Ell (Ellipsoid Statistics)

Statistical ellipsoid and ellipse representations with lazy-loaded coordinate conversions and Mahalanobis distance calculations.

Overview

gri-ell provides the Ell class, which pairs a geodetic position (inherited from Pos) with a statistical uncertainty region -- an ellipsoid in 3D or ellipse in 2D. Internally, Ell can store covariance matrices, information matrices, or 2D ellipse parameters, and lazily converts between them as needed.

EllVel and EllAcc extend Ell with velocity and acceleration (via diamond inheritance with gri-pos's Vel/Acc), carrying full 6x6 or 9x9 covariance matrices with position-velocity-acceleration cross-terms.

The primary use cases are geolocation error analysis, ellipsoid convolution (via gri-convolve), Kalman filter output, and statistical distance calculations between position estimates.

Requires Python 3.12+.

Mathematical Background

Covariance and information matrices. An Ell object encapsulates two dual representations of the same uncertainty:

• Covariance matrix C (ENU coordinates, m^2, 1-sigma): describes spread of the position estimate
• Information matrix I = R C^{-1} R^T (XYZ coordinates, 1/m^2, 1-sigma): the inverse covariance rotated into ECEF, used in fusion algorithms

where R is the ENU-to-XYZ rotation matrix at the position.

2D ellipse. The 2D ellipse is the East-North projection of the 3D ellipsoid, parameterized by semi-major axis (SMA), semi-minor axis (SMI), orientation angle (degrees clockwise from North), and altitude uncertainty. Ellipse parameters are reported at 95% confidence.

Mahalanobis distance. The statistical distance from a point x to an ellipsoid centered at mu with information matrix I:

d_M = sqrt((x - mu)^T I (x - mu))

This is unitless: 0 at the center, 1 on the ellipsoid surface, 2 at twice the ellipsoid boundary. It generalizes Euclidean distance to account for the shape and orientation of the uncertainty region.

Norm distance. Mahalanobis distance scaled to 95% confidence intervals by dividing by sqrt(SIG_TO_95). After scaling, a norm distance of 1.0 means the point lies on the 95% confidence boundary.

Scaling factors from the chi-squared distribution:

Reference: Mahalanobis, P.C. (1936). "On the generalized distance in statistics."

Installation

pip install gri-ell

For development:

git clone https://gitlab.com/geosol-foss/python/gri-ell.git
cd gri-ell
uv sync

Quick Start

from gri_ell import Ell
from gri_pos import Pos

# Create from 2D ellipse parameters (SMA, SMI, orientation in deg, all 95%)
ell = Ell.from_2d(Pos.LLA(40.0, -105.0, 1600), sma_95_m=100, smi_95_m=50, ori_deg=45)

# Access ellipse properties
print(ell.ellipse.sma_95)   # 100.0 (semi-major axis, meters, 95%)
print(ell.ellipse.smi_95)   # 50.0
print(ell.ellipse.ori_deg)  # 45.0 (degrees clockwise from North)

# Statistical distance to another point
other = Pos.LLA(40.001, -105.001, 1610)
d = ell.dist_norm(other)    # Normalized 95% distance
print(f"Norm distance: {d:.2f}")

Creating Ell Objects

Ell provides several static constructors. Choose based on what data you have:

import numpy as np

# From 2D parameters (most common)
ell = Ell.from_2d(pos, sma_95_m=100, smi_95_m=50, ori_deg=45)

# From a 3x3 covariance matrix (ENU, m^2, 1-sigma)
cov = np.array([[100, 10, 0], [10, 50, 0], [0, 0, 25]])
ell = Ell.COV(pos, cov)

# From an information matrix (XYZ, 1/m^2, 1-sigma)
info = np.linalg.inv(cov_xyz)
ell = Ell.INFO(pos, info)

Matrix Representations

Each Ell stores one representation and lazily computes the others on first access:

The Ellipse object provides the 2D parameters directly:

ell.ellipse.sma_95    # Semi-major axis (meters, 95%)
ell.ellipse.smi_95    # Semi-minor axis (meters, 95%)
ell.ellipse.ori_deg   # Orientation (degrees CW from North)
ell.ellipse.alt_95    # Altitude uncertainty (meters, 95%)
ell.ellipse.ori_rad   # Orientation in radians

# CEP (Circular Error Probable) - radius of circle containing conf% of 2D probability
ell.ellipse.cep_pct()       # CEP-50 (default conf=0.50)
ell.ellipse.cep_pct(0.95)   # CEP-95
ell.ellipse.cep_sigma()     # CEP at 1-sigma (default sigma=1.0)
ell.ellipse.cep_sigma(2.0)  # CEP at 2-sigma

Statistical Distances

All distance methods return unitless values: 0 at center, 1 on the boundary, >1 outside.

# 3D Mahalanobis distance (sigma units)
d = ell.dist_mahalanobis(other_pos)

# 3D normalized distance (95% confidence units)
d = ell.dist_norm(other_pos)

# 2D variants (East-North projection only, faster but less precise)
d = ell.dist_mahalanobis_2d(other_pos)
d = ell.dist_norm_2d(other_pos)

# Combined distance: accounts for uncertainty of both ellipsoids
d = ell.dist_norm(other_ell, combined=True)

# SEP (Spherical Error Probable) - radius of sphere containing conf% of 3D probability
ell.sep_pct()       # SEP-50 (default conf=0.50)
ell.sep_pct(0.95)   # SEP-95
ell.sep_sigma()     # SEP at 1-sigma (default sigma=1.0)
ell.sep_sigma(2.0)  # SEP at 2-sigma

Dependencies

• gri-memoize: Per-instance caching of lazy-loaded matrix conversions
• gri-pos: Position class (Pos) that Ell inherits from
• gri-utils: Coordinate conversions, constants, ellipsoid math
• numpy: Matrix operations

Other Projects

Current list of other GRI FOSS Projects we are building and maintaining.

License

MIT License. See LICENSE for details.