Examples

We will now discuss several examples of identifiability in Ω from b for various sets Ω in parameter space and initial conditions b. We start by discussing certain special choices

of Ω selected based on the properties we have established as important for identifiability.

Subsequently, we consider how the behaviors of trajectories of (2.3) for some particular matrices relate to our results.

As we have discussed, Ω_b “ tA P R^nˆn : tb, Ab, ..., A^n´1bu are linearly independentu “ tA P R^nˆn : w^TA “ λw ñ w^Tb ‰ 0u is the largest open set in which the model is identifiable from b. Thus, identifiability from b holds in any subset of Ω_b, regardless of whether that set is open. Also, we can define a larger (non-open) set Ω in which we have identifiability from b by extending Ω_b in trivial ways, such as by combining Ω_b with a single matrix A₀ P Ωbc

. For Ω “ Ωb Y tA0u, the model will be identifiable in Ω from b because the orbit γpA0, bq is confined to a proper subspace of Rⁿ and will not coincide with any orbit for a matrix in Ω_b (which cannot be so confined). This type of trivial extension could be continued with a sequence of matrices that have confined trajectories for the initial condition b, but with no two solutions that are the same. On the other hand, define Ω “ R^nˆnzΩb, which is not open, such that Theorem5and Corollary12do not apply. It is clear that any two distinct matrices A, B with the same eigenvalues that share b as a eigenvector for the same eigenvalue will yield the same solution, xpt; A, bq “ xpt; B, bq for all t P R, and such matrices can be found in Ω. Hence, by Definition 3, model (2.3) is not identifiable in Ω from b. This argument shows that we do not have identifiability from b on the complement of Ω_b.

Next, define Ω_J “ tA P R^nˆn : A has more than one Jordan block for some eigenvalueu.

From Theorem15we know that for any A P ΩJ, tb, Ab, ..., A^n´1bu are linearly dependent for any b P Rⁿ. Ω_J is a set of measure zero and is not open in R^nˆn, however, so Theorem 5 does not apply to the identifiability of ΩJ. We can again appeal directly to Definition3 to show that the model is not identifiable in Ω_J from b for any choice of b P Rⁿ. For example, in R^3ˆ3, fix b “ r1, 1, 1s^T. Let

A and B both have the Jordan matrix Definition3, model (2.3) is not identifiable in Ω_J from b. An analogous pair of matrices that violate identifiability can be obtained for any b P Rⁿ by simply choosing two matrices with the same Jordan form, having more than one Jordan block for some eigenvalue and sharing b as an eigenvector for that eigenvalue. In fact, Ω_J X Ωb “ H. So it is straightforward to construct sets Ω Ď R^nˆn on which the model is not identifiable from b, using elements of Ω_J. To illustrate the application of Theorem 13, we will now discuss several examples of 2 ˆ 2 and 3 ˆ 3 matrices. For each, we consider under what conditions they have confined trajectories and correspondingly, for what initial conditions b we have A P Ωb.

Figure 3 shows the phase plane for each of four matrices with trajectories plotted for a few different initial conditions. Matrix

has distinct eigenvalues. For any initial condition b lying on an eigenvector, the correspond-ing orbit will be confined to a proper subspace of R². These are the only initial conditions that lead to confined trajectories, so A_a P Ωb for all b not on an eigenvector. This observa-tion is consistent with the requirements on the structure of b for identifiability, as given in Proposition16and the associated discussion. That is, if we consider A_a and b written in the basis in which Aa is in Jordan form (diagonalized), then b would have a zero component if and only if it were an eigenvector.

Matrix

(a) (b)

(c) (d)

Figure 3: Phase portraits for system (2.3) generated by four matrices with different eigen-value structures. In this and subsequent figures, red curves are trajectories from which the corresponding model is not identifiable while blue curves are trajectories that yield identifi-ability. Matrices A_a, A_b, A_c, A_d, as given in the text, were used to generate panels (a), (b), (c), and (d) respectively.

has a repeated eigenvalue of ´2 and is not diagonalizable. Since it only has one Jordan block, the requirement on b for the orbit to not be confined is that in the basis in which A_b is given in Jordan form, b must have a nonzero component in the last entry. Equivalently, this requirement means that b cannot lie on the genuine eigenvector r1, 1s^T. From the phase plane

it is easy to see that the orbit arising from any initial condition lying on this eigenvector will be confined to a proper subspace of R². So, A_b P Ωb for all b not on the genuine eigenvector.

Matrix

A_c“

» –

´2 ´3 3 ´2

fi fl

has a complex conjugate pair of eigenvalues. In this circumstance, R² has no nontrivial A_c-invariant proper subspaces. Therefore, the orbit from any nonzero initial condition is not confined to a proper subspace of R². This conclusion is clear from the phase plane. Hence, A_cP Ωb for all b P R²zt0u.

Finally, matrix

A_d “

» –

´2 0 0 ´2

fi fl

represents the case of a repeated Jordan block for λ “ ´2. This is a star shaped system in which every initial condition lies on an eigenvector and hence every orbit is confined to a proper subspace. Thus, in this case, there exists no b such that A_d P Ωb.

Figure 4: Phase space structures for matrix A_e.

Figure 4corresponds to the matrix

which has three distinct real eigenvalues. The corresponding eigenvectors are plotted in black and the planes represent the two-dimensional invariant subspaces spanned by pairs of eigenvectors. Any initial condition lying in one of these planes will have a zero component (in the basis in which A is diagonalized) and the corresponding orbit will be confined to that plane. Any initial condition outside of these planes will have a orbit that is not confined to a proper subspace. Hence, A_e P Ωb for all b not lying in one of the planes. This example is the three-dimensional analogue of A_a but is more interesting because for A_e to be in Ω_b, not only must b not lie on an eigenvector of A_e, but it also may not land on any two-dimensional plane spanned by two eigenvectors of Ae.

Figure 5: Phase space structures for matrix A_f.

Figure 5was generated from

Figure 6: Phase space structures for matrix A_g.

which has a complex conjugate pair of eigenvalues and one real eigenvalue: λ_1,2 “ ´0.2 ˘ i and λ₃ “ ´0.3. Any orbit with an initial condition on the xy plane will stay confined to that plane. This is the only proper A_f-invariant subspace of R³. It is clear from the phase plane that any orbit from an initial condition outside of this plane will not be confined to a proper subspace. Hence, A_f P Ωb for all b not lying in the xy plane.

Figure 6corresponds to

A_g “

»

—

—

— –

´1 1 0

0 ´1 0

0 0 ´1

fi ffi ffi ffi fl ,

which has two Jordan blocks for the eigenvalue λ “ ´1. Because of the repeated block, the orbit from any initial condition is confined to a proper subspace of R³. Hence, there is no b P R³ such that A_g P Ωb. This conclusion can also be verified from Theorem 17. The left eigenvectors of Ag are: x1 “ r0, 1, 0s, x2 “ r0, 0, 1s. For an arbitrary b “ rb1, b2, b3s^T, there exists a left eigenvector, namely, x “ r0, ´b3, b₂s, such that xb “ 0. Thus, Theorem 17implies that tb, Agb, A²_gbu are linearly dependent for all b P R³.