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On local identifiability of linear systems
On Local Identifiability of Linear Systems
JACQUES DELFORGE Dipartement de Biologie, Commissariat b I’Energie Atomique, Centre d’Etudes Nuckaires de Saclay, 91191 Gif-sur- Yvette, France Received 4 Ju& 1983; revised I9 January I984
ABSTRACT The identifiability of parameters in a model with known structure and no noise is a problem of accurate determination of the number of parametric space points which are solutions to the identification problem. Mathematicians usually make a distinction between local and global uniqueness. This distinction is very useful in practice as well as in theory, because it relates to difficulties at two distinct levels. This paper establishes a new set of necessary and sufficient conditions for local identifiability, with very broad-based validity: first, a condition for the model to be locally identifiable almost everywhere; second, a condition leading to conclusions as regards any particular point in the parametric space. From the above criteria, we derive several conditions which, although necessary only, are very useful in practice for the rapid detection of certain nonidentifiable models. The above criteria are of very general significance, easily programmable, and readily applied, as demonstrated by the many examples given.
I.
INTRODUCTION
The identifiability of parameters in a model with known structure and no noise is a problem of great importance, since it involves the accurate determination of the number of parametric space points which are solutions to the identification problem. Investigations [l-4] have shown this number to be widely variable; although the model is linear, except in very special cases we met with the well-known difficulties associated with minimization problems when attempting numerical identification of model parameters. In particular, whenever a numerical or analog program provides a solution, it is not known whether the solution is unique (in which case the solution found is fully justified), or if a finite (but more than one) or infinite number of solutions exist. In the latter case, the solution found is only one of many, and there is no reason why it should be preferred. One should then either determine the set of possible solutions and conclude that the actual solution MATHEMATICAL
BIOSCIENCES
70:1-37
QElsevier Science Publishing Co., Inc., 1984 52 Vanderbilt Ave., New York, NY 10017
(1984)
1 0025-5564/84/$03.00
2
JACQUES
DELFORGE
is one of the mathematical solutions found, or, whenever possible, propose experimental procedures leading to a unique solution. Mathematicians usually make a distinction between local and global uniqueness. This distinction is very useful in practice as well as in theory, because it relates to difficulties at two distinct levels. Irrespective of the method used to investigate identifiability, we always arrive at a system of nonlinear equations, and attempt to find the exact number of solutions. Studies of local uniqueness, an obviously necessary condition for global uniqueness, are usually based on expansions, which may be either numerical (involving numerical computation of the solution; such methods are therefore not applicable to structural investigations) or analytical through computation of the Jacobian matrix (the functional matrix in which element ij is the derivative of the i th equation with respect to the jth unknown). The mathematical tools are therefore well known. In practice, however, the various methods proposed are found to differ widely, especially as regards convenience. In particular, the methods based on Iacobian matrix computation show limitations due to the analytical computations required, which grow very rapidly in complexity as the model size increases. Global uniqueness investigations are even more difficult, because usable mathematical results are few and very limited in scope. In most cases, solving the identifiability problem requires explicit analytical computation of the solution, which makes most of these methods practically inoperative as soon as the number of compartments exceeds three or four (except for highly exceptional model structures [4]). It thus appears essential that identifiability investigations be directed toward the development of methods applicable to complex models. A specific requirement is for such methods to be readily programmable, so that model complexity only affects computing time. With that intent, we proposed in a previous publication [6] a necessary and sufficient uniqueness condition which is readily programmable, because it is based solely on model structure and on the characteristics of injection and observation matrices. However, this result included certain inaccuracies, as pointed out in [5, 181, and its validity was subject to a number of restrictive conditions, especially as regards the absence of zero eigenvalues and zero elements in CA2B. This paper states new necessary and sufficient conditions for local identifiability, with much broader validity than the above; a condition for the model to be locally identifiable almost everywhere; and a condition which permits drawing conclusions for each specific point in the parametric space (although this particular condition uses structural properties only). From the above criteria, we derive a number of conditions which, although necessary only, are very useful in practice for detecting nonidentifiable models. The criteria are of very general significance, easily programmable, and readily applied, as demonstrated by the many examples given.
LOCAL IDENTIFIABILITY
II.
REVIEW DATA
1.
EQUATIONS
3
OF EQUATIONS
VERIFIED
DERIVED
BY A LINEAR
FROM
EXPERIMENTAL
MODEL
Time-dependent variations of a linear system with n homogeneous compartments (or equivalence classes in the more general context of transformation systems [7]) are given by the following linear differential system:
dN(t) =AN(t)+Bu(t),
-
dt
tE
N(0) = 0,
y(t) = CN(t),
[O,TI, (2) (3)
where: N(t) is a vector function with dimension n, the i th component of which represents variations of the i th compartment. A is an n x n square matrix. The elements of that matrix have well-known significance [8,9]. However, it should be recalled that a,,, i # j, is a positive number relating to transfers from compartment j to compartment i, and is a positive number relating to exits from compartment j a = 4:&2,,, ti’the outside of the system. It will be assumed that matrix A has distinct eigenvalues. B is a matrix with dimensions n X h. It represents the possibilities of injection into the system. We assume that b compartments can be injected separately through an arbitrary function u(t). We denote by 1, the set of suffixes of such compartments, which will be known as injectable compartments. B is thus defined as B= {e,],,,s,
(4)
where e, is the vector all the components of which are zero except the i th component, which is unity. C is a matrix with dimensions c X n, known as the observation matrix. It represents the possibilities of observation of the system. It will be assumed here that c compartments can be observed. We denote by I, the set of suffixes of such compartments, which will be known as observable compartments. The matrix C is thus defined by
(m’ is the conventional
notation
for the transpose).
Last, we assume the system to be observable and controllable as defined by Kalman [lo], which is equivalent to assuming that the fitting of experi-
4
JACQUES DELFORGE
mental data requires n distinct exponential terms (representing the n eigenvalues). Since in most cases the number of compartments is determined by the number of exponential terms, this condition is not very restrictive [ll]. The solution of the system (l)-(3) is given by a(f-7)B~( T) d7. If Y(t) is the functional
matrix with dimensions
(6)
c x b, defined as
Y(t)= CeA’B,
(7)
we therefore have y(t) = /,jY( t - T)U(T)dT. Y(t) is a transfer function, i.e., Y(t) can be identified from experimental data [using suitably selected functions u(t)]; conversely, if Y(t) is known, experimental observations for any given vector u(t) can be predicted. Investigating the identifiability of the matrix A on the basis of experimental data is therefore equivalent to investigating the determination of A from Y(t), which is assumed to be known through experimental procedures.
2.
EQUATIONS
VERIFIED
BY M
We denote by A,, X*, . . , A, the n eigenvalues distinct; by A the matrix defined as
and M the matrix A. eigenvalues be inverted. well-known
of matrix A, assumed to be
n x n square matrix in which the columns are the eigenvectors of The n rows of M are designated as { l/;}r=1,2 ,,,,, n. Since the are distinct, the eigenvectors generate R” and the matrix M can We denote by { W, }, _ 1,2,, , n the n columns of M-‘. Based on a relationship, we may write matrix A in the form A = MAM-‘.
Since we have assumed that all terms {~?‘k’}~=i,~,,,~,, fitting of experimental data, numerical eigenvalues can be data. We therefore assume A to be known experimentally. ship (9), it is apparent that the problem of identifiability equivalent to that of the matrix M.
(9) are present in the derived from these From the relationof the matrix A is
LOCAL IDENTIFIABILITY
5
Based on a detailed analytical study of (7), it can be shown [ll] that the set of data on M which can be derived from experimental data is summarized by the two equation systems: E=CM
(10)
D=M-‘B,
(11)
and
where E and D are two matrices derived from the experimental data. If c < n or b < n, it is obvious that Equations (lo)-(11) alone do not permit identification of the matrix M. Since these equations contain all the information available from experimental data, more information is clearly required for identification. Such information is provided by the model structure, which is assumed to be known-or, more accurately, by the constraints imposed on the model structure through not allowing certain transfers between compartments (nonexistent arcs). Structural constraints on the matrix A can be defined by two vectors F and G, and written as F’AG = 0. Only the following two types of constraints
(12) will be considered
here:
(a) Type A constraint:
a'J = 0,
i.e.,
F’ = e,,
This constraint represents the impossibility ment j to compartment i. (b) Type B constraint: a eJ
=-
t
a,,=O,
i.e.,
G = eJ,
with i f j.
of direct transfer from compart-
F’= (l,l,...,
1) and
G=e,.
i=l
This constraint represents the impossibility ment j to the outside of the system.
of direct transfer from compart-
From (9), the conversion of constraints on A to constraints is straightforward, since (12) can also be written F’MAM-‘G The set of equations
verified
by M and
on (M, M-‘)
=0 M-’
(13) therefore
consists
of: the
JACQUES
6
DELFORGE
systems (lo)-(11); a set of equations of type (13) representing structural constraints on A and, by definition, the system MM-’ Id is the identity matrix of rank n.
the various = Id, where
III.
1)
EXAMPLE
OF LOCAL INVESTIGATION
(EXAMPLE
Difficulties in understanding the method proposed here are mainly due to the serious notation problems which arise when an attempt is made to find a general solution. For a clear presentation of the principles involved, this section gives the details of a specific example. Extension to the general case is straightforward, and covered in Section IV and in the appendix. Let us consider the model of Figure 1, where n = 3, c = 1, and h = 1:
-(a,,+ -c 121) A=
a21
al2 -(Q12
0
B=
The model is observable and a,,#O.
a13
+a,,)
0
a32
1 0 7 0I
-
a13
c = [O,l,O].
if azl # 0 and ui3 # 0. It is controllable
3
1
I
I -
-.
I I --
FIG. 1.
-
INJECTION
\
t
, \.
if azl f 0
OBSER\‘AT IO?:
LOCAL IDENTIFIABILITY The following
equations
are verified by M and M- ‘:
(a) The equation CM=
which corresponds
E.
(14)
to Vz = E, where E is known.
(b) The equation M-‘B=
corresponding
D.
05)
to W, = D, where D is known.
(c) The equation MM-’ (d) The equations
representing
= Id.
(16)
the constraints:
Impossibility of direct transfer from compartment 1 to compartment 3: the equation with M and M-’ is given by (13) with F = e3 and G = e,. We thus obtain the expression &AW, = 0. Impossibility
of direct transfer
from compartment
(17) 3 to compartment
08)
V*hW, = 0. Impossibility system:
of direct exit from compartment
-
i
2:
2 to the outside
y:nw*=o.
of the
(19)
i=l
Impossibility system:
of direct exit from compartment
-
t yllw,=o. r=l
3 to the outside
of the
(20)
The method is based on the well-known principle of taking the derivatives of all equations with respect to all unknowns, to form the Jacobian matrix Z, which by definition is the matrix where element (i, j) is the derivative of the ith equation with respect to the jth unknown. A well-known sufficient
8
JACQUES
DELFORGE
condition for local uniqueness is that the determinant of ~9’2 is different from zero. In the example given in this section, the Jacobian is given by the matrix of Table 1 (where the derivative of an equation with respect to a vector is defined as the vector formed by the derivatives of the equation with respect to each of the vector components). In general, the Jacobian is thus a matrix with dimension (cn + bn + n2 + r) X2n2, where r is the number of constraints. Based on a well-known rule, the determinant of ZrX is different from zero only if at least 2n2 rows of A“ are linearly independent. Also, it can be
TABLE The Jacobian
1 2
Derivative with respect to: Equation
4
(14)
0
vz 100 010
v,
Wl
w2
w3
0
0
0
0
Row No. 1 2 3
001 100 (15)
0
(20)
-(AW,)’
0
-(AW,)’
0
m-(AW,)’
010 001
4 0
0
5 6
VI
0
0
7
0
0
-2
v,‘l r=l
Column
No.
123
456
789
10 11 12
13 14 15
16 17 18
19
9
LOCAL IDENTIFIABILITY
proven in a general way [ll] that, within the equations CM = E, M-‘B = D, and MM-’ = Id, at least cb rows are linearly dependent on the others. For the Jacobian matrix to contain at least 2n2 linearly independent rows, it is therefore necessary that at least (n - c)( n - b) constraints exist (a detailed proof of this necessary condition for identifiability is given in [12]). In our example, we must therefore find in the rows of 3? corresponding to Equations (14), (15), and (16) an equation showing dependence on the others (since c = b = 1). We actually find that row 10 is dependent on the first six equations. Since M and M-’ can be inverted, it can be readily verified that, following elimination of row 10, the other rows associated with Equations (14), (15), and (16) are indeed linearly independent. Since the model under investigation involves four constraints [i.e., the exact number (n - c)( n - b) required], a sufficient condition for identifiability is for the determinant of the square matrix X1 derived from 3 through elimination of row 10 to be different from zero. Based on a well-known rule of determinant computation, it is immediately apparent that the determinant of X1 is identical to that of 3t02, derived from 2’ by eliminating the first six rows and columns 4, 5, 6, 10, 11, and 12. It will be observed that we thus eliminate all the columns corresponding to derivatives with respect to vectors V, and Wj, which are known from Equations (14) and (15). From the relationship (9), we conclude that AM=MA
(21)
and
We can therefore write the vectors AW, and ?$A in the following form: AK = AMPie, = M-‘Ae,
(23)
v/A = eJM.11= e:AM.
