Title: Spectral Density Estimation and Comparison for Functional Time Series
Version: 1.0.0
Author: Shahin Tavakoli [aut, cre]
Maintainer: Shahin Tavakoli <s.tavakoli@statslab.cam.ac.uk>
Description: Functions for estimating spectral density operator of functional time series (FTS) and comparing the spectral density operator of two functional time series, in a way that allows detection of differences of the spectral density operator in frequencies and along the curve length.
Depends: R (≥ 3.2.0)
Imports: sna (≥ 2.3-2)
License: GPL-2
LazyData: true
NeedsCompilation: no
Packaged: 2015-09-08 09:02:39 UTC; st624
Repository: CRAN
Date/Publication: 2015-09-08 13:13:41

ftsspec: collection of functions for estimating spectral density operator of functional time series (FTS) and comparing the spectral density operator of two functional time series, in a way that allows detection of differences of the spectral density operator in frequencies and along the curve length.

Description

ftsspec: collection of functions for estimating spectral density operator of functional time series (FTS) and comparing the spectral density operator of two functional time series, in a way that allows detection of differences of the spectral density operator in frequencies and along the curve length.

References

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

The Epanechnikov weight function, with support in [-1,1]

Description

The Epanechnikov weight function, with support in [-1,1]

Usage

Epanechnikov_kernel(x)

Arguments

x

argument at which the function is evaluated

Generate the Filter of a multivariate MA process

Description

Generate the Filter of a multivariate MA process

Usage

Generate_filterMA(d.ts, d.n, MA.len = 3, ma.scale = rep(1, MA.len),
  a.smooth.coef = 0, seed = 1)

Arguments

d.ts

dimension of the (output) time series

d.n

dimension of the noise that is filtered

MA.len

Length of the filter. Set to 3 by default.

ma.scale

scaling factor of each lag matrix. See details.

a.smooth.coef

A coefficient to shrink coefficients of filter. Set to 0 by default.

seed

The random seed used to generate the filter. Set to 1 by default.

Value

A d.ts x d.n x MA.len array

Details

Generates a filter (i.e. a d.ts x d.n x MA.len array) for a moving average process. The entries of the filter are generate randomly, but can be reproduced by specifying the random seed seed.

The ma.scale parameter should be a vector of length MA.len, and corresponds to a scaling factor applied to each lag of the filter of the MA process that is generated.

Examples

ma.scale1=c(-1.4,2.3,-2)
a1=Generate_filterMA(10, 10, MA.len=3, ma.scale=ma.scale1, seed=10)
str(a1)
rm(a1)

Get the square root of the covariance matrix associated to a noise type

Description

Get the square root of the covariance matrix associated to a noise type

Usage

Get_noise_sd(noise.type, d.n)

Arguments

noise.type

the type of noise that is driving the MA process. See Details section.

d.n

dimension of the noise that is filtered

Compute the marginal p-values at each basis coefficients of for testing the equality of two spectral density kernels

Description

Compute the marginal p-values at each basis coefficients of for testing the equality of two spectral density kernels

Usage

Marginal_basis_pval(spec1, spec2, m, kappa.square, is.pi.multiple)

Arguments

spec1

The two sample spectral densities (at the same frequency \omega) to be compared.

spec2

The two sample spectral densities (at the same frequency \omega) to be compared.

m

The number of Fourier frequencies over which the periodogram operator was smoothed.

kappa.square

the L2-norm of the weight function used to estimate the spectral density operator

is.pi.multiple

A logical variable, to specify if \omega = {0, \pi} or not.

Generic function to adjust pvalues

Description

Generic function to adjust pvalues

function to adjust pvalues for class SampleSpecDiffFreq

Usage

PvalAdjust(sample.spec.diff, method)

## S3 method for class 'SampleSpecDiffFreq'
PvalAdjust(sample.spec.diff, method)

Arguments

sample.spec.diff

Object of the class SampleSpecDiffFreq

method

method used to adjust p-values

See Also

Spec_compare_localize_freq

Simulate a new Moving Average (MA) vector time series and return the time series

Description

Simulate a new Moving Average (MA) vector time series and return the time series

Usage

Simulate_new_MA(a, T.len, noise.type, DEBUG = FALSE)

Arguments

a

Array, returned by Generate_filterMA, containing the filter of the MA process

T.len

Numeric, the length of the time series to generate

noise.type

the type of noise that is driving the MA process. See Details section.

