Reference

CurrentModule = PortfolioOptimisers

PortfolioOptimisers

Documentation for PortfolioOptimisers.

abstract type DBHTRootMethod end

struct EqualRoot <: DBHTRootMethod end

struct UniqueRoot <: DBHTRootMethod end

AdjCliq(A::AbstractMatrix{<:Real}, CliqList::AbstractMatrix{<:Real},
        CliqRoot::AbstractVector{<:Real})

Find adjacent clique to the root candidates.

Inputs

• A: N×N adjacency matrix.
• CliqList: Nc×3 matrix. Each row vector lists the three vertices consisting of a 3-clique in the MPG.
• CliqRoot: Nc×1 vector of root cliques.

Outputs

• Adj: Nc×Nc adjacency matrix of the cliques with the root cliques.

BubbleCluster8s(Rpm::AbstractMatrix{<:Real}, Dpm::AbstractMatrix{<:Real},
                Hb::AbstractMatrix{<:Real}, Mb::AbstractMatrix{<:Real},
                Mv::AbstractMatrix{<:Real}, CliqList::AbstractMatrix{<:Real})

Obtains non-discrete and discrete clusterings from the bubble topology of the Planar Maximally Filtered Graph (PMFG).

Inputs

• Rpm: N×N sparse weighted adjacency matrix of the PMFG.
• Dpm: N×N shortest path lengths matrix of the PMFG.
• Hb: undirected bubble tree of the PMFG.
• Mb: Nc×Nb bubble membership matrix for 3-cliques. Mb[n, bi] = 1 indicated that 3-clique n belongs to bubble bi.
• Mv: N×Nb bubble membership matrix for vertices.
• CliqList: Nc×3 matrix. Each row vector lists the three vertices consisting of a 3-clique in the MPG.

Outputs

• Adjv: N×Nk cluster membership matrix for vertices for non-discrete clustering via the bubble topology. Adjv[n, k] = 1 indicates cluster membership of vertex n to the k'th non-discrete cluster.
• Tc: N×1 cluster membership vector. Tc[n] = k indicates cluster membership of vertex n to the k'th discrete cluster.

BubbleHierarchy(Pred::AbstractVector{<:Real}, Sb::AbstractVector{<:Real})

Build the bubble hierarchy.

Inputs

• Pred: Nc×1 vector of predicted hierarchies.
• Sb: Nc×1 vector. Sb[n] = 1 indicates 3-clique n is separating.

Outputs

• Mb: Nc×Nb bubble membership matrix for 3-cliques. Mb[n, bi] = 1 indicated that 3-clique n belongs to bubble bi.
• H2: Nb×Nb adjacency matrix for the bubble hierarchical tree where Nb is the number of bubbles.

BubbleMember(Rpm::AbstractMatrix{<:Real}, Mv::AbstractMatrix{<:Real},
             Mc::AbstractMatrix{<:Real})

Assigns each vertex in the to a specific bubble.

Inputs

• Rpm: N×N sparse weighted adjacency matrix of the PMFG.
• Mv: N×Nb bubble membership matrix. Mv[n, bi] = 1 means vertex n is a vertex of bubble bi.
• Mc: Matrix of the bubbles which coincide with the cluster.

Outputs

• Mvv: Matrix of the vertices belonging to the bubble.

BuildHierarchy(M::AbstractMatrix{<:Real})

Builds the predicted hierarchy.

Inputs

• M: N×Nc matrix of nodes and 3-cliques.

Outputs

• Pred: Nc×1 vector of predicted hierarchies.

CliqHierarchyTree2s(Apm::AbstractMatrix{<:Real}, type::Symbol = :Unique)

Looks for 3-cliques of a Maximal Planar Graph (MPG), then construct a hierarchy of the cliques with the definition of "inside" a clique being a subgraph of smaller size when the entire graph is made disjoint by removing the clique [1].

Inputs

• Apm: N×N adjacency matrix of an MPG.

• type: type for finding the root of the graph DBHTRootMethod. Uses Voronoi tesselation between tiling triangles.

  • UniqueRoot: create a unique root.
  • EqualRoot: the root is created from the candidate's adjacency tree.

Outputs

• H1: Nc×Nc adjacency matrix for 3-clique hierarchical tree where Nc is the number of 3-cliques.
• H2: Nb×Nb adjacency matrix for the bubble hierarchical tree where Nb is the number of bubbles.
• Mb: Nc×Nb bubble membership matrix for 3-cliques. Mb[n, bi] = 1 indicated that 3-clique n belongs to bubble bi.
• CliqList: Nc×3 matrix. Each row vector lists the three vertices consisting of a 3-clique in the MPG.
• Sb: Nc×1 vector. Sb[n] = 1 indicates 3-clique n is separating.

