Principal-Manifold-Estimation

• Description: This repository is devoted to the principal manifold estimation (PME) algorithm. The PME algorithm comes from the following paper by Kun Meng and Ani Eloyan (hereafter ME).

Meng K, Eloyan A. Principal manifold estimation via model complexity selection. Journal of the Royal Statistical Society: Series B (Statistical Methodology). 2021;83:369–394. https://doi.org/10.1111/rssb.12416

• Depends R (>= 3.0)

• Maintainer: Kun Meng kun_meng@brown.edu

In addition to this "README" file, there are four files in this repository.

1. "Principal_Manifold_Estimation.R": This file contains the functions "hdmde" and "PME." The "hdmde" function implements the high-dimensional mixture density estimation (HDMDE) presented in Algorithm 1 of ME, and the "PME" function implements the principal manifold estimation presented in Algorithm 2 of ME.

The syntax of PME:

result=PME(x.obs=data_points, d=intrinsic_dimension, N0=0, tuning.para.seq=exp((-15:5)), alpha=0.05, max.comp=100, epsilon=0.05, max.iter=100, print.MSDs=TRUE)
# "data_points" is an n-by-D matrix, each row of which denotes a D-dimensional data point.
# "intrinsic_dimension": is the dimension of the potentially interested underlying manifold.
# More details are provided in the "Principal_Manifold_Estimation.R" file.

2. "Examples_for_PME.R": This file presents six simulated examples with dimension pairs (d=1, D=2), (d=1,D=3), and (d=2,D=3) to demonstrate the performance of PME.

3. "Graphical_Output_of_Examples.pdf": This file shows the graphical output of "Examples_for_PME.R."

4. "Comparison_between_PME_and_existing_method.R": This file reproduces the simulation study in Figure 6 (b) of ME, comparing PME, HS, and ISOMAP. We measure the performances of these three methods using mean squared distance (MSD). In this file, for simplicity, we do only ten simulations. The corresponding output is the ten MSDs of each method, say 30 numbers in a 10-by-3 matrix. HS stands for the HS principal curve algorithm in Hastie and Stuetzle (1989). In this file, we apply the "principal_curve" function to perform HS. ISOMAP is based on Tenenbaum et al. (2000) and is implemented using the "isomap" function.

Kun Meng, Ph.D.
(Michael)
Department of Biostatistics,
Brown University

References

T. Hastie and W. Stuetzle. Principal curves. Journal of the American Statistical Association, 84(406): 502-516, 1989

J. B. Tenenbaum, V. De Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. science, 290(5500):2319-2323, 2000.