Calculate the sensitivity of the dominant eigenvalue of a population matrix projection model using differentiation of the transfer function.
tfs_lambda(A, d=NULL, e=NULL, startval=0.001, tolerance=1e-10,
return.fit=FALSE, plot.fit=FALSE)
tfsm_lambda(A, startval=0.001, tolerance=1e-10)a square, nonnegative numeric matrix of any dimension.
numeric vectors that determine the perturbation structure (see details).
tfs_lambda calculates the limit of the derivative of the
transfer function as lambda of the perturbed matrix approaches the dominant
eigenvalue of A (see details). startval provides a starting
value for the algorithm: the smaller startval is, the quicker the
algorithm should converge.
the tolerance level for determining convergence (see details).
if TRUE the lambda and sensitivity values obtained
from the convergence algorithm are returned alongside the sensitivity at the
limit.
if TRUE then convergence of the algorithm is plotted
as sensitivity~lambda.
For tfs_lambda, the sensitivity of lambda-max to the specified
perturbation structure. If return.fit=TRUE a list containing
components:
the sensitivity of lambda-max to the specified perturbation structure
the lambda values obtained in the fitting process
the sensitivity values obtained in the fitting process.
tfs_lambda and tfsm_lambda differentiate a transfer function to
find sensitivity of the dominant eigenvalue of A to perturbations.
This provides an alternative method to using matrix eigenvectors to
calculate the sensitivity matrix and is useful as it may incorporate a
greater diversity of perturbation structures.
tfs_lambda evaluates the transfer function of a specific perturbation
structure. The perturbation structure is determined by d%*%t(e).
Therefore, the rows to be perturbed are determined by d and the
columns to be perturbed are determined by e. The values in d and e
determine the relative perturbation magnitude. For example, if only entry
[3,2] of a 3 by 3 matrix is to be perturbed, then d = c(0,0,1) and
e = c(0,1,0). If entries [3,2] and [3,3] are to be perturbed with the
magnitude of perturbation to [3,2] half that of [3,3] then d = c(0,0,1)
and e = c(0,0.5,1). d and e may also be expressed as
numeric one-column matrices, e.g. d = matrix(c(0,0,1), ncol=1),
e = matrix(c(0,0.5,1), ncol=1). See Hodgson et al. (2006) for more
information on perturbation structures.
tfsm_lambda returns a matrix of sensitivity values for observed
transitions (similar to that obtained when using sens to
evaluate sensitivity using eigenvectors), where a separate transfer function
for each nonzero element of A is calculated (each element perturbed
independently of the others).
The formula used by tfs_lambda and tfsm_lambda cannot be
evaluated at lambda-max, therefore it is necessary to find the limit of the
formula as lambda approaches lambda-max. This is done using a bisection
method, starting at a value of lambda-max + startval. startval
should be small, to avoid the potential of false convergence. The algorithm
continues until successive sensitivity calculations are within an accuracy
of one another, determined by tolerance: a tolerance of 1e-10
means that the sensitivity calculation should be accurate to 10 decimal
places. However, as the limit approaches lambda-max, matrices are no longer
invertible (singular): if matrices are found to be singular then
tolerance should be relaxed and made larger.
For tfs_lambda, there is an extra option to return and/or plot the above
fitting process using return.fit=TRUE and plot.fit=TRUE
respectively.
Hodgson et al. (2006) J. Theor. Biol., 70, 214-224.
Other TransferFunctionAnalyses: tfa_inertia,
tfa_lambda, tfam_inertia,
tfam_lambda, tfs_inertia
Other PerturbationAnalyses: elas,
sens, tfa_inertia,
tfa_lambda, tfam_inertia,
tfam_lambda, tfs_inertia
# NOT RUN {
# Create a 3x3 matrix
( A <- matrix(c(0,1,2,0.5,0.1,0,0,0.6,0.6), byrow=TRUE, ncol=3) )
# Calculate the sensitivity matrix
tfsm_lambda(A)
# Calculate the sensitivity of simultaneous perturbation to
# A[1,2] and A[1,3]
tfs_lambda(A, d=c(1,0,0), e=c(0,1,1))
# Calculate the sensitivity of simultaneous perturbation to
# A[1,2] and A[1,3] and return and plot the fitting process
tfs_lambda(A, d=c(1,0,0), e=c(0,1,1),
return.fit=TRUE, plot.fit=TRUE)
# }
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