Kinds and complexity of models

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Kinds and complexity of models
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Many kinds

• remember,can be named after.
• the approach (individual-based models)
• the assumption (null models)
• the math (matrix models)
• the relationship (non-linear models)
• the process (stochastic models)
• the application (management models)
• as a result (and as we discussed) succinct model
  descriptions are an exercise in adjective use

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Keep in mind what characterizes your model

• Is time an important variable?
• Do you need to monitor the behaviors of
  individuals?
• Are you trying to capture stochastic processes?
• Can your submodels be modelled in different ways?

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Selecting modelling type

• goes hand-in-hand with mathematical /
  quantitative formulation
• Are mean parameter values sufficient? How is
  parameter variation handled? (static v.
  individual-based models)
• some model structures mirror the systems they are
  intended to mimic
• today, introduce individual-based models (brief
  example of static model)
• model complexity should scale with the amount of
  available data
• dynamic model complexity may prevent its use in
  data poor situations
• static models better in data poor situations

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Selecting model complexity or articulation

• Articulation determined by the number of model
  components and use of space and time
• Level of complexity determined by.
• knowledge of system processes
• Ask yourself Which model components are most
  relevant to the problem at hand?
• comprehensiveness of supporting data set
• Naturally, certainty of the outcome proportional
  to quality of the data
• error propagation
• The adage applies Garbage in, garbage out
• Of course even making garbage can be useful
• Promotes better understanding of the system
• Highlights areas where data are poor or lacking
  altogether

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Model complexity

• In some circumstances, sound theory can offset a
  poor data set
• can be infinitely reductionist promotes model
  complexity
• resort to higher order more holistic phenomena
• more on this later
• Recall the pollen model
• Interested in proportion released that arrives at
  some downstream habitat, absolute numbers dont
  matter (for now)
• mechanism much less important than pollen
  production (most will be released)

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Model complexity

• Leads to many possible models of varying
  complexity gtgt model selection and AIC objective
  basis for identifying the best among competing
  candidate models
• AIC nlog(RSS/n)2 2K, where K is the number
  of model parameters (1)
• want low AIC
• Increasing number of model parameters must
  sufficiently improve performance (reduce residual
  sum of squares) to offset its own negative
  impact on AIC
• more on this next year
• Aggregation unification of system components
  that can be treated as homogenous and modelled as
  one component
• continuum of aggregations possible
• truly individual-based models and many static
  models represent 2 ends of that continuum
• Example
• IBM each individual in the population described
  by unique values
• many static the population is represented by one
  set of values
• Complexity decreases with aggregation

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Model complexity
Hypothetical relation between model aggregation
and complexity
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Individual-based models (IBM)

• simulation model
• usually dynamic and reductionist
• virtually reproduce the behavior of a system
• tuned to specific application typically not well
  suited to making generalizable ecological
  predictions (Maynard Smith 1974)
• when appropriate
• interactions between organisms or groups of
  organisms
• general models do not capture sufficient detail
  of idiosyncratic systems
• accounting for wide variability among individuals
  is important
• tools
• object-oriented programming (OOP) methods map
  well to individual-based modelling
• e.g. OOP Classes gt Objects
• IBM Populations gt Individuals

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IBM
Bird migration model example
Illinois land cover
Echo strength
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IBM gt Migration model
Basis for approach

• Patterns on radar represent emergent properties
  derived from the behaviors of many individuals
• e.g. departure time, flight speed, flight
  direction, flight altitude, habitat preference
• Through modeling, this project attempts to
  determine what behavioral, spatial, and temporal
  parameters give rise to patterns of seen on radar
• Necessary to model individual variation, hence IBM

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IBM gt Migration model
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IBM gt Migration model
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IBM gt Migration model

• Incomplete list of variables pertaining to how
  birds appear on WSR-88D
• Time
• Ti elapsed time, i0 at the moment of first
  takeoff constant
• Radar location and geometry
• xr E cartesian position of radar constant
• yr N cartesian position of radar constant
• fr angle of elevation or tilt of the radar
  reflector constant
• I index of refraction, typically 4/3 earths
  radius model (constant)
• hr height (ASL) of the radar reflector constant
• hbR height (ASL) of the beam at range,
  R calculate
• Target location
• xti E cartesian position at Ti model
• yti N cartesian position at Ti model
• Rri line of sight range from the radar at
  Ti calculate
• hti height (ASL) of the target at Ti calculate
• htbi height of the target within beam at
  Ti calculate

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So what
IBM gt Migration model

• Determine avian distributions and flight
  behaviors across many species. Results reflect
  generalizable characteristics of avian behavior
  during flight.
• Estimate numbers of migrants departing sub-pulse
  volume sized habitats essentially, address the
  displacement problem with outcomes potentially
  important for avian conservation.
• Estimate proportion of targets departing
  different habitat types. Approximate what
  proportion of migratory pool represented by
  forest species, grassland species, etc.
• Provide foundation for understanding varied radar
  phenomenology (aspect effects, ascent and
  reorientation, etc.)
• Predict behavioral responses to weather, terrain,
  etc not yet known to science

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Approach to modeling
Define problem
Conceptual Diagram
Mathematical formulation
Parameterization
Verification
Sensitivity analysis
Calibration
Validation
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Approach to modeling
Define problem
Conceptual Diagram
Mathematical formulation
Parameterization
Verification
Sensitivity analysis
Calibration
Validation
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Parameterization, Verification, and sensitivity
analysis interact

• Parameterization process of establishing
  correct values for
• model variables
• Verification a verified model behaves the
  way a modeller
• expects check parameterization
• Sensitivity analysis measures effect of changes
  in parameter
• values on response variable
• Iterative process parameterization gt
  verification gt sensitivity
• .repeat as needed refining parameters with
  each iteration.

