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Page 1: Bayesian uncertainty quantification of spatio-temporal ... · Bayesian uncertainty quantification of spatio-temporal trends in soil organic carbon using INLA and SPDE Nicolas P.A

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

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Soil properties vary in space and time dimension

How to make accurate predictions with associateduncertainty ?

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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

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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)

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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)

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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

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ResultsExample of topsoil Carbon content in agricultural soils of Brittany region

1990 - 2014

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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

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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)

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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

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Inference on space-varying linear time trend

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Uncertainty of spatio temporal maps

Median of the Carboncontents for the year 2000

5% quantile

95% quantile

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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.)

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Thank you !GIS Sol, INRAe, Chinese council


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