# Univariate Outlier Detection¶

A univariate outlier is an occurrence of an abnormal value within a single observation point.

## 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 algorithm for univariate outlier detection considers each feature independently. The univariate outlier detection 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 the univariate outlier detection:

$\text{Outlier}(\alpha_n, m_n, \sigma_n) = \{x: \frac {|x - m_n|}{\sigma_n} > g(n, \alpha_n) \}$

where $$m_n$$ and $$\sigma_n$$ are (robust) estimates of the mean and standard deviation computed for a given data set, $$\alpha_n$$ is the confidence coefficient, and $$g(n, \alpha_n)$$ defines the limits of the region and should be adjusted to the number of observations.

## Batch Processing¶

### Algorithm Input¶

The univariate 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.

Input ID

Input

data

Pointer to the $$n \times p$$ numeric table with the data for outlier detection.

Note

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.

Note

The input can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

scatter

Pointer to the $$1 \times p$$ numeric table with the vector of standard deviations.

Note

The input can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

threshold

Pointer to the $$1 \times p$$ numeric table with non-negative numbers that define the outlier region.

Note

The input can be an object of any class derived from NumericTable except PackedSymmetricMatrix and PackedTriangularMatrix.

Note

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 set of $$1.0$$ threshold A set of $$3.0$$

### Algorithm Parameters¶

The univariate outlier detection algorithm has the following parameters:

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, the only method supported by the algorithm.

### Algorithm Output¶

The univariate 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.

Result ID

Result

weights

Pointer to the $$n \times p$$ numeric table of zeros and ones. Zero in the position $$(i, j)$$ indicates an outlier in the $$i$$-th observation of the $$j$$-th feature.

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 PackedSymmetricMatrix, PackedTriangularMatrix, and СSRNumericTable.

## Examples¶

Batch Processing:

Note

There is no support for Java on GPU.

Batch Processing:

Batch Processing: