Classical and Bayesian nonlinear regression applied to hydraulic rating curve inference.

1
Classical and Bayesian nonlinear regression
applied to hydraulic rating curve inference.

• Construction and uncertainty analysis of
  stage-discharge rating curves

2
Motivation for this work

• River hydrology
• Management of fresh water resources
• Decision-making concerning flood risk
• Decision-making concerning drought
• River hydrology gt How much water is flowing
  through the rivers?
• Key definition discharge,
• amount of water passing through a
• cross-section of the river each time
• unit

3
Key problem

• Discharge is expensive. But hydrologists wants
  discharge time series!
• Solution Find a relationship between discharge
  and something that is inexpensive to measure.
• Usually, that something is water level.
• This job must be done over and over again Need
  solid tools for finding such relationships.
• Discharge measurements are uncertain gt need
  statistical tools
• Program must be easy for hydrologists to use gt
  User friendliness in statistics?

4
Water level definitions

• Stage the height of the water level at a river
  site

h
Q
h0
Datum, height0
5
Stage-discharge relationship
h
QC(h-h0)b
Q
h0
Datum, height0
Discharge, Q
6
Stage-discharge relationship

• Simple physical attributes
• Q0 for h?h0
• Q(h2)gtQ(h1) for h2gth1gth0
• Parametric form suggested by hydraulics (Lambie
  (1978) and ISO 1100/2 (1998)) QC(h-h0)b
• Parameters may be fixed only in stage intervals -
  segmentation

h
h
width
Q
7
Calibration data and statistical model

• n stage-discharge measurements.
• Discharge is error-prone.
• Statistical inference on C,b,h0 nonlinear
  regression
• QiC (hi-h0)b Ei, where EilogN(0,?2) i.i.d.
  noise and i?1,,n
• qiab log(hi-h0) ?i, where ?iN(0,?2) i.i.d.
• Problem Enable hydrological engineers to
  estimate Q(h)C(h-h0)b and evaluate the
  calibration uncertainty.

8
One segment fitting, the old way

• Guess or make approximate measurement of h0. Then
  linear regression on qi vs log(hi-h0).
• For each plausible value of h0, do linear
  regression. Choose the h0 that gives least SSE
  for the regression.
• Same as doing max likelihood inference on the
  model qiab log(hi-h0) ?i ,
  i?1,,n
• Means that uncertainty analysis becomes available
  (?).
• Studied by Venetis (1970).
• Also by Clarke (1999).

9
Problems concerning classic one segment curve
fitting

• Sometimes exhibits heteroscedasticity.
• Sometimes there's no finite solution!
• Found a set of requirements that ensures finite
  estimates.
• In practice, broken requirements means no finite
  estimates.
• The model can produce broken requirements for any
  set of stage measurements!
• Parameter estimators have infinite expectancy -gt
  Uncertainty inference becomes difficult!
• Explored in paper 1, Reitan and Petersen-Øverleir
  (2006) and in the appendix, Reitan and
  Petersen-Øverleir (2005).

10
Bayesian one segment fitting

• Based in the same data model, but with a prior
  distribution to the set of parameters. The
  Bayesian study of this resulted in paper 2,
  Reitan and Petersen-Øverleir (2008a).
• Bayesian analysis of other models done by Moyeeda
  and Clark (2005) and Árnason (2005).

11
Pros and cons

• Upsides
• Encodes hydraulic knowledge.
• Can put softer restrictions on the parameters.
• Finite estimates.
• Natural uncertainty measures.
• Downsides
• Requires heavier numerical methods (MCMC).
• Coming up with a prior distribution can be hard.
• Also sometimes exhibits heteroscedasticity.

12
Input - prior distribution

• Prior distribution form as simple as possible.
• Prior knowledge either local or regional.
• Regional knowledge can be extracted once and for
  all.
• At-site prior knowledge can be set through asking
  for credibility intervals.

13
Output estimates and uncertainties

• Estimates expectancies or medians from the
  posterior distribution.
• Uncertainty credibility intervals of parameters
  and the curve itself, Q(h)C (h-h0)b.

14
Segmentation

• Original idea Divide the stage-discharge
  measurement into sets and fit Q(h)C (h-h0)b
  separately for each segment. This can fit a wider
  range of measurement sets.