(24)
and
In the example given, we therefore have AW, = M-‘Ae,
= - (a,, + uzl) WI + a2,W2
AW, = M-‘Ae,
= a,,Wl - ( aI2 + a32) W2 + a,*W,
AW, = M- ‘Ae, = aI3 W, - aI3 W, V2A=eiAM=a21Vl-(~,~+a~2)V;
(25)
10
JACQUES DELFORGE
and 5
?A=
5
j=l
eJAM=-a,,l/,.
(26)
/=I
Based on the above equations,
we can write 3’
in the following form:
x2 = .GP344z
(27)
with (M-l)’
0
0 0
0 0
i
0
0
0
M 0
0 M
(M-l)’
0
_/@‘=
0
1 (28)
and Z3 as shown in Table 2. The determinants of matrices M and M-’ being different from zero, it is clear that the determinant of .?P2 is nil only if the determinant of Z3 is nil. We can expand the determinant again, based on rows 1, 4, 5, and 6. By deleting these rows and columns 1, 4, 8 and 11, we obtain a matrix 3P4 which has-the same determinant as Z3. It will be noted that we have thus eliminated all the columns associated with those coefficients which are factorized with vectors V, and W, known from relationships (14) and (15).
TABLE 2 The Matrix X3 1
0
0
0 0
1 0
0 1
0 0
a13
1
1
0
2 3 4
0
10
0
0
0
0
0
00
0
1
0
0 0
0 0
00 01
0
1
0
00
“21
0
1 0
0 1
-(ad
+
an)
0
-
1
0 0
0
0
t”12
0
0
0
a12
00
100 0 00
0
0
-
in (27)
+
0 2
a32
a13
3
-
a12 013
4
Cal,+
-
+
u32
ael
=13
6
0
7
a21
o
o
0 8
9
-(a,,+u,,)
1
0
0 11
8 9
0 a,1 10
5 6 I
0
0 a32)
0 5
0
0
0 a32)
0 0
10
11 0 12
12
11
LOCAL IDENTIFIABILITY
All the remaining rows initially associated with Equation (16) contain two coefficients equal to 1. Based on a well-known property, we know that the determinant is unchanged if the matrix is modified by adding or subtracting columns. The determinant of .P4 is therefore identical to that of matrix 2’ defined as
&“j1
&0s=
Id
0
(29)
K
[ with
K= %2
+
0
0
0
- a21
a32
-
-
ad
0
a13
0
a21
0
a32 -
(‘12’
a32)
0
aed
-
(30) ‘32 a13
We finally find that a sufficient identifiability condition is that the determinant of K be different from zero. This can be readily shown to be verified only if a21 + 0, (31)
a13
+
0,
ad
+
a12
+
=32.
From the above, we conclude that a sufficient condition for local uniqueness is that the model be controllable and observable and that a,, # aI2 + aj2.
IV.
1.
A SUFFICIENT
STATEMENT
CONDITION
FOR LOCAL IDENTIFIABILITY
OF RESULT
The reasoning applied to the above example is quite general, as emphasized throughout the discussion by remarks stressing the general nature of the matrices derived and operations performed. To avoid repeating that lengthy procedure in every specific case, this subsection gives the rules for writing matrix K directly. Let us consider the set of suffix pairs (p, q) such that p +ZIc and q 6 Is. The number of such suffix pairs (p, q) is therefore (n - c)( n - b). We will now write the matrix K, with dimensions r X( n - b)( n - c); each of the (n - b)( n - c) columns is associated with a suffix pair (p, q); each of the r rows is associated with a constraint on the structure of matrix A. We will write K row by row.
12
JACQUES
DELFORGE
(a) In a row corresponding to the impossibility of direct transfer from compartment j to compartment i, i.e., to a constraint a,, = 0 (type A constraint), the coefficient of column ( p, q) is given by
4P-r)aq,- %-,)a,, 1
(32)
with
4-,, =
1 0
1
if if
x=y, x#y.
(b) In a row corresponding to the impossibility of direct transfer from compartment j to the outside of the model, i.e., a constraint a,, = 0 (type B constraint), the coefficient of column (p, q) is given by (33) THEOREM
I
A sufficient condition for local identifiability is that the determinant of K’K is different from zero.
Because of its length, the proof is given in an appendix. Remark I. It is important to note that, thanks to the relationships (32) and (33), this sufficient condition is readily applicable, as the examples will show. In particular, computation of matrix K can be programmed very easily, and is therefore feasible irrespective of model complexity. On the other hand, it is well known that determinant computation can present difficulties in the case of large dimensions. In practice, however, matrix K includes a large number of zeros and computation is usually straightforward.
2.
EXAMPLES
Example 2.
In the model of Figure 2,
-(ad + a21+ a311 A=
[l,O,O],
c=[;7
;I,
a12
0
u31 B’=
0
a12 -
a21
a23
-
,
a23 I
whence
Z,=
(11,
whence
I,=
{1,2}.
The model is observable if u 23 f 0 and controllable if a31 f 0 and aT3( a21 + a,,) # a21a12. From the relationships (32) and (33), the matrix K is defined
LOCAL IDENTIFIABILITY
13
FIG.
2.
as (p, 4) = (392) (373)
constr&ts a13 =
K=
0
a 32 --0 ar3 = 0 Qe2
=
0
It can be very readily shown that the determinant of K’K is different from zero only if uZ3 # 0 (an assumption already included in the condition for observability). When it is observable and controllable, the model of Figure 2 is thus always locally identifiable. Exumple 3. Let us now consider the model of Figure 3, which has five compartments. Compartments 1 and 4 are injected (I, = {1,4}), whereas compartments 4 and 5 are observed (I, = {4,5}):
Using the relationships (32) and (33) leads directly to the matrix K shown by Table 3. This table illustrates a case where the computation of K’K is not straightforward. In such cases, there is usually an advantage in replacing the computation of the determinant of K’K by the determination of necessary and sufficient conditions for matrix K to have (n - c)( n - b) independent rows. In the example given, we can write a matrix K’ including rows 1, 2, 3, 5, 7, 9, 10, 11, and 12 of matrix K. It is obvious (through computation of the
14
JACQUES DELFORGE
4
a 02
L
FIG. 3.
determinant, which is straightforward) that the matrix K’ is invertible if u51a34u25 # 0. Conversely, we can verify that if a5i = 0 or u34 = 0 or uZ5 = 0, the matrix K has not 9 independent rows. We conclude that if the model of Figure 3 is controllable and observable and if a51u34a25 # 0, then the model is locally identifiable. V.
1.
EXAMPLE OF A LOCALLY IDENTIFIABLE NONINVERSIBLE JACOBIAN MATRIX A COMMON
MODEL
WITH
ERROR
In the introduction, we mentioned that most approaches to the problem of linear model identifiability lead to a system of non-linear equations, the number of which is to be determined. It is well known that this problem has no general mathematical solution. Whatever the approach, it is always possible to investigate local uniqueness using the Jacobian matrix, as we did in Sections III and IV. However, many authors have stated erroneously that a necessary and sufficient condition for local uniqueness at a point P is for the determinant of the matrix H’H to be different from zero at P (H being the Jacobian matrix associated with the system of nonlinear equations resulting from the approach used).
LOCAL IDENTIFIABILITY
15
16
JACQUES
DELFORGE
Although this condition is indeed sufficient for local uniqueness, it is important to point out that it is not always necessary, as shown by S. Vajda [13] on the basis of a nonlinear example (see also References [19] and [ZO]). It will now be shown that this remark also holds in the case of linear models, and that its significance is much greater than it might appear. First, let us recall the conventional one-dimensional results aimed at finding the solutions of the equation f(x) = 0. If u is a solution, i.e., if f(a) = 0, the condition f’(a) f 0 is not a necessary condition for solution a to be locally unique. It can be seen that, if f’(a) = 0, solution a is also locally unique if one of the higher derivatives is different from zero at point a. Conversely, a necessary and sufficient condition for the solution u not to be locally unique is that the derivatives of f(x) of all orders are zero at point a. Since the amount of computation required may be large, the application of such a criterion may present problems. However, there is an important particular case; if f’(x) = 0 everywhere in the neighborhood of a (i.e., over one dimension through the interval [a - E, a + r]), then the solution a is not locally unique (it can then be shown that all derivatives of higher order are equal to zero). Similar results are obtained in the case of equation systems. The Jacobian matrix H relates to the first derivative of the equations with respect to the unknowns. If the determinant of H’H is zero, it is necessary to expand the system further in the neighborhood of the solution by computing the matrix equivalents of higher derivatives, which would be very difficult in practice. We thus note the following results, which, although fractional, are valid irrespective of the system of equations involved: (a) if the determinant of H’H is different from zero at P, solution P is locally unique (a well-known result). (b) if the determinant of H’H is zero at P and everywhere in the neighborhood of P, then the solution P is not locally unique. This is the case when the system has fewer equations than unknowns. As suggested by S. Vajda in [13], this result can be derived from DieudonnC’s rank theorem [14]. The proof, however, is lengthy and elaborate. The results of F. M. Fisher [20] can also be used. (c) If the determinant of H’H is zero at P and in a subspace of measure zero in the neighborhood of P, but differs from zero in the rest of this neighborhood (i.e., almost everywhere), then there is no way to arrive at a conclusion without a detailed study of higher order derivatives (this is a case which, although rather frequent, has been omitted by some authors). 2. EXAMPLE
( RECONSIDERATION
OF EXAMPLE
1)
We will now reconsider Example 1 (discussed in Section III and illustrated by Figure l), using the transfer method proposed by R. Bellman and K. J.
17
LOCAL IDENTIFIABILITY
Astrijm [15] and developed since by a number of authors, especially by C. Cobelli et al. [4] and by S. Audoly and L. D’Angio [16]. If Y(s) is the Laplace transform of y(t), and U(s) that of u(t), we have Y(s) = G(s,A)U(s), G(s, A) being the transfer matrix defined as G(s,A)=C[sZ,-A]-+.
(34)
For a given matrix A, the elements of the matrix G(s, A) appear as the ratio of two polynomials in s, of degree less than or equal to n. We denote by {a,} the set of parameters included in the polynomials. The transfer matrix being known, the behavior of the system can be fully described. The problem of the identifiability of the matrix A from experimental data y(t) can thus be reduced to a study of the identifiability of A based on the transfer matrix. Every parameter (Y, (which is then assumed to be known, since it is determined from experimental data) is equal to a nonlinear combination, of degree less than or equal to n, of the unknown elements in the matrix A. We thus obtain a system of nonlinear equations. The matrix A is therefore identifiable when this system has a unique solution. In the example of Figure 1, the following system of equations is obtained after simplification: (35)
~,1+~21+~12+~32+~13=~1,
-(ad+a21)(%2+a32)+al2a21
_
(36)
~13b,l+~21+~,2+~32)=~2~
a13a,,(a12
+
032)
=
a39
a21=a4, u21”13
=
a5.
(37) (38) (39)
The Jacobian matrix can be computed. Leaving the equations in the order of (35)-(39) and taking the derivatives with respect to the unknowns written in the order a,l, %l? a12, u329 013, the Jacobian matrix J is given by
18
JACQUES
DELFORGE
It can be shown that the determinant of J’J is different from zero only if u2i # 0, ui3 # 0, and a,i + ui2 + u3z. This result is obviously identical to that derived in Section III [Equation (31)] from the matrix K defined by (30). However, it should be noted that computation by the transfer method does not provide relationships such as (32) and (33) from which the matrix K can be written directly. The cases a zi = 0 and ui3 = 0 are excluded, owing to our controllability and observability assumptions. We must however investigate the particular case where U el =
If the above relationship become
u12
+
u32.
(41)
is taken into consideration,
2a,i + a,, + -(a,,
+ %i)%
+
%2U21-
Ul3(2a,l+
equations
U13
a211
=
q,
(42)
=
a29
(43)
a3 9
(44)
=
a49
(45)
=
a5.
(46)
2_
Q13Ue1a21 u21”13
(35)-(39)
The following solution can be readily derived: a21
=
a4
[Equation
(45)],
[combination
of (45) and (46)],
[combination
of (44)) (45), and (46)].
ui2 is then derived easily from (43), and u32 from (41). We have thus shown that, if the relation (41) is verified, the identification problem has a unique local (or even global) solution. It has also been shown that the determinant of J’J is zero in this case. We therefore draw the conclusion that, contrary to published conclusions, for the determinant of J’J to be different from zero is not a necessary condition for local uniqueness. VI.
1.
