// CovPCA performs principal component analysis on an m-by-m covariance matrix
// Σ. The principal components U and their variances Λ are returned in the
// descending order of the variances.
//
// By definition, the variances should be nonnegative. Due to finite-precision
// arithmetics, however, some close-to-zero variances might turn out to be
// negative. If the absolute value of a negative variance is smaller than the
// tolerance ε, the function nullifies that variance and proceeds without any
// errors; otherwise, an error is returned.
func CovPCA(Σ []float64, m uint, ε float64) (U []float64, Λ []float64, err error) {
U = make([]float64, m*m)
Λ = make([]float64, m)
if err = decomposition.SymmetricEigen(Σ, U, Λ, m); err != nil {
return nil, nil, err
}
for i := uint(0); i < m; i++ {
if Λ[i] < 0.0 {
if -Λ[i] < ε {
Λ[i] = 0.0
} else {
return nil, nil, errors.New("the matrix should be positive semidefinite")
}
}
}
// Λ is in the ascending order. Reverse!
for i, j := uint(0), m-1; i < j; i, j = i+1, j-1 {
Λ[i], Λ[j] = Λ[j], Λ[i]
for k := uint(0); k < m; k++ {
U[i*m+k], U[j*m+k] = U[j*m+k], U[i*m+k]
}
}
return
}
func TestInverse(t *testing.T) {
m := uint(3)
A := []float64{
1.0, 2.0, 3.0,
2.0, 4.0, 5.0,
3.0, 5.0, 6.0,
}
U := make([]float64, m*m)
Λ := make([]float64, m)
err := decomposition.SymmetricEigen(A, U, Λ, m)
assert.Equal(err, nil, t)
err = matrix.Invert(A, m)
assert.Equal(err, nil, t)
I, err := invert(U, Λ, m)
assert.Equal(err, nil, t)
assert.EqualWithin(A, I, 1e-14, t)
}