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A log-Euclidean mean needs an uncertainty language built in the same geometry; log-tensor dispersion separates a central positive-definite tensor from the spread and sampling precision around it.
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Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The teaching chapters remain open; these grouped routes form the index.
Route 1 · Define geometric dispersion
Route 2 · Work the log-tensor ledger
Route 3 · Transfer to uncertainty
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE GEOMETRIC DISPERSION
1. A tensor mean needs a matching uncertainty language
Positive-definite conductivity tensors describe both the strength and direction of transport. A log-Euclidean mean gives one central tensor by averaging matrix logarithms and mapping back with the matrix exponential. The next question is not another mean: it is how far the observed tensors spread around that centre.
Entrywise standard deviations are difficult to interpret because tensor entries depend on coordinates and because additive differences do not match the multiplicative contrasts that motivated the log transformation. Log-tensor variance measures dispersion in the same space used to form the mean.
This creates a coherent descriptive system, but it does not turn spatially dependent soil samples into independent replicates or make a geometric mean a physical upscaling law.
CHAPTER 2 OF 12 · DEFINE GEOMETRIC DISPERSION
2. Log coordinates and geodesic distance
For a symmetric positive-definite tensor K, the matrix logarithm log K is symmetric. Under the log-Euclidean metric, the distance between K1 and K2 is the Frobenius norm of log K1 minus log K2.
Given tensors K1 through Kn, their log-Euclidean mean is exp of the arithmetic mean of their logarithms. A natural sample dispersion is the average or degrees-of-freedom-adjusted sum of squared log distances to that mean.
Because the calculation occurs in a flat vector space of symmetric matrices, covariance matrices, standard errors and many familiar multivariate tools can be applied to vectorised log tensors. The interpretation must still return to conductivity ratios and orientations.
CHAPTER 3 OF 12 · WORK THE LOG-TENSOR LEDGER
3. Worked example: crossed diagonal tensors
Consider two diagonal conductivity tensors in common units: K1 = diag(1, 4) and K2 = diag(4, 1). Their log tensors are diag(0, ln 4) and diag(ln 4, 0). The log mean is diag(ln 2, ln 2), so the log-Euclidean mean tensor is diag(2, 2).
Each deviation from the log mean is diag(−ln 2, ln 2) or its reverse. Its squared Frobenius length is 2(ln 2)^2, about 0.961.
| Tensor | Log principal values | Squared log distance to mean |
|---|---|---|
| K1 = diag(1, 4) | (0, 1.386) | 0.961 |
| K2 = diag(4, 1) | (1.386, 0) | 0.961 |
| Mean = diag(2, 2) | (0.693, 0.693) | 0 |
The example exposes an important limit. A nearly isotropic mean does not prove that individual samples are nearly isotropic. Dispersion and orientation structure must accompany the centre.
CHAPTER 4 OF 12 · WORK THE LOG-TENSOR LEDGER
4. From variance to interpretable contrasts
A log difference of 0.693 in one principal direction corresponds to a conductivity ratio of exp(0.693), or two. Reporting only “variance 0.961” loses that physical intuition.
Translate uncertainty back through exponentiation. Intervals symmetric in log space become multiplicative intervals in conductivity space. A component with log standard error 0.1 corresponds roughly to a factor exp(0.1), about 1.105, on either side of its central value.
For full tensors, simultaneous regions are not obtained by exponentiating separate entrywise intervals. Covariance among log entries and the chosen vectorisation convention matter.
Sample covariance in log space
Vectorise each symmetric log tensor without duplicating off-diagonal information. The sample covariance of those vectors describes joint variation in scale, anisotropy and orientation as represented in the chosen frame.
Diagonal elements give marginal log-coordinate variances; off-diagonal elements show coordinated changes. An eigendecomposition of this covariance can identify dominant modes of tensor variation, but these statistical modes are not automatically physical principal flow directions.
Record whether off-diagonal terms are scaled by square root of two in the vectorisation. That convention preserves the Frobenius inner product and avoids silent weighting errors.
