R/stability_functions_adjusted.R
stabilityIntersectionMBM.Rd
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.
stabilityIntersectionMBM( features, sim.mat, threshold = 0.9, correction.for.chance = "estimate", N = 10000, impute.na = NULL )
features 


sim.mat 

threshold 

correction.for.chance 

N 

impute.na 

numeric(1)
Stability value.
The stability measure is defined as (see Notation) $$\frac{2}{m(m1)}\sum_{i=1}^{m1} \sum_{j=i+1}^{m} \frac{I(V_i, V_j)  E(I(V_i, V_j))}{\sqrt{V_i \cdot V_j}  E(I(V_i, V_j))}$$ with $$I(V_i, V_j) = V_i \cap V_j + \mathop{\mathrm{MBM}}(V_i \setminus V_j, V_j \backslash V_i).$$ \(\mathop{\mathrm{MBM}}(V_i \setminus V_j, V_j \backslash V_i)\) denotes the size of the maximum bipartite matching based on the graph whose vertices are the features of \(V_i \setminus V_j\) on the one side and the features of \(V_j \backslash V_i\) on the other side. Vertices x and y are connected if and only if \(\mathrm{Similarity}(x, y) \geq \mathrm{threshold}.\) Requires the package igraph.
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 \(V = \bigcup_{i=1}^m V_i\).
Bommert A, Rahnenführer J (2020). “Adjusted Measures for Feature Selection Stability for Data Sets with Similar Features.” In Machine Learning, Optimization, and Data Science, 203214. doi: 10.1007/9783030645830_19 .
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, "")) stabilityIntersectionMBM(features = feats, sim.mat = mat, N = 1000)#>#> [1] 0.5646009