Mahalanobis Distance

Unlike Euclidean distance, the Mahalanobis distance is scale-invariant and accounts for correlations between variables. It standardizes the data using the inverse of the covariance matrix (Σ⁻¹). This means a unit change in one variable is weighted according to its variance and its relationship with other variables. For example, in a robot's state vector where position (meters) and velocity (m/s) are correlated, the distance correctly measures deviations without being skewed by differing units or scales.

The squared Mahalanobis distance follows a chi-squared (χ²) distribution when the data point is drawn from the assumed multivariate normal distribution. This provides a direct statistical test for outlier rejection. In a Kalman filter, the innovation (difference between predicted and actual measurement) is often evaluated using its Mahalanobis distance. If the squared distance exceeds a threshold based on the χ² distribution (e.g., for 3 degrees of freedom, a 95% confidence threshold is ~7.81), the measurement is flagged as an outlier and can be rejected.

Points with an equal Mahalanobis distance from the distribution mean form ellipsoids in the variable space. The shape and orientation of these ellipsoids are defined by the eigenvectors and eigenvalues of the covariance matrix. This geometrically represents regions of equal probability density for a multivariate Gaussian.

• The principal axes of the ellipsoid align with the directions of maximum variance.
• This visualizes why a point can be close in Euclidean terms but far in Mahalanobis terms if it lies in a direction of low variance.

The Mahalanobis distance is fundamental to the Kalman filter update step. The filter's Kalman gain is computed to minimize the posterior error covariance, which is intrinsically linked to this distance metric. The innovation covariance (S = HPHᵀ + R) acts as the covariance matrix for calculating the Mahalanobis distance of the incoming measurement. This ensures the filter update is optimally weighted based on the current state uncertainty (P) and sensor noise (R).

In multi-object tracking and SLAM data association, the Mahalanobis distance is used within the nearest neighbor or global nearest neighbor methods to assign new sensor observations to existing tracks or map landmarks. It compares a predicted measurement (from a track) to a candidate observation. The assignment that minimizes the Mahalanobis distance is statistically the most likely correct match, as it considers the uncertainty in the prediction.

A key property is its sensitivity to the shape of the underlying distribution. It effectively measures distance in terms of standard deviations from the mean along the principal components. This means:

• A small shift in a direction of high variance results in a small Mahalanobis distance.
• The same magnitude shift in a direction of low variance results in a large Mahalanobis distance. This property is critical for detecting anomalies that are unusual given the observed correlation structure, not just large in a raw numerical sense.

A covariance matrix is a square matrix that quantifies the uncertainty of a state vector and the correlations between its components. In state estimation, it represents the estimated error of the system.

• Diagonal elements represent the variance (uncertainty) of each state variable.
• Off-diagonal elements represent the covariance (correlation) between different state variables.
• The Mahalanobis distance calculation directly uses the inverse of the covariance matrix to account for these correlations when measuring distance.
• In a Kalman filter, the covariance matrix is propagated and updated alongside the state estimate.

Outlier rejection is the process of identifying and discarding sensor measurements that are statistically inconsistent with the current state estimate. The Mahalanobis distance is the primary metric for this task.

• A measurement's innovation (difference from prediction) is computed.
• The Mahalanobis distance of this innovation is calculated using the innovation covariance.
• If the squared Mahalanobis distance exceeds a threshold (e.g., based on a chi-squared distribution), the measurement is flagged as an outlier.
• This is critical for maintaining robustness against erroneous data from sensor failures or environmental noise.

A Kalman filter is an optimal recursive algorithm for estimating the state of a linear dynamic system from noisy measurements. The Mahalanobis distance is implicitly used within its update step.

• The filter maintains a state estimate and a covariance matrix representing estimation uncertainty.
• It operates in a predict-update cycle: predict the state forward, then update with a new measurement.
• The Kalman gain determines how much to trust the new measurement versus the prediction.
• The innovation and its covariance are used to compute a form of Mahalanobis distance to assess measurement consistency.

Gating is a technique in multi-target tracking and SLAM used for data association—the problem of determining which sensor observation corresponds to which tracked object or map feature. Mahalanobis distance defines the validation gate.

• A validation region (gate) is established around a predicted measurement.
• The gate is typically an ellipsoid defined by a Mahalanobis distance threshold.
• Only observations falling within this ellipsoid are considered potential matches for the track.
• This efficiently reduces combinatorial complexity by pruning unlikely associations before more expensive matching algorithms run.

The multivariate normal (Gaussian) distribution is a generalization of the one-dimensional normal distribution to multiple, potentially correlated variables. The Mahalanobis distance is the distance measure native to this distribution.

• It is fully characterized by a mean vector (center) and a covariance matrix (spread and orientation).
• Contours of constant probability density are ellipsoids centered at the mean.
• The squared Mahalanobis distance from a point to the distribution's mean is proportional to the negative log-likelihood of that point.
• Most probabilistic state estimators (KF, EKF, UKF) assume state uncertainty is modeled by this distribution.

The chi-squared distribution is a theoretical probability distribution that arises in statistical hypothesis testing. It provides the statistical basis for setting thresholds when using the Mahalanobis distance for outlier detection.

• If a random vector x is distributed according to a multivariate normal distribution, then the squared Mahalanobis distance of x follows a chi-squared distribution.
• The degrees of freedom equal the dimensionality of x.
• A common practice is to reject a measurement as an outlier if its squared Mahalanobis distance exceeds the 95th or 99th percentile of the corresponding chi-squared distribution.
• This provides a statistically principled method for threshold selection.