Geometric data analysis is an approach that uses geometric and topological concepts to analyze and interpret data. This method is particularly useful in fields like statistics, machine learning, and data science, where complex datasets can often be visualized or understood better through geometric perspectives. Key concepts and elements of geometric data analysis include: 1. **Geometric Structures**: Data can be represented in various geometric forms, such as points, curves, surfaces, or more complex shapes like manifolds.
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Geometric data analysis represents multivariate observations as weighted clouds of points, then studies their Euclidean geometry through distances, orthogonal projections and principal components. Principal component analysis applies this approach to numerical variables; correspondence analysis adapts it to a table of counts with its own weights and metric. Choosing the weights and metric is part of the analysis, since these choices decide which differences between observations count as large.[ref]
Geometric Data Analysis by Brigitte Le Roux and Henry Rouanet. Chapters 1–4, overview, correspondence analysis, Euclidean clouds and principal component analysis
For a column-centered data matrix with singular value decomposition , the first principal components give coordinates . They are the orthogonal projection onto the chosen principal directions. Rescaling one variable changes the distances and can change those directions: measurements in metres and millimetres should not receive different influence merely because the units were changed. The geometry must therefore be interpreted together with the scaling convention, rather than treating the resulting plot as an intrinsic picture of the observations.