CliqueRoot(::UniqueRoot, Root, Pred, Nc, args...)

DBHTs(D::AbstractMatrix{<:Real}, S::AbstractMatrix{<:Real}; branchorder::Symbol = :optimal,
      type::Symbol = :Unique)

Perform Direct Bubble Hierarchical Tree clustering, a deterministic clustering algorithm [2]. This version uses a graph-theoretic filtering technique called Triangulated Maximally Filtered Graph (TMFG).

Arguments

• D: N×N dissimilarity matrix, e.g. a distance matrix.

• S: N×N non-negative sim matrix, examples include:

  • $\mathbf{S} = \mathbf{C} + \lvert \min \mathbf{C} \rvert$.
  • $\mathbf{S} = \lceil\max \left(\mathbf{D}^{\odot 2}\right)\rceil - \mathbf{D}^{\odot 2}$.
  • $\mathbf{S} = \exp \odot (-\mathbf{D})$.

  Where $\mathbf{C}$ is the correlation matrix, $\mathbf{D}$ the dissimilarity matrix D, and $\odot$ the Hadamard (elementwise) operator.

• branchorder: parameter for ordering the final dendrogram's branches accepted by Clustering.jl.

• type: type for finding the root of a Direct Bubble Hierarchical Clustering Tree in case there is more than one candidate DBHTRootMethod.

  • :Unique: create a unique root.
  • :Equal: the root is created from the candidate's adjacency tree.

Outputs

• T8: N×1 cluster membership vector.
• Rpm: N×N adjacency matrix of the Planar Maximally Filtered Graph (PMFG).
• Adjv: Bubble cluster membership matrix from BubbleCluster8s.
• Dpm: N×N shortest path length matrix of the PMFG.
• Mv: N×Nb bubble membership matrix. Mv[n, bi] = 1 means vertex n is a vertex of bubble bi.
• Z: (N-1)×3 linkage matrix in the same format as the output from Matlab.
• Z_hclust: Z matrix in Clustering.Hclust format.

DendroConstruct(Zi::AbstractMatrix{<:Real}, LabelVec1::AbstractVector{<:Real},
                LabelVec2::AbstractVector{<:Real},
                LinkageDist::Union{<:Real, <:AbstractVector{<:Real}})

Construct the linkage matrix by continuially adding rows to the matrix.

Inputs

• Zi: Linkage matrix at iteration i in the same format as the output from Matlab.
• LabelVec1: label vector for the vertices in the bubble for the previous valid iteration.
• LabelVec2: label vector for the vertices in the bubble for the trial iteration.

Outputs

• Z: Linkage matrix at iteration i + 1 in the same format as the output from Matlab.

DirectHb(Rpm::AbstractMatrix{<:Real}, Hb::AbstractMatrix{<:Real},
         Mb::AbstractMatrix{<:Real}, Mv::AbstractMatrix{<:Real},
         CliqList::AbstractMatrix{<:Real})

Computes the directions on each separating 3-clique of a Maximal Planar Graph (MPH), hence computes the Directed Bubble Hierarchy Tree (DBHT).

Inputs

• Rpm: N×N sparse weighted adjacency matrix of the Planar Maximally Filtered Graph (MPFG).
• Hb: Undirected bubble tree of the PMFG.
• Mb: Nc×Nb bubble membership matrix for 3-cliques. Mb[n, bi] = 1 indicated that 3-clique n belongs to bubble bi.
• Mv: N×Nb bubble membership matrix for vertices.
• CliqList: Nc×3 matrix. Each row vector lists the three vertices consisting of a 3-clique in the MPG.

Outputs

• Hc: Nb×Nb unweighted directed adjacency matrix of the DBHT. Hc[i, j]=1 indicates a directed edge from bubble i to bubble j.

FindDisjoint(Adj::AbstractMatrix{<:Real}, Cliq::AbstractVector{<:Real})

Finds disjointed cliques in adjacency matrix.

Inputs

• Adj: N×N adjacency matrix.
• Cliq: 3×1 vector of 3-cliques.

Outputs

• T: N×1 vector containing the adjacency number of each node.
• IndxNot: N×1 vector of nodes with no adjacencies.

HierarchyConstruct4s(Rpm::AbstractMatrix{<:Real}, Dpm::AbstractMatrix{<:Real},
                     Tc::AbstractVector{<:Real}, Mv::AbstractMatrix{<:Real})

Constructs the intra- and inter-cluster hierarchy by utilizing Bubble Hierarchy structure of a Maximal Planar graph, in this a Planar Maximally Filtered Graph (PMFG).

