Module: dimred

Description	Usage
tools for performing dimensionality reduction
Authors: Gareth Tribello

Details

This module contains implementations of various dimensionality reduction algorithms. This tutorial provides an introduction to the ways in which these tools have been used in the papers outlined below.

In general, all these actions take a collection of vectors in input. You can use values from any action that outputs a vector as input. Most commonly, however, the input to these actions is the output from a COLLECT_FRAMES shortcut or a subset of the points that were collected with a COLLECT_FRAMES shortcut that have been selected with one of the tools from the landmarks module.

Many dimensionality reduction algorithms then work in a manner similar to the way we use when we make maps. You start with distances between London, Belfast, Paris and Dublin and then you try to arrange points on a piece of paper so that the (suitably transformed) distances between the points in your map representing each of those cities are related to the true distances between the cities. Stating this more mathematically dimensionality reduction algorithms endeavor to find an isometry between points distributed in a high-dimensional space and a set of points distributed in a low-dimensional plane. In other words, if we have $M$ $D$ -dimensional points, $\mathbf{X}$ , and we can calculate dissimilarities between pairs them, $D_{ij}$ , we can, with an MDS calculation, try to create $M$ projections, $\mathbf{x}$ , of the high dimensionality points in a $d$ -dimensional linear space by trying to arrange the projections so that the Euclidean distances between pairs of them, $d_{ij}$ , resemble the dissimilarities between the high dimensional points. In short we minimize:

$\chi^2 = \sum_{i \ne j} w_i w_j \left( F(D_{ij}) - f(d_{ij}) \right)^2$

where $F(D_{ij})$ is some transformation of the distance between point $X^{i}$ and point $X^{j}$ and $f(d_{ij})$ is some transformation of the distance between the projection of $X^{i}$ , \f $x^i\f$ , and the projection of $X^{j}$ , $x^j$ . $w_i$ and $w_j$ are the weights of configurations $X^i$ and $^j$ respectively.

Actions

The following actions are part of this module

Name	Description	Tags
ARRANGE_POINTS	Arrange points in a low dimensional space so that the (transformed) distances between points in the low dimensional space match the dissimilarities provided in an input matrix.	DIMRED
CLASSICAL_MDS	Create a low-dimensional projection of a trajectory using the classical multidimensional	DIMRED
PCA	Perform principal component analysis (PCA) using either the positions of the atoms a large number of collective variables as input.	DIMRED
PROJECT_POINTS	Find the projection of a point in a low dimensional space by matching the (transformed) distance between it and a series of reference configurations that were input	DIMRED
SKETCHMAP	Construct a sketch map projection of the input data	DIMRED
SKETCHMAP_PROJECTION	Read in a sketch-map projection	DIMRED

References

More information about this module is available in the following articles: