# Towards Understanding Clustering Problems and Algorithms: An Instance Space Analysis
## 4 Building an Instance Space from Artificial Datasets
### Feature Selection and Projection in the Instance Space
The characteristics have been \(log_{10}\)–scaled. Figure 5: Distribution of selected meta–features on the projected instance space. * avg_abs_cor: the values of this meta-feature increase from left to right in the instance space. This meta-feature computes the correlations between all input attributes a dataset has and averages their absolute values. Higher values are obtained for datasets with more redundant input attributes;
* avg_nnd: the values of this meta-feature tend to increase from right to left in the instance space. The degree captures more precisely the effective level of cohesiveness and affinity due to the actual interaction strength between nodes of the proximity graph built from a dataset [44]. A higher degree is expected for datasets with more examples near each other, so that the proximity graph shows more connections;
* clust_coef: the values of this meta-feature increase from bottom-left to the top-right of the instance space. The clustering coefficient of a vertex measures the ratio of the number of edges between its neighbors and the maximum number of edges that could possibly exist between them. A higher value is obtained for datasets with a more dense clustering structure;
* intr_dim: this meta-feature is characterized by having higher values for the G2 datasets. It measures the squared average distance between all points divided by the variance of the distances [45]. G2 datasets have a certain degree of overlap between clusters, justifying the higher values achieved;
* kurtosis: the values of this meta-feature tend to increase smoothly towards the bottom-left of the instance space. It calculates the Mardia's multivariate kurtosis coefficient [43] of the dataset's input attributes. A higher value reflects more complex datasets in terms of data distribution;
* low_dist: the values of this meta-feature tend to increase towards the bottom-right of the instance space. This measure computes the percentage of points with a distance lower than the mean distance between all data points in a dataset, minus its standard deviation. Higher values tend to be obtained for datasets with a greater degree of overlap between clusters;
* perc_out: the values of this meta-feature increase from bottom-right to the top-left of the instance space. In this work, outliers are considered points whose distance is less than \(q_{1}-1.5d\) and higher than \(q_{3}+1.5d\), where \(q_{1}\) and \(q_{3}\) are the first and third quartiles, respectively, and \(d=q_{3}-q_{1}\). Essentially, this is an indicator of data quality that shows the ability of algorithms to deal with datasets that have non-uniform distribution;
* ratio_ex_ftr: the values of this meta-feature increase from bottom-left to the top-right of the instance space. This measure reflects the ratio between the number of examples and the number of attributes. It gives an indication of data sparsity, with lower values obtained for sparse datasets;
* ratio_pca: the values of this meta-feature tend to increase towards the up-right of the instance space. It is a rough measure of the proportion of relevant dimensions the dataset has, when compared to the number of principal components needed to represent 95% of data variability. High values of this measure indicate more complex relationships of the input variables [60];
* skewness: the values of this meta-feature tend to increase sharply towards the bottom-left of the instance space. It calculates the Mardia's multivariate skewness coefficient [43]. A higher value reflects more complex datasets in terms of data distribution.