intrinsic_location {GeodRegr}

Gradient descent for location based on M-type estimators

Description

Finds \mathrm{argmin}_{p\in M}\sum_{i=1} ^ {N} \rho(d(p,y_i)) through a gradient descent algorithm.

Usage

intrinsic_location(
  manifold,
  y,
  estimator,
  c = NULL,
  p_tol = 1e-05,
  V_tol = 1e-05,
  max_iter = 1e+05
)

Arguments

manifold

Type of manifold ('euclidean', 'sphere', 'hyperbolic', or 'kendall').

y

A matrix or data frame whose columns represent points on the manifold.

estimator

M-type estimator ('l2', 'l1', 'huber', or 'tukey').

c

Multiplier of \sigma, the square root of the variance, used in the cutoff parameter for the 'huber' and 'tukey' estimators; should be NULL for the 'l2' or 'l1' estimators.

p_tol

Termination condition for the distance between consecutive updates of p.

V_tol

Termination condition for the distance between columns of consecutive updates of V, parallel transported to be in the same tangent space. Can be a vector of positive real numbers for each independent variable or a single positive number.

max_iter

Maximum number of gradient descent steps before ending the algorithm.

Details

In the case of the 'sphere', an error will be raised if all points are on a pair of antipodes.

Value

A vector representing the location estimate

Author(s)

Ha-Young Shin

References

Fletcher, P. T. (2013). Geodesic regression and the theory of least squares on Riemannian manifolds. International Journal of Computer Vision, 105, 171-185.

Kim, H. J., Adluru, N., Collins, M. D., Chung, M. K., Bendin, B. B., Johnson, S. C., Davidson, R. J. and Singh, V. (2014). Multivariate general linear models (MGLM) on Riemannian manifolds with applications to statistical analysis of diffusion weighted images. 2014 IEEE Conference on Computer Vision and Pattern Recognition, 2705-2712.

Shin, H.-Y. and Oh H.-S. (2020). Robust Geodesic Regression. <arXiv:2007.04518>

See Also

geo_reg, rbase.mean, rbase.median.

Examples

y <- matrix(runif(100, 1000, 2000), nrow = 10)
intrinsic_location('euclidean', y, 'l2')