Using the Maryland Biological Stream Survey Data to Test Spatial Statistical Models

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Using the Maryland Biological Stream Survey Data
to Test Spatial Statistical Models

• A Collaborative Approach to Analyzing Stream
  Network Data

Andrew A. Merton
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Overview

• The material presented here is a subset of the
  work done by Erin Peterson for her Ph.D.
• Interested in developing geostatistical models
  for predicting water quality characteristics in
  stream segments
• Data Maryland Biological Stream Survey (MBSS)
• The scope and nature of the problem requires
  interdisciplinary collaboration
• Ecology, geoscience, statistics, others

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Stream Network Data

• The response data is comprised of observations
  within a stream network
• What does it mean to be a neighbor in such a
  framework?
• How does one characterize the distance between
  neighbors?
• Should distance measures be confined to the
  stream network?
• Does flow (direction) matter?

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Stream Network Data

• Potential explanatory variables are not
  restricted to be within the stream network
• Topography, soil type, land usage, etc.
• How does one sensibly incorporate these
  explanatory variables into the analysis?
• Can we develop tools to aggregate upstream
  watershed covariates for subsequent downstream
  segments?

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Competing Models

• Given a collection of competing models, how does
  one select the best model?
• Is one subset of explanatory variables better or
  closer to the true model?
• Should one assume correlated residuals and, if
  so, what form should the correlation function
  take?
• How does the distance measure impact the choice
  of correlation function?

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Functional Distances Spatial Relationships
Geostatistical models are based on straight-line
distance
Straight-line Distance (SLD) Is this an
appropriate measure of distance? Influential
continuous landscape variables geology type or
acid rain (As the crow flies)
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Functional Distances Spatial Relationships
Distances and relationships are represented
differently depending on the distance measure
Symmetric Hydrologic Distance (SHD) Hydrologic
connectivity (As the fish swims)
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Functional Distances Spatial Relationships
Distances and relationships are represented
differently depending on the distance measure
Asymmetric Hydrologic Distance (AHD) Longitudinal
transport of material (As the sht flows)
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Candidate Models

• Restrict the model space to general linear
  models
• Look at all possible subsets of explanatory
  variables X (Hoeting et al)
• Require a correlation structure that can
  accommodate the various distance measures
• Could assume that the residuals are spatially
  independent, i.e., S ?2I (probably not best)
• Ver Hoef et al propose a better solution

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Asymmetric Autocovariance Models for Stream
Networks

• Weighted asymmetric hydrologic distance (WAHD)
• Developed by Jay Ver Hoef, National Marine Mammal
  Laboratory, Seattle
• Moving average models
• Incorporates flow and uses hydrologic distance
• Represents discontinuity at confluences

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Exponential Correlation Structure

• The exponential correlation function can be used
  for both SLD and SHD
• For AHD, one must multiply ? (element-wise) by
  the weight matrix A, i.e., ?ij aij ?ij, hence
  WAHD
• The weights represent the proportion of flow
  volume that the downstream location receives from
  the upstream location
• Estimating the aij is non-trivial Need special
  GIS tools (Theobald et al)

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GIS Tools
Theobald et al have created automated tools to
extract data about hydrologic relationships
between sample pointsVisual Basic for
Applications programs that

• Calculate separation distances between sites
• ? SLD, SHD, Asymmetric hydrologic distance (AHD)
• Calculate watershed covariates for each stream
  segment
• ? Functional Linkage of Watersheds and Streams
  (FLoWS)
• Convert GIS data to a format compatible with
  statistics software

SLD
AHD
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Spatial Weights for WAHD

• Proportional influence influence of each
  neighboring sample site on a downstream sample
  site
• Weighted by catchment area Surrogate for flow

• Calculate influence of each upstream segment on
  segment directly downstream
• Calculate the proportional influence of one
  sample site on another
• Multiply the edge proportional influences
• Output
• nn weighted incidence matrix

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Spatial Weights for WAHD

• Proportional influence influence of each
  neighboring sample site on a downstream sample
  site
• Weighted by catchment area Surrogate for flow

• Calculate influence of each upstream segment on
  segment directly downstream
• Calculate the proportional influence of one
  sample site on another
• Multiply the edge proportional influences
• Output
• nn weighted incidence matrix

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Spatial Weights for WAHD

• Proportional influence influence of each
  neighboring sample site on a downstream sample
  site
• Weighted by catchment area Surrogate for flow

• Calculate influence of each upstream segment on
  segment directly downstream
• Calculate the proportional influence of one
  sample site on another
• Multiply the edge proportional influences
• Output
• nn weighted incidence matrix

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Spatial Weights for WAHD

• Proportional influence influence of each
  neighboring sample site on a downstream sample
  site
• Weighted by catchment area Surrogate for flow

• Calculate influence of each upstream segment on
  segment directly downstream
• Calculate the proportional influence of one
  sample site on another
• Multiply the edge proportional influences
• Output
• nn weighted incidence matrix

survey sites stream segment
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Spatial Weights for WAHD

• Proportional influence influence of each
  neighboring sample site on a downstream sample
  site
• Weighted by catchment area Surrogate for flow

• Calculate influence of each upstream segment on
  segment directly downstream
• Calculate the proportional influence of one
  sample site on another
• Multiply the edge proportional influences
• Output
• nn weighted incidence matrix

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Parameter Estimation

• Maximize the (profile) likelihood to obtain
  estimates for ?, ?, and ?2

Profile likelihood
MLEs
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Model Selection

• Hoeting et al adapted the Akaike Information
  Corrected Criterion for spatial models
• AICC estimates the difference between the
  candidate model and the true model
• Select models with small AICC

where n is the number of observations, p-1 is the
number of covariates, and k is the number of
autocorrelation parameters
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Spatial Distribution of MBSS Data
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Summary Statistics for Distance Measures

• Distance measure greatly impacts the number of
  neighboring sites as well as the median, mean,
  and maximum separation distance between sites

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Comparing Distance Measures

• The selected models (one for each distance
  measure) were compared by computing the mean
  square prediction error (MSPE)
• GLM Assumed independent errors
• Withheld the same 100 (randomly) selected records
  from each model fit
• Want MSPE to be small

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Comparing Distance MeasuresPrediction
Performance for Various Responses
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Maps of the Relative Weights

• Generated maps by kriging (interpolation)
• Predicted values are linear combinations of the
  observed data, i.e.,

Z1 is the observed data, Z2 is the predicted
value, ?11 is the correlation matrix for the
observed sites, and ? is the correlation matrix
between the prediction site and the observed sites
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Relative Weights Used to Make Prediction at Site
465
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Residual Correlations for Site 465
Straight-line
General Linear Model
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Some Comments on the Sampling Design

• Probability-based random survey design
• Designed to maximize spatial independence of
  survey sites
• Does not adequately represent spatial
  relationships in stream networks using hydrologic
  distance measures

244 sites did not have neighbors Sample Size
881 Number of sites with 1 neighbor 393 Mean
number of neighbors per site 2.81
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Conclusions

• A collaborative effort enabled the analysis of a
  complicated problem
• Ecology Posed the problem of interest, provides
  insight into variable (model) selection
• Geoscience Development of powerful tools based
  on GIS
• Statistics Development of valid covariance
  structures, model selection techniques
• Others e.g., very understanding (and
  sympathetic) spouses