Graphical drawbacks; statistical approaches

Graphical approach and its drawbacks

In the graphical approach, data are drawn as box plots (outliers beyond 1.5 × IQR), scatter plots, histograms or Q-Q plots, and a person visually identifies the points that lie apart from the rest.

Drawbacks

• Subjective: depends on the person's judgment; no clear formal definition of an outlier.
• Not scalable: impractical for large data sets, and not automatic.
• Limited to low dimensions (1-3 attributes), where we can plot.
• Not usable for real-time/streaming detection, and cannot give a numeric degree of outlierness.
• Different people may give different results.

Statistical approaches

These assume a statistical model (distribution) for the normal data. An object that has a low probability under the model, i.e. does not fit it, is an outlier.

1. Parametric methods
   • Univariate normal distribution: compute mean μ and standard deviation σ; objects with |x − μ|/σ > 3 (z-score) lie outside μ ± 3σ and are outliers (about 0.3% of normal data).
   • Multivariate: use the Mahalanobis distance or χ² statistic to the mean; large distance means an outlier.
   • Mixture of distributions: normal data is from distribution F_N, anomalies from a different distribution F_A; the object is assigned to the more probable one (likelihood approach).
   • Grubbs' test for one outlier at a time.
2. Non-parametric methods: no assumed distribution; the model is learnt from the data, e.g. a histogram or kernel density estimate. An object falling in a bin with a very low frequency is an outlier.

Strengths: well founded in theory, give a confidence level; efficient when the distribution is known. Weaknesses: most tests are for a single attribute; the data distribution is often not known; poor for high dimensions.