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Bayesian uncertainty quantification of spatio-temporal
trends in soil organic carbon using INLA and SPDE
Nicolas P.A. Saby1, Thomas Opitz2, Bifeng Hu1,3, Blandine Lemercier4 and Hocine Bourennane3
1: Unité Infosol, INRAe, Orléans, France
2: Unité BioSp, INRAe, Avignon, France
3: Unité UR Sols , INRAe, Orléans France
4: UMR SAS, Institut Agro, Rennes, France
Soil properties vary in space and time dimension
How to make accurate predictions with associateduncertainty ?
Motivations
1. Perfect data are rarely available1. Missing values in time series
2. Standard approaches need good data for reliable inferences
2. The estimation of space-varying regression coefficients is challenging (spatial smoothness of the estimated trend surface)
3. A precise assessment of estimation and prediction uncertainty is challenging with multi-steps approaches
4. Space-time covariance matrices become increasingly intractable
Objectives
• Develop a fully Bayesian estimation framework based on• the integrated nested Laplace approximation (INLA),• combined with the so-called stochastic partial differential
equation (SPDE) approach providing numerically convenient representations of Gaussian processes over continuous space.
• INLA allows joint estimation of all model components, including smoothness parameters (Issues 2, 3, 4) for any sampling design (Issue 1)
Statistical framework for space-time variation
𝑌 𝒔, 𝑡 = 𝛽0 +
𝑖
𝛽𝑖𝑧𝑖(𝒔) +𝑊0 𝒔 + ǁ𝑡 − 0.5 𝑊𝟏 𝒔 +
𝑙=1
𝐾
𝐵𝑙(𝑡)𝑊𝑙 𝒔 + 𝜀(𝒔, 𝑡)
𝛽𝑖 : fixed effects of covariates 𝑧𝑖(𝒔) (precision)
𝑊0 𝒔 : Space-varying regression trend (range, precision)
𝑊𝟏 𝒔 : Space-varying linear time trends with normalized time ǁ𝑡 (range, precision)
𝑊𝑙 𝒔 : Space-time residual process (range, precision)
𝜀(𝒔, 𝑡) i.i.d process (precision)
The French Soil testing databaseNational project for soil monitoring of agricultural soils in the framework of the GIS Sol
• Data Characteristics:
• Georeferencing: imprecise – municipality
• Sampling: no control on the strategy - sampling year
• Standardized analytical procedures for the selectedlaboratories
• Available soil parameters:
• Particle size analysis
• Organic C and total N, pH, CEC
• Macronutrients (P K Ca Mg)
• Micronutrients
• Basic information on land use bdat.gissol.frSaby et al, 2017, Soil Use and Management, https://doi.org/10.1111/sum.12369
ResultsExample of topsoil Carbon content in agricultural soils of Brittany region
1990 - 2014
Uncertainty of categorical effects
𝑌 𝒔, 𝑡 = 𝛽0 +
𝑖
𝛽𝑖𝑧𝑖(𝒔) +𝑊0 𝒔 + ǁ𝑡 − 0.5 𝑊𝟏 𝒔 +
𝑙=1
𝐾
𝐵𝑙(𝑡)𝑊𝑙 𝒔 + 𝜀(𝒔, 𝑡)
• Mean and 95 % credibility interval
• Uncertainty estimatesare based on Posteriordistributions
Uncertainty of the hyper parameters
𝑌 𝒔, 𝑡 = 𝛽0 +
𝑖
𝛽𝑖𝑧𝑖(𝒔) +𝑊0 𝒔 + ǁ𝑡 − 0.5 𝑊𝟏 𝒔 +
𝑙=1
𝐾
𝐵𝑙(𝑡)𝑊𝑙 𝒔 + 𝜀(𝒔, 𝑡)
Uncertainty of space-varying linear time trends
𝑌 𝒔, 𝑡 = 𝛽0 +
𝑖
𝛽𝑖𝑧𝑖(𝒔) +𝑊0 𝒔 + ǁ𝑡 − 0.5 𝑊𝟏 𝒔 +
𝑙=1
𝐾
𝐵𝑙(𝑡)𝑊𝑙 𝒔 + 𝜀(𝒔, 𝑡)
Uncertainty based on Posterior distributions
Median
5% quantile
95% quantile
Inference on space-varying linear time trend
Uncertainty of spatio temporal maps
Median of the Carboncontents for the year 2000
5% quantile
95% quantile
Discussion
• Fast computation with package PARDISO https://pardiso-project.org/r-inla/
• Possible to compute the CRPS to compare models taking into accountthe uncertainty
• By default INLA provides univariate posterior distributions of latent variables (for example W0(si), W1(si)) and fitted values
• Posterior simulation is necessary to provide posterior distributions for other quantities that combine several latent variables (interpolation in space and time for example, etc.)
Thank you !GIS Sol, INRAe, Chinese council