Datamodel integration:

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Data-model integration Examples from belowground
ecosystem ecology Kiona Ogle University of
Wyoming Departments of Botany
Statistics www.uwyo.edu/oglelab
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Todays Task

• What are some ecological questions to which
  sensor network data could be applied?
• How would those data be used in models?
• Overview modeling of ecological data and
  processes.

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Types of Questions

• What are some ecological questions to which
  sensor network data could be applied?
• Spatial temporal processes
• Improved ecological understanding
• More accurate prediction forecasting
• Example problems
• Biogeochemical exchanges between the atmosphere
  biosphere
• How do environmental perturbations affect carbon
  water exchange?
• Partitioning ecosystem processes components
• Linking processes mechanisms operating at
  multiple temporal spatial scales

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How to Address Such Questions?

• Couple data and models
• Sensor network data
• Very rich
• Real-time large datasets spatially extensive
  and/or temporally intensive
• Heterogeneous
• Different locations, processes, and conditions
• Models data analysis
• Less appropriate
• Classical analyses that assume linearity and
  normality of data
• Design-based inference about patterns
• More appropriate
• Coupling of process-based models with diverse and
  rich datasets
• Model-based inference about patterns and
  mechanisms

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Why Couple Data Process Models?

• Parameter estimation (or model
  parameterization)
• Quantification of uncertainty
• Improved predictions and forecasts
• Decision support, management, conservation
• Synthesize multiple types of data
• Relate different system components to each other
• Learn about important mechanisms
• Hypothesis generation
• Use data-informed models to generate testable
  hypotheses
• Inform sampling and network design
• Data analysis
• Go beyond simple classical analyses
• Explicit integration of multiple data types,
  diverse scales, and nonlinear and non-Gaussian
  processes

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How to Couple Data Process Models?

• Multiple approaches, for example
• Maximum likelihood-based models
• Least squares, minimization of objective
  functions
• Hierarchical Bayesian models
• Hierarchical Bayesian approach
• Recall, from Jennifers talk

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Outline

• The process model
• Types of ecological models
• Building process models
• Examples from belowground ecosystem ecology
• Motivating issues
• Ex 1 Estimating components of soil organic
  matter decomposition
• Ex 2 Deconvolution of soil respiration (i.e.,
  CO2 efflux)
• In both examples, highlight
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
  network data applications

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Hierarchical Bayesian Model
Data model (likelihood)
Probabilistic process model
The process model
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The Process Model

• Conceptual model
• Systems diagrams
• Graphical models

• Model formulation
• Explicit, mathematical eqns
• Systems equations
• State-space equations

Inputs
Outputs
Compare
Conceptual model
Mathematical model
Unobserved quantities (parameters)
Observed quantities (data)
Analytical output
Observed quantities (driving variables)
Numerical/ simulation output
Predict
Simulation model
Unobserved or latent quantities
The process model
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Types of Process Models
Jorgensen (1986) Fundamentals of Ecological
Modelling. 389 pp. Elsevier, Amsterdam.
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Upcoming Example Soil Carbon Cycle Model
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Example Process Model
Pools or state variables
Simplified systems diagram of the soil carbon
cycle in a temperate forest
Flows of carbon
Source Xu et al. (2006) Global Biogeochemical
Cycles Vol. 20 GB2007.
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Model Formulation

• A matrix of flux rates or carbon transfer
  coefficients (parameters)
• u(t) flux of carbon into the system(e.g.,
  photosynthetic flux) (driving variable or modeled
  quantity)
• B vector of allocation fractions (parameters)
• X vector of state variables (unobservable latent
  quantities, outputs)

Source Xu et al. (2006) Global Biogeochemical
Cycles Vol. 20 GB2007.
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Model Formulation
Observable (data)
Source Xu et al. (2006) Global Biogeochemical
Cycles Vol. 20 GB2007.
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How to Couple Data Process Models?

