update_step_gamma

Performs one update step for the <code>gamma_k</code> parameter in an iterative optimization routine, potentially applying L1 shrinkage.

Implements statistical methods for detecting evolutionary shifts
in both the optimal trait value (mean) and evolutionary diffusion variance.
The method uses an L1-penalized optimization framework to identify branches
where shifts occur, and the shift magnitudes. It also supports the
inclusion of measurement error. For more details, see Zhang, Ho, and Kenney (2023) <doi:10.48550/arXiv.2312.17480>.

Wensha Zhang

ShiVa

Detection of Evolutionary Shifts in Both Optimal Value and
Variance

Lam Si Tung Ho

Toby Kenney

update_step_gamma function

<dl><dt>gamma_k</dt>
<dd>Current value of the <code>gamma_k</code> parameter to be updated.</dd>
<dt>X_k</dt>
<dd>The \(k\)th column of the design matrix <code>X</code>.</dd>
<dt>Sigma</dt>
<dd>Current covariance matrix.</dd>
<dt>r</dt>
<dd>Residual vector, typically <code>Y - X %*% beta</code>.</dd>
<dt>lambda2</dt>
<dd>Non-negative tuning parameter controlling the degree of L1 shrinkage applied to <code>gamma_k</code>.</dd>
<dt>t</dt>
<dd>Step size for the gradient update.</dd>
<dt>penalty</dt>
<dd>Penalty type; either <code>"L1"</code> for soft thresholding or <code>"None"</code> for unpenalized updates.</dd>
<dt>V</dt>
<dd>Baseline covariance matrix when \(\sigma^2 = 1\).</dd>
<dt>q_k</dt>
<dd>The \(k\)th element of the design vector <code>q</code>, used in the update.</dd></dl>

Arguments

Update Step for Gamma — update_step_gamma

<dl>

<dt>gamma_k</dt>
<dd>Current value of the <code>gamma_k</code> parameter to be updated.</dd>


<dt>X_k</dt>
<dd>The \(k\)th column of the design matrix <code>X</code>.</dd>


<dt>Sigma</dt>
<dd>Current covariance matrix.</dd>


<dt>r</dt>
<dd>Residual vector, typically <code>Y - X %*% beta</code>.</dd>


<dt>lambda2</dt>
<dd>Non-negative tuning parameter controlling the degree of L1 shrinkage applied to <code>gamma_k</code>.</dd>


<dt>t</dt>
<dd>Step size for the gradient update.</dd>


<dt>penalty</dt>
<dd>Penalty type; either <code>"L1"</code> for soft thresholding or <code>"None"</code> for unpenalized updates.</dd>


<dt>V</dt>
<dd>Baseline covariance matrix when \(\sigma^2 = 1\).</dd>


<dt>q_k</dt>
<dd>The \(k\)th element of the design vector <code>q</code>, used in the update.</dd>

</dl>

update_step_gamma: Update Step for Gamma

Description

Usage

Value

Arguments