Bayesian modeling for Bayesian modeling for ordinal substrate size ordinal substrate size using EPA stream data using EPA stream data Megan Dailey Higgs Megan Dailey Higgs Jennifer Hoeting Jennifer Hoeting Brian Bledsoe* Brian Bledsoe* Department of Statistics, Colorado State University Department of Statistics, Colorado State University *Department of Civil Engineering, Colorado State *Department of Civil Engineering, Colorado State University University A spatial model for ordered A spatial model for ordered categorical data categorical data
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Bayesian modeling for ordinal substrate size using EPA stream data
Bayesian modeling for ordinal substrate size using EPA stream data. A spatial model for ordered categorical data. Megan Dailey Higgs Jennifer Hoeting Brian Bledsoe* Department of Statistics, Colorado State University *Department of Civil Engineering, Colorado State University. - PowerPoint PPT Presentation
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Bayesian modeling for Bayesian modeling for ordinal substrate size ordinal substrate size
Department of Statistics, Colorado State UniversityDepartment of Statistics, Colorado State University*Department of Civil Engineering, Colorado State University*Department of Civil Engineering, Colorado State University
A spatial model for ordered A spatial model for ordered categorical datacategorical data
Substrate size in streamsSubstrate size in streams► Influences in-stream physical habitatInfluences in-stream physical habitat► Often indicative of stream healthOften indicative of stream health► EPA collected data at 485 sites in Washington and EPA collected data at 485 sites in Washington and
Oregon between 1994 and 2004Oregon between 1994 and 2004
Data Collection ProtocolData Collection Protocol►At a site:At a site:
11 transects x 5 points along each 11 transects x 5 points along each transect transect
Choose particle under the sharp end of a Choose particle under the sharp end of a stickstick
Visually Visually estimate and classifyestimate and classify size size
Creating the responseCreating the response► For a site:For a site:
Transform the original size classes to Transform the original size classes to
loglog1010(Geometric Mean) for all sample points(Geometric Mean) for all sample points Find the median for the siteFind the median for the site
►Geometric meanGeometric mean
The responseThe response► YYii = median[ = median[loglog1010((geometric meangeometric mean)] for site i)] for site i
► Transformation provides a more symmetric, Transformation provides a more symmetric, continuous-like variablecontinuous-like variable Typically modeled as a continuous variableTypically modeled as a continuous variable Predictive models have performed poorlyPredictive models have performed poorly
► Response is an ordered categorical variableResponse is an ordered categorical variable 12 categories (6 with very few observations)12 categories (6 with very few observations)
Ordered categorical dataOrdered categorical data
►YYii is a categorical response variable is a categorical response variable with K ordered values: {1,…,K}with K ordered values: {1,…,K}
►Modeling objectives:Modeling objectives: Explain the variation in the ordered
response from covariate(s) Incorporate the spatial dependence Estimate, predict, and create maps of Pr(Yi
≤ k) and Pr(Yi = k)
Formulating the spatial modelFormulating the spatial model
Spatial model for ordered categorical data
+ =Non-spatial model for ordered categorical data
Albert & Chib (1993, 1997)
Spatial model for binary and count data
• Diggle, Tawn, & Moyeed (1998)
• Gelfand & Ravishanker (1998)
•Generalized geostatistical models with a Generalized geostatistical models with a latent latent
Gaussian processGaussian process•Metropolis Hastings within Gibbs sampling Metropolis Hastings within Gibbs sampling
Metropolis-Hastings within Gibbs samplingMetropolis-Hastings within Gibbs sampling Prior: Prior:
► – – flat and conjugate Normalflat and conjugate Normal► 22 and and – Independent uniform priors – Independent uniform priors► multivariate normalmultivariate normal
Simulated dataSimulated data► Simulated data at a subset of the original Simulated data at a subset of the original
locations (n = 82)locations (n = 82) Cluster infill around the 82 sites (n=120)Cluster infill around the 82 sites (n=120)
Spatial process:Spatial process:►W is a stationary Gaussian process with W is a stationary Gaussian process with
E[W(E[W(ss)]=0 and Cov[W()]=0 and Cov[W(ssii),W(),W(ssjj)] = )] = 22((ssii--ssjj;;))►Exponential correlation function: Exponential correlation function: (d) = exp(-(d) = exp(-
dd))
Covariate:Covariate:►Distance weighted stream powerDistance weighted stream power
Preliminary ResultsPreliminary Results
►Posterior quantitiesPosterior quantities Based on 1000 iterations (burn-in = 1000)Based on 1000 iterations (burn-in = 1000)
Posterior mean of the spatial Posterior mean of the spatial processprocess
Posterior SD of the spatial processPosterior SD of the spatial process
Posterior mean and SD forPosterior mean and SD for Pr(Y Pr(Yii = 2) = 2)
Posterior mean and SD forPosterior mean and SD for Pr(Y Pr(Yii = 5) = 5)
Posterior mean and SD forPosterior mean and SD for Pr(Y Pr(Yii ≤≤ 5) 5)
Future WorkFuture Work► Convergence and mixing for the spatial model Convergence and mixing for the spatial model
► Models and methods for large data setsModels and methods for large data sets Spectral parameterization of the spatial processSpectral parameterization of the spatial process
► Investigate different spatial correlation functions Investigate different spatial correlation functions and distance metricsand distance metrics TraditionalTraditional Stream basedStream based
► Model selection for the spatial modelModel selection for the spatial model
Funding and AffiliationsFunding and Affiliations
FUNDING/DISCLAIMERThe work reported here was developed under the STAR Research Assistance Agreement CR-829095 awarded by the U.S. Environmental Protection Agency (EPA) to Colorado State University. This presentation has not been formally reviewed by EPA. The views expressed here are solely those of the authors and STARMAP, the Program they represent. EPA does not endorse any products or commercial services mentioned in this presentation.
Megan’s research is also partially supported by the PRIMES National Science Foundation Grant DGE-0221595003.
CR-829095
Thank youThank you
Subset of data Subset of data ((nnsmall small = 82)= 82)
Sample path plot - ExampleSample path plot - Example
Surface for estimating Surface for estimating =(=(22,,))