Transcript Modelling dendritic ecological networks Erin E. Peterson| Research Scientist 31 January 2012 CSIRO DIGITAL PRODUCTIVITY FLAGSHIP.

Modelling dendritic ecological networks
Erin E. Peterson| Research Scientist
31 January 2012
CSIRO DIGITAL PRODUCTIVITY FLAGSHIP
Spatial Ecology
Geographic
Information
Science
Statistics
Landscape
Ecology
Aquatic
Ecology / Biology
NCEAS Spatial Stats for Streams Working Group
Overview
• Background
• Dendritic ecological network (DEN)
• Stream networks as DENs
• Challenges of modelling DEN data
• Taxonomy of network analyses
• Spatial statistical modelling on
stream networks: A brief overview
• Making methods accessible
• Where to from here
Dendritic ecological networks
Naturally exhibit a dendritic network topology
Dendritic ecological networks
Dendrites and dendritic spines
Dendritic ecological networks
Plant architecture
Dendritic ecological networks
Metallic dendrites
Dendritic ecological networks
River and stream networks
Some Definitions…
Network
Confluence
Outlet
Segment
Segment
or
Edge
Tributary
Watershed
Stream networks are unique
Streams form directed, dendritic networks
 Directional water flow is a fundamental component of
the system
 Form semi-restrictive corridors
 Active versus passive movement
Physical network structure matters
 May be almost linear to highly-complex, with many
branches
 Affects physical, chemical & biological processes
• Food web structure and interactions with terrestrial
environment
• Genetic diversity
• Species assemblages
Stream networks are unique
Sometimes Euclidean distance may not make sense…
Stream networks are unique
Disjunct conditions may occur upstream of confluences
Burned & debris flow
Stream networks are unique
Streams are embedded in 2-D terrestrial
environment
• Integrative reflection of the surrounding
terrestrial landscape
• Laterally connected: Watershed-FloodplainChannel
• Sub-watersheds are nested within one
another
• Physically connected by stream flow
• Neighboring watersheds may have similar
characteristics
Streams are unique
Mobile organisms respond to spatial arrangement of habitat
within or along the network, e.g. Fish and amphibians
 Temporally variable
Schlosser’s dynamic landscape model of
stream fish life history (Figure 2 from
Schlosser and Angermeier 1995)
Figure 5a from Isaak et al. 2010
Streams are unique
• Complex processes and landscape-scale patterns
• Represent heterogeneity or variability
• Produced by
• Biotic and abiotic conditions
• Unique spatial characteristics of the stream
• Ecological processes: hydrologic, physical, chemical and biotic
• Past conditions
• Disturbance
• Multi-scale processes & interactions within and between aquatic and
terrestrial environments produce complex, multi-scale patterns
Challenges of Modelling Streams Data
Conceptualisation of streams as DENs
Most statistical models
•
Ignore spatial relationships altogether
•
Ignore network relationships
Create a mismatch between analytical
approaches and our conceptualization of
the system as a DEN
• Limits our understanding of how streams
function
• Weakens our ability to make accurate and
unbiased predictions
• Reduces the effectiveness of management
actions
Challenges of Modelling Streams Data
Fundamental stream characteristics
• Dendritic network structure
• Connectivity
• Directional flow
• Spatio-temporal variability of habitat and
flow
• Dual spatial representation
Spatial Representation
Dual
2-D
Network
≠
=
(a)
(b)
(c)
Challenges of modelling streams data
Lack of data
• Spatially dense or continuous data is
expensive to collect
Unfamiliarity with statistical or mathematical
methods
Application to stream environments may not be
intuitive
• Graph-theoretic and metapopulation models
− Nodes represent habitat patches or
populations
− Edges represent functional
relationships: rate of dispersal
• In streams processes occur ON the network
Node
Node
Node
Node
C. Torgersen
http://www.upperdesplainesriver.org/profile.htm
Taxonomy of Network Analyses
Purpose: Provide a pragmatic conceptual framework
to help researchers
1. Understand the similarities & differences between the
processes of interest and the analytical methods
available
2. Select the most suitable method for their study
3. Integrate network analyses to acquire a more coherent
whole-of-system understanding
Peterson et al. (2013), Ecology Letters
Taxonomy of Network Analyses
River Network
NON
x1
(a)
ABOUT
x2
x3
x4
(b)
(c)
OVER
ON
ACROSS
x2
x1
x
x3
x4
(d)
