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Non-Euclidean statistics for covariance matrices, with applications to diffusion tensor imaging
Ian L. Dryden, Alexey Koloydenko, Diwei Zhou
TL;DR
Covariance-matrix analysis requires methods that respect the non-Euclidean geometry of positive semi-definite symmetric matrices, especially for diffusion tensor imaging. The paper develops Procrustes size-and-shape methods for mean estimation, compares them with alternative decompositions and metrics, and reports applications including a Procrustes anisotropy measure with improved contrast in one example.
Problem
The paper addresses how to estimate and analyze covariance matrices while accounting for their non-Euclidean geometry, motivated particularly by diffusion tensor imaging.
Method
The paper develops Fréchet-mean and Procrustes size-and-shape approaches, compares them with matrix-logarithm, matrix-square-root, Cholesky, and other metric-based methods.
Results
In the diffusion-tensor example, Procrustes Anisotropy offers more contrast than Fractional Anisotropy in a highly anisotropic corpus-callosum region and is judged slightly preferable.
Takeaways & Limitations
The Procrustes approach is particularly suited to covariance matrices close to deficient rank and connects covariance-matrix methodology with statistical shape analysis.
Takeaways & Limitations
The Fréchet mean need not be unique in general, although uniqueness is guaranteed under stated support and curvature conditions.
Abstract
from arXiv · showhide
The statistical analysis of covariance matrix data is considered and, in particular, methodology is discussed which takes into account the non-Euclidean nature of the space of positive semi-definite symmetric matrices. The main motivation for the work is the analysis of diffusion tensors in medical image analysis. The primary focus is on estimation of a mean covariance matrix and, in particular, on the use of Procrustes size-and-shape space. Comparisons are made with other estimation techniques, including using the matrix logarithm, matrix square root and Cholesky decomposition. Applications to diffusion tensor imaging are considered and, in particular, a new measure of fractional anisotropy called Procrustes Anisotropy is discussed.
1. Introduction.
The paper addresses statistical analysis of covariance matrices when their positive semi-definite symmetric matrix space is non-Euclidean. It develops and compares mean-estimation approaches, emphasizing Procrustes size-and-shape space and applications to diffusion tensor imaging.
- Covariance-matrix data arise in applications including diffusion tensor imaging and longitudinal data analysis, motivating estimation and inference for population covariance matrices.
- With sample covariance matrices, researchers may estimate an average matrix, interpolate between matrices, or test equality of group mean matrices.
- The usual Wishart-based maximum likelihood estimator is the arithmetic mean, but non-Euclidean geometry motivates alternative distances and Fréchet means.
- The paper introduces covariance-matrix analysis in Kendall’s size-and-shape space, including distances, geodesics, Fréchet means, tangent spaces, Procrustes estimators, and consistency properties.
- It compares metric choices, studies anisotropy and deficient-rank cases, and applies the methods to diffusion tensor images and simulation data.
2. Diffusion tensor imaging.
Diffusion tensor imaging models molecular displacement at each brain voxel with a covariance-based diffusion tensor estimated from MRI signals. The tensors can be visualized geometrically and summarized using anisotropy measures for brain-image analysis and interpolation.
- A diffusion tensor is a 3 × 3 covariance matrix estimated at each brain voxel from a physically motivated model fitted to diffusion-weighted measurements.
- The diffusion tensor model represents water-molecule displacement with a multivariate normal distribution centered at the voxel and covariance matrix Σ.
- The tensor is conventionally D = Σ/2, a symmetric positive semi-definite matrix estimated from MRI signals collected under magnetic-field gradients in axial directions.
- An ellipsoid visualizes a diffusion tensor, with principal axes determined by the eigenvectors of D and lengths related to its eigenvalues.
- The paper uses diffusion-tensor samples to estimate average tensors, study variability, and interpolate tensor images at higher spatial resolution.
- Fractional Anisotropy measures directional diffusion; FA is near 1 for a strong principal axis and equals 0 for isotropy.
3. Covariance matrix estimation.
The section frames covariance estimation as a choice of distance: Euclidean averaging is standard, while non-Euclidean alternatives define means through Fréchet minimization and include log-, square-root-, Cholesky-, and Procrustes-based approaches.
- Euclidean estimation: Under a scaled Wishart model, the maximum likelihood estimator is the arithmetic mean, equivalently minimizing squared Euclidean distances.
- Euclidean estimation: Euclidean distance can produce estimates with negative eigenvalues when extrapolating beyond the observed data.
- Fréchet means: The Fréchet mean minimizes expected squared distance under the selected Riemannian metric and may require conditions for uniqueness.
- Alternative estimators: The paper compares logarithm-based, Riemannian, Cholesky, square-root, and Procrustes formulations for estimating covariance means.
- Alternative estimators: The Procrustes metric avoids extrapolation into negative-eigenvalue matrices but cannot handle positive semi-definite matrices with deficient rank.
- Alternative estimators: Procrustes estimation optimizes rotations and reflections to match covariance decompositions in Euclidean distance, relaxing fixed triangular or square-root parameterizations.
4. Procrustes size-and-shape analysis.
The paper represents covariance matrices as reflection size-and-shapes, using Procrustes matching to define distances, geodesics, means, tangent-space inference, and scale-invariant estimators.
- Non-Euclidean size-and-shape metric: Each covariance matrix is represented by decompositions whose equivalence classes identify factors differing by orthogonal rotations or reflections.
- Non-Euclidean size-and-shape metric: The resulting covariance distance is a Riemannian metric equivalent to reflection size-and-shape distance between Helmert-transformed configurations.
