// Decompose computes an m-by-n matrix M for an m-by-m covariance matrix Σ such
// that, for an n-element vector X with uncorrelated components, M * X is an
// m-element vector whose components are correlated according to Σ. The
// function reduces the number of dimensions from m to n such that a certain
// portion of the variance is preserved, which is controlled by λ ∈ (0, 1].
func Decompose(Σ []float64, m uint, λ float64) ([]float64, uint, error) {
U, Λ, err := decomposition.CovPCA(Σ, m, math.Sqrt(math.Nextafter(1, 2)-1))
if err != nil {
return nil, 0, err
}
n := m
// NOTE: Λ is in the descending order and nonnegative.
var cum, sum float64
for i := uint(0); i < m; i++ {
sum += Λ[i]
}
for i := uint(0); i < m; i++ {
cum += Λ[i]
if cum/sum >= λ {
n = i + 1
break
}
}
for i := uint(0); i < n; i++ {
coef := math.Sqrt(Λ[i])
for j := uint(0); j < m; j++ {
U[i*m+j] *= coef
}
}
return U[:m*n], n, nil
}
func TestCorrelateLarge(t *testing.T) {
_, application, _ := system.Load("fixtures/016_160.tgff")
ε := math.Sqrt(math.Nextafter(1.0, 2.0) - 1.0)
R := Compute(application, index(160), 5)
_, _, err := decomposition.CovPCA(R, 160, ε)
assert.Success(err, t)
}
func TestCorrelateSmall(t *testing.T) {
_, application, _ := system.Load("fixtures/002_020.tgff")
R := Compute(application, index(20), 2)
_, _, err := decomposition.CovPCA(R, 20, 0)
assert.Success(err, t)
R = Compute(application, index(1), 2)
assert.Equal(R, []float64{1.0}, t)
}
// Decompose computes an m-by-n matrix C and an n-by-m matrix D given an m-by-m
// covariance matrix Σ such that (a) for an n-element vector Z with uncorrelated
// components, C * X is an m-element vector whose components are correlated
// according to Σ, and (b) for an m-element vector X with correlated components
// according to Σ, D * X is an n-element vector with uncorrelated components.
// The function reduces the number of dimensions from m to n such that a certain
// portion of the variance is preserved, which is controlled by λ ∈ (0, 1].
//
// Internally the function relies on decomposition.CovPCA and returns its
// outputs as well (the principal components and vectors of Σ).
func Decompose(Σ []float64, m uint, λ, ε float64) (C []float64, D []float64,
U []float64, Λ []float64, err error) {
U, Λ, err = decomposition.CovPCA(Σ, m, ε)
if err != nil {
return nil, nil, nil, nil, err
}
n := m
var cum, sum float64
for i := uint(0); i < m; i++ {
sum += Λ[i]
}
for i := uint(0); i < m; i++ {
cum += Λ[i]
if cum/sum >= λ {
n = i + 1
break
}
}
C = make([]float64, m*n)
D = make([]float64, n*m)
for i := uint(0); i < n; i++ {
σ := math.Sqrt(Λ[i])
for j := uint(0); j < m; j++ {
ρ := U[i*m+j]
C[i*m+j] = ρ * σ
D[j*n+i] = ρ / σ
}
}
return
}
