ABSTRACT
Traditional theories of geometric representations for stimulus spaces (see, e.g., Shepard, 1962a, 1962b) rely on the notion of a “distance” in some multidimensional vector space ℝn to describe the subjective proximity between various stimuli whose features are represented by the axes of the space. Such a distance is often viewed as induced by a “norm” of the vector space, defined as a real-valued function ℝn → ℝ and denoted ||·||, that satisfies the following conditions for all x, y ∈ R n and α ∈ ℝ: (i) ||x|| ≥ 0 with equality holding if and only if x = 0; (ii) ||αx|| = |α|·||x||; (iii) ||x + y|| ≤ ||x|| + ||y||. The distance measure or “metric” induced by such a norm is defined as https://www.w3.org/1998/Math/MathML"> Δ ( x , x ′ ) = ‖ x − x ′ ‖ . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203837610/be0849bc-52af-4cab-a8c1-376880320eb3/content/math_181_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> Such a metric is continuous with respect to (x, x′) and it satisfies the axioms of (i) non-negativity: Δ(x, x′) ≥ 0, with 0 attained if and only if x = x′; (ii) symmetry: Δ(x, x′) = Δ(x′, x); and (iii) triangle inequality: Δ(x, x′) + Δ(x′, x″) ≥ Δ(x, x″), for any triplet x, x′, x″. The norm ||·|| may also be used to define an inner product 〈·,·〉: https://www.w3.org/1998/Math/MathML"> 〈 x , x ′ 〉 = 1 2 ( ‖ x + x ′ ‖ 2 − ‖ x ‖ 2 − ‖ x ′ ‖ 2 ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203837610/be0849bc-52af-4cab-a8c1-376880320eb3/content/math_182_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The inner product operation on the vector space allows one to define the angle between two vectors (and hence orthogonality). which allows one to project one vector onto another (and hence onto a subspace).
