Spec_compare_localize_freq function

Compare the spectral density operator of two Functional Time Series and localize frequencies at which they differ.

Spec_compare_localize_freq(X, Y, B.T = (dim(X)[1])^(-1/5), W, autok = 2,
  subgrid.density, verbose = 0, demean = FALSE, K.fixed = NA,
  subgrid.density.relative.to.bandwidth)

Arguments

• X,Y: The $T \times nbasis$ matrices of containing the coordinates, expressed in some functional basis, of the two FTS that to be compared. expressed in a basis.
• B.T: The bandwidth of frequencies over which the periodogram operator is smoothed. If B.T=0, the periodogram operator is returned.
• W: The weight function used to smooth the periodogram operator. Set by default to be the Epanechnikov kernel
• autok: A variable used to specify if (and which) pseudo-AIC criterion is used to select the truncation parameter $K$.
• subgrid.density: Only used if subgrid=TRUE. Specifies the approximate number of frequencies within the bandwidth over which the periodogram operator is smoothed.
• verbose: A variable to show the progress of the computations. By default, verbose=0.
• demean: A logical variable to choose if the FTS is centered before computing its spectral density operator.
• K.fixed: The value of K used if autok=0.
• subgrid.density.relative.to.bandwidth: logical parameter to specify if subgrid.density is specified relative to the bandwidth parameter B.T

Details

X,Y must be of equal size $T.len \times d$, where T.len is the length of the time series, and $d$ is the number of basis functions. Each row corresponds to a time point, and each column corresponds to the coefficient of the corresponding basis function of the FTS.

autok=0 returns the p-values for $K=1, \ldots, \code{K.fixed}$. autok=1 uses the AIC criterion of Tavakoli & Panaretos (2015), which is a generalization of the pseudo-AIC introduced in Panaretos et al (2010). autok=2 uses the AIC* criterion of Tavakoli & Panaretos (2015), which is an extension of the AIC criterion that takes into account the difficulty associated with the estimation of eigenvalues of a compact operator.

Examples

ma.scale2=ma.scale1=c(-1.4,2.3,-2)
ma.scale2[3] = ma.scale1[3]+.0
a1=Generate_filterMA(10, 10, MA.len=3, ma.scale=ma.scale1)
a2=Generate_filterMA(10, 10, MA.len=3, ma.scale=ma.scale2)
X=Simulate_new_MA(a1, T.len=512, noise.type='wiener')
Y=Simulate_new_MA(a2, T.len=512, noise.type='wiener')
ans0=Spec_compare_localize_freq(X, Y, W=Epanechnikov_kernel, autok=2,
subgrid.density=10, verbose=0, demean=FALSE,
subgrid.density.relative.to.bandwidth=TRUE)
plot(ans0)
plot(ans0, method='fdr')
PvalAdjust(ans0, method='fdr') ## print FDR adjusted p-values
abline(h=.05, lty=3)
ans0=Spec_compare_localize_freq(X, Y, W=Epanechnikov_kernel, autok=0,
subgrid.density=10, verbose=0, demean=FALSE,
subgrid.density.relative.to.bandwidth=TRUE, K.fixed=4) ## fixed values of K
plot(ans0)
plot(ans0, 'fdr')
plot(ans0, 'holm')
PvalAdjust(ans0, method='fdr')
rm(ans0)

References

Tavakoli, Shahin and Panaretos, Victor M. "Detecting and LocalizingDifferences in Functional Time Series Dynamics: A Case Study inMolecular Biophysics", 2014, under revision

Panaretos, Victor M., David Kraus, and John H. Maddocks. "Second-ordercomparison of Gaussian random functions and the geometry of DNAminicircles." Journal of the American Statistical Association 105.490(2010): 670-682.