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Parameterization
Parameterization, Verification, Sensitivity

• parameter data sources
• literature
• usually results in a range of values
• e.g. migrant airspeeds (Larkin 1991)

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PVS gt Parameterization

• Sources (cont)
• data collection
• and if you forget your parametersnever fear
• Handbook of Ecological Parameters and
  Ecotoxicology Jorgensen et al. (2000)
• 120,000 ecological parameters
• parameter value selection not a random process
• e.g. If you compare all possible outcomes of 10
  values each for 10 variables 1010 possibilities
• systematic trial and error
• test submodels independently when possible
• systematically vary parameters for one or two
  variables within a prescribed range
• will involve many model runs

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Back to migration model.
PVS gt Parameterization

• Incomplete list of variables pertaining to how
  birds appear on WSR-88D
• Time
• Ti elapsed time, i0 at the moment of first
  takeoff constant
• Radar location and geometry
• xr E cartesian position of radar constant
• yr N cartesian position of radar constant
• fr angle of elevation or tilt of the radar
  reflector constant
• I index of refraction, typically 4/3 earths
  radius model (constant)
• hr height (ASL) of the radar reflector constant
• hbR height (ASL) of the beam at range,
  R calculate
• Target location
• xti E cartesian position at Ti model
• yti N cartesian position at Ti model
• Rri line of sight range from the radar at
  Ti calculate
• hti height (ASL) of the target at Ti calculate
• htbi height of the target within beam at
  Ti calculate

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PVS gt Parameterization gt Migration model
IBM sampling from a normal distribution Normal
distribution determined from pub. data used to
assign speeds to individuals
Diehl Yufang least squares
This approach used to assign values for other
flight behaviors (direction, height, rate of
climb, etc).
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Migrant habitat selection
PVS gt Parameterization gt Migration model

• Simplify landscape
• Aggregate similar habitat types
• groupings share similar migrant preference
• emphasize most dominant types
• Decompose 17 habitat types into 3 forest,
  grassland, agriculture
• Assign relative migrant abundances, partially
  from literature
• Forest preferred over agriculture 1001
• Grassland preferred over agriculture 331
• (Develop algorithm to populate landscape with
  birds in stopover based on those abundances)

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10 km
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PVS gt Parameterization

• parameter selection can be automated
• requires some sophistication
• must meet objective criteria e.g. Y similar to
  SD (from Jorgensen and Bendoriccho 2001)
• Y ?((xc-xm)2/xma)/n1/2
• where xc is the computed value, xm is the
  measured value (from literature, data
  collection), xma is mean measured value, n is the
  number of measured or computed values. Goal is
  to automatically re-compute xc to minimize Y.

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PVS gt Parameterization gt Automated

• beware converging on local minimum
• Model can be solved for a range of values until
  model converges on an optimal solution, i.e.
  reaches some minimum (as in minimizing Y above)
• generally only a problem in complex automated
  parameterization procedures involving several
  variables
• risk increases if range of values too narrow
• e.g. testing xc between ranges indicated in red
  may cause an algorithm to converge on a local
  minimum

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PVS gt Parameterization gt Automated
Migration model iterative reparameterization
Bird model may iteratively optimize correlation
between observed (radar) and expected (model).
Other spatially explicit metrics also being
considered.
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PVS gt Parameterization

• sound application of ecological theory can offset
  lack of data when parameterizing a model
• thermodynamic laws dictate heat loss based on
  surface area and temperature difference
• remember allometric relations many variables
  scale with body size in predictable ways
• these scaling relations proven remarkably
  generalizable

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Scaling laws
PVS gt Parameterization

• Generality has been called the holy grail of
  ecology. (TREE 99-14)
• Physics envy
• Generality in ecology has been elusive, though
  recent work in allometric scaling laws come as
  close as any.
• What are scaling laws?

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PVS gt Parameterization gt Scaling laws

• Body size affects rates of many biological
  processes. Most biological phenomena scale with
  body mass raised to the quarter power (e.g.,
  M1/4). These include metabolic rates, population
  growth, embryonic growth, life span, etc.
• Yet no theory explains why so many biological
  processes exhibit quarter power scaling.

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PVS gt Parameterization gt Scaling laws
West et al. (1997) beautifully explain metabolic
scaling relations in plants and animals through
branching transport models.
.quarter power scaling is perhaps the single
most pervasive theme underlying all biological
diversity. Notice broad generality of this
static model as opposed to specificity of IBM.
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Allometric scaling models show remarkable
predictive power
PVS gt Parameterization gt Scaling laws
from West et al. 1997
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Success of recent scaling models
PVS gt Parameterization gt Scaling laws

• Seminal paper West et al. 1997, Science
  276122-126.
• The theory of West, Brown Enquist and its
  extensions quantitatively explain and predict a
  large body of empirical measurements taken across
  broad scales for a variety of biological
  phenomena this includes not only quarter power
  allometric exponents but, just as importantly,
  details of hierarchical branching and
  hydrodynamic flow.
• - Savage et al. 2004

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PVS gt Parameterization

• If parameterization fails to produce acceptable
  correspondence
• some important aspect of the biology is missing
• boundaries on parameters are too narrow
• observed data is questionable
• observations do not reflect the dynamics of the
  model
• e.g. resolution is not matched in space or time
• You want to know fish growth rates in different
  parts of a lake but only have measurements for
  the entire lake without regard to location
  within.