02a-pca-introduction.qmd

title	PCA: Overview
subtitle	Introduction to Statistical Modelling
author	Prof. Joris Vankerschaver

# What is dimensionality reduction? ## What and why - Reduce the number of variables ("dimensionality") in a dataset **in a principled way**. - Useful for - Visualization - Data preprocessing - Computational efficiency - Many different approaches - Principal component analysis (this course) - Multidimensional scaling - t-SNE, UMAP, ... ## Visualization {height=75% fig-align="center"} From: Lever et al., *Principal component analysis*, Nature Methods, Vol. 14, p. 641–642, 2017. ## Visualization Genotype data 197,146 loci in 1387 Europeans, summarized in two principal components (left) and compared to geographical origin (right). {height=50% fig-align="center"} From: Novembre et al., *Genes mirror geography within Europe*, Nature, Vol. 456, 6 November 2008. ## Data preprocessing Bodyfat dataset: - Suffered from high multicollinearity. - Conclusions from regression model are doubtful. {height=50% fig-align="center"} ## Computational efficiency - A 250 x 250 image consists of 250$^2$ = 62,500 pixels. - Not all pixels are equally informative. - Extract signal that is maximally informative, discard rest. {fig-align="center"} ## Principal component analysis - Covered in this course. - Works by finding directions in which **variance is maximized**. - Good first choice, not so good if patterns are highly nonlinear. {height=50% fig-align="center"} ## Other dimensionality reduction methods t-SNE, UMAP: - Useful for highly nonlinear relations between features. - "Deforms" data so that local structure is maintained. - Frequently used in single-cell RNA sequencing analysis. {height=80% fig-align="center"} From: \url{https://www.cancer.gov/ccg/blog/2020/interview-t-sne}