(e)
1
2
3
(f)
Taxonomy of Network Analyses
Range of data types used to describe the physical structure of streams and the
structure/function of ecological processes
Fall into three general categories
1. Physical structure
2. Biological or physicochemical processes
3. Aggregation of (1) or (2)
Data type limits the type of analysis you can use
Taxonomy of Network Analyses
NON Network Analysis
• Ignore spatial relationships altogether
NON
• Ignore network-based spatial relationships
• Data: Physical sub-network structure, physicochemical
or biological processes
• Use spatially explicit covariates to represent:
– Sub-network structure, connectivity, or direction
– Habitat quality, proximity, connectivity, arrangement
– Least-cost path analyses and moving window
approaches
– Patch size, composition and distance measures
x
2
x
x
1
3
x
4
Questions: influence of physical network structure and in-stream habitat on
physicochemical and biological stream processes
Difficult to describe complex, multi-scale spatial processes/interactions within
the dual coordinate system through spatially explicit covariates alone
Taxonomy of network analyses
ABOUT Network Analysis
• Primary focus: physical network structure & connectivity of network itself
• Traditionally non-spatial: e.g. food webs at one location
• Spatially structured graphs
− Modified to represent the dual coordinate system
•
Temporally dynamic graphs: time-ordered networks
ABOUT
• Data:
– Whole-of-network: drainage density, fractal dimension
– Sub-network: confluence angle, stream order
Questions: Influence of structure on biological or
physicochemical processes in streams
•
e.g. population structure, habitat connectivity,
prioritization of conservation areas
Dale & Fortin (2010), Annu. Rev. Ecol. Evol. Syst.; Blonder et al. (2012), Methods Ecol. Evol.
Taxonomy of network analyses
ON Network Analysis (1)
•
•
•
Point data: physical sub-network structure
Raster-based metapopulation models
Account for dual spatial representation, network structure,
connectivity, and directionality
ON
x2
x3
x1
Questions: Investigate influence of network structure on
•
•
•
•
Fragmentation
Movement behaviour of organisms
Population distribution
Metapopulation persistence
x4
Advanced theoretical understanding of network structure, connectivity, and function
• Not typically used to understand impacts of land management on in-stream habitat and
organismal distributions
•
Lateral connectivity, in-stream movement is a function of distance, & in-stream habitat
heterogeneity is usually ignored
Fagan (2002), Ecology; Grant et al. (2007), Ecology Letters; Carrara et al. (2012), PNAS
Taxonomy of network analyses
ON Network Analysis (2)
• Point data: Physicochemical or biological processes
• Account for dual spatial representation, network
structure, connectivity, & directionality
• Spatial statistical models on stream networks
ON
x2
x3
x1
x4
Questions:
• Investigate influence of land-use on in-stream processes
• Study the influence of in-stream processes on another physicochemical or
biological response
• Make predictions at unobserved locations, with estimates of uncertainty
Not typically used to investigate the influence of network structure on stream
processes
Ver Hoef et al. (2006), Environ. Ecol. Stat.; Ver Hoef & Peterson (2010), JASA
Taxonomy of network analyses
OVER Network Analysis
• Data: Sub-network structure, physicochemical and
biological processes, or aggregation of those
characteristics
OVER
x
Questions: Summarizing over a network(s)
ACROSS Network Analysis
ACROSS
• Data: Whole-of-network structure or aggregation of
physicochemical/biological processes or sub-network
structure
1
2
3
Questions: Compare or contrast across networks or sets
of networks
Taxonomy of Network Analyses
Each method and taxonomic class accounts for structure,
connectivity, and directionality to some degree
Choice of method reflects
• Type of data
• Ecological question
Spatial Statistical Modelling on Stream Networks
A Brief Overview
 A moving-average approach for spatial
statistical models of stream network
data based on three spatial
relationships
For additional information
• Ver Hoef, Peterson, and Theobald (2006) Spatial statistical models that use flow and
stream distance. Environmental & Ecological Statistics, 13: 449-464.
• Ver Hoef and Peterson (2010) A moving average approach to spatial statistical
models of stream networks. The Journal of the American Statistical Association,
489: 6-18.
