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.
Details
We will presume that each curve is observed on a grid of T points with 0≤q t_1<t_2…<t_T≤q τ.
Thus, the raw data set (X_1,X_2,…,X_n) of n observations will consist of an n by T data matrix.
By applying the singular value decomposition, X_1,X_2,…,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.
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.
Author(s)
Han Lin Shang
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.
See Also
fbootstrap
Examples
# Bootstrapping the distribution of a summary statistics of functional data.
boot1 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "st")
boot2 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "sm", smo = 0.05)
boot3 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "mvn")
boot4 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "stiefel")
boot5 = pcscorebootstrapdata(ElNino$y, 400, "mean", bootmethod = "meboot")