EquivSystem: Equivalent system matrices for an unidentifiable linear ODE...

In qiuxing/ode.ident: Identifiability Analysis for Linear ODE systems

Description

This function generates an equivalent system matrix for a linear ODE system that is unidentifiable at x0 based on a given coefficient matrix D.

Usage

EquivSystem(A, x0, Dmat, ...)

Arguments

A	The original system matrix.
x0	The initial condition.
Dmat	When A is not identifiable at x0, the class of all equivalent system matrices is an affine space with nonzero degrees of freedom. Dmat is the coefficient matrix that determines the value of one specific equivalent system matrix in that affine space.
...	Additional parameters used by function ICISAnalysis(A, x0, ...).

Details

EquivSystem computes [A, x0], the equivalence class of A given x0, and then creates an equivalent system A (for the same x0) based on the specified difference matrix Dmat (with the same dimensionality of A). Remarks: a) when Dmat is the zero matrix, EquivSystem(A, x0, 0)=A; b) when A is *identifiable* at x0, EquivSystem(A, x0, Dmat)=A for all input Dmat. For unidentifiable cases, using a random Dmat a.s. will lead to a different system Atilde that is equivalent to A.

Value

Atilde

A system matrix that is equivalent to A at x0.

Author(s)

Xing Qiu

References

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

See Also

ICISAnalysis, PISAnalysis

Examples

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)

## Using a random D matrix to create an equivalent system matrix
D1 <- matrix(rnorm(3*3), 3)

## Note that, because A1 is identifiable at x0.a, adding a random D1
## matrix will not change A1 at all.
B1 <- EquivSystem(A1, x0.a, Dmat=D1)
all.equal(A1, B1) #yes

## Now try x0.b. This time, we will get a system matrix B2 that is
## totally different from A1.
B2 <- EquivSystem(A1, x0.b, Dmat=D1)
B2; A1

qiuxing/ode.ident documentation built on Sept. 30, 2020, 11:17 a.m.