KPCRKHS

Compute estimate of Kernel partial correlation (KPC) coefficient using conditional mean embeddings in the reproducing kernel Hilbert spaces (RKHS).

Implementations of two empirical versions the kernel partial correlation (KPC) coefficient
and the associated variable selection algorithms. KPC is a measure of the strength of conditional
association between Y and Z given X, with X, Y, Z being random variables taking values in
general topological spaces. As the name suggests, KPC is defined in terms of kernels on
reproducing kernel Hilbert spaces (RKHSs). The population KPC is a deterministic number
between 0 and 1; it is 0 if and only if Y is conditionally independent of Z given X, and it is 1 if
and only if Y is a measurable function of Z and X. One empirical KPC estimator is based on
geometric graphs, such as K-nearest neighbor graphs and minimum spanning trees, and is
consistent under very weak conditions. The other empirical estimator, defined using conditional
mean embeddings (CMEs) as used in the RKHS literature, is also consistent under suitable
conditions. Using KPC, a stepwise forward variable selection algorithm KFOCI (using the graph
based estimator of KPC) is provided, as well as a similar stepwise forward selection algorithm
based on the RKHS based estimator. For more details on KPC, its empirical estimators and its
application on variable selection, see Huang, Z., N. Deb, and B. Sen (2022). “Kernel partial
correlation coefficient – a measure of conditional dependence” (URL listed below). When X is
empty, KPC measures the unconditional dependence between Y and Z, which has been described
in Deb, N., P. Ghosal, and B. Sen (2020), “Measuring association on topological spaces using
kernels and geometric graphs” <arXiv:2010.01768>, and it is implemented in the functions
KMAc() and Klin() in this package. The latter can be computed in near linear time.

Zhen Huang

Kernel Partial Correlation Coefficient

Nabarun Deb

Bodhisattva Sen

KPCRKHS function

<dl><dt>Y</dt>
<dd>a matrix (n by dy)</dd>
<dt>X</dt>
<dd>a matrix (n by dx) or <code>NULL</code> if \(X\) is empty</dd>
<dt>Z</dt>
<dd>a matrix (n by dz)</dd>
<dt>ky</dt>
<dd>a function \(k(y, y')\) of class <code>kernel</code>. It can be the kernel implemented in <code>kernlab</code> e.g., Gaussian kernel: <code>rbfdot(sigma = 1)</code>, linear kernel: <code>vanilladot()</code>.</dd>
<dt>kx</dt>
<dd>the kernel function for \(X\)</dd>
<dt>kxz</dt>
<dd>the kernel function for \((X, Z)\) or for \(Z\) if \(X\) is empty</dd>
<dt>eps</dt>
<dd>a small positive regularization parameter for inverting the empirical cross-covariance operator</dd>
<dt>appro</dt>
<dd>whether to use incomplete Cholesky decomposition for approximation</dd>
<dt>tol</dt>
<dd>tolerance used for incomplete Cholesky decomposition (implemented by the function <code>inchol</code> in the package <code>kernlab</code>)</dd></dl>

Arguments

Kernel partial correlation with RKHS method — KPCRKHS

<dl>

<dt>Y</dt>
<dd>a matrix (n by dy)</dd>


<dt>X</dt>
<dd>a matrix (n by dx) or <code>NULL</code> if \(X\) is empty</dd>


<dt>Z</dt>
<dd>a matrix (n by dz)</dd>


<dt>ky</dt>
<dd>a function \(k(y, y')\) of class <code>kernel</code>. It can be the kernel implemented in <code>kernlab</code> e.g., Gaussian kernel: <code>rbfdot(sigma = 1)</code>, linear kernel: <code>vanilladot()</code>.</dd>


<dt>kx</dt>
<dd>the kernel function for \(X\)</dd>


<dt>kxz</dt>
<dd>the kernel function for \((X, Z)\) or for \(Z\) if \(X\) is empty</dd>


<dt>eps</dt>
<dd>a small positive regularization parameter for inverting the empirical cross-covariance operator</dd>


<dt>appro</dt>
<dd>whether to use incomplete Cholesky decomposition for approximation</dd>


<dt>tol</dt>
<dd>tolerance used for incomplete Cholesky decomposition (implemented by the function <code>inchol</code> in the package <code>kernlab</code>)</dd>

</dl>

KPCRKHS: Kernel partial correlation with RKHS method

Description

Usage

Value

Arguments

Details

See Also

Examples