twostage2: The two-stage estimator of a linear ordinary differential...
In qiuxing/ode.ident: Identifiability Analysis for Linear ODE systems

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

View source: R/twostage.R

This function takes the discrete temporal data as input, and first apply roughness penalized smoothing spline to represent these data as smooth functions (basis splines). It then solves an algebraic equation based on the pairwise inner products to estimate A, the system matrix.

1	twostage2(Y, Ts, nord=4, rough.pen=1e-3)

`Y`	An `dxn`-dimensional matrix of discrete observations. Each row is a dimension and each column is a timepoint.
`Ts`	An `n`-dimensional vector of time points in ascending order. Our recommendation is to standardize those time points so that `Ts[1]`=0 and `Ts[length(Ts)]`=1.
`nord`	The order of b-splines, which is one higher than their degree. The default of 4 gives cubic splines.
`rough.pen`	Roughness penalty used in smoothing spline. Its default value is 0.001, which is a reasonable value if: (a) the range of`Ts` is relatively small (e.g., from 0 to 1 as we recommended), and (b) the signal-to-noise level of the data is relatively small, i.e., it is easy to see the overall temporal trend from the discrete data by visual examination.

This twostage method is called the functional two-stage method in our manuscript. We also implemented a simpler two-stage method based on finite difference as twostage1 in this R package.

`Ahat`	The estimated system matrix.
`S`	An `nxn`-dimensional matrix that is needed by `PISAnalysis`. It represents the pairwise inner product between the solution curves and themselves.
`L`	An `nxn`-dimensional matrix that is needed by `PISAnalysis`. It represents the pairwise inner product between the derivative of solution curves and the solution curves.
`xt.hat`	The smoothed curves used by the functional two-stage method.
`x0`	The estimated initial condition.

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.

fsmooth, twostage1

## load Example 3.1. In this example, yy1 are discrete and noisy
## observations of (A2, x0.A), and yy2 are observations of (A2,x0.B).
## The first case is practically identifiable but the second is not.
data("example3.1")

## To demonstrate that PIS is useful, we will use the functional
## two-stage method to estimate A2
myfit1 <- twostage2(yy1, tt)
S <- myfit1$S; L <- myfit1$L
myfit2 <- twostage2(yy2, tt)

## PISs of the first case are all relatively large
## (good practical identifiability) 
PISs1 <- PISAnalysis(yy1, S, L); PISs1
## PISs of the second case are all relatively small
## (bad practical identifiability) 
PISs2 <- PISAnalysis(yy2, S, L); PISs2

## A2 is the true system matrix
round(A2,2)
## The first case is a good estimate of A2
round(myfit1$Ahat,2); round(sum((myfit1$Ahat - A2)^2),2)
## The second case is a bad estimate of A2 due to
## identifiability issues
round(myfit2$Ahat,2); round(sum((myfit2$Ahat - A2)^2),2)