Stability Measure Zucknick

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.

stabilityZucknick(
  features,
  sim.mat,
  threshold = 0.9,
  correction.for.chance = "none",
  N = 10000,
  impute.na = NULL
)

Arguments

features: list (length >= 2)
Chosen features per dataset. Each element of the list contains the features for one dataset. The features must be given by their names (character) or indices (integerish).
sim.mat: numeric matrix
Similarity matrix which contains the similarity structure of all features based on all datasets. The similarity values must be in the range of [0, 1] where 0 indicates very low similarity and 1 indicates very high similarity. If the list elements of features are integerish vectors, then the feature numbering must correspond to the ordering of sim.mat. If the list elements of features are character vectors, then sim.mat must be named and the names of sim.mat must correspond to the entries in features.
threshold: numeric(1)
Threshold for indicating which features are similar and which are not. Two features are considered as similar, if and only if the corresponding entry of sim.mat is greater than or equal to threshold.
correction.for.chance: character(1)
Should a correction for chance be applied? Correction for chance means that if features are chosen at random, the expected value must be independent of the number of chosen features. To correct for chance, the original score is transformed by $(score - expected) / (maximum - expected)$. For stability measures whose score is the average value of pairwise scores, this transformation is done for all components individually. Options are "none", "estimate" and "exact". For "none", no correction is performed, i.e. the original score is used. For "estimate", N random feature sets of the same sizes as the input feature sets (features) are generated. For "exact", all possible combinations of feature sets of the same sizes as the input feature sets are used. Computation is only feasible for very small numbers of features (p) and numbers of considered datasets (length(features)).
N: numeric(1)
Number of random feature sets to consider. Only relevant if correction.for.chance is set to "estimate".
impute.na: numeric(1)
In some scenarios, the stability cannot be assessed based on all feature sets. E.g. if some of the feature sets are empty, the respective pairwise comparisons yield NA as result. With which value should these missing values be imputed? NULL means no imputation.

Value

numeric(1) Stability value.

Details

The stability measure is defined as $$\frac{2}{m(m-1)}\sum_{i=1}^{m-1} \sum_{j=i+1}^{m} \frac{|V_i \cap V_j| + C(V_i, V_j) + C(V_j, V_i)}{|V_i \cup V_j|}$$ with $$C(V_k, V_l) = \frac{1}{|V_l|} \sum_{(x, y) \in V_k \times (V_l \setminus V_k) \ \mathrm{with Similarity}(x,y) \geq \mathrm{threshold}} \mathop{\mathrm{Similarity}}(x,y).$$ Note that this definition slightly differs from its original in order to make it suitable for arbitrary similarity measures.

Notation

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 $V = \bigcup_{i=1}^m V_i$.

References

Zucknick M, Richardson S, Stronach EA (2008). “Comparing the Characteristics of Gene Expression Profiles Derived by Univariate and Multivariate Classification Methods.” Statistical Applications in Genetics and Molecular Biology, 7(1). doi:10.2202/1544-6115.1307 .

Bommert A, Rahnenführer J, Lang M (2017). “A Multicriteria Approach to Find Predictive and Sparse Models with Stable Feature Selection for High-Dimensional Data.” Computational and Mathematical Methods in Medicine, 2017, 1--18. doi:10.1155/2017/7907163 .

Bommert A (2020). Integration of Feature Selection Stability in Model Fitting. Ph.D. thesis, TU Dortmund University, Germany. doi:10.17877/DE290R-21906 .

Examples