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### Parametric Methods II: Detection of Multivariate Outliers

- Multivariate data: A data set involving two or more attributes or variables
- Transform the multivariate outlier detection task into a univariate outlier detection problem
- Method 1. Compute Mahalaobis distance
- Let ō be the mean vector for a multivariate data set. Mahalaobis distance for an object o to ō is

\[ MDist(o, \bar{o}) = (o-\bar{o})^{T} S^{(-1)}(o-\bar{o}) \]

where S is the covariance matrix - Use the Grubb's test on this measure to detect outliers
- Method 2. Use χ2 –statistic:

\[ X^{2} = \sum_{i=1}^{n}\frac{(o_{i}-E_{i})^{2}}{E_{i}} \]

- where
*Ei*is the mean of the*i*-dimension among all objects, and n is the dimensionality - If χ2 –statistic is large, then object
*oi*is an outlier

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