MAHALANOBIS DISTANCE (Mahalanobis Distance)

Mahalanobis distance is a distance measure that, in measuring the difference between two numerical vectors, also takes into account how the criteria are related (correlated) to one another. Its input is two vectors of equal length and a covariance matrix summarising the relationship between criteria; its output is a single number. If the criteria are completely independent of one another (if the covariance matrix is the identity matrix), Mahalanobis distance equals Euclidean distance; its real difference appears when the criteria are correlated. It is a building block feeding the "how far is the alternative from the ideal" step in correlation-sensitive versions of ranking methods such as Hellwig.

Prasanta Chandra Mahalanobis introduced it to statistics in 1936, to compare skull measurements across different population groups in India; his question was "how unusual is this group of measurements relative to the usual pattern formed by the other measurements." This question works directly in decision analysis when there is a strong relationship between criteria (such as "large-budget projects generally take longer").

Mahalanobis distance is built around one question: does the raw gap between two points fit the criteria's usual pattern of moving together, or does it break that pattern? If two criteria generally rise and fall together (positive correlation), an alternative that deviates in the same direction on both is counted as "usual" and treated as close; an alternative that deviates on only one but not the other is counted as "unusual" and treated as more distant, even when the raw gaps are identical.

This reveals information that Euclidean distance cannot see. Euclidean distance treats every criterion as independent and of equal importance; if two alternatives have the same total deviation, they are equidistant, regardless of whether that deviation fits the criteria's usual relationship. Mahalanobis distance asks exactly this, and can sometimes turn an exact tie under Euclidean distance into a clear difference. If the decision-maker's question is "the criteria are independent, each matters on its own," Euclidean distance suffices; if it is "the criteria are related, and a deviation that runs against this relationship should be treated as riskier," Mahalanobis distance is required.

The calculation consists of two steps.

First, establishing the relationship (covariance). The relationship between criteria is summarised in a covariance matrix; this matrix is either computed from existing observations or set from expert judgement (knowledge of which criteria move together). If the criteria are independent of one another, this matrix is the identity matrix (zero off the diagonal), and Mahalanobis distance reduces to Euclidean distance.

Second, the relationship-weighted difference. The raw difference between the two vectors is taken, but this difference is not summed directly; it is weighted by the inverse of the covariance matrix. This weighting shrinks deviations that fit the correlation and inflates deviations that go against it. The square root of the result is the Mahalanobis distance.

If the number of observations is smaller than the number of criteria (for instance, when only two alternatives are being compared), the covariance matrix can be singular (non-invertible); in that case a generalised inverse (Moore-Penrose) is used, and this requires careful interpretation of the results. The formula behind these steps and its implementation in DecisionMind are given on the method page; this card carries no formulas.

Mahalanobis distance is a deviation magnitude weighted by correlation; it is not in the same unit as raw Euclidean distance and cannot be compared directly with it. A value of zero means the two vectors are identical. When the covariance matrix is the identity matrix (when the criteria are independent), Mahalanobis and Euclidean distance are numerically equal; this does not mean the two measures say the same thing, but that the relationship happens to be switched off in this particular case.

Mahalanobis distance's most important property is that it can separate two alternatives with the same raw (Euclidean) distance: one that deviates in a way that fits the criteria's usual relationship gets a small Mahalanobis value, while one that deviates in a way that breaks that relationship gets a large one. This distinction depends on the accuracy of the covariance matrix; a wrongly or poorly estimated covariance matrix produces a wrong distinction.

For this reason, instead of writing:

"By Euclidean distance, these two alternatives are equally far from the ideal"

the report should read:

"Although their raw deviations are equal, once the known relationship between the criteria is taken into account, these two alternatives are not equally distant; one fits the usual pattern while the other breaks it"

Mahalanobis distance works with crisp numerical data. DM3 carries only the base Mahalanobis member under this name; there is no separate extension. You need two numerical vectors of equal length and a covariance matrix showing the relationship between the criteria; this matrix is either computed from a sufficient number of past observations or set from expert judgement. If the number of observations is smaller than the number of criteria, the reliability of the covariance matrix falls and a generalised inverse may be needed; this must be stated in the report. A minimum of two criteria is required; as the number of criteria grows, a reliable covariance matrix requires more observations. Mahalanobis does not produce weights and does not take weights; its input is not criterion weights but the relationship between criteria.

Mahalanobis is a suitable choice if there is a known or measurable strong relationship between your criteria (two cost items generally rising together, or a performance measure and a risk measure moving in opposite directions), and you consider this relationship important to the decision. Usability increases when there are enough reliable observations to compute the covariance matrix meaningfully.

It should not be used where the criteria are genuinely independent of one another, or where the relationship cannot be reliably estimated (too few observations, a singular covariance matrix); in that case Euclidean distance is both simpler and more reliable. Where criteria are binary or categorical, Hamming distance should be used rather than Mahalanobis. Where robustness against outliers is the priority, Manhattan distance is more suitable.

A known, strong relationship between criteria that should enter the decision → Mahalanobis

Criteria are independent, or the relationship cannot be reliably estimated → Euclidean

Criteria are binary/categorical → Hamming

Robustness to outliers is the priority → Manhattan

The relationship is disregarded, only a single p setting is wanted → Minkowski

Mahalanobis distance's principal strength is that it is the only widely used distance measure that does not disregard the relationship between criteria; none of Euclidean, Manhattan or Minkowski distance does this. It is also naturally insensitive to differences in scale, because the covariance matrix already accounts for each criterion's own variability; it is thereby less dependent on a separate prior normalisation step. On correlated criteria it can reveal a distinction that Euclidean distance conceals (as shown in the Aviation case); this is why it is widely used in statistics for outlier detection (De Maesschalck, Jouan-Rimbaud and Massart, 2000).

Its limitations depend on the quality of its input, the covariance matrix. First, reliably computing the covariance matrix requires more observations than criteria; with too few observations the matrix is singular (non-invertible) and a generalised inverse must be used, which makes the result harder to interpret. Second, if the covariance matrix has been computed from incorrect or outdated data (if the relationship has changed over time), Mahalanobis distance produces a wrong distinction; the measure is only as good as its input. Third, its calculation and meaning are more abstract than Euclidean or Manhattan distance; explaining the idea of a "correlation-weighted difference" to a decision-maker is harder than explaining "total difference." Fourth, where DM3 compares only two alternatives (as in the J.expected example), a sample covariance cannot be computed; in that case the covariance must be supplied from outside, either from expert judgement or from past data (Roszkowska, 2024).

The most common mistake is computing the covariance matrix from a small number of observations and using it without questioning its reliability; a relationship drawn from a handful of observations may not be a genuine pattern. A second mistake is applying Mahalanobis where there is no relationship between the criteria at all (independent criteria) and expecting a result different from Euclidean; for independent criteria the two measures are already equal. A third mistake is computing the covariance matrix once and using it for years without ever updating it; the relationship between criteria can change over time. A fourth mistake is presenting a result from a singular covariance matrix with the same reliability as other analyses, without reporting that a generalised inverse was used.

The governing principle is this:

Mahalanobis distance adds something only when the relationship between criteria is reliably known; if the relationship is unknown or unreliable, Euclidean distance is both simpler and more honest.