- Geodesics: Minimal geodesics are constructed through optimally matched decompositions and support covariance interpolation, regression, extrapolation, and prediction.
- Means and inference: Sample Fréchet means are computed with the Generalized Procrustes Algorithm, which has a global unique minimum under a regular geodesic-ball condition.
- Means and inference: Tangent-space coordinates approximate the non-Euclidean metric and enable multivariate-normal inference, principal components, and large-sample asymptotic analysis.
- Scale invariance: Scale-invariant analysis uses reflection shape space and full Procrustes mean shapes, with analogous tangent-space inference.
- Scale invariance: Table 1 summarizes the paper’s covariance distances and corresponding estimators.
5. Comparison of approaches.
The paper compares covariance-matrix metrics and estimators, emphasizing invariance properties, anisotropy measures, rank-deficient handling, and interpolation behavior. Procrustes approaches are particularly advantageous for deficient-rank matrices and are preferred over Cholesky-based inference in the stated context.
- Metrics and estimators: The considered estimators include arithmetic-average methods and Procrustes-based methods computed with the Generalized Procrustes Algorithm.The Procrustes estimators are reported to work very well in practice.
- Metric properties: Distances dE, dL, dR, and dF are invariant under simultaneous rotation and reflection, while dL, dR, and dF are also invariant under simultaneous scaling.The Riemannian metric dR additionally has affine invariance.
- Anisotropy: Procrustes Anisotropy satisfies 0 ≤ PA ≤ 1, with PA = 0 indicating isotropy and PA ≈ 1 indicating a very strong principal axis.It is defined using the full Procrustes shape distance from isotropy with a scale factor.
- Deficient-rank case: Procrustes metrics can handle deficient-rank covariance matrices, unlike dL, dR, and dF, which are not valid for such comparisons.This is presented as a strong advantage because covariance matrices may be close to rank deficient.
- Metrics and estimators: Distances dE, dL, dC, and dS differ in the geodesic paths they produce between covariance tensors.Figure 3 compares four paths obtained from these metrics.
- Comparison with Cholesky methods: Procrustes approaches are preferred over Cholesky-based inference because Cholesky methods can be unreliable when particular covariance-matrix diagonal elements vary little and can induce biased covariance-structure estimation.For small variability, baseline registration and Procrustes superimposition are similar, but the paper favors Procrustes methods generally.
6. Applications.
The applications evaluate Procrustes methods for covariance-matrix principal components, diffusion-tensor interpolation, anisotropy visualization, and estimator efficiency. Across these applications, Procrustes size-and-shape methods support geodesic analysis and provide competitive or preferable results under several settings.
- Principal components analysis: Procrustes analysis estimates a mean and principal component from noisy covariance-matrix paths, recovering variation around a true geodesic.The experiment uses 11 equally spaced covariance matrices, three noisy path realizations, and a Procrustes size-and-shape mean.
- Principal components analysis: The Procrustes representation registers configurations by removing translation, rotation, and reflection before displaying principal-component directions and scores.Figure 5 shows the noisy configurations, registered data, Procrustes mean size-and-shape, first three PCs, and PC scores.
- Interpolation: Size-and-shape interpolation produces smoother FA and PA brain images, while preserving a visible distinction between the cingulum and corpus callosum.The interpolation inserts two equally spaced points between voxels.
- Anisotropy: PA, FA, and GA are broadly similar, but PA provides more contrast in the highly anisotropic corpus-callosum region and is judged slightly preferable in this example.The tensors in the corresponding visualization are scaled to have volume proportional to the displayed quantity.
- Simulation study: Simulation efficiency depends strongly on the population covariance matrix and error distribution, with H, S, and L estimators performing consistently well overall.For a nearly deficient-rank covariance, H, S, and L are generally better, while E is slightly inferior and C and R can perform poorly under some metrics.
- Simulation study: For a well-conditioned mean covariance, H performs best under square-root Gaussian errors, C performs best under Cholesky Gaussian errors, and H or S perform better at n = 30.The comparison uses k = 3, σ = 0.1, sample sizes n = 10,30, and RMSE and Stein-loss measures.
7. Discussion.
The paper extends covariance-matrix estimation through non-Euclidean and shape-analysis methods, with applications beyond diffusion tensor imaging. It also identifies tunable power metrics and connections to high-dimensional and deficient-rank covariance problems.
- 7. Discussion.: Power Euclidean metrics interpolate between matrix-power choices, approaching the log-Euclidean metric as α → 0.The discussion considers α ∈ {1/2,1} and allows any nonzero real α depending on the application.
- 7. Discussion.: For positive α, smaller α produces estimators more resistant to outliers, whereas larger α produces estimators less resistant to outliers.Negative α instead uses powers of the inverse covariance matrix, and Procrustes registration can also be incorporated.
- 7. Discussion.: Varying α provides a practical visualization tool for helping neurologists interpret white fiber tracts in diffusion tensor images.
- 7. Discussion.: The paper introduces new covariance-matrix estimation methods rooted in statistical shape analysis.This connection may also transfer methodology between covariance-matrix analysis and shape analysis.
- 7. Discussion.: The methods may also support high-dimensional covariance analysis, averaging affine transformations, and posterior summaries from covariance-matrix Markov chain Monte Carlo output.High-dimensional settings may involve k ≫ n, while sparsity and banding are commonly used to improve covariance or inverse-covariance estimation.
- 7. Discussion.: The Procrustes size-and-shape and matrix square root metrics may benefit longitudinal-data modeling and applications with covariance matrices close to deficient rank.The paper highlights structure tensors associated with image surfaces as another deficient-rank application.