Periodic Demmler-Reinsch Basis

Description

Calculates the periodic Demmler-Reinsch basisfor a given smoothness and a given vector of grid points. For details see (Schwarz, Krivobokova; 2016).

Usage

DR.basis(x, n, q)

Arguments

Value

mxn dimensional matrix with the n DR basis functions evaluated at grid points x

Examples

DR.basis(seq(1,10)/10,5,2)

Data Examples

Description

example1, example2 and example3 generate i.i.d. vectors from a given distribution with different Toeplitz covariance matrices. The covariance function \sigma of the Toeplitz covariance matrix of

• example1: has a polynomial decay, \sigma(\tau)= sd^2(1+|\tau|)^{-gamma},

• example2: follows an ARMA(2,2) model with coefficients (0.7,-0.4,-0.2,0.2) and innovations variance sd^2,

• example3: yields a Lipschitz continuous spectral density f that is not differentiable, i.e. f(x)= sd^2({|\sin(x+0.5\pi)|^{gamma}+0.45})

Usage

example1(p, n, sd, gamma, family = "Gaussian")

example2(p, n, sd, family = "Gaussian")

example3(p, n, sd, gamma, family = "Gaussian")

Arguments

Value

A list containing the following elements:

• Y: pxn dimensional data matrix

• sdf: true spectral density function

• acf: true covariance function

Examples

example1(p=10, n=1, sd=1, gamma=1.2, family="Gaussian")
example2(p=10,n=1,sd=1,family="Gaussian")
example3(p=10, n=1, sd=1, gamma=2,family="Gaussian")

Data Transformation

Description

Applies the Discrete Cosine I transform, data binning and the variance stabilizing transform function to the data.

Usage

Data.trafo(y, Te, dct.out = FALSE)

Arguments

Value

A list containing the following elements:

• m: number of data points per bin, that is m=n*round(p/Te). If p/Te is not an integer, the first/last bin may contain more than m data points.

• y.star: 2Te-2 dimensional vector with binned, variance stabilized and mirrowed data. The bin number Te may be modified to guarantee at least two data points per bin. If p/Te is not an integer, the vector dimension is 2*floor(p/round(p/Te))-2.

• dct.matrix: p-dim. DCT-I matrix (if dct.out=TRUE)

Toeplitz Covariance and Precision Matrix Estimator

Description

Estimates the Toeplitz covariance matrix, the inverse matrix and the spectral density from a sample of n i.i.d. p-dimensional vectors with mean zero.

Usage

Toep.estimator(y, Te, q, method, f.true = NULL)

Arguments

Value

A list containing the following elements:

• toep: p-dim. Toeplitz covariance matrix

• toep.inv: p-dim. precision matrix

• acf: p-dim. vector with the covariance function

• sdf: p-dim. vector with the spectral density in the interval [0,1]

Examples

#EXAMPLE 1: Simulate Gaussian ARMA(2,2)
library(nlme)
library(MASS)
p=100
n=1
Sigma=1.44*corMatrix(Initialize(corARMA(c(0.7, -0.4,-0.2, 0.2),p=2,q=2),data=diag(1:p)))
Y=matrix(mvrnorm(n, mu=numeric(p), Sigma=Sigma),n,p)
fit.toep=Toep.estimator(y=Y,Te=10,q=2,method="GCV")$toep


#EXAMPLE 2: AQUAPORIN DATA
data(aquaporin)
n=length(aquaporin$Y)
y.train=aquaporin$Y[1:(0.01*n)]
y.train=y.train-mean(y.train)
fit.toep=Toep.estimator(y=y.train,Te=10,q=1,method="ML")$toep

Aquaporin Dataset

Description

Dataset with molecular dynamics simulations for the yeast aquaporin (Aqy1) - the gated water channel of the yeast Pichi pastoris. The dataset contains only the diameter Y of the channel which is used in the data analysis in (Klockmann and Krivobokova, 2023). The diameter Y is measured by the distance between two centers of mass of certain residues of the protein. The dataset includes a 100 nanosecond time frame, split into 20000 equidistant observations. The full dataset, including the Euclidean coordinates of all 783 atoms, is available from the authors. For more details see (Klockmann, Krivobokova; 2023).

Usage

aquaporin

Format

A data frame with 20000 rows and 1 variable:

• Y: the diameter of the channel

Source

see (Klockmann, Krivobokova; 2023).

Examples

data(aquaporin)