Inputs

• Rpm: N×N sparse weighted adjacency matrix of the PMFG.
• Dpm: N×N shortest path lengths matrix of the PMFG.
• Tc: N×1 cluster membership vector. Tc[n] = k indicates cluster membership of vertex n to the k'th discrete cluster.
• Mv: N×Nb bubble membership matrix. Mv[n, bi] = 1 means vertex n is a vertex of bubble bi.

Outputs

• Z: (N-1)×3 linkage matrix in the same format as the output from Matlab.

J_LoGo(sigma, separators, cliques)

Compute the sparse inverse covariance from a clique tree and separators [3].

Inputs

• sigma: N×N covariance matrix.
• separators: list of 3-cliques that are not triangular faces.
• cliques: list of all 4-cliques.

Outputs

• jlogo: J_LoGo covariance matrix.

LinkageFunction(d::AbstractMatrix{<:Real}, labelvec::AbstractVector{<:Real})

Looks for the pair of clusters with the best linkage.

Inputs

• d: Nv×Nv distance matrix for a list of vertices assigned to a bubble.
• labelvec: label vector for the vertices in the bubble.

Outputs

• PairLink: pair of links with the best linkage.
• dvu: value of the best linkage.

PMFG_T2s(W::AbstractMatrix{<:Real}, nargout::Integer = 3)

Constructs a Triangulated Maximally Filtered Graph (TMFG) starting from a tetrahedron and recursively inserting vertices inside existing triangles (T2 move) in order to approximate a Maximal Planar Graph with the largest total weight, aka Planar Maximally Filtered Graph (PMFG). All weights are non-negative [4].

Arguments

• W: N×N matrix of non-negative weights.
• nargout: number of output arguments, the same arguments are always returne, this only controls whether some arguments are empty or not.

Outputs

• A: adjacency matrix of the PMFG with weights.
• tri: list of triangles (triangular faces).
• clique3: list of 3-cliques taht are not triangular faces, all 3-cliques are given by [tri; clique3].
• cliques: list of all 4-cliques, if nargout <= 3, this will be returned as an empty array.
• cliqueTree: 4-cliques tree structure (adjacency matrix), if nargout <= 4, it is returned as an empty array.

breadth(CIJ::AbstractMatrix{<:Real}, source::Integer)

Breadth-first search.

Inputs

• CIJ: binary (directed/undirected) connection matrix.
• source: source vertex.

Outputs

• distance: distance between source and i'th vertex (0 for source vertex).
• branch: vertex that precedes i in the breadth-first search tree (-1 for source vertex).

build_link_and_dendro(rg::AbstractRange, dpm::AbstractMatrix{<:Real},
                       LabelVec::AbstractVector{<:Real}, LabelVec1::AbstractVector{<:Real},
                       LabelVec2::AbstractVector{<:Real}, V::AbstractVector{<:Real},
                       nc::Real, Z::AbstractMatrix{<:Real})

Computes iterates over the vertices to construct the linkage matrix iteration by iteration.

Inputs

• rg: range of indices of the vertices in a bubble.
• dpm: Nv×Nv distance matrix for a list of vertices assigned to a bubble.
• LabelVec: vector labels of all vertices.
• LabelVec1: label vector for the vertices in the bubble for the previous valid iteration.
• LabelVec2: label vector for the vertices in the bubble for the trial iteration.

Outputs

• Z: updated linkage matrix in the same format as the output from Matlab.
• nc: updated inverse of the linkage distance.
• LabelVec1: updated LabelVec1 for the next iteration.

clique3(A::AbstractMatrix{<:Real})

Computes the list of 3-cliques.

Inputs

• A: N×N adjacency matrix of a Maximal Planar Graph (MPG).

Outputs

• K3: vector of vectors with the corresponding indices of candidate cliques.
• E: matrix with non-zero indices and entries of candidate cliques.
• CliqList: Nc×3 matrix. Each row vector lists the three vertices consisting of a 3-clique in the MPG.

distance_wei(L::AbstractMatrix{<:Real})

The distance matrix contains lengths of shortest paths between all node pairs. An entry [u, v] represents the length of the shortest path from node u to node v. The average shortest path length is the characteristic path length of the network. The function uses Dijkstra's algorithm.

Inputs

• L: Directed/undirected connection-length matrix.

  • Lengths between disconnected nodes are set to Inf.
  • Lengths on the main diagonal are set to 0.

Outputs

• D: distance (shortest weighted path) matrix.
• B: number of edged in the shortest weigthed path matrix.

turn_into_Hclust_merges(Z::AbstractMatrix{<:Real})

Turns a Matlab-style linkage matrix to a useable format for Hclust.

Inputs

• Z: Matlab-style linkage matrix.

Outputs

• Z: Hclust-style linkage matrix.