Algorithm 2: Wishart clustering

Input: {x1…xl}—objects, d(xi, xj)—distance function, k, h

Output: ${\left\{y_i=y\left(x_i\right)\right\}}_{i=1}^l$—cluster labels

dk(xi)← distance to k-th nearest neighbour of xi

sort objects so that $d_k\left(x_{\left(1\right)}\right)⩽\dots ⩽d_k\left(x_{\left(l\right)}\right)$) the cluster c is defined to be a height-significant one, with respect to height value h > 0 if ${max}_{x_i{x}_j\in c}\left(p\left(xi\right)-p\left(xj\right)\right)\ge h.p\left(x\right)=\frac{k}{V_k\left(x\right)l},V_k\left(x\right)$—volume of the minimum hypersphere with its centre at point x containing at least k observations

fori←1 to l do

Vi←x ∈ x(1)…x(i − 1)|d(x(i), x) ⩽ dk(x(i))

Ci←y(x)|x ∈ Vi

ifCi = ∅ then

generate new cluster c

y(i)← c

if|Ci| = 1 then

let Ci = {c}

ifcompleted(c) then

$y_{\left(i\right)}$← noise

else

$y_{\left(i\right)}$←c

if$\left|C_i\right|>1$ then

let {Ci = c1, …, ct}

ifcompleted(cj)∀j then

$y_{\left(i\right)}$← noise

Si←c ∈ Ci|cissignificant(h)

if|Si| > 1 then

completed(c)← True ∀c ∈ Si

y(x)← noise ∀x ∈ c ∈ Ci∖Si

$y_{\left(i\right)}$← noise

else

merge all clusters from Ci and update labels in merged clusters

return y