A NECESSARY AND IDENTIFIABILITY MAIN
SUFFICIENT
CONDITION
FOR
LOCAL
RESULT
To investigate the local identifiability of a solution P, we first compute the matrix K, which is straightforward on the basis of the relationships (32)
LOCAL IDENTIFIABILITY
19
and (33). We then use the general results stated in the previous section: if the determinant of K’K is not zero, it follows that solution P is locally unique. However, if the determinant of K’K is zero at P and everywhere in a neighborhood to P, it follows that the solution P is not locally unique. If an intermediate situation prevails, i.e., if the determinant of K’K is zero at P and in a subset Yi of a neighborhood Y of P, although different from zero through the rest of this neighborhood, then no conclusion can be drawn for the time being. We can, however, make an important remark. If the solution P is not locally unique, all other solutions P’ near P are also not locally unique and must therefore be located in the subset Sq of Y. It therefore follows that if the solution P is not locally unique within V, it is also not locally unique within subset 9,. Since the converse obviously holds, investigating the local identifiability of solution P in V is equivalent to investigating the local identifiability of P in subset Y;. Since the subset Yi is a set of points in V where the determinant of K’K is zero, it can be defined by one or more equations verified by the coefficients of the matrix K, i.e., by certain parameters a,, which are present in K. Equations thus obtained can be used to express new constraints on A. We thus write Kc’), a matrix derived from matrix K, to which we append new rows representing the new constraints. The above reasoning applied to K in neighborhood Y also holds as regards Kc’), considering the subspace Yi only: (a) If the determinant of K (‘)‘K(‘) is different from zero, it follows that the solution P is locally unique within Yi and therefore within V. (b) If the determinant of K(*)‘K(‘) is zero throughout the neighboring space Vi of the solution P (VI being the intersection of Y with Yi), then it follows that the solution P is not locally unique. (c) If we return to the intermediate situation in which the determinant of K(l)rK(l) is zero at P and in a subset Y2 of Vl;, but different from zero throughout the rest of “v, (i.e., almost everywhere within Y’i), no conclusion can be drawn and further investigation is required. The reasoning continues along similar lines; a subspace Y; is defined through equations which must be verified by parameters a,, for the determinant of K(‘)‘K(‘) to be zero; these equations are regarded as new constraints applied to the matrix A. We thus define Kc*‘, a matrix derived from Kc’) by appending the rows representing the new constraints. The determinant of Kc2)‘K(*) is then investigated in the subspace Y2. If required, we can thus define a series of matrices Kck) and subspaces &. Since the dimension of the subspaces YA decreases by at least one unit at each step, the series must be finite, and we can always determine that the solution P is either locally identifiable or nonidentifiable. Hence the follow-
20
JACQUES
DELFORGE
ing theorems: THEOREM
2
A necessary and sufficient condition for a model to be locally identifiable almost everywhere is that the determinant of K’K is different from zero almost everywhere. THEOREM
3
A necessary and sufficient condition for local identifiability of a solution P is the existence of a stage k at which the determinant of KCk”KCk) is different from zero. 2.
EXAMPLES
Example 4.
In the model of Figure 4, we have
-(u31 + 041) A=
012
-((I,?+"12+"3z+Q)
0
-toe3
031
*32
041
O42
0
0
0
0
10’
0
1 1
0
014
0
024
0
whence
lB=
{1.3},
01’
whence
I,=
{3,4}.
FIG.4.
0
-+ 043) 043
-(a,4
+
014
+
024)
1%
21
LOCAL IDENTIFIABILITY
From the relationships
(32) and (33), the matrix K is defined by:
(p, q) = (1,2)
K=
(1,4),(2,2)
(2,4)
0
Constraints a 21 -0 -
0
0
0
a43
0
0
a13
=
0
0
0
0
a43
a23
=
0
a34
=
0
a rl
=
0
0
-
0
- a4l
a31
a41
0
-
0
- a4l
a32
It is obvious that the determinant of K’K is always zero, since the columns (1,2) and (2,2) are zero. The model of Figure 4 is thus never identifiable. Example I (continued). Let us now reexamine the example investigated in Section III, and shown by Figure 1. We have seen that if a el
al2 - aj2 = 0,
-
(47)
the determinant of K’K is zero. However, by using the transfer method, we have shown in Section V.2 that local uniqueness nevertheless existed. This result will now be derived very easily by using Theorem 3. We follow the procedure explained in Section VI to write the matrix K(l), appending a row representing the constraint (47) to matrix K as defined by (30). This row is readily derived from the relationships (32) and (33) by subtracting the rows associated with a,, and a32 from the row associated with a,,. We thus obtain
0
-
0
apl - al3
0 0 K(l)
=
-2a21
0
a21
a21
a32
-
a32
-
tal2’+
0321
0 al2
+
a32
-
a21
Since azl, a13, and a32 are assumed to be different from zero (controllability and observability assumptions), it is obvious that the determinant of KC1)‘KC1) is never zero. It therefore follows, from Theorem 3, that if (47) is verified, the identification problem has a locally unique solution although the determinant of K’K is zero. This result is the same as derived in Section V.2 by the transfer method; the method proposed here is however much faster, since it requires no analytical computation,
22
JACQUES
DELFORGE
Exumple 3 (continued). Let us reconsider Example 3 of Section IV.2, shown by Figure 3. From Theorem 1 we drew the conclusion that if u51u34uz5 differs from zero, local uniqueness exists. Theorem 3 permits investigating local identifiability in the following particular cases:
(a) a51 = 0. From the relationship (32), the condition u5, = 0 results in an all-zero row. In this case, the matrix K (l) is thus identical to K. Since we know that the determinant of K’K is zero, it follows from Theorem 3 that local uniqueness does not exist. (b) 025 = 0. The condition a 25 = 0 results in the following row:
(0,0,0>0,0, u55 - u2*,0,0, - u23).
(48)
We observe immediately that, in the matrix K(l) [written by appending the row (48) to matrix K shown by Table 31, the column associated with ( p, q) = (2,2) is all zero. The determinant of K ‘(‘)K(l) is therefore zero, and there is no local uniqueness in this particular instance. (c) a34 = 0. The condition a 34 = 0 results in the following row: (0,0,0,0,0,0,0,0,
(49)
a54).
This case is similar to the above: it can be seen that the column associated with ( p, q) = (3,3) is all zero. Example 5.
-
A=
(a21
In the model of Figure 5, we have +
a31
+
0
a12
041) -
a21
a12
a31
0
a41
0 IB= {I),
u23 -(a23+ue3)
0 Ic = {2,3}.
The computation of K from relationships we arrive at the matrix of Table 4. Assuming 041
we observe determinant
of KC”
f
0,
u,,#O,
(32) and (33) is straightforward;
and
a,,-ua,fO, of K’K
immediately that the determinant of the matrix K’, defined as a33
a14 -
a44
1 3
(50) is zero only if the
(51)
LOCAL
23
IDENTIFIABILITY
FIG. 5.
is
zero, i.e., if (a33 - Q)(%
-
h)-Q41a14
=
(52)
0.
To determine whether local uniqueness exists when (52) is verified, we may use Theorem 3 and derive the matrix KC”. A new feature, as compared with the previous examples, is that the new constraint defined by (52) is a nonlinear equation. This however presents no special problem, since the corresponding row of KC’) is derived from the relationships (32) and (33) using the well-known properties of derivatives. If L,, is the row defined by (32), and therefore represents the derivatives of a,, with respect to the unknowns (which, as mentioned earlier, are elements of the matrices M and M-l), the row of KC” associated with constraint (52) is (4,
- ‘L)(%
- %)+(a,3
- %)(J%3
- L-
L4lQl4
-
a41L14.
(53) TABLE 4
LJ,4)=(1.2) a23
(1.3) a33 -
0 0 K=
-
a31
-
041
all
0 0 0 0
0
- a4
-
a21
-
-
a22
031
0
(472) 0 0 0 0
(194) 0 -
a21 031
0 0 0 -
-
a23 -
a33 -
a21
a22 -
014
0 0 0 0
aZ1 - a44
041
0
(4.3)
a,4
a44 031
0
(434) 0 0 0 0 0 0 -
041
0
Constraints a,3 = 0 a 24 --0 cl 34 -0 a 32 --0 aq2 = 0 a 43 --0 ad = 0 “e2 = 0
JACQUES DELFORGE
24 From (32), rows Lij are written as
L,, = (0, - ~31,0,0,0,0)~ L,,
=
(ax,
a31,
L44
=
(WA
-
u41,0,0,0),
a41,0,0,0),
L4,=(0,0~0~~,,,~3,,~4,), L,,
=
(O,O,
a44
-
q,,O,O,
From (53), the row of K(l) associated
-
q4).
with the constraint
(52) is then given
by (- %1(%
- a4419 -
a31ca33
-
@l,)YO,
-
%a149
-
a,,a,,,O).
Based on assumption (50), and therefore assuming the relationship (52) to be verified, it can readily be shown that the determinant of K(‘)‘K(‘) is zero only if the determinant of matrix K”” defined as K(l)’
(a33
= -
-
a31(033
Qll) -
d
-
a14 a31u14
(54 I
is zero. The determinant of K(l)’ is equal to - 2u,,u3,(u3, - all). Since u3i, are assumed to be different from zero (controllability and ~147 and ~41 observability assumption), we deduce that u33 = ali is impossible [Equation (52)] and that the determinant of K (l)rK(l) is never zero. It therefore follows that, based on the assumption (50), the solution to the identification problem is locally unique if the relationship (52) holds. This is another example of a model locally identifiable with a noninvertible Jacobian matrix. VII.
NECESSARY STRUCTURAL IDENTIFIABILITY
CONDITIONS
FOR
LOCAL
From the necessary and sufficient condition for local identifiability, valid almost everywhere, as stated by Theorem 2, we will now derive several conditions for local identifiability which are only necessary (not sufficient), but show the significant advantage of being directly related to the model structure. Should one of these necessary conditions fail to be verified, it immediately follows that the case is one of nonidentifiability. (In this section it is assumed that none of the eigenvalues of matrix A is zero.) 1.
REQUIREMENT
FOR (n - c)(n - h) USEFUL
CONSTRAINTS
In an earlier publication [6]. we defined a useful constraint as a constraint for which the associated equation [of the type (13)] was independent of
25
LOCAL IDENTIFIABILITY
equations derived from experimental data, i.e., the equation systems (lo), (11). It will now be shown that, as part of the necessary and sufficient condition stated in Section VI, useful constraints are actually those for which the associated rows of the matrix K as defined by (32) or (33) are different from zero. Conversely, the useless constraints defined in [6] lead to all-zero rows in K, providing no additional data for the determination of identifiability. Let us consider the row of the matrix K associated with the impossibility of transfer between a compartment j and a compartment i. From (32), the elements of this row are given by (55) We will now examine all possible cases, with compartment noninjected and compartment i observed or nonobserved.
j injected
or
(a) Let us assume that j E 1,. Since q 4 I,, we cannot have q = j, and therefore the second term of (55) is always zero, irrespective of p and q. If i E I,, then since p 4 I,, we can never have p = i. The first term also is thus always zero. The associated constraint is therefore of the useless type. However, if i E I,, there is a p value such that p = i. Thus, irrespective of q e Is, we find, again from (55), terms u4, in columns (i, q). In this case, the row associated with the constraint is nonzero only if there is a term a4, different from zero, i.e., if there is nonzero transfer from compartment j to a noninjected compartment q. (b) Assuming that j E ZB and i cZI,, the associated row in the matrix K is always nonzero because the column associated with q = j and p = i contains the element a,,, which in turn is nonzero because, from basic assumptions, none of the A eigenvalues is zero (due to the structure of matrix A, the only case where a diagonal element a,, is zero is that in which a full column of A is zero, which implies that one of the eigenvalues is zero). (c) Let us assume that j 4 IB and i E Z,. Since p E I,, we cannot have p = i, and the first term of (55) is thus always zero. However, as j E Is, there exists a value of q such that q = j. The associated row of K therefore includes all the u,~ terms with p +ZI,. This row is nonzero only if one of the U terms is nonzero, i.e., if a nonzero direct transfer exists between a &observed compartment p and compartment i. We finally obtain Table 5, which is of course reminiscent of Table 1 in [6]. It should however be pointed out that conditions are now less restrictive, because certain assumptions made in [6] are no longer necessary (e.g., the assumption that all elements in CA’B are nonzero). Further, conditions C, and C, no longer include the restrictive condition expressed by the relationship (40) of [6]. Last, we have added a column giving the type of equation
26
JACQUES TABLE
DELFORGE
5
Type A Constraints: Impossibility of Transfer from Compartment From compartment
To compartment
i
.i
Nonobserved
Injected
j to Compartment
Added
type
C, verified
Al,
C, not verified Noninjected
Observed
C2 verified C, not verified
Noninjected
Nonobserved
Injected
Observed
“Condition Condition
Constraint
condition’l
-
Al2
A2, A22
I Useful constraint?
Equation type
Yes No
Linear in v
Yes No
Linear in W,
A3
Yes
A4
No
C,: Nonzero transfer from compartment j to a noninjected C,: Nonzero transfer to compartment i from a nonobserved
v,AW,=O
compartment. compartment.
associated with constraint type. This information, although difficult to obtain directly from (32), can be readily derived from the computations in Sections 3.C and 3.D of [6]. In the same way as before, Table 6 can be derived from the relationship (33); it gives the necessary and sufficient conditions for a B-type constraint to be useful, i.e., for it to lead to a nonzero row in K. Since the matrix K has (n - c)( it - b) columns, a necessary condition for K’K to have a nonzero determinant is that matrix A contains at least (n - c)( n - b) nonzero rows. Hence, the following theorem: THEOREMS
A necessary condition for identifiability (n - c)( n - b) useful constraints.
is the existence
of no less th’an
Remark 2. This necessary condition is, in practice, highly selective, since the vast majority of models where it is verified are indeed locally identifiable. Following an incomplete proof, this condition had been regarded as sufficient in [6]. In [5], J. P. Norton gives a counterexample where, although (n - c)( n - b) useful constraints exist, the model is still not identifiable. In [7], it is shown that nonidentifiability results from a linear dependence between two constraints of the Al, type. The precise conditions of independence between two constraints of the Al, type or A2, type are specified in [7]
LOCAL
27
IDENTIFIABILITY TABLE 6 Type B Constraint: Impossibility
Compartment J Injected, nonobserved
of Exit from Compartment
1st added condition”
2nd added condition’
C, verified
-
Constraint type
C, not verified Noninjected, observed
Equation
Bl,
Yes
Bl,
No
type’ Linear in Vr.
