# Simultaneous Instance and Attribute Selection for Noise Filtering
## 3 Results
The complexity of the proposed ROFS depends on several factors: (a) the complexity of the method to obtain candidate feature sets (\(CF\)), (b) the complexity of the condensing method (\(D\)), (c) the complexity of the supervised classifier used (\(S\)), and (d) the complexity of the sorting algorithm (\(R\)). Let \(m\) be the number of instances, and let \(f\) be the number of candidate attribute sets. The complexity of phase 1 comprises the computation of the candidate attribute sets and the computation of the Adaptive All-kNN algorithm and, therefore, is bounded by \(O\big{(}CF+m^{2}\big{)}\). Phase 2 comprises the computation of the small pieces, and its complexity is bounded by the execution of the condensing algorithm over each projection as \(O(f*D)\). Finally, the third phase consists of the fusion procedure and is bound by the sorting procedure and the complexity of the supervised classifier used over each small piece, being \(O(R+f*S)\). Finally, the complexity of ROFS is given by \(O\big{(}CF+m^{2}+f*D+R+f*S)\). Figure 2: Second phase of ROFS. For each feature set, it will obtain a noise-free selected set by projecting the filtered instances using only the current attributes, and then it will obtain relevant instances by applying a condensation method, finally obtaining a small piece. In addition, its ability to handle diverse types of data (e.g., numeric, categorical, missing, multiclass) will depend on the method used to obtain candidate feature sets, the condensation method used, and the supervised classifier selected. ```
Inputs: Training set \(T\) Method to compute candidate attribute sets: \(CA\) Method to condense instances: \(Cond\) Supervised classifier: classifier \(classif\) Selected instances and attributes: \(E\) Initialization
1. Obtain a validation set \(V\) from \(T\)
2. Compute the initial classifier accuracy with respect to the validation set as \(init\_acc=classif(V,T)\) Phase 1
3. Apply the method to compute candidate attribute sets, as \(CAS=CA(T)\)
4. Apply the All-KNN method for noise filtering as \(F=AdaAllKNN(T)\) Phase 2
5. \(SP=\varnothing\)
6. For each candidate attribute set \(ca_{i}\in CAS\) 6.1. Project the instances in \(F\) using the features in to obtain an initial small piece \(ca_{i}\) to obtain an initial small piece \(sp_{i}\) 6.2. Apply the condensation method as \(csp_{i}=Cond(sp_{i})\) 6.3. Add the current condensed small piece to \(SP\) as \(SP=SP\cup\{csp_{i}\}\) Phase 3
7. Sort the set \(SP\) according to the classifier accuracy with respect to the validation set \(V\) of each of the small pieces
8. Consider the best small piece as the current solution \(E\)
9. While the accuracy of the current solution is lower than the initial accuracy and there exist small pieces that can improve the accuracy 9.1. Find the instances misclassified by the current solution (_Miss_) 9.2. Find the small pieces \(csp_{i}\in SP\) that correctly classify the misclassified instances (only consider the small pieces that contribute with at least one attribute) 9.3. Consider the candidate integration \(EI=E\cup\{csp_{i}\}\) 9.4. If the fusion of the current solution with one of the found small pieces leads to better accuracy, consider it as the current solution \(E=EI\). 10. Return \(E\)
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**Algorithm 2** Robust Objective Filtering Selection  
The advanced nature of the proposed ROFS is supported by the incorporation of the Adaptive All-kNN algorithm for noise filtering, the computation of candidate attribute sets, and the iterative process to fuse small pieces of instances and features. These novel characteristics make ROFS suitable for obtaining high-quality subsets of attributes and instances. The theoretical justification of the design choices made in ROFS is supported by the following:  
1. The use of candidate attribute sets is inspired by Voting Algorithms [26] and has obtained good results by supporting multiple views of the existing data. 2.