ftsa (version 5.5)

pcscorebootstrapdata: Bootstrap independent and identically distributed functional data or functional time series

Description

Computes bootstrap or smoothed bootstrap samples based on either independent and identically distributed functional data or functional time series.

Usage

pcscorebootstrapdata(dat, bootrep, statistic, bootmethod = c("st", "sm", 
	"mvn", "stiefel", "meboot"), smo)

Arguments

dat

An object of class matrix.

bootrep

Number of bootstrap samples.

statistic

Summary statistics.

bootmethod

Bootstrap method. When bootmethod = "st", the sampling with replacement is implemented. To avoid the repeated bootstrap samples, the smoothed boostrap method can be implemented by adding multivariate Gaussian random noise. When bootmethod = "mvn", the bootstrapped principal component scores are drawn from a multivariate Gaussian distribution with the mean and covariance matrices of the original principal component scores. When bootmethod = "stiefel", the bootstrapped principal component scores are drawn from a Stiefel manifold with the mean and covariance matrices of the original principal component scores. When bootmethod = "meboot", the bootstrapped principal component scores are drawn from a maximum entropy algorithm of Vinod (2004).

smo

Smoothing parameter.

Value

bootdata

Bootstrap samples. If the original data matrix is \(p\) by \(n\), then the bootstrapped data are \(p\) by \(n\) by \(bootrep\).

meanfunction

Bootstrap summary statistics. If the original data matrix is \(p\) by \(n\), then the bootstrapped summary statistics is \(p\) by \(bootrep\).

Details

We will presume that each curve is observed on a grid of \(T\) points with \(0\leq t_1<t_2\dots<t_T\leq \tau\). Thus, the raw data set \((X_1,X_2,\dots,X_n)\) of \(n\) observations will consist of an \(n\) by \(T\) data matrix. By applying the singular value decomposition, \(X_1,X_2,\dots,X_n\) can be decomposed into \(X = ULR^{\top}\), where the crossproduct of \(U\) and \(R\) is identity matrix.

Holding the mean and \(L\) and \(R\) fixed at their realized values, there are four re-sampling methods that differ mainly by the ways of re-sampling U.

(a) Obtain the re-sampled singular column matrix by randomly sampling with replacement from the original principal component scores.

(b) To avoid the appearance of repeated values in bootstrapped principal component scores, we adapt a smooth bootstrap procedure by adding a white noise component to the bootstrap.

(c) Because principal component scores follow a standard multivariate normal distribution asymptotically, we can randomly draw principal component scores from a multivariate normal distribution with mean vector and covariance matrix of original principal component scores.

(d) Because the crossproduct of U is identitiy matrix, U is considered as a point on the Stiefel manifold, that is the space of \(n\) orthogonal vectors, thus we can randomly draw principal component scores from the Stiefel manifold.

References

H. D. Vinod (2004), "Ranking mutual funds using unconventional utility theory and stochastic dominance", Journal of Empirical Finance, 11(3), 353-377.

A. Cuevas, M. Febrero, R. Fraiman (2006), "On the use of the bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(2), 1063-1074.

D. S. Poskitt and A. Sengarapillai (2013), "Description length and dimensionality reduction in functional data analysis", Computational Statistics and Data Analysis, 58, 98-113.

H. L. Shang (2015), "Re-sampling techniques for estimating the distribution of descriptive statistics of functional data", Communications in Statistics--Simulation and Computation, 44(3), 614-635.

H. L. Shang (2018), "Bootstrap methods for stationary functional time series", Statistics and Computing, 28(1), 1-10.

See Also

fbootstrap

Examples

# NOT RUN {
# Bootstrapping the distribution of a summary statistics of functional data.	
boot1 = pcscorebootstrapdata(ElNino_ERSST_region_1and2$y, 400, 
	"mean", bootmethod = "st")
boot2 = pcscorebootstrapdata(ElNino_ERSST_region_1and2$y, 400, 
	"mean", bootmethod = "sm", smo = 0.05)
boot3 = pcscorebootstrapdata(ElNino_ERSST_region_1and2$y, 400, 
	"mean", bootmethod = "mvn")
boot4 = pcscorebootstrapdata(ElNino_ERSST_region_1and2$y, 400, 
	"mean", bootmethod = "stiefel")
boot5 = pcscorebootstrapdata(ElNino_ERSST_region_1and2$y, 400, 
	"mean", bootmethod = "meboot")
# }