rainbow (version 3.8)

fdepth: Compute functional depth.

Description

Compute functional depth.

Usage

fdepth(data, type = c("FM", "mode", "RP", "RPD", "radius"), trim = 0.25, alpha, weight)

Value

A list containing the following components is returned.

median

Median curve (highest depth).

lmed

Index of median curve.

ltrim

Indexes of the trimmed curves.

prof

Functional depth for each curve.

mtrim

Mean of trimmed curves.

weight

Weight values for all observations, when type = "radius".

Arguments

data

An object of class fds or fts.

type

Type of functional depth. See detail section below.

trim

Percentage of trimming.

alpha

Tuning parameter used when type = "radius".

weight

Hard-thresholding or soft-thresholding when type = "radius".

Author

Han Lin Shang

Details

If type = "FM", it computes the functional depth of Fraiman and Muniz (2001), which is considered as the first functional depth.

If type = "mode", it computes the functional depth of Cuevas et al. (2006). A functional mode is defined as the curve most densely surrounded by the rest of curves of the dataset.

If type = "RP" and type = "RPD", it computes random projection functional depth of Cuevas et al. (2007). Cuevas et al. (2007) considered the random projection depth based on measuring the depth of the functional data under projections and taking additional information of their derivatives. The basic idea is to project each functional curve, along a random direction, defining a point in \(R^2\). A data depth in \(R^2\) provides an order of the projected points.

If type = "radius", it ranks the curves according to alpha-radius. Then using the hard thresholding or soft thresholding, trimmed mean and median can be computed.

The argument trim = 0.25 first order curves by depth, and then trim 25 percent curves that have comparably lower depth.

References

R. Fraiman and G. Muniz (2001) "Trimmed means for functional data", Test, 10(2), 419-440.

A. Cuevas and M. Febrero and R. Fraiman (2001) "Cluster Analysis:a further approach based on density estimation", Computational Statistics and Data Analysis, 36(4), 441-456.

A. Cuevas and M. Febrero and R. Fraiman (2006) "On the use of bootstrap for estimating functions with functional data", Computational Statistics and Data Analysis, 51(10), 1063-1074.

A. Cuevas and M. Febrero and R. Fraiman (2007) "Robust estimation and classification for functional data via projection-based depth notions", Computational Statistics, 22(3), 481-496.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2007) "A functional analysis of NOx levels: location and scale estimation and outlier detection", Computational Statistics, 22(3), 411-427.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2008) "Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels", Environmetrics, 19(4), 331-345.

M. Febrero and P. Galeano and W. Gonzalez-Manteiga (2010) "Measures of influence for the functional linear model with scalar response", Journal of Multivariate Analysis, 101(2), 327-339.

J. A. Cuesta-Albertos and A. Nieto-Reyes (2010) "Functional classification and the random Tukey depth. Practical issues", Combining Soft Computing and Statistical Methods in Data Analysis, Advances in Intelligent and Soft Computing, Volume 77, 123-130.

D. Gervini (2012) "Outlier detection and trimmed estimation in general functional spaces", Statistica Sinica, 22(4), 1639-1660.

P. Rana, G. Aneiros and J. M. Vilar (2015) "Detection of outliers in functional time series", Environmetrics, 26, 178-191.

Examples

fdepth(data = ElNino_OISST_region_1and2, type = "FM")
fdepth(data = ElNino_OISST_region_1and2, type = "mode")
fdepth(data = ElNino_OISST_region_1and2, type = "RP")
fdepth(data = ElNino_OISST_region_1and2, type = "RPD")
fdepth(data = ElNino_OISST_region_1and2, type = "radius", trim = 0.25, 
	alpha = 0.5, weight = "hard")
fdepth(data = ElNino_OISST_region_1and2, type = "radius", trim = 0.25, 
	alpha = 0.5, weight = "soft")