-
B21
Yes
C, not verified
C, verified
B2?
Yes
C4 not verified
B2,
No
B3
Yes
VXl1W, = 0
C, verified
B4,
Yes
Linear in Vx
C, not verified
B42
No
B43
No
C, verified
C, not verified “Condition
Useful constraint?
C, verified
Noninjected nonobserved Injected, observed
j to the Outside
C,: As in Table 1. Condition
1 to a nonobserved compartment. nonobserved compartment either compartments of nonzero transfers hV~=X;=,l+
C,: There is a transfer
vxizw,
Linear in W,
from compartment
Condition Cq: There is a transfer from a to compartment J or to one of the arrival originating in compartment j.
and [17]. It can be readily shown from (32) that such a dependence identical rows in the matrix K. 2.
NECESSARY
THEOREM
CONDITIONS
= 0
RELATING
TO TYPE
leads to
B CONSTRAINTS
6
(a) A necessary condition for identifiability is the existence of a number of type B constraints at least equal to the number of compartments simultaneously noninjected and nonobserved. (b) It is further necessary that, in each of such simultaneously noninjected and nonobserved compartments, at least one of the following two conditions be
JACQUES DELFORGE
28 verified:
direct exit from the compartment to the outside is impossible; the compartment is the arrival compartment of a nonzero transfer originating from a compartment where direct exit to the outside is impossible. Proof. Let us assume that compartment p is neither injected nor observed. Therefore, there exists, among the (n - c)(n - b) columns of K, a column associated with the suffix pair (p, p). But, from the relationship (32), the element in this column associated with the constraint ai, = 0 is given by
8cP-l)aP, - 8(P-J)aw This element is always zero, since when i = p or j = p, we obtain a,,, which, from basic assumptions, is nil. Thus, only type B constraints can lead to nonzero elements in column ( p, p). Let us assume the existence of k noninjected and nonobserved compartments. From the above, the matrix K can be written in the following form:
where K’ is a matrix with dimension e X k, e being the number of type B constraints. A necessary condition for the determinant of K’K to be nonzero is that the determinant of K”K’ is also nonzero. It immediately follows that emust be greater than k; hence the first proposition in the theorem. It is further necessary that none of the columns in K’ be all zero. And, from (33), the element of column ( p, p) associated with the impossibility of direct exit from a compartment is given by -
aPJ
-
%Y)%
If there is no direct exit from the compartment P to the outside, a row exists where j =p. We thus obtain app, which is always nonzero from basic assumptions (nonzero eigenvalues). In the opposite case, it is necessary for j to exist such that ap, # 0. Hence the second proposition in the theorem. Example 6. Figure 6 shows the same model, although with different observable compartments. In the first case, there is only one nonobserved and noninjected compartment (compartment 3) and a single type B constraint (impossibility of exit from compartment 3). The necessary conditions for identifiability given in the theorem can be readily shown to hold (there actually is local uniqueness but two distinct solutions, and therefore no global uniqueness). In the
29
LOCAL IDENTIFIABILITY
FIG. 6
second case, however, although compartments 2 and 3 are neither injected nor observed, there still is only one type B constraint (exit from compartment 3). The first necessary condition of the theorem is not verified; therefore, there is no local uniqueness and the identification problem has an infinite number of solutions. Example 7. The model in Figure 7 has five compartments, two of which are simultaneously noninjected and nonobserved (compartments 2 and 3). Since there are three type B constraints, the first part of Theorem 3 is verified. However, the second condition in the theorem is not met, because there is nonzero transfer from compartment 2 to the outside, and no direct transfer between a compartment with no exit to the outside (compartment 1, 4, or 5) and compartment 2. The necessary condition of Theorem 5 is thus not verified; the identification problem has an infinite number of solutions.
JACQUES
DELFORGE
FIG. 7
However, if we add a direct transfer, for instance between compartment 5 (with no possibility of direct exit to the outside) and compartment 2, all the necessary conditions in the theorem are verified. It will be observed that the model is then as shown by Figure 3 and that, as shown earlier in Section IV.2, a necessary condition for identifiability is for uZ5 to be different from zero. Example 8. The compartmental system in Figure 8 is a catenary model with input and output in the same compartment. In this model, there are n - 1 nonobserved and noninjected compartments. The first necessary condition of the Theorem 6 is verified if there are n - 1 type B constraints. Thus,
FIG. 8
31
LOCAL IDENTIFIABILITY
the necessary condition for identifiability of this model is that the number of excretions to the environment is zero or one. This result is already known, but this example shows the directness of the conclusion. VIII.
CONCLUSIONS
Identifiability is a major problem in model justification. In this paper, we have proposed a number of criteria applicable to the investigation of local identifiability. The many examples given have shown these criteria to be easy to use, even in the case of complex models. Recent progress made in the field, and the results presented in this paper, which are valid although the assumptions include very few restrictions, lead to the conclusion that studies of local identifiability are now, in most cases, exempt from difficulties. It is important to note that the criteria presented here are not limited to giving results in the neighborhood of a particular solution; they also permit the investigation of local identifiability throughout the parametric space. They may thus be regarded as a good starting point for global identifiability studies. APPENDIX.
PROOF
Let us designate
OF THEOREM
1
as
(56)
cp(M,M_‘)=.F the system of equations
verified by M and M-‘.
This system includes
CM=E, M-‘B= MM-’ and a set of equations
representing
{ F,‘MAM-‘G,}
(57)
D
(58)
= Id;
(59)
the various structural = 0,
k=1,2
,..., r.
constraints
on A: (60)
Y is the vector formed by the elements of M and Mp ‘, the elements of M being written row by row and those of Mm1 column by column:
v=(v;,v; For a variation
,..., v;,w,
)...,
WJ.
A+‘- or Y, we have, by definition, A9
= #A-Y,
where &’ is the Jacobian matrix given by the Table 7.
(62)
(59)
(57)
Derivative with respect to Equation
(M-1)’
0
0
0
&Id 0
C,, Id
0
(AC’)’
C,; Id
Cl, Id .,:
..
(AC’)
0
0
Cc,:Id
C,nId
The Jacobian X a
TABLE 1
0
Y2
0
vz
0
“1
B,; Id
81, Id
Y,
Vz
“1
B,,Id
B,, Id
0
.,:
..’
Y,
0
v2
0
VI
0
4, Id
B,,, Id
33
LOCAL IDENTIFIABILITY
c 1:
+..q =
34
JACQUES
We suppose that AF If the compartment i (there exists k with C,, injected, we deduce that
DELFORGE
= 0, and we want to show that AY = 0. is observed, we deduce from Table 7 that A< = 0 = 0, except for p = i), and if compartment j is AW, = 0 (there exists k with Bqk = 0 except for
4 = _i). By deleting the columns corresponding to y, i E I,, and to W,, j E I,, and by deleting the rows corresponding to Equations (57), (58), we obtain the matrix Z’, the vectors A.F’ and AV’, and the simplified equation A.F’ = Z’AY’. This matrix X” can always be written in the form .P=X.M;
(63)
thus A9 J? is a block-type matrices M:
diagonal
= X.-MAY’.
matrix, consisting
(64) of n matrices ( M-l)r
and n
(M-l)’ 0
(M-l)’
“M=
(65)
(M-l)' M 0
M M
We want to study the structure of the matrix Xx. Based on relationships (21) and (22), we have
(Aw,)‘=
(W'e,)'
= (M-RAE,)' = ( A~,)'( and V,A = e:MA = e:AM.
M-~)'
35
LOCAL IDENTIFIABILITY
Thus, in a row of the equation (64) associated (KAY = 0), the transposed vector corresponding given by
and the transposed
vector corresponding
with a type A constraint to (MP’)‘Al$, p P I,, is
to MA WY, q t-i Is, is given by (67)
&&A.
In a row associated with a type B constraint (C:,,~.hF$$ = 0), the transposed vector corresponding to (M-l)’ A?$, p 4 I,, is given by - (Ae,)‘, and the transposed
vector corresponding
(68) to MAW,, q P ZB, is given by
(69) Let us consider
a row associated with Equation As W4 + SAW,
= AVp M-‘e,
(59) (VPW, = 0 or 1). Then + epMAW4.
We shall consider the four following types: (a) p E I,. q E IB. This row is nil (AC E Y’, AW, E A%‘-‘). (b) P 4 I,, qE 1s. The row contains only one coefficient not zero: a 1 in the column corresponding to the 9th component of (M-l)’ AV;, denoted {(M-‘)‘AV&. The row contains only one 1 in the column corre(c) PEG, 441,. sponding to { MA W4 }p. (4 P @ I,, q @ IB. The row contains two coefficients equal to 1: in the column corresponding to ((M- ‘)l AV; }4 and in the column corresponding to { MAW4},. Thus ((M-l)‘A~~],+{~A~,},=~
‘jp E I,
We finally find that the only components
and Vq P IB,
of MA+‘-’ different
((M-‘)‘AV,‘)
4
and {MAW,),
with
pPZ,
and
qEI,.
(70)
from zero are
36
JACQUES DELFORGE
We can deduce from (66) and (67) that the component of X associated to a type A constraint (YAW, = 0) and corresponding to {( M-‘)rAV~}4 is
and that the component
corresponding Sc,-,&4e,
to (MAW,},
is (72)
= aC,-4ja,P.
From (68) and (69), we deduce that the component type B constraint (C:,,V,AW, = 0) and corresponding
of X associated to a to {(M-l)' AV;}, is
-(Ae,)‘e,=-a,, and that the component
corresponding
(73)
to { MAW, }p is
(74) Because the equation
(70) is always verified, we can consider as unknowns
and define the system of linear equations
with K an r X(n - c)(n - b) matrix, defined on the basis of the relationships (32) and (33), deduced from (71)-(74). Based on a well-known property, we deduce that if the determinant of K’K is different from zero, there is no variation AV” # 0 such that AY” = 0. In this case, we conclude that if Y defined by (61) is one solution, then it is locally unique, because there no variation AV # 0 of ^Y such that A9 = 0. REFERENCES 1
.I. Delforge, The problem of structural identifiability of a linear compartmental system: Solved or not? Muth. Biosci. 36:119-125 (1977). 2 J. P. Norton, An investigation of the sources of nonuniqueness in deterministic identifiability, Math. Biosci. 60:89-108 (1982). 3 E. Walter, Identifiahiliq of State Space Models. Lecture Notes in Biomathematics. 46, Springer, 1982. 4 C. Cobelli, A. Lepschy, and G. Romanin Jacur, Identifiability results on some constrained compartmental systems, M&h. Biosci. 47:173-195, (1979).
37
LOCAL IDENTIFIABILITY
5 J. P. Norton, Letter to the editor, Math. Biosci. 61:296-298 (1982). 6
11 12
J. Delforge, Necessary and sufficient structural condition for local identifiability of a system with linear compartments, Math. Biosci. 54:159-180 (1981). P. Delattre, Transformation system with time-dependent characteristics and population theory, Math. Biosci. 32:239-274 (1976). J. A. Jacqua, Compartmental Ana!)% in Biology and Medicine, Elsevier, Amsterdam, 1972. A. Rescigno and G. Segre, Drug and Tracer Kinetics, Waltham, 1966. R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical Sysrem Theory, McGraw-Hill, 1969. W. D. Smith and R. R. Mohler, J. Theoref. Biol. 57:1-21 (1976). J. Delforge, New results on the problem of identifiability of a linear system, Math.
13
Biosci. 52:73-96 (1980). S. Vajda, Comments on structural identifiability
7 8 9 10
IEEE
Trans. Automat. Control AC24:495-496 J. Dieudonne, Fondations of Modern Analysis,
in linear time-invariant
systems,
(1979).
Academic, New York, 1960. 15 R. Bellman and K. J. Astrom, On structural identifiability, Marh. Biosci. 7:329-339 14
16
(1970). S. Audoly and L. D’Angio, On the identifiability of linear compartmental systems: A revisited transfer function approach based on topological properties, Math. Biosci.
66:201-228 (1973). 17 J. Delforge, An approach to the problem of linear system investigation of structural properties of the connection, matrices, Math. Comput. Simulation 24~466-471 (1982). 18 J. Delforge, Letter to the Editor, Math. Biosci. 61:299-305 19 A. Thowsen. Identifiability of dynamics systems, Internat.
identifiability through the injection and observation (1982). J. Systems
Sci. 9:813-825
(1978). 20
F. M. Fisher, The Identification
Problem in Economics, McGraw-Hill, New York, 1966.
