adelie.matrix.standardize#

adelie.matrix.standardize(mat: ndarray | MatrixNaiveBase32 | MatrixNaiveBase64, centers: ndarray | None = None, scales: ndarray | None = None, ddof: int = 0, *, n_threads: int = 1)[source]#

Creates a standardized matrix.

Given a matrix \(Z \in \mathbb{R}^{n \times p}\), the standardized matrix \(X \in \mathbb{R}^{n \times p}\) of \(Z\) centered by \(c \in \mathbb{R}^{p}\) and scaled by \(s \in \mathbb{R}^p\) is defined by

\[\begin{align*} X = (Z - \mathbf{1} c^\top) \mathrm{diag}(s)^{-1} \end{align*}\]

We define the column means \(\overline{Z} \in \mathbb{R}^p\) to be

\[\begin{align*} \overline{Z} = \frac{1}{n} Z^\top \mathbf{1} \end{align*}\]

Lastly, we define the column standard deviations \(\hat{\sigma} \in \mathbb{R}^p\) to be

\[\begin{align*} \hat{\sigma}_j = \frac{1}{\sqrt{n - \mathrm{df}}} \|Z_{\cdot j} - c_j \mathbf{1} \|_2 \end{align*}\]

where \(\mathrm{df}\) is the degrees of freedom given by ddof.

Note

This matrix only works for naive method!

Parameters:

matUnion[ndarray, MatrixNaiveBase32, MatrixNaiveBase64]: The underlying matrix \(Z\) to standardize.
centersndarray, optional: The center values \(c\) for each column of mat. If None, the column means \(\overline{Z}\) are used as centers. Default is None.
scalesndarray, optional: The scale values \(s\) for each column of mat. If None, the column standard deviations \(\hat{\sigma}\) are used as scales. Default is None.
ddofint, optional: The degrees of freedom used to compute scales. This is only used if scales is None. Default is 0.
n_threadsint, optional: Number of threads. Default is 1.

Returns:

wrap: Wrapper matrix object.