# Multivariate Outlier Detection¶

In multivariate outlier detection methods, the observation point is the entire feature vector.

## Details¶

Given a set $$X$$ of $$n$$ feature vectors $$x_1 = (x_{11}, \ldots, x_{1p}), \ldots, x_n = (x_{n1}, \ldots, x_{np})$$ of dimension $$p$$, the problem is to identify the vectors that do not belong to the underlying distribution (see [Ben2005] for exact definitions of an outlier).

The multivariate outlier detection method takes into account dependencies between features. This method can be parametric, assumes a known underlying distribution for the data set, and defines an outlier region such that if an observation belongs to the region, it is marked as an outlier. Definition of the outlier region is connected to the assumed underlying data distribution.

The following is an example of an outlier region for multivariate outlier detection:

$\text{Outlier}(\alpha_n, M_n, \Sigma_n) = \{x: \sqrt{(x - M_n) \sum _{n}{-1} (x - M_n)} > g(n, \alpha_n) \}$

where $$M_n$$ and Sigma_n are (robust) estimates of the vector of means and variance-covariance matrix computed for a given data set, $$\alpha_n$$ is the confidence coefficient, and $$g(n, \alpha_n)$$ defines the limit of the region.

## Batch Processing¶

### Algorithm Input¶

The multivariate outlier detection algorithm accepts the input described below. Pass the Input ID as a parameter to the methods that provide input for your algorithm. For more details, see Algorithms.

Algorithm Input for Multivariate Outlier Detection (Batch Processing)

Input ID

Input

data

Pointer to the $$n \times p$$ numeric table with the data for outlier detection. The input can be an object of any class derived from the NumericTable class.

location

Pointer to the $$1 \times p$$ numeric table with the vector of means. The input can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

scatter

Pointer to the $$p \times p$$ numeric table that contains the variance-covariance matrix. The input can be an object of any class derived from NumericTable except PackedTriangularMatrix.

threshold

Pointer to the $$1 \times 1$$ numeric table with the non-negative number that defines the outlier region. The input can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

If you do not provide at least one of the location, scatter, threshold inputs, the library will initialize all of them with the following default values:

 location A set of $$0.0$$ scatter A numeric table with diagonal elements equal to $$1.0$$ and non-diagonal elements equal to $$0.0$$ threshold $$3.0$$

### Algorithm Parameters¶

The multivariate outlier detection algorithm has the following parameters:

Algorithm Parameters for Multivariate Outlier Detection (Batch Processing)

Parameter

Default Value

Description

algorithmFPType

float

The floating-point type that the algorithm uses for intermediate computations. Can be float or double.

method

defaultDense

Performance-oriented computation method.

### Algorithm Output¶

The multivariate outlier detection algorithm calculates the result described below. Pass the Result ID as a parameter to the methods that access the results of your algorithm. For more details, see Algorithms.

Algorithm Output for Multivariate Outlier Detection (Batch Processing)

Result ID

Result

weights

Pointer to the $$n \times 1$$ numeric table of zeros and ones. Zero in the $$i$$-th position indicates that the $$i$$-th feature vector is an outlier.

Note

By default, the result is an object of the HomogenNumericTable class, but you can define the result as an object of any class derived from NumericTable except the PackedSymmetricMatrix, PackedTriangularMatrix, and CSRNumericTable.

## Examples¶

Batch Processing:

## Performance Considerations¶

• If input data is homogeneous, provide input data and store results in homogeneous numeric tables of the same type as specified in the algorithmFPType class template parameter.
• For the default outlier detection method (defaultDense), you can benefit from splitting the input data set into blocks for parallel processing.