Spatial Statistical Modelling on Stream Networks
Distance Measures & Spatial Relationships
B
Euclidean distance
A
• As the crow flies
C
B
Euclidean
Hydrologic distance
Flow-unconnected
A
• As the fish swims
C
B
Hydrologic distance
A
Flow-connected
• As the water flows
C
Spatial statistical modelling on streams
Challenge: Spatial autocovariance models developed for
Euclidean distance may not be valid for hydrologic
distances
Not to scale. All lengths = 1
Ver Hoef et al. (2006), Environ. Ecol. Stat.
Spatial statistical modelling on streams
Autocovariances can be developed by creating random variables as the
integraton of a moving-average (MA) function over a white-noise random
process:
Where x and s are locations on the real line and g(x|θ) is the MA function
defined on R1, which is square integral. W(x) is Brownian motion,
Based on (1), the expression of a valid autocovariance between Z(s) and
Z(s+h):
Yaglom (1987)
Tail-up Models
where CTU is the covariance matrix
CTU  [ 2R(D /  )   2I]
 2  partial sill,   range,  2  nugget
W
R is matrix spatial correlation function
e.g., spherical, exponential, etc.
D is all pairwise stream distances
Ver Hoef & Peterson (2010), JASA
is the Hadamard (element-wise) product
W is "Weight" matrix
Tail-Down Models
if site 1 = site 2,
1
 (h, )
if flow-connected and h  α2 ,
2
2  1
CTD (a, b, h |  ,  )   
 2 (a, b, ) if flow-unconnected and max(a, b)  α2 ,
0
if h or max(a, b) >  2
where:
CTD is the covariance matrix,
a and b are the distance from each site to the common downstream junction
and h = a + b,
δ2 is the partial sill, α is the range, and
κ1 and κ2 are the matrix correlation functions for flow-connected and flowunconnected cases.
Mixture Models
Variance component approach
• Single model fit using a mixture of covariances
based on different spatial relationships
• Sum of positive-definite covariance matrices
• Models: Tail-up, Tail-down, Euclidean
Flexible Modelling Approach
• Measured and unmeasured variables at multiple
scales
• Spatial-weighting schemes for Tail-up models
2
2
2
2
Σ   Euc
REuc   down
Rdown   up
Rup   nug
I
where
REuc , Rdown , Rup
matrices of autocovariance values for Euclidean
(Euc), tail-down (down), and tail-up (up) models.
2
2
2
2
 Euc
, down
, up
, nug
are the variance components.
Spatial Statistical Modelling On Stream Networks
Provides a semi-continuous view of conditions across relatively
broad spatial scales
Making Methods Accessible
Multidisciplinary skills are required
 Knowledge of aquatic ecology
 Specialised skills using geographic information
systems (GIS)
 Spatial statistics
One person rarely has all of these skills
Knowledge transfer & methodological
uptake requires software/tools
http://ddsgeo.com
 Journal articles are not enough
 z observed 

  Xβ  ε, var(ε)  Σ(θ)
 z unobserved 
Making Methods Accessible
Suite of GIS and Statistical tools
ArcGIS
STARS
Custom Toolset
R
.ssn
SSN
Package
Peterson & Ver Hoef (2014), JSS; Ver Hoef et al. (2014), JSS
Making Methods Accessible – GIS Tools
Spatial Tools for the Analysis of River
Systems (STARS)
•
Geoprocessing Toolbox written in Python for
ArcGIS v10.2
Toolsets
•
Pre-processing: Identify unique topological
errors and relationships that are prohibited
•
Calculate: Derive spatial data needed to fit a spatial statistical model to
stream network data
•
Export: Export the spatial features, topological relationships, and
attribute information to a format that can be easily accessed using R
Making Methods Accessible – GIS Tools
Export – Create .ssn Object
.ssn directory:
• Contains topological, spatial, and attribute information needed to fit a spatial
statistical model
.ssn directory components:
• Sites & edges shapefiles
• Text file containing binary IDs for
each network in LSN
• Prediction sites shapefile: optional
Easily transferable
• If it’s in the .ssn folder, you need it
.ssn
edges.shp
net1.dat
sites.shp
net2.dat
preds.shp
net3.dat
Making Methods Accessible – R package
Conventions for spatial classes set out by Bivand et al. (2008, 2013)
• Co-authored sp package to extend R classes and methods for spatial data
