Hotelling T2 Outlier

Detect PCA outliers by covariance-adjusted Hotelling T², not Euclidean radius.

Description

Hotelling T² measures a two-component score’s quadratic distance under a supplied covariance model.

Hotelling T² measures a two-component score’s quadratic distance under a supplied covariance model. The maximum is a caller-selected outlier criterion.^[1]

Detect PCA outliers by covariance-adjusted Hotelling T², not Euclidean radius.

When to use Hotelling T2 Outlier

Use this tool to detect PCA outliers by covariance-adjusted Hotelling T², not Euclidean radius.

Prepare the measured profile, formula counts, upstream model outputs or caller-defined reference criteria requested. Keep paired entries aligned and units, sample basis and preprocessing consistent.

How to use Hotelling T2 Outlier and understand its results

Prepare the measured profile, formula counts, upstream model outputs or caller-defined reference criteria requested. Keep paired entries aligned and units, sample basis and preprocessing consistent.
Enter pcOne, pcTwo, meanPcOne, meanPcTwo, covariance, maximumT2 using the field table and calculation rules, then run the tool.
Read tSquared, outlierIndexes. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
pcOne	Pc one	list of number	Required
pcTwo	Pc two	list of number	Required
meanPcOne	Mean pc one	number	Required
meanPcTwo	Mean pc two	number	Required
covariance	Covariance	list of list of number	Required
maximumT2	Maximum t 2	number	Required

Enter nonempty aligned PC1/PC2 lists, finite corresponding centering means, a 2×2 covariance matrix and nonnegative maximum. Require positive diagonal entries and determinant. Return T² values and indexes strictly exceeding the maximum.

T²=(d x²−(b+c)xy+a y²)/(ad−bc)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
tSquared	T squared	list of number
outlierIndexes	Outlier indexes	list of number

json

{
  "input": {
    "pcOne": [
      0,
      2,
      4
    ],
    "pcTwo": [
      0,
      0,
      0
    ],
    "meanPcOne": 0,
    "meanPcTwo": 0,
    "covariance": [
      [
        1,
        0
      ],
      [
        0,
        1
      ]
    ],
    "maximumT2": 4
  },
  "output": {
    "tSquared": [
      0,
      4,
      16
    ],
    "outlierIndexes": [
      2
    ]
  }
}

Limitations and assumptions

Symmetry is not checked, so provide a valid symmetric covariance yourself. No centering, model fit or distribution-based cutoff is estimated.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Matrices are limited to 128 rows and 128 entries per row; other lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Q Residual Gate: Flag samples exceeding a caller-selected Q-residual model-fit limit. For a complementary task, use VIP Threshold Gate: Gate variable-importance scores against a supplied VIP minimum.

References

Hotelling T2 Outlier — tool.io

Similar or alternative tools

Q Residual Gate
Flag samples exceeding a caller-selected Q-residual model-fit limit.
VIP Threshold Gate
Gate variable-importance scores against a supplied VIP minimum.