popdemo (version 1.3-1)

tfs_lambda: Calculate sensitivity using transfer functions

Description

Calculate the sensitivity of the dominant eigenvalue of a population matrix projection model using differentiation of the transfer function.

Usage

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)

Arguments

A

a square, nonnegative numeric matrix of any dimension.

d, e

numeric vectors that determine the perturbation structure (see details).

startval

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.

tolerance

the tolerance level for determining convergence (see details).

return.fit

if TRUE the lambda and sensitivity values obtained from the convergence algorithm are returned alongside the sensitivity at the limit.

plot.fit

if TRUE then convergence of the algorithm is plotted as sensitivity~lambda.

Value

For tfs_lambda, the sensitivity of lambda-max to the specified perturbation structure. If return.fit=TRUE a list containing components:

sens

the sensitivity of lambda-max to the specified perturbation structure

lambda.fit

the lambda values obtained in the fitting process

sens.fit

the sensitivity values obtained in the fitting process.

For tfsm_lambda, a matrix containing sensitivity of lambda-max to each element of A.

Details

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.

References

• Hodgson et al. (2006) J. Theor. Biol., 70, 214-224.

See Also

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()

Examples

# 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)

# }