Making rating curves the Bayesian approach

1
Making rating curves - the Bayesian approach
2
Rating curves what is wanted?

• A best estimate of the relationship between stage
  and discharge at a given place in a river.
• The relationship should be on the form
  QC(h-h0)b or a segmented version of that.
  Qdischarge, hstage.
• It should be possible to deal with the
  uncertainty in such estimates.
• There should also be other statistical measures
  of the quality of such a curve.
• These measures should be easy to interpret by
  non-statisticians.

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Making rating curves the old fashioned way

• For a known zero-stage, the rating curve can be
  written as qabx, where qlog(Q), xlog(h-h0)
  and alog(C).
• For a set of measurements, one can then do linear
  regression with q as response, x as covariate and
  a and b as unknown linear parameters. Minimize SS
  analytically (standard linear regression).

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The old approach handling c-h0

• The problem is that the effective bottom level,
  h0-c, is not known.
• Solution Minimize SS by stepping through all
  possible values of c.
• The advantage This is the same as maximizing the
  likelihood for the regression problem qiab
  log(hic)?i or QiC (hi-h0)b Ei where ?i
  N(0,?2) is iid noise and Ei e?i .
• This model makes hydraulic and statistical sense!

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Problems with the old approach

• We have prior information about curves that we
  would like to use in the estimation.
• Inference and statistical quality measures are
  difficult to interpret.
• Difficult to get a grip on the discharge estimate
  uncertainty.
• There is a chance that one gets infinite
  parameter estimates using this method!

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Bayesian statistics

• Frequentistic treats the parameters as fixed and
  finds estimators that will catch their values
  approximately.
• Bayesian treats the parameters as having a
  stochastic distribution which is derived from the
  observations and to prior knowledge.
• Bayes theorem f( ? D) f( D ?)f(?)/f(D)
  where f stands for a distribution, D is the data
  set and ? is the parameter set.

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Prior knowledge

• Prior info about a and b can be obtained from
  already generated rating curves (using the
  frequentistic approach) or by hydraulic
  principles.
• Prior info about the noise can be obtained from
  knowledge about the measurements.
• Problem Difficult to set the prior for the
  location parameter h0-c, but we know it will not
  be far below the stage measurements.

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Prior knowledge of a and b from the database
Histogram of generated as from the database.
Normal approximation seems ok.
Histogram of generated bs from the database.
Normal approximation seems less fine, but is used
for practical reasons.
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Bayesian regression

• Data given parameters is the same here
  qiab log(hic)?i . Dhi, qii1n
• Problem even though we have prior info, this
  does not give us the form of the prior f(?),
  ?(a,b,c,?2).
• If the priors are on a certain form, one can do
  Bayesian linear regression analytically qiab
  xi?i for xilog(hic) for a given c.
• Same thought as for the frequentistic approach,
  handle a,b and ?2 using a linear model, and
  handle c using discretization.

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Problems with Bayesian regression

• While this gives us the form of f(a,b,?2), it
  does not give us the form of f(c).
• We know that the stage levels are not too far
  above the zero-level. Wed like to code this
  prior info but we dont want to use the stage
  measurement (using them both in the prior and the
  likelihood).
• Jeffreys priors containing the covariates is a
  general problem with the Bayesian regression
  approach! Ok, if you really are in a regression
  setting, but this is not the case here.

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Problems with the first Bayesian approach

• The form that makes the linear regression
  analytical is rather strange.
• It requires the form of the prior for ?2 which
  influences the priors for (a,b). However, prior
  info about these two would be better kept
  separate.
• Difficult to set the prior info for users.
• Expected discharge is infinite in this approach!
  (Median will be finite.)

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A new Bayesian regression approach

• Using a semi conjugate prior, (a,b)N2,
  independent of ?2IG, we separate prior
  knowledge about a,b and ?2.
• We can no longer handle (a,b,?2) analytically for
  known c.
• However, (a,b,c,?2) can be sampled using MCMC
  methods.
• The sampling method must be effective, since
  users do not want to wait to long for the results.