• 130 other packages Depend or Import sp
•
Spatial regression, spatial point processes, spatial extremes modelling, disease mapping &
aerial data analysis, habitat selection, spatial ordination methods, spatial analysis of marker
data, population graphs network analysis of spatial conditional genetic covariance, etc…
sp classes and methods: S4 object
• Formally defined class structure for points, lines, polygons, grids
• Ensures data is in the “right” format
Following the sp conventions makes it easier to
• Move spatial data between R packages
• Use plotting, printing, subsetting, summarising, etc. tools in sp
• Interface directly between R and GIS
• spgrass6, RPyGeo, RSAGA
Bivand, Pebesma & Gomez-Rubio (2008, 2013), Applied Spatial Data Analysis with R
Software / Tools – SSN Package
SpatialStreamNetwork Class
• SpatialStreamNetwork is a unique spatial data structure
• Contains both point and line features: edges, sites, prediction sites
Extends SpatialLines: edges
• Same naming conventions for slots
• Network.line.coords: Unique coordinate system used to navigate
network
SSNPoints Class: observed sites & multiple sets of prediction points
• Similar to SpatialPointsDataFrame
• Different naming conventions for slots
• network.point.coords:
Making Methods Accessible
http://www.fs.fed.us/rm/boise/AWAE/projects/SpatialStreamNetworks.shtml
Tools:
• STARS: Spatial tools for the Analysis of River Systems
• SSN: R package for spatial statistical stream-network modelling
Learning materials
• Tutorials, vignettes, example datasets, papers
Example datasets for statisticians
• Promote the development of new methods for stream networks
>45,000,000 hourly records
>15,000 unique stream sites
>70 agencies
$10,000,000 data value
NorWeST Stream Temperature Project
The BLOB…it just keeps growing…
 40,397 summers of data swallowed
 380,000 stream kilometers of thermal ooze
The National Stream Internet Project
An analytical framework for creating new
information from old stream data
Dan Isaak, Erin Peterson, Dave Nagel, Jay Ver Hoef, Jeff Kershner
BIG DATA =
BIG POSSIBILITIES
Where to from here…
Where to from here…
Where to from here…
Where to from here…
Where to from here…
• How connected/fragmented populations?
• Is this connectivity likely to change through time?
• If there is fragmentation, is it due to physical barriers? Or
physiological barriers?
• What is the likely role that competition will play?
• How do we spatially prioritize restoration efforts now and into
the future?
Where to from here…
River Network
NON
x1
(a)
ABOUT
x2
x3
x4
(b)
(c)
OVER
ON
ACROSS
x2
x1
x
x3
x4
(d)
(e)
1
2
3
(f)
Need to integrate methods if we want to answer these questions
Where to from here…
Make methods more accessible to freshwater ecologists
Bring analytical methods in line with ecological
conceptualisation of stream networks as DENs
Provide a better scientific understanding of processes
Provide spatially and temporally explicit information to manage
vulnerable species now and into the future
Spatial Statistical Modelling On Stream Networks
Tail-down model
FU
• Drawback: Correlation between flow-connected cannot be
much > flow-unconnected locations at a fixed distance
• MA functions with heavy shoulders have > FU autocorrelation
FC
• MA with heavier tails have slightly less correlation for FU
sites
Ordered values of Ruc
Full discussion provided in Ver Hoef & Peterson (2010)
RUC 
Autocorrelation(FU)
Autocorrelation(FC)
FU=FC
Software / Tools – SSN Package
Binary ID storage & access is an issue
• ID length depends on network size and configuration
• long sequences of 1’s & 0’s
Relational database: SQLite
 Easy to install and access data using RSQLite Package & SQL queries
 Available for Windows, Unix, Linux, and Mac
 Free!
binaryID.db automatically created when .ssn folder imported &
SpatialStreamNetwork object generated
Where to from here…
Spatial statistical stream-network models
Engage with statisticians to develop/adapt spatial statistical methods for
streams
•
Occupancy models
•
Models for extremes
•
Big data
•
Spatio-temporal models
•
Multivariate data
•
Nonstationarity
•
Visualisation tools