• Hierarchical Bayesian approach

Data model (likelihood)
Probabilistic process model
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Outline

• The process model
• Types of ecological models
• Building process models
• Examples from belowground ecosystem ecology
• Motivating issues
• Ex 1 Estimating components of soil organic
  matter decomposition
• Ex 2 Deconvolution of soil respiration (i.e.,
  CO2 efflux)
• In both examples, highlight
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
  network data applications

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Ecosystem Processes
Emphasis on aboveground
What about belowground?
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Biogeochemical Cycles
H20
N
H20
N
H20
C
C
P
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Biogeochemical Cycles
Belowground system is critical Tightly linked to
aboveground system
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Belowground Issues

• Aboveground
• Lots of info
• Easy to measure

• Belowground
• Little info
• Difficult to measure
• Aboveground measurements (helpful but limited)

• Outstanding issues
• Partitioning above- belowground
• Quantifying partitioning belowground
• Implications for ecosystem function
• Examples arid semiarid systems

Figure from Kieft et al. (1998) Ecology 79671-683
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Motivating Questions Soil Carbon Cycle

• From where in the soil is CO2 coming from?
• What are the relative contributions of autotrophs
  vs. heterotrophs?
• What factors control decomposition rates
  heterotrophic activity?
• How does pulseprecipitationaffect sourcesof
  respiredCO2?
• Implications ofclimate changefor desert
  soilcarbon cycling?

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Integrative Approach

• Diverse data sources
• Experimental observational
• Lab field studies
• Multiple scales
• Varying amounts completeness
• Process-based models
• Key mechanisms, processes, components
• Balance detail simplicity
• Multiple scales interactions
• Statistical models data-model integration
• Hierarchical Bayesian framework
• Mark chain Monte Carlo

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Examples Presented Today
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Ex 1 Soil organic matter decomposition

• Objectives
• Identify soil microbial processes affecting
  decomposition
• Learn how vegetation (i.e., microsite) controls
  these processes

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Experimental Design

• Mesquite shrubland
• in southern Arizona
• Microsite types
• bare ground
• grass
• small mesquite
• big mesquite

Bare ground
Grass
Small mesquite
Big mesquite
3 cores (reps)
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Experimental Design
...
Add water
...
Add sugar water
Incubate at 25 oC
CO2
Measure CO2 efflux (soil respiration rate) at 24
48 hours
CO2
8 depths (layers)
CO2
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Experimental Design
...
Add water
...
Add sugar water
Measure Microbial biomass Soil organic
carbon Soil nitrogen
Incubate at 25 oC
CO2
Measure CO2 efflux (soil respiration rate) at 24
48 hours
CO2
8 depths (layers)
CO2
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Design Data Overview

• Full-factorial design
• Microsite
• 4 levels bare, grass, small mesq, big mesq
• Soil layer
• 8 levels 0-2, 2-5, ..., 40-50 cm
• Substrate addition type
• 2 levels water only, sugar water
• Incubation time
• 2 levels 24, 48 hrs
• Soil core or rep
• 3 cores per microsite

• Stochastic data
• Soil respiration rate
• N 359 (25 missing)
• Microbial biomass
• N 18 (14 missing)
• Soil organic carbon
• N 89 (7 missing)

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Some Data
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Analysis Objectives
microbes
soil C
CO2 flux
Estimate microbial respiration (decomposition)
parameters (i.e., process parameters)
?
Soil depth
?
data
Respiration
biomass activity
Microbial biomass
Carbon substrate
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Process Model Soil Respiration
Estimate microbial respiration (decomposition)
parameters (i.e., process parameters)
Respiration
Microbial biomass
Carbon substrate
Michaelis-Menton type model
microbial base-line metabolic rate microbial
carbon-use efficiency
Assume Ac related to substrate quality
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Data-Model Integration

• Full-factorial design
• Microsite
• Soil layer
• Substrate addition type
• Incubation time
• Soil core or rep

B C R N

• Stochastic data
• Soil respiration rate
• Microbial biomass
• Soil organic carbon

• Things to consider
• Multiple data types
• Nonlinear model
• Missing data
• Experimental design

some data
some data
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Data Model (Likelihood)

• Let LR log(R)
• For microsite m, soil depth d, soil core r,
  substrate-addition type s, and time period t

Mean (truth) (latent process)
Observationprecision ( 1/variance)
Observed rate
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Data Model (Likelihood)

