/// <summary>
/// Multidimensional scaling/PCoA: transform distances to points in a coordinate system.
/// </summary>
/// <param name="input">A matrix of pairwise distances. Zero indicates identical objects.</param>
/// <returns>A matrix, the columns of which are coordinates in the nth dimension.
/// The rows are in the same order as the input.</returns>
public static ILArray <double> Scale(ILArray <double> input)
{
int n = input.Length;
ILArray <double> p = ILMath.eye <double>(n, n) - ILMath.repmat(1.0 / n, n, n);
ILArray <double> a = -.5 * ILMath.multiplyElem(input, input);
ILArray <double> b = ILMath.multiply(p, a, p);
ILArray <complex> V = ILMath.empty <complex>();
ILArray <complex> E = ILMath.eig((b + b.T) / 2, V);
ILArray <int> i = ILMath.empty <int>();
ILArray <double> e = ILMath.sort(ILMath.diag(ILMath.real(E)), i);
e = ILMath.flipud(e);
i = ILMath.toint32(ILMath.flipud(ILMath.todouble(i)));
ILArray <int> keep = ILMath.empty <int>();
for (int j = 0; j < e.Length; j++)
{
if (e[j] > 0.000000001)
{
keep.SetValue(j, keep.Length);
}
}
ILArray <double> Y;
if (ILMath.isempty(keep))
{
Y = ILMath.zeros(n, 1);
}
else
{
Y = ILMath.zeros <double>(V.S[0], keep.Length);
for (int j = 0; j < keep.Length; j++)
{
Y[ILMath.full, j] = ILMath.todouble(-V[ILMath.full, i[keep[j]]]);
}
Y = ILMath.multiply(Y, ILMath.diag(ILMath.sqrt(e[keep])));
}
ILArray <int> maxind = ILMath.empty <int>();
ILMath.max(ILMath.abs(Y), maxind, 0);
int d = Y.S[1];
ILArray <int> indices = maxind + ILMath.toint32(ILMath.array <int>(SteppedRange(0, n, (d - 1) * n)));
ILArray <double> colsign = ILMath.sign(Y[indices]);
for (int j = 0; j < Y.S[1]; j++)
{
Y[ILMath.full, j] = Y[ILMath.full, j] * colsign[j];
}
return(Y);
}