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

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

The centers \(c\) and scales \(s\) are either user-provided or deduced from \(Z\) via mean() and var(), respectively, with equal sample weights \(1/n\). The degrees of freedom \(\mathrm{df}\) can be specified for \(s\) so that the sample weights are \(1 / (n-\mathrm{df})\) instead. Therefore, \(c\) will usually coincide with

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

and \(s\) with

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

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 implied column means are used via mean(). Default is None.
scalesndarray, optional: The scale values \(s\) for each column of mat. If None, the implied column standard deviations are used via var(). 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.