• Now, for the covariates...
• For microsite m, soil depth d, and soil core r
• Note the likelihoods are for both the observed
  and missing data

Observed
Mean (truth) (latent process)
Observation precision ( 1/variance)
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Data Model (Likelihood)
Likelihood components
Data parameters
Latent processes
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Probabilistic Process Model
Latent processes
Deterministic model for soil microbes carbon
contents
Stochastic model for latent respiration
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Probabilistic Process Model
Stochastic model for latent respiration
Specify expected process Michaelis-Menten
(process) model
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Probabilistic Process Model
Process components
Process parameters
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Parameter Model (Priors)
Data parameters
Process parameters
Conjugate, relatively non-informative priors for
precision terms
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Parameter Model (Priors)
Data parameters
Process parameters
Non-informative Dirichlet priors for relative
distributions of microbes and carbon
Multivariate version of the beta
distribution (with all parameters set to 1
multidimensional uniform)
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Parameter Model (Priors)
Data parameters
Process parameters
Relatively non-informative (diffuse) normal
priors for the rest
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The Posterior
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The Posterior
No analytical solution for the joint posterior
distribution No analytical solution for most of
the marginal distributions Approximate the
posterior Markov chain Monte Carlo
methods,implemented in WinBUGS
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Model Implementation WinBUGS
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Model Goodness-of-fit
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Example Results
C (total soil carbon, g C/m2)
B (microbial biomass, g dw/m2)
Bare
Bigmesq.
Med.Mesq.
Grass
Bare
Bigmesq.
Med.Mesq.
Grass
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Example Results
Bare ground
Big mesquite
Relative amount ofmicrobial biomass
Surface Deep
Surface Deep
Soil depth (or layer)
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Sensitivity to Data Sources
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Ex 2 Deconvolution of Soil Respiration

• From where in the soil is CO2 coming from?
• What are the relative contributions of autotrophs
  vs. heterotrophs?
• What factors control decomposition rates
  heterotrophic activity?
• How does pulseprecipitationaffect sourcesof
  respiredCO2?
• Multiple datasources
• lots
• limited

data
data
data
data
data
data
data
data
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The Field Sites
San Pedro River Basin
Santa Rita Experimental Range
Sonoran Desert
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Stable Isotope Tracers
Important data source facilitates partitioning
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Data Source Examples
Datasets field/lab pubs
Potential sensor network data
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Example Data
Santa Rita pulse experiment
San Pedro automated flux measurements
San Pedro incubation experiment
Santa Rita pulse experiment d13C
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• Hierarchical Bayesian ModelDeconvolution
  Approach
• Integrate multiple sources of information
• Diverse data sources
• Different temporal spatial scales
• Literature information
• Lab field studies
• Detailed flux models
• Respiration rates by source type soil depth
• Dynamic models
• Mechanistic isotope mixing models
• Multiple sources

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Data Source Examples
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Bayesian Deconvolution
The Hierarchical Bayesian Model
Some Likelihood Components
Likelihood of data (isotopes soil flux)
Observations (data)
Latent processes from isotope mixing model
flux models
Functions of parameters ?
Define process models
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The Deconvolution Problem
Theory Process Models
Isotope mixing model (multiple sources depths)
Contributions by source (i ) and depth (z )?
Temporal variability?
Relative contributions (by source depth)
Source-specific respiration? Spatial temporal
variability?
Total flux (at soil surface)
Flux model (source- depth- specific)
(Q10 Function, Energy of Activation)
From previous incubation/decomposition study
(Ex 1)
Mass profiles (substrate, microbes, roots)
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The Deconvolution Problem
Objectives
Flux model (source- depth- specific)
Covariate data
What is ?i? (source-specific parameters)
? Component fluxes
? Total soil flux
? Contributions
How to estimate ?i?
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Bayesian Deconvolution
The Parameter Model (Priors)
Example Lloyd Taylor (1994) model
Informative priors for Eo and To
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• Implementation
• Markov chain Monte Carlo (MCMC)
• Sample parameters (?i ) from posterior
• Posteriors for ?is, ri(z,t)s, pi(z,t)s, etc.
• Means, medians, uncertainty
• WinBUGS

