Run PCA and classical MDS on the same observations. PCA protects variance. MDS protects pairwise distances. The difference stops being theoretical when you can see and measure both.
label, group, feature 1, feature 2, feature 3
The representative sample is computed on load.
| Observation | PCA x | PCA y | MDS x | MDS y |
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PCA rotates standardized features toward directions with the greatest spread. Classical MDS starts from every pairwise distance and searches for a lower-dimensional map that preserves those relationships. Compare the same labels across both panels: similar geometry means the dominant variance also carries most of the distance structure.
The first two PCA eigenvalues divided by all eigenvalues tell you how much standardized variance survives the projection. Higher is better, but it does not guarantee every local distance survives.
Normalized stress compares original distances with distances in the MDS map. Near zero means the 2D geometry faithfully preserves the original relationships.
Replace the sample with measurements from a real portfolio dataset. Keep labels and groups, add three numeric features, then export the two embeddings as analysis-ready evidence.
The dimensionality reduction lab computes PCA and classical multidimensional scaling from the same standardized data. Each feature is centered and divided by its sample standard deviation; a constant feature uses one as the fallback divisor. For the saved eight observation sample, the standardized covariance eigenvalues are about two point nine six four seven two one, zero point zero three five zero three two, and zero point zero zero zero two four seven. Their sum is three. PCA retains the largest two, about ninety nine point nine nine one eight percent of variance. The page rounds this to one hundred point zero percent, which does not mean nothing was lost. All bars use two hundred pixels per covariance eigenvalue unit. The smallest bar is deliberately tiny; its numeric label gives its value. Try four observations with features one zero zero, zero one zero, minus one zero zero, and zero minus one zero. After sample standardization, their coordinates are plus or minus square root of one point five on the two varying axes. The third axis is constant zero. The exported PCA and MDS maps both contain these four points, up to the chosen eigenvector orientation, and preserve their pairwise distances. Adjacent points are square root of three apart; opposite points are square root of six apart. This diagram uses one hundred twenty pixels per standardized coordinate unit. Zero stress here follows from this planar Euclidean dataset; it is not a promise for every three dimensional input. Retention measures the sum of the two kept PCA eigenvalues divided by all nonnegative eigenvalues. Stress is the square root of squared pairwise distance errors divided by squared original distances. For this source, classical MDS takes Euclidean distances of the same standardized observations, so its two dimensional map agrees with PCA up to orientation. The sample retention is about ninety nine point nine nine one eight percent, while the page displays one hundred point zero after rounding. The planar example retains exactly all varying dimensions. These bars use six pixels per percentage point. Export gives every label and group and six decimal coordinates for both methods. The original full CSV remains evidence; a rounded summary alone does not prove zero distortion.