Bootstrap independent and identically distributed functional data or functional time series
Computes bootstrap or smoothed bootstrap samples based on either independent and identically distributed functional data or functional time series.
pcscorebootstrapdata(dat, bootrep, statistic, bootmethod = c("st", "sm",
"mvn", "stiefel", "meboot"), smo)dat |
An object of class |
bootrep |
Number of bootstrap samples. |
statistic |
Summary statistics. |
bootmethod |
Bootstrap method. When |
smo |
Smoothing parameter. |
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.
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. |
Han Lin Shang
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.
# Bootstrapping the distribution of a summary statistics of functional data. boot1 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, statistic = "mean", bootmethod = "st") boot2 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, statistic = "mean", bootmethod = "sm", smo = 0.05) boot3 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, statistic = "mean", bootmethod = "mvn") boot4 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, statistic = "mean", bootmethod = "stiefel") boot5 = pcscorebootstrapdata(dat = ElNino_ERSST_region_1and2$y, bootrep = 200, statistic = "mean", bootmethod = "meboot")
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