The use of likelihood functions to fit conceptual models with more than one dependent variable

252

160

A Bayesian methodology is developed to evaluate parameter uncertainty in catchment models fitted to a hydrologic response such as runoff, the goal being to improve the chance of successful regionalization. The catchment model is posed as a nonlinear regression model with stochastic errors possibly being both autocorrelated and heteroscedastic. The end result of this methodology, which may use Box‐Cox power transformations and ARMA error models, is the posterior distribution, which summarizes what is known about the catchment model parameters. This can be simplified to a multivariate normal provided a linearization in parameter space is acceptable; means of checking and improving this assumption are discussed. The posterior standard deviations give a direct measure of parameter uncertainty, and study of the posterior correlation matrix can indicate what kinds of data are required to improve the precision of poorly determined parameters. Finally, a case study involving a nine‐parameter catchment model fitted to monthly runoff and soil moisture data is presented. It is shown that use of ordinary least squares when its underlying error assumptions are violated gives an erroneous description of parameter uncertainty.

Section: The Most Likely Values Of the Parameters /• And D For Run 1 mentioning

confidence: 72%

Improved parameter inference in catchment models: 1. Evaluating parameter uncertainty

Kuczera¹

1983

252

160

“…Also such information has been pooled with rainfallrunoff data. For example, Douglas et al [1976] fitted their model jointly to runoff and soil moisture data and observed that improvement in goodness of fit was not realized; they did not consider improvement in the precision of parameter estimates. However, Kuczera [1982a] showed for a simple linear catchment model that joint fitting based on generalized least squares can considerably improve the precision of inferred parameters.…”

Section: Introductionmentioning

confidence: 99%

Improved parameter inference in catchment models: 2. Combining different kinds of hydrologic data and testing their compatibility

Kuczera¹

1983

Often some of the parameters of catchment models fitted to runoff data are poorly determined thereby making the task of developing useful regionalization relationships more difficult. The Bayesian methodology developed in part 1 is extended to utilize several kinds of hydrologic data in parameter inference, the goal being to improve the precision of poorly determined parameters. The concept of compatibility is developed using statistical hypothesis tests. Different kinds of data are said to be compatible if differences between their fitted parameters are not statistically significant. The pooling of incompatible data may undermine the model's ability to predict runoff and also induce bias in the parameters. A hierarchy of three levels of information is introduced to enable systematic checking for compatibility. Finally, a case study is presented. Using data on runoff, soil moisture, and interception, it is shown that substantial reductions in parameter uncertainty can be realized; also the importance of compatibility testing is demonstrated.

“…We assume that errors of streamflow and snow models are uncorrelated and therefore the log likelihood function is

l true(θ {| \overset{Y}{true}}_{1}, {\overset{Y}{true}}_{2} true) = - \frac{n_{1}}{2} \ln true(2 π true) - \frac{n_{1}}{2} \ln true(σ_{e 1}^{2} true) - \frac{1}{2 σ_{e 1}^{2}} \sum_{j = 1}^{n_{1}} {[y_{1, j} true(θ true) - {\overset{y}{true}}_{1, j}]}^{2} + n_{2} \log \frac{2 σ_{ξ} ω_{β}}{ξ + ξ^{- 1}} - \sum_{i = 1}^{n_{2}} \log true(σ_{e 2, i} true) - c_{β} \sum_{i = 1}^{n_{2}} {| a_{ξ, i} |}^{2 / true(1 + β true)},

where

{\overset{Y}{true}}_{1}

and

{\overset{Y}{true}}_{2}

are measured SWE and streamflow observations and θ contains all the parameters. Douglas et al [1976] pointed out that it is possible that the other response (e.g., y 1 ) will dominate the parameter estimation procedure if n 1 is much greater than n 2 in . Whether the larger number of streamflow observations dominates the parameter estimation in our case study is discussed later.…”

Section: Methodsmentioning

confidence: 99%

Bayesian inference of uncertainties in precipitation‐streamflow modeling in a snow affected catchment

Koskela

Croke

Koivusalo

et al. 2012

.[1] Bayesian inference is used to study the effect of precipitation and model structural uncertainty on estimates of model parameters and confidence limits of predictive variables in a conceptual rainfall-runoff model in the snow-fed Rudbäck catchment (142 ha) in southern Finland. The IHACRES model is coupled with a simple degree day model to account for snow accumulation and melt. The posterior probability distribution of the model parameters is sampled by using the Differential Evolution Adaptive Metropolis (DREAM (ZS) ) algorithm and the generalized likelihood function. Precipitation uncertainty is taken into account by introducing additional latent variables that were used as multipliers for individual storm events. Results suggest that occasional snow water equivalent (SWE) observations together with daily streamflow observations do not contain enough information to simultaneously identify model parameters, precipitation uncertainty and model structural uncertainty in the Rudbäck catchment. The addition of an autoregressive component to account for model structure error and latent variables having uniform priors to account for input uncertainty lead to dubious posterior distributions of model parameters. Thus our hypothesis that informative priors for latent variables could be replaced by additional SWE data could not be confirmed. The model was found to work adequately in 1-day-ahead simulation mode, but the results were poor in the simulation batch mode. This was caused by the interaction of parameters that were used to describe different sources of uncertainty. The findings may have lessons for other cases where parameterizations are similarly high in relation to available prior information.