ProjectionPursuit

gensphere(N::Int64, d::Int64)

Generate low discrepancy sequences of length N on a d dimension unit sphere.

Roughly speaking, a low discrepancy sequence is a sequence of samples of a random variable has the property that the number of points within a region is proportional to the size of the region.

This function returns a N-by-d matrix, where each row represents a sample of a random vector that is uniformly distributed on a d-dimensional unit sphere.

See Brauchart, Johann S., Josef Dick, and Lou Fang. "Spatial low-discrepancy sequences, spherical cone discrepancy, and applications in financial modeling." Journal of Computational and Applied Mathematics 286 (2015): 28-53.

fastgensphere(N::Int64, d::Int64; par::Bool=true)

A faster way to generate lds of length N on a d dimension unit sphere.

Similar to gensphere(). This function is much faster and support parallelism, but may degenarate when d is large.

fromsphere(theta::Vector{Float64})

Transform a d-dimensional spherical coordinate to the corresponding d+1 Cartesian coordinate.

See also tosphere.

tosphere(s::Vector{Float64})

Transform a d-dimensional Cartesian coordinate to the corresponding d-1 spherical coordinate.

See also fromsphere.

ProjectionPursuitRes

Returned object of projection_pursuit or sphere_optimize.

Fields

• object_vals::Vector{Float64}: projetion pursuit index. Analogous to eigenvalues in PCA.
• proj::Matrix{Float64}: projetion pursuit direction. Analogous to eigenvectors in PCA.

sphere_optimize

Optimize a given objective fucntion constrained on a unit sphere.

See http://hdl.handle.net/10012/16710 for more detail.

Arguments

• data::Matrix{Float64}: each row contains an observation.
• object_fun::Function: an arbitrary objective function defined on unit sphere.
• n_of_candidate::Int64=5: number of candidate directions in the fine search step.
• unit_sphere=nothing: the unit spehre can be pre-generated. By default a unit sphere will be generated automatically.
• proj=nothing: previously found projection directions. See projection_pursuit.
• fnscale::Int64=-1: maximize or minimize the objective function. By default equals -1, which means maximize objective function. If the objective function needs to be minimized, set to fnscale=1.
• par::Bool=true: run the program in parallel.
• fast::Bool=true: generate the unit sphere using fast algorithm. See gensphere and fastgensphere.

projection_pursuit

Conduct dimension reduction corresponding to specified objective function.

See http://hdl.handle.net/10012/16710 for more detail.

Arguments

• data::Matrix{Float64}: each row contains an observation.
• object_fun::Function: an arbitrary objective function defined on unit sphere.
• outdim::Int64: target dimension of the output.
• n_of_candidate::Int64=5: number of candidate directions in the fine search step.
• unit_sphere=nothing: the unit spehre can be pre-generated. By default a unit sphere will be generated automatically.
• fnscale::Int64=-1: maximize or minimize the objective function. By default equals -1, which means maximize objective function. If the objective function needs to be minimized, set to fnscale=1.
• par::Bool=true: run the program in parallel.
• fast::Bool=true: generate the unit sphere using fast algorithm. See gensphere and fastgensphere.