ˆ
ÿ /3 ‡ /1 /5  …/3 ÿ /1 /5 † ÿ 4/21 /6 ;
2/1

…43†

and so
1
a04 ˆ
2/1



q
2
ÿ /3 ‡ /1 /5  …/3 ÿ /1 /5 † ÿ 4/21 /6 ;

…44†

whereby
1
t4 ˆ
2/3




q
2
2
ÿ /3 ‡ /1 /5  …/3 ÿ /1 /5 † ÿ 4/1 /6 ÿ 2/1 a42 :

Therefore a possible reparameterisation of (23)±(29) is obtained with B ˆ B, C ˆ C and

…45†

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

0

ÿ/4
B 0
AˆB
@ /4
0

0
ÿa42
0
a42

/2
0
ÿ…/1 ‡ /2 †
/1

153

1

0
C
/3 =/1
C;
A
0
ÿ…a04 ‡ …/3 =/1 ††

where a04 and a42 are given by Eqs. (44) and (43) respectively, and
/1 ˆ a03 ‡ a43 ; /2 ˆ a13 ; /3 ˆ a24 a43 ; /4 ˆ a31 ;
/5 ˆ a04 ‡ a24 ‡ a42 ; /6 ˆ a04 a42 :
The corresponding state space transformation T is given by (33) where t4 is given by (45) and
t6 ˆ a24 . For certain ranges of parameter values aij the reparameterised model is also compartmental 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 reparameterisations 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.
Although the extensions to the reparameterisation procedure presented make the process more
formal and constructive, there still remain steps or choices that are not entirely systematic. For
example, the choice of complementary vector ®elds and initial state h a€ects the di€eomorphism
W. In the examples presented, the simplest and most natural choice for these was made. Moreover,
determining a particular coordinate transformation in the last step is equivalent to constructing a
reparameterised model (in terms of the parameters of the original) and so, since there is a wide
range of possible ideas as to what properties an identi®able model should possess, the process can
never be fully automated. The resulting ¯exibility within the methodology allows the modeller to

154

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

tailor the choice of reparameterisation with respect to the new identi®able parameter combinations 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 Professor 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

MATHEMATICA

Within M A T H E M A T I C A , the model characterised by the triple …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 output y) and appends the corresponding row of the Jacobian to a submatrix J:

Applying this function to both outputs of Example 2 gives rise to the submatrix J which is, after
row reduction, given by:

155

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

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 vector n…p† as a vector ®eld f…
†:

T

The corresponding ordinary di€erential equation given by z_ …t† ˆ f…z…t††, z…0† ˆ …h1 ; . . . ; h7 † , is
solved:

to obtain the ¯ow w1t …z† (in

MATHEMATICA,

psi1[t,z]) given by

156

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

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 state h ˆ …h1 ; . . . ; h7 †T is arbitrary, provided that W is well-de®ned. To obtain the
new parameter combinations the inverse mapping of W is calculated:

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

157

Since the hi , i ˆ 1; . . . ; 7, are scalars, to simplify this mapping we set hi ˆ 0, i 6ˆ 7, and h7 ˆ 1.
The new parameter combinations are obtained from the last 6 rows:

Hence the new parameter combinations are given by
/1 ˆ a03 ‡ a43 ;

/2 ˆ a13 ;

/5 ˆ a04 ‡ a24 ‡ a42 ;

/3 ˆ a24 a43 ;

/4 ˆ a31 ;

/6 ˆ a04 a42 :

Appendix B. Application of geometric rules [22] to Example 2
Rule 1. The length of the shortest path from a perturbed compartment to an observed compartment is preserved.

158

N.D. Evans, M.J. Chappell / Mathematical Biosciences 168 (2000) 137±159

The ®rst compartment is perturbed and observed while the second compartment is also observed. The shortest path from the ®rst to the third compartments must have length 3. Therefore
a21 ˆ 0 and either
a31 ˆ 0;

a41 6ˆ 0;

a24 ˆ 0;

a34 6ˆ 0;

a23 6ˆ 0

or a41 ˆ 0;

a31 6ˆ 0;

a23 ˆ 0;

a43 6ˆ 0;

a24 6ˆ 0:

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 appropriate to label the shortest path as 1 ! 3 ! 4 ! 2. This rules out the case above where
a31 ˆ 0.
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 compartment 1. This can only occur if
a13 6ˆ 0;

a12 ˆ a14 ˆ a32 ˆ a34 ˆ 0:

Applying this rule to the other observed compartment (compartment 2) yields no further information.
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 for a02 ˆ a42 ˆ 0, or a02 ˆ a04 ˆ 0.

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