Aggregate representation of genetic soil horizons via proportional-odds logistic regression

D.E. Beaudette, P. Roudier, J.M. Skovlin









This document is based on aqp version 1.8-8 and soilDB version 1.5-5.

Describing soil morphology in aggregate is hard

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  • hz depths / designations: overlap, consistency, frequency
  • style and convention: variation over time and by describer
  • transition and infrequent horizons: BA, AB, BCt, etc.
  • lumpers vs. spliters: A-Bt1-Bt2-R vs. A1-A2-AB-Bt1-Bt2-Bt3-Cr-R

can we do better than selecting a “representative pedon” from a collection?

Aggregation over generalized horizon labels

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  1. determine the core concept: e.g. A-Bt1-Bt2-Bt3-Cr-R
  2. assess existing data, relevant management or scientific needs
  3. assign generalized horizon labels (GHL)
  4. aggregate over GHL, in R / AQP syntax:
    • empirical: slice() →  slab() →  probability depth-functions
    • model-based: slice() →  orm() →  probability model
  5. determine most-likely horizonation

generalized horizon labels are expert-guided, “micro-correlation” decisions

Quick detour: definitions

proportional-odds (PO) logistic regression

\[ P[Y \geq j | X] = \frac{1}{1 + exp[-(\alpha_{j} + X \beta]} \] Extension of logistic regression model; predictions constrained by horizon designation and order. RCS basis functions accommodate non-linearity.

Shannon Entropy (H index)

\[ H = -\sum_{i=1}^{n}{p_{i} * log_{n}(p_{i})} \] \( H \) is an index of uncertainty associated with predicted probabilities, \( \mathbf{p} \), of encountering horizons \( i \) through \( n \) at some depth. Larger values suggest more confusion.

Brier scores

\[ B = \frac{1}{n} \sum_{i=1}^{n}{ ( p_{i} - y_{i} )^{2} } \] \( B \) is an index of agreement between predicted probabilities, \( \mathbf{p} \), and horizons, \( \mathbf{y} \), over depth-slices \( i \) through \( n \) associated with a specific horizon. Larger values suggest less agreement between probabilities and observed horizon labels.

Examples using 54 profiles correlated to Loafercreek soil series

  • fine-loamy, mixed, super-active, thermic ultic haploxeralfs
  • extent: foothills of the Sierra Nevada Mountains (MLRA 18)
  • uses: recreation, range, vineyard, low-density residential

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Examples using 54 profiles correlated to Loafercreek soil series

  • fine-loamy, mixed, super-active, thermic ultic haploxeralfs
  • extent: foothills of the Sierra Nevada Mountains, MLRA 18
  • uses: recreation, range, vineyard, low-density residential plot of chunk plot-sample-data

colors represent generalized horizon labels (GHL)

Assignment of GHL: expert knowledge + data

plot of chunk plot-sample-data-zoom

colors represent generalized horizon labels (GHL)

slice(): resample along 1-cm increments

plot of chunk slice-data-1

colors represent generalized horizon labels (GHL)

slab(): slice-wise probability calculation

plot of chunk slice-data-2

no assumptions, simple interpretation, directly tied to the original data; but over-fit

slice() and fit PO-logistic regression model

plot of chunk slice-and-fit-1

proportional-odds logistic regression generalizes the process

Probabilistic representation of genetic horizons

plot of chunk compare-proportions

empirical probabilities when data are sparse, PO-LR when data are available

Quantifying uncertainty

plot of chunk shannon

Shannon entropy: continuous metric of confusion; Brier scores: agreement by GHL

Quantifying uncertainty

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model stability: iterative re-fitting (n=25, reps=250), mean R2 = 0.89

Most-likely (ML) horizonation

plot of chunk plot-ml-hz

most-likely horizon boundaries determined by probability depth-functions

Conclusions: aggregate soil morphology

plot of chunk ml-hz-conclusions

  • fact: sampling by genetic horizon is efficient and will continue to be important
  • we can do better than picking a single, representative profile
  • soil series defined by GHL rules, PO-LR model, and properties aggregated by GHL
  • variability between descriptions smoothed as sample size increases– given thoughtful correlation
  • continuous depth-functions of genetic, or diagnostic horizons
  • most-likely horizonation, based on depth-function crossings
  • quantitative estimates of uncertainty: Brier scores, Shannon Entropy, etc.

Conclusions: further work

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  • minimum sample sizes, model diagnostics, best-practice guidelines, etc.
  • more realistic estimates of SE, e.g. correlation structure via GEE
  • pedogenic interpretation of model coefficients

Thank You!

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