In this article, I am going to show you how to choose the number of principal components when using principal component analysis for dimensionality reduction.
In the first section, I am going to give you a short answer for those of you who are in a hurry and want to get something working. Later, I am going to provide a more extended explanation for those of you who are interested in understanding PCA.
At the very beginning of the tutorial, I’ll explain the dimensionality of a dataset, what dimensionality reduction means, the main approaches to dimensionality reduction, the reasons for dimensionality reduction and what PCA means. Then, I will go deeper into the topic of PCA by implementing the PCA algorithm with the Scikit-learn machine learning library. This will help you to easily apply PCA to a real-world dataset and get results very fast.
Uniform Manifold Approximation and Projection (UMAP) is a dimension reduction technique that can be used for visualisation similarly to t-SNE, but also for general non-linear dimension reduction. The algorithm is founded on three assumptions about the data
The data is uniformly distributed on Riemannian manifold;
The Riemannian metric is locally constant (or can be approximated as such);
The manifold is locally connected.
From these assumptions it is possible to model the manifold with a fuzzy topological structure. The embedding is found by searching for a low dimensional projection of the data that has the closest possible equivalent fuzzy topological structure.
G. zheng Li, X. qiang Zeng, and J. Yang. In Proceedings of the 7th IEEE International Conference on Bioinformatics and Bioengineering, page 1439--1444. (2009)