JACQUES DELFORGE Dipartement de Biologie, Commissariat b I’Energie Atomique, Centre d’Etudes Nuckaires de Saclay, 91191 Gif-sur- Yvette, France Received 4 Ju& 1983; revised I9 January I984
ABSTRACT The identifiability of parameters in a model with known structure and no noise is a problem of accurate determination of the number of parametric space points which are solutions to the identification problem. Mathematicians usually make a distinction between local and global uniqueness. This distinction is very useful in practice as well as in theory, because it relates to difficulties at two distinct levels. This paper establishes a new set of necessary and sufficient conditions for local identifiability, with very broad-based validity: first, a condition for the model to be locally identifiable almost everywhere; second, a condition leading to conclusions as regards any particular point in the parametric space. From the above criteria, we derive several conditions which, although necessary only, are very useful in practice for the rapid detection of certain nonidentifiable models. The above criteria are of very general significance, easily programmable, and readily applied, as demonstrated by the many examples given.
I.
INTRODUCTION
The identifiability of parameters in a model with known structure and no noise is a problem of great importance, since it involves the accurate determination of the number of parametric space points which are solutions to the identification problem. Investigations [l-4] have shown this number to be widely variable; although the model is linear, except in very special cases we met with the well-known difficulties associated with minimization problems when attempting numerical identification of model parameters. In particular, whenever a numerical or analog program provides a solution, it is not known whether the solution is unique (in which case the solution found is fully justified), or if a finite (but more than one) or infinite number of solutions exist. In the latter case, the solution found is only one of many, and there is no reason why it should be preferred. One should then either determine the set of possible solutions and conclude that the actual solution MATHEMATICAL
BIOSCIENCES
70:1-37
QElsevier Science Publishing Co., Inc., 1984 52 Vanderbilt Ave., New York, NY 10017
(1984)
1 0025-5564/84/$03.00
2
JACQUES
DELFORGE
is one of the mathematical solutions found, or, whenever possible, propose experimental procedures leading to a unique solution. Mathematicians usually make a distinction between local and global uniqueness. This distinction is very useful in practice as well as in theory, because it relates to difficulties at two distinct levels. Irrespective of the method used to investigate identifiability, we always arrive at a system of nonlinear equations, and attempt to find the exact number of solutions. Studies of local uniqueness, an obviously necessary condition for global uniqueness, are usually based on expansions, which may be either numerical (involving numerical computation of the solution; such methods are therefore not applicable to structural investigations) or analytical through computation of the Jacobian matrix (the functional matrix in which element ij is the derivative of the i th equation with respect to the jth unknown). The mathematical tools are therefore well known. In practice, however, the various methods proposed are found to differ widely, especially as regards convenience. In particular, the methods based on Iacobian matrix computation show limitations due to the analytical computations required, which grow very rapidly in complexity as the model size increases. Global uniqueness investigations are even more difficult, because usable mathematical results are few and very limited in scope. In most cases, solving the identifiability problem requires explicit analytical computation of the solution, which makes most of these methods practically inoperative as soon as the number of compartments exceeds three or four (except for highly exceptional model structures [4]). It thus appears essential that identifiability investigations be directed toward the development of methods applicable to complex models. A specific requirement is for such methods to be readily programmable, so that model complexity only affects computing time. With that intent, we proposed in a previous publication [6] a necessary and sufficient uniqueness condition which is readily programmable, because it is based solely on model structure and on the characteristics of injection and observation matrices. However, this result included certain inaccuracies, as pointed out in [5, 181, and its validity was subject to a number of restrictive conditions, especially as regards the absence of zero eigenvalues and zero elements in CA2B. This paper states new necessary and sufficient conditions for local identifiability, with much broader validity than the above; a condition for the model to be locally identifiable almost everywhere; and a condition which permits drawing conclusions for each specific point in the parametric space (although this particular condition uses structural properties only). From the above criteria, we derive a number of conditions which, although necessary only, are very useful in practice for detecting nonidentifiable models. The criteria are of very general significance, easily programmable, and readily applied, as demonstrated by the many examples given.
LOCAL IDENTIFIABILITY
II.
REVIEW DATA
1.
EQUATIONS
3
OF EQUATIONS
VERIFIED
DERIVED
BY A LINEAR
FROM
EXPERIMENTAL
MODEL
Time-dependent variations of a linear system with n homogeneous compartments (or equivalence classes in the more general context of transformation systems [7]) are given by the following linear differential system:
dN(t) =AN(t)+Bu(t),
-
dt
tE
N(0) = 0,
y(t) = CN(t),
[O,TI, (2) (3)
where: N(t) is a vector function with dimension n, the i th component of which represents variations of the i th compartment. A is an n x n square matrix. The elements of that matrix have well-known significance [8,9]. However, it should be recalled that a,,, i # j, is a positive number relating to transfers from compartment j to compartment i, and is a positive number relating to exits from compartment j a = 4:&2,,, ti’the outside of the system. It will be assumed that matrix A has distinct eigenvalues. B is a matrix with dimensions n X h. It represents the possibilities of injection into the system. We assume that b compartments can be injected separately through an arbitrary function u(t). We denote by 1, the set of suffixes of such compartments, which will be known as injectable compartments. B is thus defined as B= {e,],,,s,
(4)
where e, is the vector all the components of which are zero except the i th component, which is unity. C is a matrix with dimensions c X n, known as the observation matrix. It represents the possibilities of observation of the system. It will be assumed here that c compartments can be observed. We denote by I, the set of suffixes of such compartments, which will be known as observable compartments. The matrix C is thus defined by
(m’ is the conventional
notation
for the transpose).
Last, we assume the system to be observable and controllable as defined by Kalman [lo], which is equivalent to assuming that the fitting of experi-
4
JACQUES DELFORGE
mental data requires n distinct exponential terms (representing the n eigenvalues). Since in most cases the number of compartments is determined by the number of exponential terms, this condition is not very restrictive [ll]. The solution of the system (l)-(3) is given by a(f-7)B~( T) d7. If Y(t) is the functional
matrix with dimensions
(6)
c x b, defined as
Y(t)= CeA’B,
(7)
we therefore have y(t) = /,jY( t - T)U(T)dT. Y(t) is a transfer function, i.e., Y(t) can be identified from experimental data [using suitably selected functions u(t)]; conversely, if Y(t) is known, experimental observations for any given vector u(t) can be predicted. Investigating the identifiability of the matrix A on the basis of experimental data is therefore equivalent to investigating the determination of A from Y(t), which is assumed to be known through experimental procedures.
2.
EQUATIONS
VERIFIED
BY M
We denote by A,, X*, . . , A, the n eigenvalues distinct; by A the matrix defined as
and M the matrix A. eigenvalues be inverted. well-known
of matrix A, assumed to be
n x n square matrix in which the columns are the eigenvectors of The n rows of M are designated as { l/;}r=1,2 ,,,,, n. Since the are distinct, the eigenvectors generate R” and the matrix M can We denote by { W, }, _ 1,2,, , n the n columns of M-‘. Based on a relationship, we may write matrix A in the form A = MAM-‘.
Since we have assumed that all terms {~?‘k’}~=i,~,,,~,, fitting of experimental data, numerical eigenvalues can be data. We therefore assume A to be known experimentally. ship (9), it is apparent that the problem of identifiability equivalent to that of the matrix M.
(9) are present in the derived from these From the relationof the matrix A is
LOCAL IDENTIFIABILITY
5
Based on a detailed analytical study of (7), it can be shown [ll] that the set of data on M which can be derived from experimental data is summarized by the two equation systems: E=CM
(10)
D=M-‘B,
(11)
and
where E and D are two matrices derived from the experimental data. If c < n or b < n, it is obvious that Equations (lo)-(11) alone do not permit identification of the matrix M. Since these equations contain all the information available from experimental data, more information is clearly required for identification. Such information is provided by the model structure, which is assumed to be known-or, more accurately, by the constraints imposed on the model structure through not allowing certain transfers between compartments (nonexistent arcs). Structural constraints on the matrix A can be defined by two vectors F and G, and written as F’AG = 0. Only the following two types of constraints
(12) will be considered
here:
(a) Type A constraint:
a'J = 0,
i.e.,
F’ = e,,
This constraint represents the impossibility ment j to compartment i. (b) Type B constraint: a eJ
=-
t
a,,=O,
i.e.,
G = eJ,
with i f j.
of direct transfer from compart-
F’= (l,l,...,
1) and
G=e,.
i=l
This constraint represents the impossibility ment j to the outside of the system.
of direct transfer from compart-
From (9), the conversion of constraints on A to constraints is straightforward, since (12) can also be written F’MAM-‘G The set of equations
verified
by M and
on (M, M-‘)
=0 M-’
(13) therefore
consists
of: the
JACQUES
6
DELFORGE
systems (lo)-(11); a set of equations of type (13) representing structural constraints on A and, by definition, the system MM-’ Id is the identity matrix of rank n.
the various = Id, where
III.
1)
EXAMPLE
OF LOCAL INVESTIGATION
(EXAMPLE
Difficulties in understanding the method proposed here are mainly due to the serious notation problems which arise when an attempt is made to find a general solution. For a clear presentation of the principles involved, this section gives the details of a specific example. Extension to the general case is straightforward, and covered in Section IV and in the appendix. Let us consider the model of Figure 1, where n = 3, c = 1, and h = 1:
-(a,,+ -c 121) A=
a21
al2 -(Q12
0
B=
The model is observable and a,,#O.
a13
+a,,)
0
a32
1 0 7 0I
-
a13
c = [O,l,O].
if azl # 0 and ui3 # 0. It is controllable
3
1
I
I -
-.
I I --
FIG. 1.
-
INJECTION
\
t
, \.
if azl f 0
OBSER\‘AT IO?:
LOCAL IDENTIFIABILITY The following
equations
are verified by M and M- ‘:
(a) The equation CM=
which corresponds
E.
(14)
to Vz = E, where E is known.
(b) The equation M-‘B=
corresponding
D.
05)
to W, = D, where D is known.
(c) The equation MM-’ (d) The equations
representing
= Id.
(16)
the constraints:
Impossibility of direct transfer from compartment 1 to compartment 3: the equation with M and M-’ is given by (13) with F = e3 and G = e,. We thus obtain the expression &AW, = 0. Impossibility
of direct transfer
from compartment
(17) 3 to compartment
08)
V*hW, = 0. Impossibility system:
of direct exit from compartment
-
i
2:
2 to the outside
y:nw*=o.
of the
(19)
i=l
Impossibility system:
of direct exit from compartment
-
t yllw,=o. r=l
3 to the outside
of the
(20)
The method is based on the well-known principle of taking the derivatives of all equations with respect to all unknowns, to form the Jacobian matrix Z, which by definition is the matrix where element (i, j) is the derivative of the ith equation with respect to the jth unknown. A well-known sufficient
8
JACQUES
DELFORGE
condition for local uniqueness is that the determinant of ~9’2 is different from zero. In the example given in this section, the Jacobian is given by the matrix of Table 1 (where the derivative of an equation with respect to a vector is defined as the vector formed by the derivatives of the equation with respect to each of the vector components). In general, the Jacobian is thus a matrix with dimension (cn + bn + n2 + r) X2n2, where r is the number of constraints. Based on a well-known rule, the determinant of ZrX is different from zero only if at least 2n2 rows of A“ are linearly independent. Also, it can be
TABLE The Jacobian
1 2
Derivative with respect to: Equation
4
(14)
0
vz 100 010
v,
Wl
w2
w3
0
0
0
0
Row No. 1 2 3
001 100 (15)
0
(20)
-(AW,)’
0
-(AW,)’
0
m-(AW,)’
010 001
4 0
0
5 6
VI
0
0
7
0
0
-2
v,‘l r=l
Column
No.
123
456
789
10 11 12
13 14 15
16 17 18
19
9
LOCAL IDENTIFIABILITY
proven in a general way [ll] that, within the equations CM = E, M-‘B = D, and MM-’ = Id, at least cb rows are linearly dependent on the others. For the Jacobian matrix to contain at least 2n2 linearly independent rows, it is therefore necessary that at least (n - c)( n - b) constraints exist (a detailed proof of this necessary condition for identifiability is given in [12]). In our example, we must therefore find in the rows of 3? corresponding to Equations (14), (15), and (16) an equation showing dependence on the others (since c = b = 1). We actually find that row 10 is dependent on the first six equations. Since M and M-’ can be inverted, it can be readily verified that, following elimination of row 10, the other rows associated with Equations (14), (15), and (16) are indeed linearly independent. Since the model under investigation involves four constraints [i.e., the exact number (n - c)( n - b) required], a sufficient condition for identifiability is for the determinant of the square matrix X1 derived from 3 through elimination of row 10 to be different from zero. Based on a well-known rule of determinant computation, it is immediately apparent that the determinant of X1 is identical to that of 3t02, derived from 2’ by eliminating the first six rows and columns 4, 5, 6, 10, 11, and 12. It will be observed that we thus eliminate all the columns corresponding to derivatives with respect to vectors V, and Wj, which are known from Equations (14) and (15). From the relationship (9), we conclude that AM=MA
(21)
and
We can therefore write the vectors AW, and ?$A in the following form: AK = AMPie, = M-‘Ae,
(23)
v/A = eJM.11= e:AM.
(24)
and
In the example given, we therefore have AW, = M-‘Ae,
= - (a,, + uzl) WI + a2,W2
AW, = M-‘Ae,
= a,,Wl - ( aI2 + a32) W2 + a,*W,
AW, = M- ‘Ae, = aI3 W, - aI3 W, V2A=eiAM=a21Vl-(~,~+a~2)V;
(25)
10
JACQUES DELFORGE
and 5
?A=
5
j=l
eJAM=-a,,l/,.
(26)
/=I
Based on the above equations,
we can write 3’
in the following form:
x2 = .GP344z
(27)
with (M-l)’
0
0 0
0 0
i
0
0
0
M 0
0 M
(M-l)’
0
_/@‘=
0
1 (28)
and Z3 as shown in Table 2. The determinants of matrices M and M-’ being different from zero, it is clear that the determinant of .?P2 is nil only if the determinant of Z3 is nil. We can expand the determinant again, based on rows 1, 4, 5, and 6. By deleting these rows and columns 1, 4, 8 and 11, we obtain a matrix 3P4 which has-the same determinant as Z3. It will be noted that we have thus eliminated all the columns associated with those coefficients which are factorized with vectors V, and W, known from relationships (14) and (15).
TABLE 2 The Matrix X3 1
0
0
0 0
1 0
0 1
0 0
a13
1
1
0
2 3 4
0
10
0
0
0
0
0
00
0
1
0
0 0
0 0
00 01
0
1
0
00
“21
0
1 0
0 1
-(ad
+
an)
0
-
1
0 0
0
0
t”12
0
0
0
a12
00
100 0 00
0
0
-
in (27)
+
0 2
a32
a13
3
-
a12 013
4
Cal,+
-
+
u32
ael
=13
6
0
7
a21
o
o
0 8
9
-(a,,+u,,)
1
0
0 11
8 9
0 a,1 10
5 6 I
0
0 a32)
0 5
0
0
0 a32)
0 0
10
11 0 12
12
11
LOCAL IDENTIFIABILITY
All the remaining rows initially associated with Equation (16) contain two coefficients equal to 1. Based on a well-known property, we know that the determinant is unchanged if the matrix is modified by adding or subtracting columns. The determinant of .P4 is therefore identical to that of matrix 2’ defined as
&“j1
&0s=
Id
0
(29)
K
[ with
K= %2
+
0
0
0
- a21
a32
-
-
ad
0
a13
0
a21
0
a32 -
(‘12’
a32)
0
aed
-
(30) ‘32 a13
We finally find that a sufficient identifiability condition is that the determinant of K be different from zero. This can be readily shown to be verified only if a21 + 0, (31)
a13
+
0,
ad
+
a12
+
=32.
From the above, we conclude that a sufficient condition for local uniqueness is that the model be controllable and observable and that a,, # aI2 + aj2.
IV.
1.
A SUFFICIENT
STATEMENT
CONDITION
FOR LOCAL IDENTIFIABILITY
OF RESULT
The reasoning applied to the above example is quite general, as emphasized throughout the discussion by remarks stressing the general nature of the matrices derived and operations performed. To avoid repeating that lengthy procedure in every specific case, this subsection gives the rules for writing matrix K directly. Let us consider the set of suffix pairs (p, q) such that p +ZIc and q 6 Is. The number of such suffix pairs (p, q) is therefore (n - c)( n - b). We will now write the matrix K, with dimensions r X( n - b)( n - c); each of the (n - b)( n - c) columns is associated with a suffix pair (p, q); each of the r rows is associated with a constraint on the structure of matrix A. We will write K row by row.
12
JACQUES
DELFORGE
(a) In a row corresponding to the impossibility of direct transfer from compartment j to compartment i, i.e., to a constraint a,, = 0 (type A constraint), the coefficient of column ( p, q) is given by
4P-r)aq,- %-,)a,, 1
(32)
with
4-,, =
1 0
1
if if
x=y, x#y.
(b) In a row corresponding to the impossibility of direct transfer from compartment j to the outside of the model, i.e., a constraint a,, = 0 (type B constraint), the coefficient of column (p, q) is given by (33) THEOREM
I
A sufficient condition for local identifiability is that the determinant of K’K is different from zero.
Because of its length, the proof is given in an appendix. Remark I. It is important to note that, thanks to the relationships (32) and (33), this sufficient condition is readily applicable, as the examples will show. In particular, computation of matrix K can be programmed very easily, and is therefore feasible irrespective of model complexity. On the other hand, it is well known that determinant computation can present difficulties in the case of large dimensions. In practice, however, matrix K includes a large number of zeros and computation is usually straightforward.
2.
EXAMPLES
Example 2.
In the model of Figure 2,
-(ad + a21+ a311 A=
[l,O,O],
c=[;7
;I,
a12
0
u31 B’=
0
a12 -
a21
a23
-
,
a23 I
whence
Z,=
(11,
whence
I,=
{1,2}.
The model is observable if u 23 f 0 and controllable if a31 f 0 and aT3( a21 + a,,) # a21a12. From the relationships (32) and (33), the matrix K is defined
LOCAL IDENTIFIABILITY
13
FIG.
2.
as (p, 4) = (392) (373)
constr&ts a13 =
K=
0
a 32 --0 ar3 = 0 Qe2
=
0
It can be very readily shown that the determinant of K’K is different from zero only if uZ3 # 0 (an assumption already included in the condition for observability). When it is observable and controllable, the model of Figure 2 is thus always locally identifiable. Exumple 3. Let us now consider the model of Figure 3, which has five compartments. Compartments 1 and 4 are injected (I, = {1,4}), whereas compartments 4 and 5 are observed (I, = {4,5}):
Using the relationships (32) and (33) leads directly to the matrix K shown by Table 3. This table illustrates a case where the computation of K’K is not straightforward. In such cases, there is usually an advantage in replacing the computation of the determinant of K’K by the determination of necessary and sufficient conditions for matrix K to have (n - c)( n - b) independent rows. In the example given, we can write a matrix K’ including rows 1, 2, 3, 5, 7, 9, 10, 11, and 12 of matrix K. It is obvious (through computation of the
14
JACQUES DELFORGE
4
a 02
L
FIG. 3.
determinant, which is straightforward) that the matrix K’ is invertible if u51a34u25 # 0. Conversely, we can verify that if a5i = 0 or u34 = 0 or uZ5 = 0, the matrix K has not 9 independent rows. We conclude that if the model of Figure 3 is controllable and observable and if a51u34a25 # 0, then the model is locally identifiable. V.
1.
EXAMPLE OF A LOCALLY IDENTIFIABLE NONINVERSIBLE JACOBIAN MATRIX A COMMON
MODEL
WITH
ERROR
In the introduction, we mentioned that most approaches to the problem of linear model identifiability lead to a system of non-linear equations, the number of which is to be determined. It is well known that this problem has no general mathematical solution. Whatever the approach, it is always possible to investigate local uniqueness using the Jacobian matrix, as we did in Sections III and IV. However, many authors have stated erroneously that a necessary and sufficient condition for local uniqueness at a point P is for the determinant of the matrix H’H to be different from zero at P (H being the Jacobian matrix associated with the system of nonlinear equations resulting from the approach used).
LOCAL IDENTIFIABILITY
15
16
JACQUES
DELFORGE
Although this condition is indeed sufficient for local uniqueness, it is important to point out that it is not always necessary, as shown by S. Vajda [13] on the basis of a nonlinear example (see also References [19] and [ZO]). It will now be shown that this remark also holds in the case of linear models, and that its significance is much greater than it might appear. First, let us recall the conventional one-dimensional results aimed at finding the solutions of the equation f(x) = 0. If u is a solution, i.e., if f(a) = 0, the condition f’(a) f 0 is not a necessary condition for solution a to be locally unique. It can be seen that, if f’(a) = 0, solution a is also locally unique if one of the higher derivatives is different from zero at point a. Conversely, a necessary and sufficient condition for the solution u not to be locally unique is that the derivatives of f(x) of all orders are zero at point a. Since the amount of computation required may be large, the application of such a criterion may present problems. However, there is an important particular case; if f’(x) = 0 everywhere in the neighborhood of a (i.e., over one dimension through the interval [a - E, a + r]), then the solution a is not locally unique (it can then be shown that all derivatives of higher order are equal to zero). Similar results are obtained in the case of equation systems. The Jacobian matrix H relates to the first derivative of the equations with respect to the unknowns. If the determinant of H’H is zero, it is necessary to expand the system further in the neighborhood of the solution by computing the matrix equivalents of higher derivatives, which would be very difficult in practice. We thus note the following results, which, although fractional, are valid irrespective of the system of equations involved: (a) if the determinant of H’H is different from zero at P, solution P is locally unique (a well-known result). (b) if the determinant of H’H is zero at P and everywhere in the neighborhood of P, then the solution P is not locally unique. This is the case when the system has fewer equations than unknowns. As suggested by S. Vajda in [13], this result can be derived from DieudonnC’s rank theorem [14]. The proof, however, is lengthy and elaborate. The results of F. M. Fisher [20] can also be used. (c) If the determinant of H’H is zero at P and in a subspace of measure zero in the neighborhood of P, but differs from zero in the rest of this neighborhood (i.e., almost everywhere), then there is no way to arrive at a conclusion without a detailed study of higher order derivatives (this is a case which, although rather frequent, has been omitted by some authors). 2. EXAMPLE
( RECONSIDERATION
OF EXAMPLE
1)
We will now reconsider Example 1 (discussed in Section III and illustrated by Figure l), using the transfer method proposed by R. Bellman and K. J.
17
LOCAL IDENTIFIABILITY
Astrijm [15] and developed since by a number of authors, especially by C. Cobelli et al. [4] and by S. Audoly and L. D’Angio [16]. If Y(s) is the Laplace transform of y(t), and U(s) that of u(t), we have Y(s) = G(s,A)U(s), G(s, A) being the transfer matrix defined as G(s,A)=C[sZ,-A]-+.
(34)
For a given matrix A, the elements of the matrix G(s, A) appear as the ratio of two polynomials in s, of degree less than or equal to n. We denote by {a,} the set of parameters included in the polynomials. The transfer matrix being known, the behavior of the system can be fully described. The problem of the identifiability of the matrix A from experimental data y(t) can thus be reduced to a study of the identifiability of A based on the transfer matrix. Every parameter (Y, (which is then assumed to be known, since it is determined from experimental data) is equal to a nonlinear combination, of degree less than or equal to n, of the unknown elements in the matrix A. We thus obtain a system of nonlinear equations. The matrix A is therefore identifiable when this system has a unique solution. In the example of Figure 1, the following system of equations is obtained after simplification: (35)
~,1+~21+~12+~32+~13=~1,
-(ad+a21)(%2+a32)+al2a21
_
(36)
~13b,l+~21+~,2+~32)=~2~
a13a,,(a12
+
032)
=
a39
a21=a4, u21”13
=
a5.
(37) (38) (39)
The Jacobian matrix can be computed. Leaving the equations in the order of (35)-(39) and taking the derivatives with respect to the unknowns written in the order a,l, %l? a12, u329 013, the Jacobian matrix J is given by
18
JACQUES
DELFORGE
It can be shown that the determinant of J’J is different from zero only if u2i # 0, ui3 # 0, and a,i + ui2 + u3z. This result is obviously identical to that derived in Section III [Equation (31)] from the matrix K defined by (30). However, it should be noted that computation by the transfer method does not provide relationships such as (32) and (33) from which the matrix K can be written directly. The cases a zi = 0 and ui3 = 0 are excluded, owing to our controllability and observability assumptions. We must however investigate the particular case where U el =
If the above relationship become
u12
+
u32.
(41)
is taken into consideration,
2a,i + a,, + -(a,,
+ %i)%
+
%2U21-
Ul3(2a,l+
equations
U13
a211
=
q,
(42)
=
a29
(43)
a3 9
(44)
=
a49
(45)
=
a5.
(46)
2_
Q13Ue1a21 u21”13
(35)-(39)
The following solution can be readily derived: a21
=
a4
[Equation
(45)],
[combination
of (45) and (46)],
[combination
of (44)) (45), and (46)].
ui2 is then derived easily from (43), and u32 from (41). We have thus shown that, if the relation (41) is verified, the identification problem has a unique local (or even global) solution. It has also been shown that the determinant of J’J is zero in this case. We therefore draw the conclusion that, contrary to published conclusions, for the determinant of J’J to be different from zero is not a necessary condition for local uniqueness. VI.
1.