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A graphical overview of the new model
?a Va ? ?a Vb ?
? ?
Hyper-parameters
a b
?2
Parameters
qi
hi
Measurements
For i in 1,,number of measurements
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Sampling methods and efficiency

• Naïve MCMC The Metropolis algorithm. Problem
  (a,b,c) are extremely mutually dependent.
• Metropolis or independence-sampler for c, Gibbs
  sampling for (a,b, ?2). Dependency of (a,b,c)
  makes trouble here, too.
• Solution Sample (a,b,c,?2) together and then do
  a Metropolis-Hastings accepting. Sample c using
  first adaptive Metropolis, then indep. sampler.
  Sample (a,b,?2 ) given c and previous ?2 using
  Gibbs-like sampling. Then accept/reject all four.

?i-12
?i2
ai,bi
ci
Iteration i-1 i
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Estimation based on simulations

• We can estimate parameters using the sampled
  parameters by either taking the mean or the
  median.
• We can estimate the discharge for a given stage
  value, either by mean or median discharge from
  the sampled parameters or by discharge from the
  mean or median parameters.
• Simulations show that median is better than mean.

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Inference based on simulations

• Uncertainty in the parameters can be established
  by looking at the variance of sampled parameters.
• Credibility intervals can be arrived at from the
  quantiles of the parameters.
• Discharge uncertainty and credibility intervals
  can be obtained by a similar approach to the
  discharge for the drawn parameters.

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Example rating curve with uncertainty
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Example prior to posterior
Prior of b.
Posterior of b.
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Example - diagnostic plots
Scatter plot of simultaneous samples from a and
b. Note the extreme correlation between the
parameters.
Residuals. Note the trumpet form. There is
heteroscedasticy here, which the model does not
catch.
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What has been achieved

• Discharge estimates with lower RMSE than
  frequentistic estimates.
• Measures of estimation uncertainty that are easy
  to interpret.
• Hopefully, quality measures should be less
  difficult to understand.
• The distribution of parameters can be used for
  decision problems. (Should we do more
  measurements?)

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What remains

• Multiple segmentation.
• Need to find good quality measures in addition to
  estimation uncertainty. Possibility Calculate
  the posterior probability of more advanced
  models.
• Learning about the priors A hierarchical
  approach.
• There is still some prior knowledge that has not
  found its way into the model namely distance
  between zero-stage and stage measurements.
• Heteroscedasticy ought to be removed.
• Should have a prior on b that closer reflects
  both prior knowledge (positive b) and the
  database collection of estimates. For example
  blogN. But this introduces problems with
  efficiency.

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A graphical view of the model and a tool for a
hierarchical approach
distribution with or without hyper-parameters
?a Va ? ?b Vb
? ?
parameters
hyper-
aj bj
parameters
For j in 1,,number of stations
?j2
hj,i qj,i
For i in 1,,number of measurements for station
j
measurements
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Solution to the prior for hc

• Possible to go from a regression situation to a
  model that has both stochastic discharge and
  stage values.
• Possibility A structural model where real
  discharge, , has a distribution. The real
  stage, , is a deterministic function of the
  curve parameters, (a, b, c). Observations, D(qi,
  hi), are the real values plus noise.
• The model gives a more realistic description of
  what happens in the real world. It also codes the
  prior knowledge about the difference between
  stage measurements and zero-stage, through the
  distribution of q and the distribution of (a, b).

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Structural model a graphical view
distribution with or without hyper-parameters
?? ??
?? ??
?q ?q
?0 ?0
parameters
?b Va ? ?b Vb
hyper-
?q ?q2
parameters
a b c
??2
?h2
latent variables
measurements
qi
hi
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Advantage and problems of a structural model

• Advantage
• More realistic modelling of the measurements and
  the underlying structure.
• Codes prior knowledge about the relationship
  between stage measurements and the zero-stage.
• Can solve heteroscedasticy.
• Gives a more detailed picture of how measurement
  errors occur.
• Since b can not be sampled using Gibbs, we might
  as well use a form that insures positive exponent.

• Problem
• Difficult to make an efficient algorithm.
• More complex. Thus even if it codes more prior
  knowledge, the estimates might be more uncertain.
  This has not been tested.