fdars (version 0.3.3)

gmed: Geometric Median of Functional Data

Description

Computes the geometric median (L1 median) of functional data using Weiszfeld's iterative algorithm. The geometric median minimizes the sum of L2 distances to all curves/surfaces, making it robust to outliers.

Usage

gmed(fdataobj, max.iter = 100, tol = 1e-06)

Value

An fdata object containing the geometric median function (1D or 2D).

Arguments

fdataobj

An object of class 'fdata'.

max.iter

Maximum number of iterations (default 100).

tol

Convergence tolerance (default 1e-6).

Details

The geometric median y minimizes: $$\sum_{i=1}^n ||X_i - y||_{L2}$$

Unlike the mean (L2 center), the geometric median is robust to outliers because extreme values have bounded influence (influence function is bounded).

The Weiszfeld algorithm is an iteratively reweighted least squares method that converges to the geometric median.

See Also

mean.fdata for the (non-robust) mean function, median for depth-based median

Examples

# Create functional data with an outlier
t <- seq(0, 1, length.out = 50)
X <- matrix(0, 20, 50)
for (i in 1:19) X[i, ] <- sin(2*pi*t) + rnorm(50, sd = 0.1)
X[20, ] <- sin(2*pi*t) + 5  # Large outlier
fd <- fdata(X, argvals = t)

# Compare mean vs geometric median
mean_curve <- mean(fd)
gmed_curve <- gmed(fd)

# The geometric median is less affected by the outlier