A NECESSARY AND IDENTIFIABILITY MAIN
SUFFICIENT
CONDITION
FOR
LOCAL
RESULT
To investigate the local identifiability of a solution P, we first compute the matrix K, which is straightforward on the basis of the relationships (32)
LOCAL IDENTIFIABILITY
19
and (33). We then use the general results stated in the previous section: if the determinant of K’K is not zero, it follows that solution P is locally unique. However, if the determinant of K’K is zero at P and everywhere in a neighborhood to P, it follows that the solution P is not locally unique. If an intermediate situation prevails, i.e., if the determinant of K’K is zero at P and in a subset Yi of a neighborhood Y of P, although different from zero through the rest of this neighborhood, then no conclusion can be drawn for the time being. We can, however, make an important remark. If the solution P is not locally unique, all other solutions P’ near P are also not locally unique and must therefore be located in the subset Sq of Y. It therefore follows that if the solution P is not locally unique within V, it is also not locally unique within subset 9,. Since the converse obviously holds, investigating the local identifiability of solution P in V is equivalent to investigating the local identifiability of P in subset Y;. Since the subset Yi is a set of points in V where the determinant of K’K is zero, it can be defined by one or more equations verified by the coefficients of the matrix K, i.e., by certain parameters a,, which are present in K. Equations thus obtained can be used to express new constraints on A. We thus write Kc’), a matrix derived from matrix K, to which we append new rows representing the new constraints. The above reasoning applied to K in neighborhood Y also holds as regards Kc’), considering the subspace Yi only: (a) If the determinant of K (‘)‘K(‘) is different from zero, it follows that the solution P is locally unique within Yi and therefore within V. (b) If the determinant of K(*)‘K(‘) is zero throughout the neighboring space Vi of the solution P (VI being the intersection of Y with Yi), then it follows that the solution P is not locally unique. (c) If we return to the intermediate situation in which the determinant of K(l)rK(l) is zero at P and in a subset Y2 of Vl;, but different from zero throughout the rest of “v, (i.e., almost everywhere within Y’i), no conclusion can be drawn and further investigation is required. The reasoning continues along similar lines; a subspace Y; is defined through equations which must be verified by parameters a,, for the determinant of K(‘)‘K(‘) to be zero; these equations are regarded as new constraints applied to the matrix A. We thus define Kc*‘, a matrix derived from Kc’) by appending the rows representing the new constraints. The determinant of Kc2)‘K(*) is then investigated in the subspace Y2. If required, we can thus define a series of matrices Kck) and subspaces &. Since the dimension of the subspaces YA decreases by at least one unit at each step, the series must be finite, and we can always determine that the solution P is either locally identifiable or nonidentifiable. Hence the follow-
20
JACQUES
DELFORGE
ing theorems: THEOREM
2
A necessary and sufficient condition for a model to be locally identifiable almost everywhere is that the determinant of K’K is different from zero almost everywhere. THEOREM
3
A necessary and sufficient condition for local identifiability of a solution P is the existence of a stage k at which the determinant of KCk”KCk) is different from zero. 2.
EXAMPLES
Example 4.
In the model of Figure 4, we have
-(u31 + 041) A=
012
-((I,?+"12+"3z+Q)
0
-toe3
031
*32
041
O42
0
0
0
0
10’
0
1 1
0
014
0
024
0
whence
lB=
{1.3},
01’
whence
I,=
{3,4}.
FIG.4.
0
-+ 043) 043
-(a,4
+
014
+
024)
1%
21
LOCAL IDENTIFIABILITY
From the relationships
(32) and (33), the matrix K is defined by:
(p, q) = (1,2)
K=
(1,4),(2,2)
(2,4)
0
Constraints a 21 -0 -
0
0
0
a43
0
0
a13
=
0
0
0
0
a43
a23
=
0
a34
=
0
a rl
=
0
0
-
0
- a4l
a31
a41
0
-
0
- a4l
a32
It is obvious that the determinant of K’K is always zero, since the columns (1,2) and (2,2) are zero. The model of Figure 4 is thus never identifiable. Example I (continued). Let us now reexamine the example investigated in Section III, and shown by Figure 1. We have seen that if a el
al2 - aj2 = 0,
-
(47)
the determinant of K’K is zero. However, by using the transfer method, we have shown in Section V.2 that local uniqueness nevertheless existed. This result will now be derived very easily by using Theorem 3. We follow the procedure explained in Section VI to write the matrix K(l), appending a row representing the constraint (47) to matrix K as defined by (30). This row is readily derived from the relationships (32) and (33) by subtracting the rows associated with a,, and a32 from the row associated with a,,. We thus obtain
0
-
0
apl - al3
0 0 K(l)
=
-2a21
0
a21
a21
a32
-
a32
-
tal2’+
0321
0 al2
+
a32
-
a21
Since azl, a13, and a32 are assumed to be different from zero (controllability and observability assumptions), it is obvious that the determinant of KC1)‘KC1) is never zero. It therefore follows, from Theorem 3, that if (47) is verified, the identification problem has a locally unique solution although the determinant of K’K is zero. This result is the same as derived in Section V.2 by the transfer method; the method proposed here is however much faster, since it requires no analytical computation,
22
JACQUES
DELFORGE
Exumple 3 (continued). Let us reconsider Example 3 of Section IV.2, shown by Figure 3. From Theorem 1 we drew the conclusion that if u51u34uz5 differs from zero, local uniqueness exists. Theorem 3 permits investigating local identifiability in the following particular cases:
(a) a51 = 0. From the relationship (32), the condition u5, = 0 results in an all-zero row. In this case, the matrix K (l) is thus identical to K. Since we know that the determinant of K’K is zero, it follows from Theorem 3 that local uniqueness does not exist. (b) 025 = 0. The condition a 25 = 0 results in the following row:
(0,0,0>0,0, u55 - u2*,0,0, - u23).
(48)
We observe immediately that, in the matrix K(l) [written by appending the row (48) to matrix K shown by Table 31, the column associated with ( p, q) = (2,2) is all zero. The determinant of K ‘(‘)K(l) is therefore zero, and there is no local uniqueness in this particular instance. (c) a34 = 0. The condition a 34 = 0 results in the following row: (0,0,0,0,0,0,0,0,
(49)
a54).
This case is similar to the above: it can be seen that the column associated with ( p, q) = (3,3) is all zero. Example 5.
-
A=
(a21
In the model of Figure 5, we have +
a31
+
0
a12
041) -
a21
a12
a31
0
a41
0 IB= {I),
u23 -(a23+ue3)
0 Ic = {2,3}.
The computation of K from relationships we arrive at the matrix of Table 4. Assuming 041
we observe determinant
of KC”
f
0,
u,,#O,
(32) and (33) is straightforward;
and
a,,-ua,fO, of K’K
immediately that the determinant of the matrix K’, defined as a33
a14 -
a44
1 3
(50) is zero only if the
(51)
LOCAL
23
IDENTIFIABILITY
FIG. 5.
is
zero, i.e., if (a33 - Q)(%
-
h)-Q41a14
=
(52)
0.
To determine whether local uniqueness exists when (52) is verified, we may use Theorem 3 and derive the matrix KC”. A new feature, as compared with the previous examples, is that the new constraint defined by (52) is a nonlinear equation. This however presents no special problem, since the corresponding row of KC’) is derived from the relationships (32) and (33) using the well-known properties of derivatives. If L,, is the row defined by (32), and therefore represents the derivatives of a,, with respect to the unknowns (which, as mentioned earlier, are elements of the matrices M and M-l), the row of KC” associated with constraint (52) is (4,
- ‘L)(%
- %)+(a,3
- %)(J%3
- L-
L4lQl4
-
a41L14.
(53) TABLE 4
LJ,4)=(1.2) a23
(1.3) a33 -
0 0 K=
-
a31
-
041
all
0 0 0 0
0
- a4
-
a21
-
-
a22
031
0
(472) 0 0 0 0
(194) 0 -
a21 031
0 0 0 -
-
a23 -
a33 -
a21
a22 -
014
0 0 0 0
aZ1 - a44
041
0
(4.3)
a,4
a44 031
0
(434) 0 0 0 0 0 0 -
041
0
Constraints a,3 = 0 a 24 --0 cl 34 -0 a 32 --0 aq2 = 0 a 43 --0 ad = 0 “e2 = 0
JACQUES DELFORGE
24 From (32), rows Lij are written as
L,, = (0, - ~31,0,0,0,0)~ L,,
=
(ax,
a31,
L44
=
(WA
-
u41,0,0,0),
a41,0,0,0),
L4,=(0,0~0~~,,,~3,,~4,), L,,
=
(O,O,
a44
-
q,,O,O,
From (53), the row of K(l) associated
-
q4).
with the constraint
(52) is then given
by (- %1(%
- a4419 -
a31ca33
-
@l,)YO,
-
%a149
-
a,,a,,,O).
Based on assumption (50), and therefore assuming the relationship (52) to be verified, it can readily be shown that the determinant of K(‘)‘K(‘) is zero only if the determinant of matrix K”” defined as K(l)’
(a33
= -
-
a31(033
Qll) -
d
-
a14 a31u14
(54 I
is zero. The determinant of K(l)’ is equal to - 2u,,u3,(u3, - all). Since u3i, are assumed to be different from zero (controllability and ~147 and ~41 observability assumption), we deduce that u33 = ali is impossible [Equation (52)] and that the determinant of K (l)rK(l) is never zero. It therefore follows that, based on the assumption (50), the solution to the identification problem is locally unique if the relationship (52) holds. This is another example of a model locally identifiable with a noninvertible Jacobian matrix. VII.
NECESSARY STRUCTURAL IDENTIFIABILITY
CONDITIONS
FOR
LOCAL
From the necessary and sufficient condition for local identifiability, valid almost everywhere, as stated by Theorem 2, we will now derive several conditions for local identifiability which are only necessary (not sufficient), but show the significant advantage of being directly related to the model structure. Should one of these necessary conditions fail to be verified, it immediately follows that the case is one of nonidentifiability. (In this section it is assumed that none of the eigenvalues of matrix A is zero.) 1.
REQUIREMENT
FOR (n - c)(n - h) USEFUL
CONSTRAINTS
In an earlier publication [6]. we defined a useful constraint as a constraint for which the associated equation [of the type (13)] was independent of
25
LOCAL IDENTIFIABILITY
equations derived from experimental data, i.e., the equation systems (lo), (11). It will now be shown that, as part of the necessary and sufficient condition stated in Section VI, useful constraints are actually those for which the associated rows of the matrix K as defined by (32) or (33) are different from zero. Conversely, the useless constraints defined in [6] lead to all-zero rows in K, providing no additional data for the determination of identifiability. Let us consider the row of the matrix K associated with the impossibility of transfer between a compartment j and a compartment i. From (32), the elements of this row are given by (55) We will now examine all possible cases, with compartment noninjected and compartment i observed or nonobserved.
j injected
or
(a) Let us assume that j E 1,. Since q 4 I,, we cannot have q = j, and therefore the second term of (55) is always zero, irrespective of p and q. If i E I,, then since p 4 I,, we can never have p = i. The first term also is thus always zero. The associated constraint is therefore of the useless type. However, if i E I,, there is a p value such that p = i. Thus, irrespective of q e Is, we find, again from (55), terms u4, in columns (i, q). In this case, the row associated with the constraint is nonzero only if there is a term a4, different from zero, i.e., if there is nonzero transfer from compartment j to a noninjected compartment q. (b) Assuming that j E ZB and i cZI,, the associated row in the matrix K is always nonzero because the column associated with q = j and p = i contains the element a,,, which in turn is nonzero because, from basic assumptions, none of the A eigenvalues is zero (due to the structure of matrix A, the only case where a diagonal element a,, is zero is that in which a full column of A is zero, which implies that one of the eigenvalues is zero). (c) Let us assume that j 4 IB and i E Z,. Since p E I,, we cannot have p = i, and the first term of (55) is thus always zero. However, as j E Is, there exists a value of q such that q = j. The associated row of K therefore includes all the u,~ terms with p +ZI,. This row is nonzero only if one of the U terms is nonzero, i.e., if a nonzero direct transfer exists between a &observed compartment p and compartment i. We finally obtain Table 5, which is of course reminiscent of Table 1 in [6]. It should however be pointed out that conditions are now less restrictive, because certain assumptions made in [6] are no longer necessary (e.g., the assumption that all elements in CA’B are nonzero). Further, conditions C, and C, no longer include the restrictive condition expressed by the relationship (40) of [6]. Last, we have added a column giving the type of equation
26
JACQUES TABLE
DELFORGE
5
Type A Constraints: Impossibility of Transfer from Compartment From compartment
To compartment
i
.i
Nonobserved
Injected
j to Compartment
Added
type
C, verified
Al,
C, not verified Noninjected
Observed
C2 verified C, not verified
Noninjected
Nonobserved
Injected
Observed
“Condition Condition
Constraint
condition’l
-
Al2
A2, A22
I Useful constraint?
Equation type
Yes No
Linear in v
Yes No
Linear in W,
A3
Yes
A4
No
C,: Nonzero transfer from compartment j to a noninjected C,: Nonzero transfer to compartment i from a nonobserved
v,AW,=O
compartment. compartment.
associated with constraint type. This information, although difficult to obtain directly from (32), can be readily derived from the computations in Sections 3.C and 3.D of [6]. In the same way as before, Table 6 can be derived from the relationship (33); it gives the necessary and sufficient conditions for a B-type constraint to be useful, i.e., for it to lead to a nonzero row in K. Since the matrix K has (n - c)( it - b) columns, a necessary condition for K’K to have a nonzero determinant is that matrix A contains at least (n - c)( n - b) nonzero rows. Hence, the following theorem: THEOREMS
A necessary condition for identifiability (n - c)( n - b) useful constraints.
is the existence
of no less th’an
Remark 2. This necessary condition is, in practice, highly selective, since the vast majority of models where it is verified are indeed locally identifiable. Following an incomplete proof, this condition had been regarded as sufficient in [6]. In [5], J. P. Norton gives a counterexample where, although (n - c)( n - b) useful constraints exist, the model is still not identifiable. In [7], it is shown that nonidentifiability results from a linear dependence between two constraints of the Al, type. The precise conditions of independence between two constraints of the Al, type or A2, type are specified in [7]
LOCAL
27
IDENTIFIABILITY TABLE 6 Type B Constraint: Impossibility
Compartment J Injected, nonobserved
of Exit from Compartment
1st added condition”
2nd added condition’
C, verified
-
Constraint type
C, not verified Noninjected, observed
Equation
Bl,
Yes
Bl,
No
type’ Linear in Vr.