DEBUG

Logical, for outputting information on the progress of the function

Value

A T.len x dim(a)[1] matrix, where each column corresponds to a coordinate of the vector time series

Details

The function simulates a moving average process of dimension dim(a)[1], defined by

X[t,] = a[,,1] * epsilon[,t-1] + a[,,2] * epsilon[,t-2] + ... + a[,,dim(a)[3]] * epsilon[t-dim(a)[3]]

noise.type specifies the nature and internal correlation of the noise that is driving the MA process. It can take the values

white-noise

the noise is Gaussian with covariance matrix identity

white-noise

the noise is Gaussian with diagonal covariance matrix, whose j-th diagonal entry is ((j - 0.5 )*pi)^(-1)

studentk

the coordinates of the noise are independent and have a student t distribution with 'k' degrees of freedom, standardized to have variance 1

Examples

ma.scale1=c(-1.4,2.3,-2)
a1=Generate_filterMA(6, 6, MA.len=3, ma.scale=ma.scale1)
X=Simulate_new_MA(a1, T.len=512, noise.type='wiener')
plot.ts(X)

Compute Spectral Density of Functional Time Series

Description

This function estimates the spectral density operator of a Functional Time Series (FTS)

Usage

Spec(X, W = Epanechnikov_kernel, B.T = (dim(X)[1])^(-1/5),
  only.diag = FALSE, trace = FALSE, demean = TRUE, subgrid = FALSE,
  subgrid.density = 10, verbose = 0,
  subgrid.density.relative.to.bandwidth = TRUE)

Arguments

X

A T \times nbasis matrix of containing the coordinates of the FTS expressed in a basis. Each row corresponds to a time point, and each column corresponds to the coefficient of the corresponding basis function of the FTS.

W

The weight function used to smooth the periodogram operator. Set by default to be the Epanechnikov kernel

B.T

The bandwidth of frequencies over which the periodogram operator is smoothed. If B.T=0, the periodogram operator is returned.

only.diag

A logical variable to choose if the function only computes the marginal spectral density of each basis coordinate (only.diag=TRUE). only.diag=FALSE by default, the full spectral density operator is computed .

trace

A logical variable to choose if only the trace of the spectral density operator is computed. trace=FALSE by default.

demean

A logical variable to choose if the FTS is centered before computing its spectral density operator.

subgrid

A logical variable to choose if the spectral density operator is only returned for a subgrid of the Fourier frequencies, which can be useful in large datasets to reduce memory usage. subgrid=FALSE by default.

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.

subgrid.density.relative.to.bandwidth

logical parameter to specify if subgrid.density is specified relative to the bandwidth parameter B.T

Value

A list containing the following elements:

spec

The estimated spectral density operator. The first dimension corresponds to the different frequencies over which the spectral density operators are estimated.

omega

The frequencies over which the spectral density is estimated.

m

The number of Fourier frequencies over which the periodogram operator was smoothed.

bw

The equivalent Bandwidth used in the weight function W(), as defined in Bloomfield (1976, p.201).

weight

The weight function used to smooth the periodogram operator.

kappa.square

The L2 norm of the weight function W.

References

spec.pgram function of R.

Bloomfield, P. (1976) "Fourier Analysis of Time Series: An Introduction", Wiley.

Panaretos, V. M. and Tavakoli, S., "Fourier Analysis of Functional Time Series", Ann. Statist. Volume 41, Number 2 (2013), 568-603.

Examples

ma.scale1=c(-1.4,2.3,-2)
a1=Generate_filterMA(10, 10, MA.len=3, ma.scale=ma.scale1)
X=Simulate_new_MA(a1, T.len=512, noise.type='wiener')
ans=Spec(X, trace=FALSE, only.diag=FALSE)
plot(ans)
plot(Spec(X, trace=FALSE, only.diag=FALSE, subgrid=TRUE, subgrid.density=10,
subgrid.density.relative.to.bandwidth=FALSE))
rm(ans)

'Spectral density operator of a MA vector process' Object

Description

'Spectral density operator of a MA vector process' Object

Usage

SpecMA(a, nfreq = 2^9, noise.type)

Arguments

a

the filter of the moving average

nfreq

the number of frequencies between 0 and pi at which the spectral density has to be computed

noise.type

the type of noise that is driving the MA process. See Simulate_new_MA

Examples

ma.scale1=c(-1.4,2.3,-2)
a1=Generate_filterMA(6, 6, MA.len=3, ma.scale=ma.scale1)
a1.spec=SpecMA(a1, nfreq=512, noise.type='wiener')
plot(a1.spec)
rm(a1, a1.spec)

Test if two spectral density operators at some fixed frequency are equal.

