To find numerical structures that are both surprising and explanatorily useful in complex organized systems, we need a general-purpose numerical pattern engine. This chapter, building on the foundation in Chapter 5, introduces a range of variations on one theme, the singular-value decomposition (SVD). Section 6.1 is about an elegant geometric diagram, the doubly ruled hyperbolic paraboloid. Section 6.2 describes the algebra of the singular-value decomposition, whose diagram this is. Sections 6.3, 6.4, and 6.5 explicate three important application contexts of this tool – principal components analysis, Partial Least Squares, and principal coordinates analysis – with variants and worked examples. Chapter 7 is occupied mainly with demonstrations of the craft by which all this may be interwoven in one context, morphometrics, that very nicely matches the theme of organized systems on which Part III concentrates. What makes morphometrics so suitable an example is its systematic attention to the parameterization of the variables that go into the SVD pattern engine, a parameterization that the SVD is specifically designed to inspect and indeed to try and simplify.

The Hyperbolic Paraboloid

How can geometry help us apprehend numerical explanations – help us turn arithmetic into understanding – in more complex systems?

Most readers who have ever taken a statistics course have seen straight lines used as if they were explanations.