Homework 9: Answer The Following Questions, 10 Points Each

Homework 9answer The Following Questions 10 Point Each

1. Consider the following definition of an anomaly: An anomaly is an object that is unusually influential in the creation of a data model. a. Compare this definition to that of the standard model-based definition of an anomaly. b. For what sizes of data sets (small, medium, or large) is this definition appropriate?

2. In one approach to anomaly detection, objects are represented as points in a multidimensional space, and the points are grouped into successive shells, where each shell represents a layer around a grouping of points, such as a convex hull. An object is an anomaly if it lies in one of the outer shells. a. To which of the definitions of an anomaly in Section 9.2 is this definition most closely related? b. Name two problems with this definition of an anomaly.

3. Consider the (relative distance) K-means scheme for outlier detection described in Section 9.5 and the accompanying figure, Figure 9.10. a. The points at the bottom of the compact cluster shown in Figure 9.10 have a somewhat higher outlier score than those points at the top of the compact cluster. Why? b. Suppose that we choose the number of clusters to be much larger, e.g., 10. Would the proposed technique still be effective in finding the most extreme outlier at the top of the figure? Why or why not? c. The use of relative distance adjusts for differences in density. Give an example of where such an approach might lead to the wrong conclusion.

4. Compare the following two measures of the extent to which an object belongs to a cluster: (1) distance of an object from the centroid of its closest cluster and (2) the silhouette coefficient described in Section 7.5.2.

5. Consider a set of points that are uniformly distributed on the interval [0,1]. Is the statistical notion of an outlier as an infrequently observed value meaningful for this data?

Paper For Above instruction

The questions presented delve into core concepts of anomaly detection and clustering analysis, with an emphasis on understanding different definitions, methodologies, and their implications. Addressing these questions requires a foundational grasp of how anomalies are characterized and detected in datasets of varying sizes and structures.

Comparison of Anomaly Definitions

The initial definition—an anomaly as an object that is unusually influential in the creation of a data model—differs from the standard model-based definition, which characteristically regards an anomaly as an outlier that deviates significantly from the majority of data points according to some statistical or distance-based measure (Chandola, Banerjee, & Kumar, 2009). The standard model-based definition focuses on the inherent properties of data points relative to a model fitted to the data, often emphasizing outliers as points with low probability under a specified distribution or those distant from cluster centers. In contrast, the influence-based definition emphasizes the structural impact of data points on the model, highlighting objects that shape the model disproportionately. This positional influence could be particularly relevant in models sensitive to certain features or in scenarios where the contribution to model parameters is critical (Aggarwal, 2017).

Regarding data set size appropriateness, this influence-centric definition tends to be more suitable for medium to large datasets where the impact of individual points on the overall model becomes significant and detectable. In small datasets, the influence of any single object is often too limited to markedly alter the model, making the definition less practical for very small samples where each data point might not exert disproportionate influence.

Density-Based Shell Approach to Anomaly Detection

The approach of grouping points into shells around a cluster, with points in the outer shells considered anomalies, is most closely aligned with the density-based definition of anomalies, which identify points residing in low-density regions of the feature space (Ester et al., 1996). This aligns with the conceptualization that outliers are sparse or isolated points relative to the main data mass, and outer shells signify points with relatively few neighbors.

However, two notable problems emerge with this shell-based approach. First, the chosen method for defining shells, such as convex hulls, can be sensitive to the shape of the data distribution; non-convex clusters might be improperly represented, leading to incorrect identification of normal points as anomalies or vice versa (Breunig et al., 2000). Second, the approach can be overly sensitive to the parameters used for shell construction, such as the number of shells or the distance thresholds, which might require extensive tuning and may not generalize well across different datasets or distributions (Markou & Singh, 2003).

K-Means Based Outlier Detection and Density Adjustment

The relative distance K-means scheme attempts to identify outliers by evaluating how far an object is from its cluster centroid relative to local density. In Figure 9.10, points at the bottom may have higher outlier scores due to being more isolated or located in less dense regions, which increases their relative distance despite their proximity to the cluster center. Conversely, points at the top might be part of a dense core, resulting in lower outlier scores even if they are distant from the centroid (Guha et al., 1999).

If the number of clusters increases (e.g., to ten), the effectiveness of this method may diminish in identifying the most extreme outliers. This is because with many clusters, the definition of an outlier becomes more localized, and points that are globally distant may be assigned to separate, small clusters, potentially complicating their detection as extreme outliers, especially if the density considerations are not appropriately tuned. The technique might also mistakenly classify points in sparse regions as outliers, which may be misleading under some circumstances (Hodge & Subramaniam, 2004).

Using relative distance to adjust for density can sometimes produce erroneous conclusions, especially in datasets with complex structures, multiple density modes, or where outliers are embedded within high-density regions. For example, in a dataset with a dense cluster surrounded by a few isolated points, the isolated points might be falsely classified as outliers or vice versa, depending on the local density assessment (Schubert et al., 2017).

Cluster Membership Measures

The distance of an object from the centroid of its closest cluster offers a straightforward measure of cluster membership, emphasizing how centrally located an object is within a cluster. Conversely, the silhouette coefficient provides a more comprehensive measure by considering both the cohesion within a cluster and the separation from other clusters (Rousseeuw, 1987). It quantifies how similar an object is to its cluster compared to other clusters, accommodating both the compactness and distinctiveness, thereby offering a richer assessment of cluster assignment quality.

Outliers in Uniform Distributions

For points uniformly distributed on the interval [0,1], the concept of an outlier based on the statistical notion of infrequent observation becomes less meaningful because, under a uniform distribution, all points are equally likely. Outliers, as defined statistically, are typically points with very low probability density, but a uniform distribution assigns equal probability to all points within its range, rendering the statistical notion of outliers less applicable or trivial (Barnett & Lewis, 1994). In such cases, outlier detection methods relying solely on statistical rarity may not identify any points as outliers, highlighting the importance of contextual or domain-specific considerations in defining outliers.

Conclusion

Overall, understanding the nuances of different anomaly and outlier detection paradigms allows for more effective data analysis tailored to specific data characteristics and analytical goals. Whether examining influence-based anomalies, density-based shell approaches, or distance metrics, the challenges of parameter tuning, dataset complexity, and appropriate definitions remain central considerations for practitioners.

References

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• Ester, M., Kriegel, H. P., Sander, J., & Xu, X. (1996). A Density-Based Algorithm for Discovering Clusters in Large Spatial Databases with Noise. KDD.
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• Schubert, E., Sander, J., Ester, M., Kriegel, H. P., & Xu, X. (2017). Local Outlier Detection — A Survey. Data Mining and Knowledge Discovery, 32(3), 393–438.