Description

A test for the null hypothesis that two spectral density operators (at the same frequency \omega) are equal, using a pseudo-AIC criterion for the choice of the truncation parameter. (used in Spec_compare_localize_freq)

Usage

Spec_compare_fixed_freq(spec1, spec2, is.pi.multiple, m, kappa.square,
  autok = 2, K.fixed = NA)

Arguments

spec1, spec2

The two sample spectral densities (at the same frequency \omega) to be compared.

is.pi.multiple

A logical variable, to specify if \omega = {0, \pi} or not.

m

The number of Fourier frequencies over which the periodogram operator was smoothed.

kappa.square

the L2-norm of the weight function used to estimate the spectral density operator

autok

A variable used to specify if (and which) pseudo-AIC criterion is used to select the truncation parameter K.

K.fixed

The value of K used if autok=0.

References

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

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

See Also

Spec_compare_localize_freq

Examples

ma.scale2=ma.scale1=c(-1.4,2.3,-2)
ma.scale2[3] = ma.scale1[3]+.3
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')
spec.X = Spec(X)
spec.Y = Spec(Y)
Spec_compare_fixed_freq(spec.X$spec[1,,], spec.Y$spec[1,,],
is.pi.multiple=TRUE, spec.X$m, spec.X$kappa.square)

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

Description

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

Usage

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.

References

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

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

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)

Compare the spectral density operator of two Functional Time Series and localize frequencies at which they differ, and (spatial) regions where they differ

Description

Compare the spectral density operator of two Functional Time Series and localize frequencies at which they differ, and (spatial) regions where they differ

Usage

Spec_compare_localize_freq_curvelength(X, Y, B.T = (dim(X)[1])^(-1/5), W,
  alpha = 0.05, accept = 0, reject = 1, verbose = 0, demean = FALSE)

Arguments

X

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.

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

alpha

level for the test

accept, reject

values for accepted, rejected regions

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.

Examples

ma.scale2=ma.scale1=c(-1.4,2.3,-2)
ma.scale2[3] = ma.scale1[3]+.4
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=2^9, noise.type='wiener')
Y=Simulate_new_MA(a2, T.len=2^9, noise.type='wiener')
ans0=Spec_compare_localize_freq_curvelength(X, Y, W=Epanechnikov_kernel, alpha=.01, demean=TRUE)
print(ans0)
plot(ans0)
rm(ma.scale1, ma.scale2, a1, a2, X, Y, ans0)

Plotting function for SampleSpecDiffFreq class

Description

Plotting function for SampleSpecDiffFreq class

Usage

## S3 method for class 'SampleSpecDiffFreq'
lines(x, method = NA, Kmax = 4, pch = 20,
  ...)

Arguments

x

object of the class SampleSpecDiffFreq

method

method used to adjust p-values

Kmax

maximum number of levels K for which the pvalues are plotted (used only if autok==0)

pch

the plot character to be used

...

additional parameters to be passed to plot()

See Also

Spec_compare_localize_freq

Plotting method for object inheriting from class SampleSpec

Description

Plotting method for object inheriting from class SampleSpec

Usage

## S3 method for class 'SampleSpec'
plot(x, ...)

Arguments

x

An object of the class SampleSpec

...

additional parameters to be passed to plot()

Plotting function for SampleSpecDiffFreq class

Description

Plotting function for SampleSpecDiffFreq class

Usage

## S3 method for class 'SampleSpecDiffFreq'
plot(x, method = NA, Kmax = 4, pch = 20, ...)

Arguments

x

object of the class SampleSpecDiffFreq

method

method used to adjust p-values

Kmax

maximum number of levels K for which the pvalues are plotted (used only if autok==0)

pch

the plot character to be used

...

additional parameters to be passed to plot()

See Also

Spec_compare_localize_freq

Plotting method for class SampleSpecDiffFreqCurvelength

Description

Plotting method for class SampleSpecDiffFreqCurvelength

Usage

## S3 method for class 'SampleSpecDiffFreqCurvelength'
plot(x, ncolumns = 3, ...)

Arguments

x

Object of the class SampleSpecDiffFreqCurvelength

ncolumns

number of columns for the plots

...

additional parameters to be passed to plot()

Plotting method for object inheriting from class SpecMA

Description

Plotting method for object inheriting from class SpecMA

Usage

## S3 method for class 'SpecMA'
plot(x, ...)

Arguments

x

A object of the class SpecMA

...

additional parameters to be passed to plot()

Printing method for class SampleSpecDiffFreqCurvelength

Description

Printing method for class SampleSpecDiffFreqCurvelength

Usage

## S3 method for class 'SampleSpecDiffFreqCurvelength'
print(x, ...)

Arguments

x

Object of the class SampleSpecDiffFreqCurvelength

...

Additional arguments for print