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Results Dynamic Source Contributions San Pedro
Site Monsoon Season
Zoom-in
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Results Root Respiration Responses Zoom-in July
27 August 4
Jul 27
Aug 4
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Results Contributions Vary by Depth
Mesquite (C3 shrub)
Soil water
Sacaton (C4 grass)
Relative contributions by depth
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• Summary
• Sources of soil CO2 efflux
• Mesquite (shrub) major contributor, stable
  source
• Sacton (grass) minor contributor, threshold
  response
• Microbes (bare) minor contributor, coupled to
  pulses
• Deconvolution data-model integration
• Soil depth (including litter)
• By species or functional groups
• Quantify spatial temporal variability
• Incorporate environmental drivers
• Implications applications
• Identify mechanisms
• Predictions forward modeling

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Outline

• The process model
• Types of ecological models
• Building process models
• Examples from belowground ecosystem ecology
• Motivating issues
• Ex 1 Estimating components of soil organic
  matter decomposition
• Ex 2 Deconvolution of soil respiration (i.e.,
  CO2 efflux)
• In both examples, highlight
• Data sources
• Process models
• Data-model integration
• Implications of data-model integration for sensor
  network data applications

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Implications for Sensor Networks

• Parameter estimation (or model
  parameterization)
• Process models related to biogeochemical
  exchanges between the atmosphere biosphere
• Quantification of uncertainty
• Improved predictions and forecasts
• Synthesize data
• Go beyond simple classical analyses
• Explicit integration of multiple data types
  scales
• Relate different system components to each other
• Learn about important mechanisms
• Hypothesis generation sampling design
• Use data-informed models to generate testable
  hypotheses
• Inform sampling and network design
• Where (spatial), when (temporal), what
  (components)?

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Questions?
Photo by Travis Huxman Monsoon flood, San Pedro
River Basin Sonoran desert
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Results Dynamic Source Contributions
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Example WinBUGS Output
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The Inverse Problem
Plant water uptake
Soil respiration
Isotope mixing model
Fractional contributions
Total flux
Flux model
(Q10 Function, Energy of Activation)
Substrate orroot profiles
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The Inverse Problem
Isotope mixing model (multiple sources depths)
Contributions by source (i ) and depth (z )?
Temporal variability?
Relative contributions (by source depth)
Source-specific respiration? Spatial temporal
variability?
Total flux (at soil surface)
Flux model (source- depth- specific)
(Q10 Function, Energy of Activation)
Mass profiles (substrate, microbes, roots)
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The Deconvolution Problem
Data-Model Integration
Flux model (source- depth- specific)
Covariate data
What is ?i? (source-specific parameters)
Likelihood of data (isotopes soil flux)
Depend on ?i
From isotope mixing model flux models
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Data Source Examples
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The Deconvolution Problem
Plant water uptake
Soil respiration
Isotope mixing model
Fractional contributions
Total flux
Flux model
(Q10 Function, Energy of Activation)
Substrate orroot profiles
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The Deconvolution Problem
Plant water uptake
Soil respiration
What is ?i?
What are?, a1, m1, a2, m2?
Likelihood of data
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Types of data provides by sensor networks

• high-frequency tunable diode laser (TDL)
  measurement of the stable isotope
• eddy covariance for measuring concentrations and
  fluxes of gases (e.g., water vapor and CO2)
• soil environmental data temperature, water
  content, water potential, etc.
• micro-met data air temp, RH, vpd, light, wind
  speed, etc.
• plant ecophys/ecosystem data sapflux, ET, albedo
  reflectance

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Key components
Data
P(q X )
q
Process models
Statistical tools data-model integration
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The Process Model

• Conceptual models
• Systems diagrams
• Graphical models

• Model formulation
• Explicit, mathematical eqns
• Systems equations
• State-space equations

Observations of real system
Conceptual model
Mathematical model
Analytical output
Compare
Observational data
Simulation model
Numerical/ simulation output
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Examples Presented Today
Jorgensen (1986) Fundamentals of Ecological
Modelling. 389 pp. Elsevier, Amsterdam.
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Data Model (Likelihood)
Likelihood components
Assuming conditional independence, likelihood of
all data is