Mahalanobis distance concept
| Definition | The Euclidean distance of a row from the mean after whitening, which needs a positive-definite covariance or a pseudoinverse on its support. Equation 0.33. Also Mahalanobis. |
|---|---|
| Example | With covariance diag(1, 4), the row (3, 4) has Euclidean distance 5 from the mean and Mahalanobis distance the square root of 13, or 3.61. |
| Book | Data Mining as Observation, draft 0.2, commit f3914f0; entry id mahalanobis-distance, kind concept. |
| Status | no ledger row names this entry. Corrections: none recorded. |
| Defining equation | Book equation 0.33. |
| Assumptions and scope |
|
| Prior art | none recorded |
| Evidence | lean/DataMiningAsObservation/Mahalanobis.lean |
| Reviewed | semantic review 2026-09-06; generated 2026-09-10 from records at the commits on the provenance page. |
Equation
Book equation 0.33.
\[d_M(x)=\sqrt{(x-\mu)^{\top}\Sigma^{-1}(x-\mu)}.\]
Book equation 10.1.
\[z(x)=\frac{x-\mu}{\sigma},\qquad d_M(x)^{2}=(x-\mu)^{\top}\Sigma^{-1}(x-\mu)=\sum_i\frac{(v_i\cdot(x-\mu))^{2}}{\lambda_i}.\]
Conditions
- The Euclidean distance of a row from the mean after whitening. In the eigenbasis its square is the sum of the squared centred coordinates over the eigenvalues, nonnegative, zero at the mean, unchanged when data and covariance are rescaled together, and a deviation along a direction of larger variance counts for less.
- The detector that scores by it treats every whitened direction equally, so it is the identity reader on whitened data, and a consumer that reads a subspace calls different rows outliers.
- It needs a positive-definite covariance, or a pseudoinverse restricted to the covariance’s support, and it equals the Euclidean distance after whitening.
Conditions are curated in entries.toml rather than read
from a record.
Ledger
none
First stated
Mahalanobis, on the generalised distance in statistics, 1936, as chapter 0 section 0.16 states it, and chapter 10 section 10.1 of Data Mining as Observation, where the statistical detector is the identity reader on whitened data.
Measurements
none
Failures and corrections
none
Invariance envelope
none declared
Machine checked
lean/DataMiningAsObservation/Mahalanobis.lean,
theorems dM2_nonneg, dM2_mean,
dM2_scale, dM2_eq_whitened,
dM2_antitone_in_variance, at observation-data-mining
f3914f0; what the check covers is stated in the book’s appendix
C.
Used in
Data Mining as Observation primer L, chapters 0, 10.
Related
whitening; identity reader; hubness; abstention.
See also
none
Status
Generated 2026-09-10 by encyclopedia/generate.py; book
at observation-data-mining f3914f0; the commit of every record is listed
in the encyclopedia’s provenance.