# DimensionalityReduction.jl

**The DimensionalityReduction package is deprecated. It is superseded by a new package MultivariateStats. **.

# 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.