CHAPTER 5 OF 12 · WORK THE LOG-TENSOR LEDGER
5. Changed context: noncommuting rotated tensors
The diagonal example is easy because the tensors share axes and commute. Rotate one anisotropic tensor and its matrix logarithm rotates with it, but averaging logs can produce a mean whose axes do not match either observation.
| Data pattern | What log dispersion captures | What still needs checking |
|---|---|---|
| Common eigenvectors | Multiplicative eigenvalue spread | Sampling and scale support |
| Rotating eigenvectors | Combined magnitude and orientation change | Frame convention and identifiability |
| Nearly repeated eigenvalues | Small anisotropy with unstable individual axes | Principal subspace rather than one axis |
The clustered-eigenspaces guide explains why a subspace can be stable even when individual eigenvectors are not.
CHAPTER 6 OF 12 · TRANSFER TO UNCERTAINTY
6. Uncertainty around the mean
If log tensors are independent and identically distributed with finite covariance, the multivariate central limit theorem supplies an approximate normal distribution for the sample log mean. The covariance of the mean scales approximately as one over n.
Soil measurements are often spatially clustered, however. Nearby cores can share structure, making the effective sample size smaller than the count. Block bootstrap, spatial covariance models or cluster-level resampling may be more appropriate than an independent bootstrap.
An uncertainty region for the estimated mean answers a different question from the spread of individual tensors. Report both when the reader needs population heterogeneity and estimation precision.
CHAPTER 7 OF 12 · TRANSFER TO UNCERTAINTY
7. Bootstrap and model-based routes
A nonparametric bootstrap resamples observational units, recomputes the log mean and maps it back. It is attractive when normal approximations are doubtful, but the resampling unit must preserve the sampling design.
A model-based route assumes a lognormal family on positive-definite matrices and estimates a covariance in log space. That provides compact inference and two-sample tests, while depending more strongly on distributional assumptions.
Compare both routes on synthetic tensors whose mean and dispersion are known. Coverage, not visual smoothness, is the decisive calibration check.
CHAPTER 8 OF 12 · TRANSFER TO UNCERTAINTY
8. Links to regularisation and scale transition
The tensor-shrinkage guide concerns stabilising noisy individual estimates. Shrinkage changes the observed dispersion and must be included in uncertainty calibration.
The log-Euclidean averaging guide explains why a geometric average is not automatically an effective conductivity tensor. A confidence region around that average still quantifies a statistical centre, not a homogenised flow response.
Keep measurement uncertainty, between-sample heterogeneity and physical upscaling error as separate ledgers.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Log-Euclidean variance depends on units only through common scale shifts in log space, but mixing tensors expressed in different units remains invalid. Every tensor must be converted before the logarithm.
A small mean standard error does not imply low soil heterogeneity; it can result from many highly variable observations. Conversely, a small sample variance can be misleading when sampling missed important directions or horizons.
Confidence ellipsoids in log coordinates are asymptotic unless a finite-sample method establishes coverage. Positive definiteness after exponentiation does not guarantee correct coverage.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Evidence and verification route
Arsigny and colleagues introduced the computational framework in Log-Euclidean Metrics for Fast and Simple Calculus on Diffusion Tensors. Schwartzman develops distributions, limiting laws and confidence regions in Lognormal Distributions and Geometric Averages of Symmetric Positive Definite Matrices.
For a broader geometric foundation, see Intrinsic statistics on Riemannian manifolds. Verification should test matrix symmetry and positive eigenvalues, reproduce log and exponential transforms, preserve the Frobenius vectorisation, and assess coverage under spatially realistic simulation.
Archive the sampling coordinates and support volumes. Tensor uncertainty without the measurement scale and spatial design can be mathematically correct yet scientifically untransferable.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not calculate variance entry by entry?
Entrywise variance ignores the geometry that motivated log-Euclidean averaging and can be hard to interpret across coordinate frames.
Is log-tensor variance dimensionless?
Log differences are dimensionless when tensors have been expressed relative to a consistent unit, but the convention must be documented.
Does an isotropic mean imply isotropic soil?
No. Opposing anisotropies can average to an isotropic centre while individual tensors remain strongly directional.
What should accompany the mean?
Report sample dispersion, uncertainty of the mean, sampling design, spatial dependence treatment and physical scale.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Final transfer check and connected reading
Add two more tensors equal to diag(1, 4). Predict how the mean and dispersion move before calculating them. Explain why the centre becomes anisotropic even though the original two-tensor mean was isotropic.
Then rotate one tensor by 45 degrees. Decide whether principal-axis intervals or a subspace summary would be more stable and justify the choice.
Continue with the Soil Science root, log-Euclidean averaging, tensor shrinkage, the Gold Standard of Education and How X Works.
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