## Dimensionality reduction 1. Principal component analysis The idea is find the direction with max variance. It first centers the data (make average 0), and compute [covariance matrix](https://en.wikipedia.org/wiki/Covariance_matrix), compute eigenvector (i.e., principal component) and eigenvalue (i.e., variance), select the principal components to keep (which correspond to max variance, therefore highest variation), then project to lower dimensions. >[!Note] >A quick reminder, although not directly related to the current topics, but covariance and correlation are related. See [[Real space correlations, radial distribution function]]. >Correlations are also extensively discussed in scattering characterizations cause it's directly related to the scattering pattern. 2. Multidimensional scaling This is a similar approach, but it intakes the dissimilar matrix, and chooses to preserve the pairwise distance (Euclidean distance). This one and principal component analysis are linear dimensionality reduction. 3. ISOMAP This is a non-linear approach and can show non-linear structures (i.e., curved surface with some pattern). It uses geodesic distance instead of Euclidean distance. Intuitively, it can "unfold" structures. ![[Drawing 2024-08-30 19.42.56.excalidraw.svg]] 4. Sketch-map Sketch map focuses on reproducing proximity information from the high-dimensional data set. It ensures that points that are closer than a characteristic distance are mapped close together, while simultaneously ensuring the far apart points are well separated in the projection. >[!Info] >See similar chapters: [[Data representation]], [[Landmarks]], [[Reweighting and enhanced sampling techniques]]].