JointMoments

Tensors and statistics for third and fourth joint central moments.

Usage

Installation:

julia> Pkg.add("JointMoments")

Use:

julia> using JointMoments

julia> data = [ 0.837698   0.49452   2.54352 
               -0.294096  -0.39636   0.728619
               -1.62089   -0.44919   1.20592 
               -1.06458   -0.68214  -1.12841 ];

Third and fourth joint central moment tensors:

julia> coskew(data)
3x3x3 Array{Float64,3}:
[:, :, 1] =
 0.294091  0.26697   0.773618
 0.26697   0.162051  0.350696
 0.773618  0.350696  0.451934

[:, :, 2] =
 0.26697   0.162051   0.350696
 0.162051  0.0852269  0.156275
 0.350696  0.156275   0.131448

[:, :, 3] =
 0.773618  0.350696   0.451934
 0.350696  0.156275   0.131448
 0.451934  0.131448  -0.645484

julia> cokurt(data)
3x3x3x3 Array{Float64,4}:
[:, :, 1, 1] =
 1.2563    0.563538  1.05898 
 0.563538  0.290737  0.643266
 1.05898   0.643266  1.68278 

[:, :, 2, 1] =
 0.563538  0.290737  0.643266
 0.290737  0.158262  0.374873
 0.643266  0.374873  0.97585 

[:, :, 3, 1] =
 1.05898   0.643266  1.68278
 0.643266  0.374873  0.97585
 1.68278   0.97585   2.69607

[:, :, 1, 2] =
 0.563538  0.290737  0.643266
 0.290737  0.158262  0.374873
 0.643266  0.374873  0.97585 

[:, :, 2, 2] =
 0.290737  0.158262   0.374873
 0.158262  0.0887859  0.218824
 0.374873  0.218824   0.58726 

[:, :, 3, 2] =
 0.643266  0.374873  0.97585
 0.374873  0.218824  0.58726
 0.97585   0.58726   1.73728

[:, :, 1, 3] =
 1.05898   0.643266  1.68278
 0.643266  0.374873  0.97585
 1.68278   0.97585   2.69607

[:, :, 2, 3] =
 0.643266  0.374873  0.97585
 0.374873  0.218824  0.58726
 0.97585   0.58726   1.73728

[:, :, 3, 3] =
 1.68278  0.97585  2.69607
 0.97585  0.58726  1.73728
 2.69607  1.73728  5.85635

Statistics:

julia> coskewness(data)
0.2838850631006579

julia> cokurtosis(data)
0.8916763961210045

coskewness and cokurtosis can use an optional weight vector, which assigns a weight to each column of the data matrix:

julia> weights = [1.0, 0.1, 0.5];

julia> coskewness(data, weights)
0.46758589701357833

julia> cokurtosis(data, weights)
1.3203902349727108

The coskew and cokurt functions can also return flattened/unfolded tensors:

julia> coskew(data, flatten=true)
3x9 Array{Float64,2}:
 0.294091  0.26697   0.773618  0.26697   0.162051   0.350696  0.773618  0.350696   0.451934
 0.26697   0.162051  0.350696  0.162051  0.0852269  0.156275  0.350696  0.156275   0.131448
 0.773618  0.350696  0.451934  0.350696  0.156275   0.131448  0.451934  0.131448  -0.645484

 julia> cokurt(data,flatten=true)
3x27 Array{Float64,2}:
 2.12678   1.11885   0.474782  1.11885    1.12294     0.187331   0.474782   0.187331   1.15524   1.11885   …  0.276558  0.474782   0.187331   1.15524    0.187331   -0.0266349  0.276558  1.15524   0.276558  0.178083
 1.11885   1.12294   0.187331  1.12294    1.40462    -0.0266349  0.187331  -0.0266349  0.276558  1.12294      0.779221  0.187331  -0.0266349  0.276558  -0.0266349  -0.517198   0.779221  0.276558  0.779221  0.218732
 0.474782  0.187331  1.15524   0.187331  -0.0266349   0.276558   1.15524    0.276558   0.178083  0.187331     0.218732  1.15524    0.276558   0.178083   0.276558    0.779221   0.218732  0.178083  0.218732  5.98947 

The coskew, cokurt, coskewness, and cokurtosis functions have standardize and bias keyword arguments. Setting standardize=true standardizes the elements of the joint moment tensors (divides by the standard deviation). Setting bias=1 uses Bessel's correction (divides by N-1 instead of N).

Tests

Unit tests can be run from the command line:

$ julia test/runtests.jl

Or from the Julia prompt:

julia> Pkg.test("JointMoments")

This package includes a rudimentary timing framework in test/timing.jl. To run the timed examples:

$ julia --color test/timing.jl