# Data Transforms

Summary

Some attribute transformations are included in the DynaML distribution, here we show how to use them. All of them inherit ReversibleScaler[I] trait. They are contained in the dynml.utils package.

## Gaussian Centering¶

Gaussian scaling/centering involves calculating the sample mean and variance of data and applying a gaussian standardization operations using the calculated statistics.

It has different implementations in slightly varying contexts.

### Univariate¶

Univariate gaussian scaling involves

\begin{align} x &\in \mathbb{R} \\ \mu &\in \mathbb{R} \\ \sigma &\in \mathbb{R} \\ \bar{x} &= \frac{x-\mu}{\sigma} \end{align}
 1 2 3 4 5 6 7 8 9 10 val mean = -1.5 val sigma = 2.5 val ugs = UnivariateGaussianScaler(mean, sigma) val x = 3.0 val xs = ugs(x) val xhat = ugs.i(xs)

### Multivariate¶

The data attributes form components of a vector, in this case we can assume each component is independent and calculate the diagonal variance or compute all the component covariances in the form of a symmetric matrix.

\begin{align} x &\in \mathbb{R}^n \\ \mu &\in \mathbb{R}^n \\ \Sigma &\in \mathbb{R}^{n \times n}\\ L L^\intercal &= \Sigma \\ \bar{x} &= L^{-1} (x - \mu) \end{align}

#### Diagonal¶

In this case the sample covariance matrix calculated from the data is diagonal and neglecting the correlations between the attributes.

\Sigma = \begin{pmatrix} \sigma^{2}_1 & \cdots & 0\\ \vdots & \ddots & \vdots\\ 0 & \cdots & \sigma^{2}_n \end{pmatrix}
 1 2 3 4 5 6 7 8 9 10 val mean: DenseVector[Double] = DenseVector(-1.5, 1.5, 0.25) val sigma: DenseVector[Double] = DenseVector(0.5, 2.5, 1.0) val gs = GaussianScaler(mean, sigma) val x: DenseVector[Double] = DenseVector(0.2, -3.5, -1.5) val xs = gs(x) val xhat = gs.i(xs)

#### Full Matrix¶

When the sample covariance matrix is calculated taking into account correlations between data attributes.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 val mean: DenseVector[Double] = DenseVector(-1.5, 1.5, 0.25) val sigma: DenseMatrix[Double] = DenseMatrix( (2.5, 0.5, 0.25), (0.5, 3.5, 1.2), (0.25, 1.2, 2.25) ) val mv_gs = MVGaussianScaler(mean, sigma) val x: DenseVector[Double] = DenseVector(0.2, -3.5, -1.5) val xs = mv_gs(x) val xhat = mv_gs.i(xs)

## Mean Centering¶

### Univariate¶

\begin{align} x &\in \mathbb{R} \\ \mu &\in \mathbb{R} \\ \bar{x} &= x-\mu \end{align}
 1 2 3 4 5 6 7 8 9 val c = -1.5 val ums = UnivariateMeanScaler(c) val x = 3.0 val xs = ums(x) val xhat = ums.i(xs)

### Multivariate¶

\begin{align} x &\in \mathbb{R}^n \\ \mu &\in \mathbb{R}^n \\ \bar{x} &= x - \mu \end{align}
 1 2 3 4 5 6 7 8 9 val mean: DenseVector[Double] = DenseVector(-1.5, 1.5, 0.25) val mms = MeanScaler(mean) val x: DenseVector[Double] = DenseVector(0.2, -3.5, -1.5) val xs = mms(x) val xhat = mms.i(xs)

## Min-Max Scaling¶

Min-max scaling is also known as $0,1$ scaling because attributes are scaled down to the domain $[0, 1]$. This is done by calculating the minimum and maximum of attribute values.

 1 2 3 4 5 6 7 8 9 10 val min: DenseVector[Double] = DenseVector(-1.5, 1.5, 0.25) val max: DenseVector[Double] = DenseVector(0.5, 2.5, 1.0) val min_max_scaler = MinMaxScaler(min, max) val x: DenseVector[Double] = DenseVector(0.2, -3.5, -1.5) val xs = min_max_scaler(x) val xhat = min_max_scaler.i(xs)

## Principal Component Analysis¶

Principal component analysis consists of projecting data onto the eigenvectors of its sample covariance matrix.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 val mean: DenseVector[Double] = DenseVector(-1.5, 1.5, 0.25) val sigma: DenseMatrix[Double] = DenseMatrix( (2.5, 0.5, 0.25), (0.5, 3.5, 1.2), (0.25, 1.2, 2.25) ) val pca = PCAScaler(mean, sigma) val x: DenseVector[Double] = DenseVector(0.2, -3.5, -1.5) val xs = pca(x) val xhat = pca.i(xs)

Slicing scalers

It is possible to slice the scalers shown above if they act on vectors. For example.

 1 2 3 4 //Slice on subset of columns val gs_sub: GaussianScaler = gs(0 to 1) //Slice on a single column val gs_last: UnivariateGaussianScaler = gs(2)