Obviously, i₀ is an important parameter since it represents the memory relating the spaces X₁ and X₂. Ξ₁ has the property that the magnitude of its projection on i₀ is a maximum while Ξ₂ to the opposite has a zero projection on i₀. γ is the detected version of the eigenvalue of Ch(X₂,X₁). In the companion paper it was shown how one can provide a Euclidean geometrical representation of the NPO. This representation is shown in Figure 7 which shows the vectors i₀, i₁, i₂, X₁, X₂, Ξ₁, Ξ₂, and the angles Θ₁, Θ₂, and γ. The length of a vector is given by |X| = κₓ(2πε)⁻¹ᐟ² ∈ H(X) and the angle between two vectors by |Θ(X₁,X₂)|-sin⁻¹ ∈ -R(X₁,X₂). The three vectors i₀, i₁, i₂ provide an orthogonal coordinate system because the corresponding signals are random, _i.e._, κ R(i₀,i₁,i₂) ≡ 0. As external observers we have a prior knowledge of this coordinate system; however, the NPO is given only the vectors X₁ and X₂ in the i₀ ⨉ i₁ and i₀ ⨉ i₂ planes. The NPO can reconstruct the entire geometry but the actual output Ξ obviously is constrained to lie in the plane of the input vector X. The following formulas are typical of the relations present. |Ξ₁| tan β = ———— |Ξ₂| cos Θ = cos 2β csc 2γ cos 2β cos 2Θ₁ = -1 + 2 ——————— 1-cos 2γ cos Θ = cos Θ₁ cos Θ₂. [Illustration: Figure 7—Geometry of the NPO] [Illustration: Figure 8—NPO run number 5] [Illustration: Figure 9—NPO run number 6] We have obtained a complete description of the NPO which involves 74 formulas. These treat the noise in the various outputs, invariances of the NPO and other interesting features. A presentation of these would be outside of the scope of this paper and would tend to obscure the main features of the NPO. Thus, we show here only a typical sample of the computer simulation, Figure 8 and Figure 9. Conditions for these runs are shown in Table I. Run No. 6 duplicates run No. 5 except for the fact that i₁ and i₂ were disabled in run No. 6. Observe that all our descriptions of the NPO and the space it is to decompose have been time invariant while the signals shown in the simulation are presented as functions of time. The conversion may be effected as follows: Given a measurable (single-valued) function x = x(t)t ∊ T where μ(T) > 0 we define the space X = {x = x(t) ∍ t ∊ T} and a probability distribution μ(x⁻¹(X′)) P(X′) = —————————— X′ open ⊂ X μ(T) on that space.