Inner Products and Cosines Between Components

Computes the inner products between the columns of a matrix of components and the cosines of the angles between them.

Usage

measureOrthogonality(components)

Arguments

components: Finite numeric matrix, samples by components. Attributes other than dim do not reach the outputs.

Details

The inner products are the Euclidean products $\sum_n v_i[n] v_j[n]$, without the sampling interval of the discrete inner product, which cancels in the cosine $$\cos\theta_{ij} = \frac{\langle v_i, v_j \rangle}{\|v_i\|_2 \|v_j\|_2}.$$ The cosine keeps its sign. The cross products are the terms by which the energy of a sum of components differs from the sum of their energies.

Value

A list of two square matrices, components by components:

innerProduct: The inner products.
cosine: The cosines.

Conditions

A column of zero norm stops the function with an error of class xemdZeroEnergy.

Examples

Time <- seq_len(1000) / 1000
Tones <- cbind(cos(2 * pi * 2 * Time), 0.25 * cos(2 * pi * 24 * Time),
               0.0625 * cos(2 * pi * 288 * Time))
# The sum of the tones and the tones.
OUT <- measureOrthogonality(cbind(rowSums(Tones), Tones))
round(OUT$cosine, 6)
#>          [,1]     [,2]     [,3]     [,4]
#> [1,] 1.000000 0.968364 0.242091 0.060523
#> [2,] 0.968364 1.000000 0.000000 0.000000
#> [3,] 0.242091 0.000000 1.000000 0.000000
#> [4,] 0.060523 0.000000 0.000000 1.000000