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Revisiting data complexity metrics based on morphology for overlap and imbalance: snapshot, new overlap number of balls metrics and singular problems prospect

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Abstract

Data Science and Machine Learning have become fundamental assets for companies and research institutions alike. As one of its fields, supervised classification allows for class prediction of new samples, learning from given training data. However, some properties can cause datasets to be problematic to classify. In order to evaluate a dataset a priori, data complexity metrics have been used extensively. They provide information regarding different intrinsic characteristics of the data, which serve to evaluate classifier compatibility and a course of action that improves performance. However, most complexity metrics focus on just one characteristic of the data, which can be insufficient to properly evaluate the dataset towards the classifiers’ performance. In fact, class overlap, a very detrimental feature for the classification process (especially when imbalance among class labels is also present) is hard to assess. This research work focuses on revisiting complexity metrics based on data morphology. In accordance to their nature, the premise is that they provide both good estimates for class overlap, and great correlations with the classification performance. For that purpose, a novel family of metrics has been developed. Being based on ball coverage by classes, they are named after Overlap Number of Balls. Finally, some prospects for the adaptation of the former family of metrics to singular (more complex) problems are discussed.

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Notes

  1. https://github.com/jdpastri/morphology-metrics.

  2. https://github.com/jdpastri/morphology-metrics.

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Acknowledgements

This work has been partially supported by the Spanish Ministry of Economy and Competitiveness under project TIN2017-89517-P, including European Regional Development Funds, and the Andalusian regional project P18-FR-4961. This work is part of the PRII2018-02 Intensification Program from the University of Granada and the FPU National Program (Ref. FPU17/04069).

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  1. Andalusian Institute of Data Science and Computational Intelligence (DASCI), University of Granada, 18071, Granada, Spain

    José Daniel Pascual-Triana, David Charte, Marta Andrés Arroyo, Alberto Fernández & Francisco Herrera

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  1. José Daniel Pascual-Triana
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Description of other complexity metrics

Description of other complexity metrics

1.1 Feature overlap

This group of metrics assesses the capability of features for the discerning of the classes. If there is at least one feature with low overlap of different classes, the classification should be easier and, thus, obtain better results. The same works for combinations of features and areas of the n-dimensional space of each dataset. Five different metrics can be enumerated:

  • F1: this is the Maximum Fisher’s Discriminant Ratio, which measures the ease to separate the classes using the features (columns) of the data. It compares the dispersion inside each class and the dispersion of the classes. Higher values indicate less overlapping features and, thus, a less complex dataset.

  • F1v: this is the Directional Vector Maximum Fisher’s Discriminant Ratio from [63]. It complements F1, looking for the hyperplane, generated by a vector, that best discerns the classes instead of using the separate features. Greater values indicate a lower complexity.

  • F2: this metric estimates the volume of the overlapping region, using the ratio between the value limits of each class for each feature and the full range of said feature, and multiplying them to get the ratio of volume overlap. Greater values indicate more overlap, which increases the complexity.

  • F3: it is the Maximum Individual Feature Efficiency, which is the biggest ratio of points outside the overlapping region of a feature and the total number of points. Greater values indicate less complexity.

  • F4: this is the Collective Feature Efficiency from [63]. It is based on an iterative use of F3 over the dataset, each time choosing the most efficient feature and setting aside the non-overlapped points of that feature, until there are no more points or features. F4 indicates the ratio of points that have been discerned over the total number of points, so greater values of F4 signal lower complexity.

1.2 Linearity

These measures assess the ease of separability of the different classes by hyperplanes, which would lead to easier classification. There are three main metrics in this category:

  • L1: it is a measure of the Sum of the Error Distance by Linear Programming. After using the linear classifier, the total error distance of the misclassified points to their closest hyperplane is computed and divided by the total number of points. The bigger this ratio, the bigger the L1 will be, indicating more complexity.

  • L2: this is the Error Rate of the Linear Classifier, that is, the number of misclassified points divided by the total number of points. The bigger the L2, the more complex a problem will be.

  • L3: it is the Non-Linearity of a Linear Classifier, from [38]. For this metric, new points are generated using pairs of points sharing a class, and they are classified using the initial data as the training set for the generation of the classification model. L3 is the ratio of the misclassified points from those interpolated. A higher value of L3 indicates more complex boundaries and problems.

1.3 Dimensionality

This set of measures reflect the data sparsity that can appear from a high dimensionality. When a dataset has low-density or even void areas, the model might fail to correctly classify new data there. Three metrics stand out:

  • T2: it is the Average Number of Features per Dimension, that is, the number of features of the dataset divided by the number of points. Greater values indicate less points per feature, which will cause sparsity and a higher complexity.

  • T3: it is the Average number of PCA Dimensions per points. This is the division of the dimensionality of the PCA selected attributes over the number of points of the dataset. Greater values signal a higher complexity.

  • T4: this is the Ratio of the PCA Dimension to the Original Dimension, which is the division of the dimensionality of the PCA selected attributes over the original dimensionality. Greater values indicate that more features are necessary to explain the data variability and, usually, higher complexity.

1.4 Class balance

These metrics measure the differences in the number of elements in each class, which could favour the classification of the predominant class. The two most common metrics are:

C1: it is the Entropy of Class Proportions. The higher the value (closer to 1), the most balanced the dataset will be, which usually indicates lower complexity.

C2: this is the Imbalance Ratio, using the multiclass modification in [72]. It has the value “0” for balanced problems, and higher values (up to 1) indicate more imbalance.

1.5 Network properties

These metrics study graph properties of the data, after using the distances between data points to generate it. To this end, each point becomes a node and the instructions in [29, 61] and [77] are followed in order to decide the edges, which only join close points that belong to the same class. The three basic metrics are the following:

  • Density: it is the Average Density of the Network, which is obtained from the ratio between the number of edges of the graph and the maximum possible amount of edges for that graph. The more edges, the lower the complexity.

  • Clustering Coefficient: this metric is derived from the mean of the ratio of edges between each point and its neighbours and the maximum number of edges between them, for every point. It signals the tendency to create cliques. The higher the value, the most complex the dataset.

  • Hubs: this is the Mean Hub Score of the graph. The hub score measures the importance of each node from both its connections and their hub scores, in an iterative way. The higher the value, the most complex the dataset.

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Pascual-Triana, J.D., Charte, D., Andrés Arroyo, M. et al. Revisiting data complexity metrics based on morphology for overlap and imbalance: snapshot, new overlap number of balls metrics and singular problems prospect. Knowl Inf Syst 63, 1961–1989 (2021). https://doi.org/10.1007/s10115-021-01577-1

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