Type that stores a partition of multiindex into subests, sizes of subests, size of original multitindex and number of subsets

accesscum(mulind::Tuple{Int, ...}, ::IndexPart, cum::SymmetricTensor{Float}...)

Returns: vector of blocks from cumulants. Each block correspond to a subests of partition (part) of multiindex (multiind).

```jldoctest julia> cum = SymmetricTensor([1.0 2.0 3.0; 2.0 4.0 6.0; 3.0 6.0 5.0]);

julia> accesscum((1,1,1,1), IndexPart(Array{Int64,1}[[1,2],[3,4]],[2,2],4,2), cum) Array{Float64,N}[ [1.0 2.0; 2.0 4.0], [1.0 2.0; 2.0 4.0]]

julia> accesscum((1,1,1,2), IndexPart(Array{Int64,1}[[1,2],[3,4]],[2,2],4,2), cum)

Array{Float64,N}[ [1.0 2.0; 2.0 4.0], [3.0 0.0; 6.0 0.0]]

julia> accesscum((1,1,1,1), IndexPart(Array{Int64,1}[[1,4],[2,3]],[2,2],4,2), cum)

Array{Float64,N}[ [1.0 2.0; 2.0 4.0], [1.0 2.0; 2.0 4.0]] ```

blockel(X::Matrix{T}, i::Tuple, j::Tuple, b::Int)

Returns Float, the element of the block (indexed by j) of the moment's tensor of X, at index cd i inside a block, where b is a standard blocks' size

julia> M = [1. 2.  5. 6. ; 3. 4.  7. 8.];

julia> blockel(M, (1,1), (1,1), 2)
5.0

julia> blockel(M, (1,1), (2,2), 2)
37.0

cumulant(X::Vector{Matrix}, cum::SymmetricTensor...)

Returns: SymmetricTensor{Float, m}, a tensor of the m'th cumulant of X, given Vector of cumulants of order 2, ..., m-2

cumulants(X::Matrix, m::Int, b::Int)

Returns [SymmetricTensor{Float, 1}, SymmetricTensor{Float, 2}, ..., SymmetricTensor{Float, m}], vector of cumulant tensors

julia> M =  [[-0.88626   0.279571];[-0.704774  0.131896]];

julia> convert(Array, cumulants(M, 3)[3])
2×2×2 Array{Float64,3}:
[:, :, 1] =
 0.0  0.0
 0.0  0.0

[:, :, 2] =
 0.0  0.0
 0.0  0.0

indpart(nind::Int, npart::Int, e::Int = 1)

Returns vector of IndexPart type, that includes partitions of set [1, 2, ..., nind] into npart subests of size != e, sizes of each subest, size of original set and number of partitions

julia>indpart(4,2)
3-element Array{Cumulants.IndexPart,1}:
 IndexPart(Array{Int64,1}[[1,2],[3,4]],[2,2],4,2)
 IndexPart(Array{Int64,1}[[1,3],[2,4]],[2,2],4,2)
 IndexPart(Array{Int64,1}[[1,4],[2,3]],[2,2],4,2)

mixel(X::Matrix{T}, ind::Tuple)

Returns Float, mixed element for cumulants 4-6 at ind multi-index

julia> M =  [[-0.88626   0.279571];[-0.704774  0.131896]];

julia> mixel(M, (1,1,1,1))
-1.232956812564408

mixel(M, (1,1,1,1,1,1))
1.015431116914347

momel(X::Matrix{Float}, ind::Tuple)

Returns Float, an element of moment's tensor at ind multiindex

julia> M =  [[-0.88626   0.279571];[-0.704774  0.131896]];

julia> momel(M, (1,1,1,1))
0.4318298020613279

momentn1c(X::Matrix{Float}, m::Int, b::Int)

Returns: SymmetricTensor{Float, m}, a tensor of the m'th moment of X, where b is a block size. Uses 1 core implementation

momentblock(X::Matrix{T}, j::Tuple, dims::Tuple, b::Int)

Returns a block of a moment's tensor of X. A block is indexed by j and if size dims, b is a standatd block size.