Segment 2
h
Segment 1
Intersection
Q
15
Problems with manual segmentation
h

1. Uncertainty analysis of manual decisions not
   statistically available.
2. Curves fitted to two neighbouring sets may not
   intersect.
3. Two such curves may have intersections only
   inside the sets.

Jump in the curve
Q
h
Q
16
Statistical model for segmentation the
interpolation model

• Idea Make a model with segments and let the data
  be attached to that model.
• Model for k segments, introduce k-1 segment
  limits parameters, hs,1, , hs,k-1. For a
  measurement hs,j-1lthilt hs,j assume
  qiajbj log(hi-h0,j) ?i.
• Make sure theres continuity by sacrificing one
  of the parameters in the upper segments (aj for
  jgt1).
• Goal Make inference on all parameters in this
  model. Also, make inference on k.

17
Frequentist inference on the interpolation model

• Segmented model first formulated and treated by
  using the maximum likelihood method in paper 3,
  Petersen-Øverleir and Reitan (2005).
• Problems
• Possibility of infinite parameter estimates
  inherited from one segment model. (Much more
  likely than usual for upper segments.)
• Multi-modality and discontinuous derivative of
  the marginal likelihood of changepoint
  parameters, hs,j.
• Inference of k through AIC or BIC?

18
Bayesian inference on the interpolation model

• Need prior distribution of changepoints, hs,j,
  and number of segments, k.
• MCMC for each sub-model characterized by k.
• Importance sampling for posterior sub-model
  probability, Pr(kD).
• Input Data, prior probability of each sub-model
  and prior for the parameter set of each
  sub-model. (Can be regional or partially
  regional. Set by asking for credibility
  intervals.)
• Output Pr(kD) and posterior dist. of all
  parameters for each k. (Summarised by estimates
  and credibility intervals.)

19
Output example for interpolation model inference
Q
20
Problems with Bayesian treatment of segmented
models

• Difficult to make efficient inference algorithms
  (but a semi-efficient one has been made).
• Changepoints only inside the dataset (thus the
  interpolation model). Extrapolation uncertainty
  underestimated because there can be changepoints
  outside the dataset.
• Solution(?) The process model, a new model where
  the segments appear through a process.
• Problems with the process model Very inefficient
  algorithms. Difficult to implement all sorts of
  relevant prior knowledge.
• Middle ground? Use changepoints of most sub-model
  from the interpolation model as data in inference
  about the changepoint process. Process model used
  for extrapolation of the curve.

21
References

• Árnason S (2005), Estimating nonlinear
  hydrological rating curves and discharge using
  the Bayesian approach. Masters Degree, Faculty of
  Engineering, University of Iceland
• Clarke, RT (1999), Uncertainty in the estimation
  of mean annual flood due to rating curve
  indefinition. J Hydrol, 222 185-190
• ISO 1100/2. (1998), Stage-discharge Relation,
  Geneva
• Lambie JC (1978), Measurement of flow -
  velocity-area methods. Hydrometry Principles and
  Practices, first edition, edited by R.W. Herschy,
  John Wiley Sons, UK.
• Moyeeda RA, Clarke RT (2005), The use of Bayesian
  methods for fitting rating curves, with case
  studies. Adv Water Res, 288807-818
• Petersen-Øverleir A, Reitan T (2005), Objective
  segmentation in compound rating curves. J Hydrol,
  311 188-201
• Reitan T, Petersen-Øverleir A (2005), Estimating
  the discharge rating curve by nonlinear
  regression - The frequentist approach.
  Statistical Research Report, University of Oslo,
  Preprint 2, 2005 Available at http//www.math.uio
  .no/eprint/stat report/2005/02-05.html
• Reitan T, Petersen-Øverleir A (2006), Existence
  of the frequentistic regression estimate of a
  power-law with a location parameter, with
  applications for making discharge rating curves.
  Stoc Env Res Risk Asses, 206 445-453
• Reitan T, Petersen-Øverleir A (2008a), Bayesian
  power-law regression with a location parameter,
  with applications for construction of discharge
  rating curves. Stoc Env Res Risk Asses, 22
  351-365
• Venetis C (1970), A note on the estimation of the
  parameters in logarithmic stage-discharge
  relationships with estimation of their error,
  Bull Inter Assoc Sci Hydrol, 15 105-111