Application of the Variable Precision Rough Sets Model to Estimate the Outlier Probability of Each Element

Table 5

Comparative table of RS-based algorithms.

Advantages	Disadvantages
Pawlak rough sets algorithm
(i) Shows the computational viability of the Pawlak rough sets-based detection method.; (ii) Linear temporal and spatial lineal complexity regarding the cardinality of the dataset.	(i) DETERMINISTIC classification.; (ii) The user must define the outlier threshold.
VPRS algorithm
(i) Shows the computational viability of the VPRS-based detection method.; (ii) Linear temporal and spatial lineal complexity regarding the cardinality of the dataset.; (iii) NONDETERMINISTIC classification.	(i) The user must define the outlier threshold and the classification error.; (ii) An inadequate selection of the error may lead to unsatisfactory results. Requires sufficient knowledge of specific aspects of the dataset.
FIND_OUTLIER_REGION algorithm
(i) Shows the computational viability of the βμ Method.; (ii) Maintains the nondeterminism of VPRS.; (iii) Any specific result that could be obtained with the Pawlak rough sets and VPRS algorithms can be determined from the result obtained.; (iv) The obtained region allows us to establish a stochastic approach to solving the problem of determining the outlier probability of a given element from a given dataset.; (v) Its use is especially feasible when needing to determine the outlier condition of the elements of the dataset for a given set of threshold values.	(i) Temporal complexity: in the worst case.; (ii) Spatial complexity: in the worst case.
βμ_PROB algorithm
(i) Shows the computational viability of the method defined.; (ii) Maintains the nondeterminism of VPRS.; (iii) Has the same advantages as the FIND_OUTLIER _REGION algorithm.; (iv) The user does not need to define the outlier threshold, the allowed classification error, or other criteria, such as distance or density.; (v) No specific knowledge of the dataset is required, such as its distribution.; (vi) The result obtained is more general than that obtained with Pawlak rough sets and VPRS.; (vii) Is valid for datasets with mixed types of attributes (continuous and discrete).	(i) Temporal complexity: in the worst case.; (ii) Spatial complexity: in the worst case.