Roman Krzysztofowicz

Bayesian theory of probabilistic forecasting via deterministic hydrologic model

Rational decision making (for flood warning, navigation, or reservoir systems) requires that the total uncertainty about a hydrologic predictand (such as river stage, discharge, or runoff volume) be quantified in terms of a probability distribution, conditional on all available information and knowledge. Hydrologic knowledge is typically embodied in a deterministic catchment model. Fundamentals are presented of a Bayesian forecasting system (BFS) for producing a probabilistic forecast of a hydrologic predictand via any deterministic catchment model. The BFS decomposes the total uncertainty into input uncertainty and hydrologic uncertainty, which are quantified independently and then integrated into a predictive (Bayes) distribution. This distribution results from a revision of a prior (climatic) distribution, is well calibrated, and has a nonnegative ex ante economic value. The BFS is compared with Monte Carlo simulation and “ensemble forecasting” technique, none of which can alone produce a probabilistic forecast that meets requirements of rational decision making, but each can serve as a component of the BFS.

The case for probabilistic forecasting in hydrology

That forecasts should be stated in probabilistic, rather than deterministic, terms has been argued from common sense and decision-theoretic perspectives for almost a century. Yet most operational hydrological forecasting systems produce deterministic forecasts and most research in operational hydrology has been devoted to finding the ‘best’ estimates rather than quantifying the predictive uncertainty. This essay presents a compendium of reasons for probabilistic forecasting of hydrological variates. Probabilistic forecasts are scientifically more honest, enable risk-based warnings of floods, enable rational decision making, and offer additional economic benefits. The growing demand for information about risk and the rising capability to quantify predictive uncertainties create an unparalleled opportunity for the hydrological profession to dramatically enhance the forecasting paradigm.

A bivariate meta-Gaussian density for use in hydrology

Convenient bivariate densities found in the literature are often unsuitable for modeling hydrologic variates. They either constrain the range of association between variates, or fix the form of the marginal distributions. The bivariate meta-Gaussian density is constructed by embedding the normal quantile transform of each variate into the Gaussian law. The density can represent a full range of association between variates and admits arbitrarily specified marginal distributions. Modeling and estimation can be decomposed into i) independent analyses of the marginal distributions, and ii) investigation of the dependence structure. Both statistical and judgmental estimation procedures are possible. Some comparisons to recent applications of bivariate densities in the hydrologic literature motivate and illustrate the model.

Hydrologic uncertainty processor for probabilistic river stage forecasting

The hydrologic uncertainty processor (HUP) is a component of the Bayesian forecasting system that produces a short-term probabilistic river stage forecast based on a probabilistic quantitative precipitation forecast (PQPF). The task of the HUP is to quantify the hydrologic uncertainty under the hypothesis that there is no precipitation uncertainty. The hydrologic uncertainty is the aggregate of all uncertainties arising from sources other than those quantified by the PQPF; these sources include the hydrologic model (model and parameter uncertainties), inputs estimated deterministically (measurement, estimation, and prediction uncertainties), and inputs not forecasted (e.g., precipitation beyond the period covered by the PQPF). Bayesian theory for the HUP is presented, and a meta-Gaussian model is developed. This parametric model allows for (1) any form of marginal distributions of river stages, (2) a nonlinear and heteroscedastic dependence structure between the model river stage and the actual river stage, and (3) an analytic solution of the Bayesian revision process. Estimation and validation of the model are described using data from the operational forecast system of the National Weather Service for a 1430-km2 headwater basin.

Bayesian system for probabilistic river stage forecasting

The purpose of this analytic-numerical Bayesian forecasting system (BFS) is to produce a short-term probabilistic river stage forecast based on a probabilistic quantitative precipitation forecast as an input and a deterministic hydrologic model (of any complexity) as a means of simulating the response of a headwater basin to precipitation. The BFS has three structural components: the precipitation uncertainty processor, the hydrologic uncertainty processor, and the integrator. A series of articles described the Bayesian forecasting theory and detailed each component of this particular BFS. This article presents a synthesis: the total system, operational expressions, estimation procedures, numerical algorithms, a complete example, and all design requirements, modeling assumptions, and operational attributes.