julia> M = [1. 2.  5. 6. ; 3. 4.  7. 8.];

julia> momentblock(M, (1,1), (2,2), 2)
2×2 Array{Float64,2}:
 5.0   7.0
 7.0  10.0

momentnc(X::Matrix}, m::Int, b::Int)

Returns: SymmetricTensor{Float, m}, a tensor of the m'th moment of X, where b is a block size. Uses multicore parallel implementation via pmap()

naivecumulant(data::Matrix, m::Int)

Returns Array{Float, m} the m'th cumulant tensor

julia> M =  [[-0.88626   0.279571];[-0.704774  0.131896]];

julia> naivecumulant(M, 3)
2×2×2 Array{Float64,3}:
[:, :, 1] =
 0.0  0.0
 0.0  0.0

[:, :, 2] =
 0.0  0.0
 0.0  0.0

naivemoment(data::Matrix{Float}, m::Int)

Returns Array{Float, m} the m'th moment tensor

julia> M =  [[-0.88626   0.279571];[-0.704774  0.131896]];

julia> naivemoment(M, 3)
2×2×2 Array{Float64,3}:
[:, :, 1] =
  -0.523092   0.142552
  0.142552  -0.0407653

[:, :, 2] =
  0.142552   -0.0407653
  -0.0407653   0.0120729

outerprodcum(retd::Int, npart::Int, cum::SymmetricTensor...; exclpartlen::Int = 1)

Returns retd dims outer products of npart cumulants in SymmetricTensor form. exclpartlen is a length of partitions to be excluded in calculations, in this algorithm exclpartlen = 1

julia> cum = SymmetricTensor([1.0 2.0 3.0; 2.0 4.0 6.0; 3.0 6.0 5.0]);

julia> outerprodcum(4,2,cum, cum)
SymmetricTensors.SymmetricTensor{Float64,4}(Union{Array{Float64,4}, Void}[[3.0 6.0; 6.0 12.0]

[6.0 12.0; 12.0 24.0]

[6.0 12.0; 12.0 24.0]

[12.0 24.0; 24.0 48.0] nothing; nothing nothing]

Union{Array{Float64,4}, Void}[nothing nothing; nothing nothing]

Union{Array{Float64,4}, Void}[[9.0 18.0; 18.0 36.0]

[18.0 36.0; 36.0 72.0] nothing; nothing nothing]

Union{Array{Float64,4}, Void}[[23.0 46.0; 46.0 92.0] [45.0; 90.0]; nothing [75.0]], 2, 2, 3, false)

outprodblocks(n::Int, part::Vector{Vector{Int}}, blocks::Vector{Array{T}}

Returns: n dims Array of outer product of blocks, given partition of indices, part.

julia> blocks = 2-element Array{Array{Float64,N},1}[[1.0 2.0; 2.0 4.0], [1.0 2.0; 2.0 4.0]];

julia> outprodblocks(IndexPart(Array{Int64,1}[[1,2],[3,4]],[2,2],4,2), blocks)

2×2×2×2 Array{Float64,4}:
[:, :, 1, 1] =
 1.0  2.0
 2.0  4.0

[:, :, 2, 1] =
 2.0  4.0
 4.0  8.0

[:, :, 1, 2] =
 2.0  4.0
 4.0  8.0

[:, :, 2, 2] =
 4.0   8.0
 8.0  16.0

usebl(bind::Tuple, n::Int, b::Int, nbar::Int)

Returns: Tuple{Int}, sizes of the last block

moment(X::Matrix}, m::Int, b::Int)

Returns: SymmetricTensor{Float, m}, a tensor of the m'th moment of X, where b is a block size. Calls 1 core or multicore moment function.