DimensionalityReduction.jl

Algorithms

• Principal Component Analysis (PCA)

PCA Usage

using DimensionalityReduction

# simulate 100 random observations
# rotate and scale as well
X = randn(100,2) * [0.8 0.7; 0.9 0.5]
Xpca = pca(X)

Rows of X each represent a data point (i.e., a different repetition of the experiment), and columns of X represent the different variables measured.

Attributes:

Xpca.rotation                # principal components
Xpca.scores                  # rotated X
Xpca.standard_deviations     # square roots of the eigenvalues
Xpca.proportion_of_variance  # fraction of variance brought by each principal component
Xpca.cumulative_variance     # cumulative proportion of variance

By default, pca() uses SVD decomposition. Alternatively, pcaeig(X) will calculate directly the eigenvectors of the covariance matrix.

pca() centers and re-scales input data by default. This is controlled by the center and scale keyword arguments:

pca(X::Matrix ; center::Bool, scale::Bool)

Centering is done by subtracting the mean, and scaling by normalizing each variable by its standard deviation.

If scale is true (default), then the principal components of the data are also scaled back to the original space and saved to Xpca.rotation

To overlay the principal components on top of the data with PyPlot

using PyPlot
plot( X[:,1], X[:,2], "r." )  # point cloud

# get data center
ctr = mean( X, 1 )

# plot principal components as lines
#  weight by their standard deviation
PCs = Xpca.rotation
for v=1:2
	weight = Xpca.standard_deviations[v]
	plot( ctr[1] + weight * [0, PCs[1,v]], 
		  ctr[2] + weight * [0, PCs[2,v]],
		  linewidth = 2)
end

To make a biplot with PyPlot

using PyPlot
scores = Xpca.scores[:,1:2]
plot( scores[:,1], scores[:,2], "r." )

To make a biplot with Gadfly:

using Gadfly
scores = Xpca.scores[:,1:2]
pl = plot(x=scores[:,1],y=scores[:,2], Geom.point)
draw(PNG("pca.png", 6inch, 6inch), pl)

Starting from a DataFrame:

using RDatasets
iris = data("datasets", "iris")
iris = convert(Array,DataArray(iris[:,1:4]))
Xpca = pca(iris)

ICA Usage

ICA has been deprecated.

t-SNE Usage

t-SNE has been deprecated.

NMF

NMF has been moved into a separate package.