ICISAnalysis: Perform identifiability analysis of an ODE system A at the...
In qiuxing/ode.ident: Identifiability Analysis for Linear ODE systems

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/IdentAnalysis.R

this function computes the initial condition-based identifiability score (ICIS) and a few related quantitative measures of identifiability of an ODE system at initial conditon x0.

1	ICISAnalysis(A, x0, n.digits=3)

`A`	The original system matrix.
`x0`	The initial condition.
`n.digits`	Number of significant digits used in computing `w0k.norm` hence `ICIS`.

The initial condition-based identifiability score ICIS is a quantitative measure of identifiability of an ODE system A at the initial value x0. If ICIS equals zero (the significant digits of ICIS is less than n.digits), A is not mathematically identifiable at x0. Besides, A may be unidentifiable due to having repeated eigenvalues, which will also be checked by ICISAnalysis.

`Identifiable`	A logic variable, TRUE means identifiable, FALSE means unidentifiable.
`Ident1`	A logic variable, TRUE means ICIS is greater than zero, FALSE means ICIS is zero (up to `n.digits` of significant digits) therefore A is not identifiable at `x0`.
`Ident2`	A logic variable, TRUE means there is no repeated eigenvalue of A (up to `n.digits` of significant digits), FALSE means there are repeated eigenvalues of A, therefore A is not identifiable (for any `x0`).
`ICIS`	The initial condition-based identifiability score. Larger value of ICIS implies better practical identifiability.
`w0k.norm`	A vector of K elements. `w0k.norm[k]` represents the projection of `x0` on the linear space spanned by the kth invariant subspace of A. Note that ICIS is defined as the minimum of `w0k.norm`.
`I0`	A `dxd`-dimensional diagonal matrix with either zero or one on the diagonal. If the ith diagonal entry is one, A is not identifiable on the invariant subspace corresponds with the ith eigenvector.
`Iplus`	A `dxd`-dimensional diagonal matrix that is the complement of `I0`. I_{+} := I_{d\times d} - I_{0}.
`Jordan`	Some useful information provided by the Jordan decomposition of A on the field of real numbers. J The block-diagonal matrix that represents real (1x1 blocks) and complex eigenvalues (2x2 blocks) of A. Qmat The matrix of eigenvectors and semi-eigenvectors (for complex eigenvalues). Qinv The inverse of `Qmat`. K1 Number of real eigenvalues. K2 Number of pairs of complex eigenvalues. So `d=K1+2*K2`. Lgap The smallest gap between eigenvalues. If this quantity is zero (up to `n.digits` of significant digits), A is not identifiable for all `x0`. RepEigenIdx A list of the indices of eigenvalues that have repeated values.

Xing Qiu

X. Qiu, T. Xu, B. Soltanalizadeh, and H. Wu. (2020+) Identifiability Analysis of Linear Ordinary Differential Equation Systems with a Single Trajectory. Submitted.

PISAnalysis

A1  <- matrix(c(0, 1, -1,
                2, 0, 0,
                3, 1, 0), 3, byrow=TRUE)
jcf1 <- JordanReal(A1); J1 <- jcf1$J; Qmat1 <- jcf1$Qmat; Qinv1 <- jcf1$Qinv

## A1 is identifiable at x0.a but not identifiable at x0.b
x0.a <- Qmat1 %*% c(2, -1, 0)
x0.b <- Qmat1 %*% c(0, -2, 3)

## now check the identifiability of A1 at x0.a and x0.b
ICISAnalysis(A1, x0.a) #yes
ICISAnalysis(A1, x0.b) #no because ICIS==0; I0 is not a zero matrix