State-Space Models for Within-Stream Network Dependence

1
State-Space Models for Within-Stream Network
Dependence
This research is funded by U.S.EPA Science To
Achieve Results (STAR) Program Cooperative Agreeme
nt
CR - 829095

• William Coar
• Department of Statistics
• Colorado State University
• Joint work with F. Jay Breidt

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Disclaimer

• The 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 presenter and STARMAP, the Program (s)he
  represents. EPA does not endorse any products or
  commercial services mentioned in this
  presentation.

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Outline

• Introduction to the problem
• Evolution of state-space models
• Likelihood
• Missing data
• Kalman recursions
• EM algorithm
• Simulation example
• Future work

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Consider a simple stream network
Y1 Y2 Y3 Y4

• Two upstream reaches merge together to create
  downstream reaches.
• Suggests a natural dependency on upstream
  reaches.
• Autocorrelation can arise from water flowing from
  reach to reach.
• Logical ordering in space.

Y5 Y6
Downstream
Y7
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The Beginnings

• Expressing a measurement on a reach in terms of
  its upstream contributors such that
• where .

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The Beginnings

• This is also the modified Cholesky decomposition
  of S-1
• For any Y(µ,?), there exists a unit lower
  triangular matrix T with corresponding diagonal D
  such that TYZ where Z(0,D).
• Simplifying T can allow for dependencies
  similar to autoregressive structures
  in time series.
• ie, a measurement depends only on its two
  immediate upstream neighbors.
• in the
  simple example.
• Suggestive of a more general state-space model.

Y1 Y2 Y3 Y4
Y5 Y6
Y7
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State-Space Model

• Define a state-space representation by
• with W(t)N(0,R(t)), V(t)N(0,Q(t)),
  and V(s) uncorrelated with W(t) for all s and t.
  Further assume that W(t) and V(t) are
  uncorrelated with all X(s1), where s1 is any
  first order reach.

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Downstream Filter

• Best mean square predictors under Normality are
• Predict X(t) given upstream information
• Update with observed information from Y(t)
• where .

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Likelihood

• Use the innovations and variances from the
  downstream filter
• In the case where data are available for every
  reach in the network, the likelihood is easily
  expressed in terms of these innovations
• where n is the total number of reaches in the
  stream network.

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EM Algorithm

• The likelihood for missing data can be difficult
  to express.
• E-Step
• Predict, update, smooth based on current
  estimates of model parameters.
• Form an approximation to the likelihood by
  filling in the missing values with smoothed
  estimates.
• The M-Step
• Maximization of the approximation to the
  likelihood in order to obtain new parameter
  estimates for the next iteration.
• Iterate until revised parameter estimates
  stabilize.
• Since log-likelihood decreases with each
  iteration, estimates should converge to MLE.

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Upstream Smoother

• Start with the very last reach in the network.
• Smooth two at a time using information from the
  filtered as well as smoothed downstream values.
• Estimate based on observations from
  the entire network with the conditional
  expectation .
• Recursive relationship results in smoothed
  estimates
• with variance
• where .

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Other Tree Type Smoothers

• Each reach as a parent that creates two children
• Existing work Huang Cressie (1997) and Chou
  (1994) for uptree filtering (fine-coarse) and
  downtree smoothing (coarse-fine)
• Model different resolutions
• Assumption that children are independent
  conditioned on the parent.
• Violated in the stream network model considered.

Parent
Child
Child
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Example
First order reaches up in the mountains
xmissing value
Fifth order reach on the plains
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Example

• Consider a network that has 39 different reaches
• 20 first order,19 higher order
• Let k be the Strahler order of reach t created by
  two reaches of order i and j.
• State-Space representation of
• with .
• Assumptions about V(t)
• Cov(V(s),V(t))0 for s ? t
• Cov(V(t),X(s1))0 for any first order reach s1

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Parameter Estimation

• Total of 12 parameters to estimate based on 33
  stream segments (6 missing values).
• 6 different ? parameters to estimate in this
  model.
• 5 different (conditional) variances to estimate.
• 1 variance parameter from first order.
• Most parameters will be estimated with few
  observations.
• Only a few reaches will contribute to estimating
  each ?.
• Suggests looking at parametric models for ?.
• Need a much larger stream network to achieve more
  reasonable parameter estimates.

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Kalman Recursions

• Downstream Filter (Y(t)X(t))
• The filtered value is either the observed Y(t),
  or its conditional expectation given the two
  immediate upstream filtered values.
• Variance is either 0 (if Y(t) is observed) or the
  prediction error variance of Y(t) given the two
  immediate upstream filtered values.
• Upstream Smoother
• Smooth two at a time, Y(u1) and Y(u2).
• Either the observed value or the conditional
  expectation of Y(ui) given all reaches with
  observed measurements.
• Need to know the logical order of flow

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Parameter Estimates
Iterate ?21 ?31 ?32 ?33 ?43 ?54
6, 0.701 -0.543 0.725 1.087 0.226 -0.526
7, 0.703 -0.550 0.723 1.069 0.247 -0.526
8, 0.705 -0.578 0.722 1.008 0.280 -0.526
True .4 .2 .55 .6 .35 .45

6, 1.245 7.761 0.0087 0.842 1.23e-32 2.951
7, 1.250 8.376 0.009 0.746 1.23e-32 2.950
8, 1.252 9.030 0.009 0.633 1.23e-32 2.949
True 3 2.5 2 3 1.5 4
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Smoothed Data Values
1
2
1 2 3 4 5 6
6, 0.759 0.759 2.891 0.676 -0.147 -1.690
7, 0.747 0.747 2.915 0.679 0.405 -1.683
8, 0.744 0.744 2.927 0.681 0.992 -1.679
True 0.946 1.029 2.994 0.382 -2.764 -2.415
3
4
6
5

• More iterations in the EM algorithm
• Better model for the coefficient parameters
• Plot estimates from regression against covariates
  (regressogram)
• Re-compute MLE based on new parametric model
  suggested by the regressogram

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Future Work

• Work with real data from larger networks.
• Obtain better initial estimates.
• Investigate EM convergence.
• Use reach-specific covariate information such as
  location within a reach, inflow from upstream
  reaches, etc.
• State space representations that allow for larger
  classes of models than the AR structure
  considered here.
• Allow for upstream measurements on the same
  reach.