The stability of feature selection is defined as the robustness of the sets of selected features with respect to small variations in the data on which the feature selection is conducted. To quantify stability, several datasets from the same data generating process can be used. Alternatively, a single dataset can be split into parts by resampling. Either way, all datasets used for feature selection must contain exactly the same features. The feature selection method of interest is applied on all of the datasets and the sets of chosen features are recorded. The stability of the feature selection is assessed based on the sets of chosen features using stability measures.
stabilitySechidis(features, sim.mat, threshold = 0.9, impute.na = NULL)
features 


sim.mat 

threshold 

impute.na 

numeric(1)
Stability value.
The stability measure is defined as
$$1  \frac{\mathop{\mathrm{trace}}(CS)}{\mathop{\mathrm{trace}}(C \Sigma)}$$ with (\(p \times p\))matrices
$$(S)_{ij} = \frac{m}{m1}\left(\frac{h_{ij}}{m}  \frac{h_i}{m} \frac{h_j}{m}\right)$$ and
$$(\Sigma)_{ii} = \frac{q}{mp} \left(1  \frac{q}{mp}\right),$$
$$(\Sigma)_{ij} = \frac{\frac{1}{m} \sum_{i=1}^{m} V_i^2  \frac{q}{m}}{p^2  p}  \frac{q^2}{m^2 p^2}, i \neq j.$$
The matrix \(C\) is created from matrix sim.mat
by setting all values of sim.mat
that are smaller
than threshold
to 0. If you want to \(C\) to be equal to sim.mat
, use threshold = 0
.
This stability measure is not corrected for chance.
Unlike for the other stability measures in this R package, that are not corrected for chance,
for stabilitySechidis
, no correction.for.chance
can be applied.
This is because for stabilitySechidis
, no finite upper bound is known at the moment,
see listStabilityMeasures.
For the definition of all stability measures in this package,
the following notation is used:
Let \(V_1, \ldots, V_m\) denote the sets of chosen features
for the \(m\) datasets, i.e. features
has length \(m\) and
\(V_i\) is a set which contains the \(i\)th entry of features
.
Furthermore, let \(h_j\) denote the number of sets that contain feature
\(X_j\) so that \(h_j\) is the absolute frequency with which feature \(X_j\)
is chosen.
Analogously, let \(h_{ij}\) denote the number of sets that include both \(X_i\) and \(X_j\).
Also, let \(q = \sum_{j=1}^p h_j = \sum_{i=1}^m V_i\) and \(V = \bigcup_{i=1}^m V_i\).
Sechidis K, Papangelou K, Nogueira S, Weatherall J, Brown G (2020). “On the Stability of Feature Selection in the Presence of Feature Correlations.” In Machine Learning and Knowledge Discovery in Databases, 327342. Springer International Publishing. doi: 10.1007/9783030461508_20 .
Bommert A (2020). Integration of Feature Selection Stability in Model Fitting. Ph.D. thesis, TU Dortmund University, Germany. doi: 10.17877/DE290R21906 .
feats = list(1:3, 1:4, 1:5) mat = 0.92 ^ abs(outer(1:10, 1:10, "")) stabilitySechidis(features = feats, sim.mat = mat)#> [1] 0.5322911