-
B21
Yes
C, not verified
C, verified
B2?
Yes
C4 not verified
B2,
No
B3
Yes
VXl1W, = 0
C, verified
B4,
Yes
Linear in Vx
C, not verified
B42
No
B43
No
C, verified
C, not verified “Condition
Useful constraint?
C, verified
Noninjected nonobserved Injected, observed
j to the Outside
C,: As in Table 1. Condition
1 to a nonobserved compartment. nonobserved compartment either compartments of nonzero transfers hV~=X;=,l+
C,: There is a transfer
vxizw,
Linear in W,
from compartment
Condition Cq: There is a transfer from a to compartment J or to one of the arrival originating in compartment j.
and [17]. It can be readily shown from (32) that such a dependence identical rows in the matrix K. 2.
NECESSARY
THEOREM
CONDITIONS
= 0
RELATING
TO TYPE
leads to
B CONSTRAINTS
6
(a) A necessary condition for identifiability is the existence of a number of type B constraints at least equal to the number of compartments simultaneously noninjected and nonobserved. (b) It is further necessary that, in each of such simultaneously noninjected and nonobserved compartments, at least one of the following two conditions be
JACQUES DELFORGE
28 verified:
direct exit from the compartment to the outside is impossible; the compartment is the arrival compartment of a nonzero transfer originating from a compartment where direct exit to the outside is impossible. Proof. Let us assume that compartment p is neither injected nor observed. Therefore, there exists, among the (n - c)(n - b) columns of K, a column associated with the suffix pair (p, p). But, from the relationship (32), the element in this column associated with the constraint ai, = 0 is given by
8cP-l)aP, - 8(P-J)aw This element is always zero, since when i = p or j = p, we obtain a,,, which, from basic assumptions, is nil. Thus, only type B constraints can lead to nonzero elements in column ( p, p). Let us assume the existence of k noninjected and nonobserved compartments. From the above, the matrix K can be written in the following form:
where K’ is a matrix with dimension e X k, e being the number of type B constraints. A necessary condition for the determinant of K’K to be nonzero is that the determinant of K”K’ is also nonzero. It immediately follows that emust be greater than k; hence the first proposition in the theorem. It is further necessary that none of the columns in K’ be all zero. And, from (33), the element of column ( p, p) associated with the impossibility of direct exit from a compartment is given by -
aPJ
-
%Y)%
If there is no direct exit from the compartment P to the outside, a row exists where j =p. We thus obtain app, which is always nonzero from basic assumptions (nonzero eigenvalues). In the opposite case, it is necessary for j to exist such that ap, # 0. Hence the second proposition in the theorem. Example 6. Figure 6 shows the same model, although with different observable compartments. In the first case, there is only one nonobserved and noninjected compartment (compartment 3) and a single type B constraint (impossibility of exit from compartment 3). The necessary conditions for identifiability given in the theorem can be readily shown to hold (there actually is local uniqueness but two distinct solutions, and therefore no global uniqueness). In the
29
LOCAL IDENTIFIABILITY
FIG. 6
second case, however, although compartments 2 and 3 are neither injected nor observed, there still is only one type B constraint (exit from compartment 3). The first necessary condition of the theorem is not verified; therefore, there is no local uniqueness and the identification problem has an infinite number of solutions. Example 7. The model in Figure 7 has five compartments, two of which are simultaneously noninjected and nonobserved (compartments 2 and 3). Since there are three type B constraints, the first part of Theorem 3 is verified. However, the second condition in the theorem is not met, because there is nonzero transfer from compartment 2 to the outside, and no direct transfer between a compartment with no exit to the outside (compartment 1, 4, or 5) and compartment 2. The necessary condition of Theorem 5 is thus not verified; the identification problem has an infinite number of solutions.
JACQUES
DELFORGE
FIG. 7
However, if we add a direct transfer, for instance between compartment 5 (with no possibility of direct exit to the outside) and compartment 2, all the necessary conditions in the theorem are verified. It will be observed that the model is then as shown by Figure 3 and that, as shown earlier in Section IV.2, a necessary condition for identifiability is for uZ5 to be different from zero. Example 8. The compartmental system in Figure 8 is a catenary model with input and output in the same compartment. In this model, there are n - 1 nonobserved and noninjected compartments. The first necessary condition of the Theorem 6 is verified if there are n - 1 type B constraints. Thus,
FIG. 8
31
LOCAL IDENTIFIABILITY
the necessary condition for identifiability of this model is that the number of excretions to the environment is zero or one. This result is already known, but this example shows the directness of the conclusion. VIII.
CONCLUSIONS
Identifiability is a major problem in model justification. In this paper, we have proposed a number of criteria applicable to the investigation of local identifiability. The many examples given have shown these criteria to be easy to use, even in the case of complex models. Recent progress made in the field, and the results presented in this paper, which are valid although the assumptions include very few restrictions, lead to the conclusion that studies of local identifiability are now, in most cases, exempt from difficulties. It is important to note that the criteria presented here are not limited to giving results in the neighborhood of a particular solution; they also permit the investigation of local identifiability throughout the parametric space. They may thus be regarded as a good starting point for global identifiability studies. APPENDIX.
PROOF
Let us designate
OF THEOREM
1
as
(56)
cp(M,M_‘)=.F the system of equations
verified by M and M-‘.
This system includes
CM=E, M-‘B= MM-’ and a set of equations
representing
{ F,‘MAM-‘G,}
(57)
D
(58)
= Id;
(59)
the various structural = 0,
k=1,2
,..., r.
constraints
on A: (60)
Y is the vector formed by the elements of M and Mp ‘, the elements of M being written row by row and those of Mm1 column by column:
v=(v;,v; For a variation
,..., v;,w,
)...,
WJ.
A+‘- or Y, we have, by definition, A9
= #A-Y,
where &’ is the Jacobian matrix given by the Table 7.
(62)
(59)
(57)
Derivative with respect to Equation
(M-1)’
0
0
0
&Id 0
C,, Id
0
(AC’)’
C,; Id
Cl, Id .,:
..
(AC’)
0
0
Cc,:Id
C,nId
The Jacobian X a
TABLE 1
0
Y2
0
vz
0
“1
B,; Id
81, Id
Y,
Vz
“1
B,,Id
B,, Id
0
.,:
..’
Y,
0
v2
0
VI
0
4, Id
B,,, Id
33
LOCAL IDENTIFIABILITY
c 1:
+..q =
34
JACQUES
We suppose that AF If the compartment i (there exists k with C,, injected, we deduce that
DELFORGE
= 0, and we want to show that AY = 0. is observed, we deduce from Table 7 that A< = 0 = 0, except for p = i), and if compartment j is AW, = 0 (there exists k with Bqk = 0 except for
4 = _i). By deleting the columns corresponding to y, i E I,, and to W,, j E I,, and by deleting the rows corresponding to Equations (57), (58), we obtain the matrix Z’, the vectors A.F’ and AV’, and the simplified equation A.F’ = Z’AY’. This matrix X” can always be written in the form .P=X.M;
(63)
thus A9 J? is a block-type matrices M:
diagonal
= X.-MAY’.
matrix, consisting
(64) of n matrices ( M-l)r
and n
(M-l)’ 0
(M-l)’
“M=
(65)
(M-l)' M 0
M M
We want to study the structure of the matrix Xx. Based on relationships (21) and (22), we have
(Aw,)‘=
(W'e,)'
= (M-RAE,)' = ( A~,)'( and V,A = e:MA = e:AM.
M-~)'
35
LOCAL IDENTIFIABILITY
Thus, in a row of the equation (64) associated (KAY = 0), the transposed vector corresponding given by
and the transposed
vector corresponding
with a type A constraint to (MP’)‘Al$, p P I,, is
to MA WY, q t-i Is, is given by (67)
&&A.
In a row associated with a type B constraint (C:,,~.hF$$ = 0), the transposed vector corresponding to (M-l)’ A?$, p 4 I,, is given by - (Ae,)‘, and the transposed
vector corresponding
(68) to MAW,, q P ZB, is given by
(69) Let us consider
a row associated with Equation As W4 + SAW,
= AVp M-‘e,
(59) (VPW, = 0 or 1). Then + epMAW4.
We shall consider the four following types: (a) p E I,. q E IB. This row is nil (AC E Y’, AW, E A%‘-‘). (b) P 4 I,, qE 1s. The row contains only one coefficient not zero: a 1 in the column corresponding to the 9th component of (M-l)’ AV;, denoted {(M-‘)‘AV&. The row contains only one 1 in the column corre(c) PEG, 441,. sponding to { MA W4 }p. (4 P @ I,, q @ IB. The row contains two coefficients equal to 1: in the column corresponding to ((M- ‘)l AV; }4 and in the column corresponding to { MAW4},. Thus ((M-l)‘A~~],+{~A~,},=~
‘jp E I,
We finally find that the only components
and Vq P IB,
of MA+‘-’ different
((M-‘)‘AV,‘)
4
and {MAW,),
with
pPZ,
and
qEI,.
(70)
from zero are
36
JACQUES DELFORGE
We can deduce from (66) and (67) that the component of X associated to a type A constraint (YAW, = 0) and corresponding to {( M-‘)rAV~}4 is
and that the component
corresponding Sc,-,&4e,
to (MAW,},
is (72)
= aC,-4ja,P.
From (68) and (69), we deduce that the component type B constraint (C:,,V,AW, = 0) and corresponding
of X associated to a to {(M-l)' AV;}, is
-(Ae,)‘e,=-a,, and that the component
corresponding
(73)
to { MAW, }p is
(74) Because the equation
(70) is always verified, we can consider as unknowns
and define the system of linear equations
with K an r X(n - c)(n - b) matrix, defined on the basis of the relationships (32) and (33), deduced from (71)-(74). Based on a well-known property, we deduce that if the determinant of K’K is different from zero, there is no variation AV” # 0 such that AY” = 0. In this case, we conclude that if Y defined by (61) is one solution, then it is locally unique, because there no variation AV # 0 of ^Y such that A9 = 0. REFERENCES 1
.I. Delforge, The problem of structural identifiability of a linear compartmental system: Solved or not? Muth. Biosci. 36:119-125 (1977). 2 J. P. Norton, An investigation of the sources of nonuniqueness in deterministic identifiability, Math. Biosci. 60:89-108 (1982). 3 E. Walter, Identifiahiliq of State Space Models. Lecture Notes in Biomathematics. 46, Springer, 1982. 4 C. Cobelli, A. Lepschy, and G. Romanin Jacur, Identifiability results on some constrained compartmental systems, M&h. Biosci. 47:173-195, (1979).
37
LOCAL IDENTIFIABILITY
5 J. P. Norton, Letter to the editor, Math. Biosci. 61:296-298 (1982). 6
11 12
J. Delforge, Necessary and sufficient structural condition for local identifiability of a system with linear compartments, Math. Biosci. 54:159-180 (1981). P. Delattre, Transformation system with time-dependent characteristics and population theory, Math. Biosci. 32:239-274 (1976). J. A. Jacqua, Compartmental Ana!)% in Biology and Medicine, Elsevier, Amsterdam, 1972. A. Rescigno and G. Segre, Drug and Tracer Kinetics, Waltham, 1966. R. E. Kalman, P. L. Falb, and M. A. Arbib, Topics in Mathematical Sysrem Theory, McGraw-Hill, 1969. W. D. Smith and R. R. Mohler, J. Theoref. Biol. 57:1-21 (1976). J. Delforge, New results on the problem of identifiability of a linear system, Math.
13
Biosci. 52:73-96 (1980). S. Vajda, Comments on structural identifiability
7 8 9 10
IEEE
Trans. Automat. Control AC24:495-496 J. Dieudonne, Fondations of Modern Analysis,
in linear time-invariant
systems,
(1979).
Academic, New York, 1960. 15 R. Bellman and K. J. Astrom, On structural identifiability, Marh. Biosci. 7:329-339 14
16
(1970). S. Audoly and L. D’Angio, On the identifiability of linear compartmental systems: A revisited transfer function approach based on topological properties, Math. Biosci.
66:201-228 (1973). 17 J. Delforge, An approach to the problem of linear system investigation of structural properties of the connection, matrices, Math. Comput. Simulation 24~466-471 (1982). 18 J. Delforge, Letter to the Editor, Math. Biosci. 61:299-305 19 A. Thowsen. Identifiability of dynamics systems, Internat.
identifiability through the injection and observation (1982). J. Systems
Sci. 9:813-825
(1978). 20
F. M. Fisher, The Identification
Problem in Economics, McGraw-Hill, New York, 1966.