Bayesian uncertainty quantification of spatio-temporal trends in soil organic carbon using INLA and SPDE Nicolas P.A. Saby 1 , Thomas Opitz 2 , Bifeng Hu 1,3 , Blandine Lemercier 4 and Hocine Bourennane 3 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
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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)
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.)