Strength of preference and risk attitude in utility measurement

Abstract A relationship between a value function v (compatible with the theory of ordered value differences) and a utility function u (compatible with the expected utility theory) is explored. According to a behavioral interpretation, v encodes the strength of preference while u encodes the strength of preference and risk attitude. The results of two experiments (one conducted in a real-world setting and another in a laboratory) involving 24 cases and the data reported in the literature involving 10 cases support the constant relavie risk attitude hypothesis. The implied unique transformation between v and u is tested as a descriptive model and as a predictive model. The descriptive model is then used for inference concerning several behavioral hypotheses.

Transformation and normalization of variates with specified distributions

Given two continuous random variables X and Y, with specified strictly increasing cumulative distribution functions F and G, respectively, the one-to-one transform t which maps one variate into another, say Y=t(X), has an analytic form, t(X)=G−1(F(X)) or t(X)=G−1(1−F(X)), depending upon whether t is increasing or decreasing. This fact of probability theory is reviewed and compared with another method for finding t that was recently proposed. Applications to system identification, normalization of a variate, and normalization of a sample are briefly discussed.

Precipitation uncertainty processor for probabilistic river stage forecasting

The precipitation uncertainty processor (PUP) is a component of the Bayesian forecasting system which produces a short-term probabilistic river stage forecast (PRSF) based on a probabilistic quantitative precipitation forecast (PQPF). The task of the PUP is to process a probability distribution of the total precipitation amount through a deterministic hydrologic model (of any complexity) into a probability distribution of the model river stage. An analytic-numerical PUP is developed based on the theory of response functions and empirical data simulated from the operational forecast system of the National Weather Service for a 1430 km2 headwater basin. The PUP outputs a five-parameter two-piece Weibull distribution of the model river stage. The corresponding response function is a two-piece power function. Structural properties of the PUP are investigated empirically, including the deterministic equivalence principle: Under certain conditions a deterministic forecast of the temporal disaggregation of the total precipitation amount is equivalent to a probabilistic forecast. This considerably simplifies the PQPF, without affecting the optimality of the PRSF.

Assessment errors in multiattribute utility functions

Abstract Three types of errors that may impair the validity of multiattribute utility functions are investigated: (1) errors in single-attribute utility functions due to effects of the range of the gamble outcomes, (2) errors in the type of multivariate risk preference due to the extreme range effects or the preference for covariance, and (3) inconsistencies between responses obtained by direct and indirect assessment techniques. The occurrences of these errors are demonstrated through the experimental assessment of a two-attribute, multiplicative utility function. This function is to be used as a criterion for real-time control of a multipurpose reservoir under conditions of uncertainty. It is shown that the range effects cause the subjects preferences to violate the transitivity axiom. Cognitive sources of this violation are hypothesized. Certain guidelines for using utility functions in decision analyses are suggested.

Bayesian Correlation Score: A Utilitarian Measure of Forecast Skill

Abstract From the theory of sufficient comparisons of experiments, a measure of skill is derived for categorical forecasts of continuous predictands. Called Bayesian correlation wore (BCS), the measure is specified in terms of three parameters of a normal-linear statistical model that combines information from two sources: a prior (climatological) record of the predictand and a verification record of forecasts. Three properties characterize the BCS: (i) It is meaningful for comparing alternative forecasts of the same predictand, as well as forecasts of different predictands, though in a limited sense; (ii) it is interpretable as correlation between the forecast and the predictand; and, most significantly, (iii) it orders alternative forecast systems consistently with their ex ante economic values to rational users (those who make decisions by maximizing the expected utility of outcomes under the posterior distribution of the predictand). Thus, by maximizing the BCS, forecasters can assure a utilitarian so...