Statistical data reduction · BEGINNER LESSON
Principal components
Principal components are new variables created by principal component analysis (PCA) that are uncorrelated with each other and ordered by how much variance they explain in the original data. They are linear combinations of the original measurements, built to preserve as much of the dataset’s variability as possible.
What they are and what is measured
Principal components are not original measurements; they are derived variables created from a dataset to summarize how the data vary. The key quantity being measured is variance, meaning how spread out the data are, and each component is built to explain as much of that spread as possible. PCA is useful when many measurements are correlated, because related variables can be compressed into a smaller set of summary axes. [0,2]
[1] Dimensionality reduction techniques in pupillometry research [3] A Principal Component Analysis - PMC - NIH
How principal components are constructed
PCA creates orthogonal, meaning mutually independent in direction, variables from the original data. Each principal component is a linear combination of the original variables, and the components are ordered so that the first explains the most variance, the next explains the next most, and so on. Eigenvectors give the directions of these components, while eigenvalues indicate how much variability each one captures. In temporal data, such as measurements across time points, this can summarize the dominant shape and strength of a signal over a trial. [0]
[1] Dimensionality reduction techniques in pupillometry research
Why they are useful
A small number of principal components can sometimes reconstruct the original dataset well enough for analysis. This is especially helpful when the original variables are strongly correlated, because shared variation can be represented more compactly. In wastewater studies, for example, a few components captured most of the variability and were interpreted as dominant environmental processes such as organic loading, nutrient enrichment, sedimentation, or microbial activity. [0,2]
[1] Dimensionality reduction techniques in pupillometry research [3] A Principal Component Analysis - PMC - NIH
Limitations and caution
Principal components summarize patterns, but they do not by themselves identify causes. Their interpretation is safest when the analyst expects one major process, because multiple processes can be mixed together in the same signal. PCA also compresses high-dimensional variation, so some information is lost, and in static bulk data it cannot reveal time change, single-cell heterogeneity, or why a pattern occurs. When several processes are suspected, methods closer to factor analysis may be more appropriate. [0,1]
[1] Dimensionality reduction techniques in pupillometry research [2] Geometric multidimensional representation of omic signatures
SEE THE IDEA
How principal components summarize data
Many measured values, often correlated with each other.
Finds directions in the data that capture the most shared variation.
New uncorrelated summary variables ordered by explained variance.
Components may reflect dominant patterns, but not direct causes.
Original variables feed PCA, which produces principal components that compress shared variance for analysis.
Principal components are summary axes built from the original measurements; they capture major variation, but they do not prove what causes that variation.
TRY IT YOURSELF
Test your understanding
Choose an answer, then check the explanation.
PUT IT TOGETHER
Match each concept to its meaning
Drag a label onto its description, or select a label and then select a description. This also works with a keyboard.
4 pairs to match
Sources and further reading
This general lesson was generated from public source material. It explains a concept; it does not interpret a private paper or provide medical advice.