Modeling Hydrologic Change: Statistical Methods. Author: Christopher K. Date: Nov. From: Technometrics Vol. Publisher: American Statistical Association. Document Type: Book review. Length: words. By Richard H. Sign In to view the full article. Beyond the visual approach, objective statistical analyses can be carried out via, e. Clearly, if a model is unable to reproduce the distributions of the observations of interest viz.
A final important issue concerns the particular structure of some data bases, which may adversely affect both the univariate and the multivariate analyses. A possible way to circumvent the problem is to use appropriate randomization techniques 49 : this may turn the discrete repeated values into continuous different ones. The idea is to add to all the repeated values a set of suitable random perturbations in the range of the instrumental resolution adopted during the data sampling: in order to avoid unverifiable assumptions, the noise is chosen to be Uniform i.
Apparently, according to the results shown in refs 28 , 45 , the randomization procedures may not significantly affect the original statistical structure of the data, while reducing the nuisance possibly introduced by Ties. In this section, the results of several univariate and multivariate analyses are presented and discussed, involving both climate and hydrological variables.
Here, three data sets are considered:. Is the model able to reproduce the probability distributions of the observed data over which it has been calibrated?
Should the check fail, then a flaw in the model would be likely, and it would not be clear what the model outputs both during the control period and the future temporal horizon would be representative of. Evidently, a positive answer to this question represents a necessary condition for the internal coherence and consistency of the models at play.
Assuming that the answer to the previous question is positive, viz. Should the future simulated projections feature a probability distribution statistically different from the one of the data simulated over the control period, a further important enquiry is as follows: Are the probability distributions of the future projections different from the ones of the past observation? A negative answer to this question might reveal possible flaws of the statistical techniques adopted. Practically, from a logical point of view, the observations cannot have, at the same time, the incompatible distributions of the control and the future simulated data, unless some of the tests failed.
In other words, the distribution of the data simulated over the control period and the distribution of the future projections cannot be, at the same time, statistically incompatible with one another, but both statistically compatible with the distribution of the observations. Considering yearly mean values, the following facts are evident. The variable P.
All the distributions are statistically compatible, both considering the body and the tails of the unknown probability laws at play. Thus, the climate model correctly reproduces the behavior of the observed yearly Precipitation over the control period, and replicates it unaltered over the future temporal horizon i. The variable T. Only the distributions of the observed and control data are statistically compatible, considering both the body and the tails of the unknown probability laws at play.
Instead, the homogeneity assumption has to be rejected considering the future projections. The variable Q. Here, the matter is unclear. In fact, considering the observed data and the future projections, the homogeneity assumption should definitely be rejected, whereas the distributions of the data simulated over the control period and the future projections turn out to be statistically compatible.
Thus, possible interpretations are as follows. The supposed compatibility between control data and future projections is wrong. Considering the time series of drought episodes, based on daily discharge values, the results are shown in Fig. The variable I. The distributions of observed and control data are statistically compatible, considering both the body and the tails of the unknown probability laws at play.
The variable D. Instead, the homogeneity test is passed by these two latter distributions. The results are unvarying over all the N R randomizations, as indicated by the corresponding box-plots. Thus, apparently, according to the model projections, the future drought durations are expected to show a probability distribution different from the one of the observed data.
Since Ties are present for D , the corresponding box-plots are over all the N R randomizations—see text. Provided that the available sample sizes are large enough, a further important information that could be drawn via the KS and AD tests is as follows. As an example, see the case of the gauge station Borgoforte for the variable D reported in the Supplementary Materials, considering the comparison between the control and the future distributions: here the sample sizes are larger than 60, and hence sensible statistical conclusions might be drawn. Such a piece of information may be quite important: on the one hand, it reveals how the model behaves, and what kind of changes it may introduce; on the other hand, should it be desirable, it provides useful indications to precisely adjust the model.
In both cases, only the pairs simulated over the control period are significantly negatively associated, being the p -Values negligible. In turn, apparently, the model introduces a dependence between P and T , and T and Q , in the control period that is neither present in the original observations nor in the future projections. Such a behavior is difficult to interpret and explain: seemingly, it may represent a flaw of the model, and no sensible indications about what is likely to be expected in the future can be achieved.
The pair P , Q. A significant positive association, with about the same magnitude, is present in all cases. Here the model correctly reproduces the concordance between Precipitation and Discharge during the control period, and replicates it in the future projections. Accordingly, no changes in the degree of association between P and Q have to be expected over the future temporal horizon i.
Considering droughts, the results are shown in Fig. Here, I and D are significantly positively associated in all cases. However, the simulated data show a degree of concordance that is about two times the one estimated for the observed data, a non-negligible increase. The fact that the actual degree of dependence between I and D is not reproduced during the control period may represent a flaw of the model, which is possibly incorrectly replicated over the future temporal horizon. A visual inspection of the available measurements is traditionally carried out via the scatter-plots of the data: these are shown in Fig.
While graphical estimates of the univariate marginals can be calculated via non-parametric empirical distribution functions not shown , the structure of the underlying copula can be guessed by plotting the corresponding non-parametric pseudo-observations, as anticipated in the Methods Section and shown in Fig. As previously pointed out, the pseudo-observations are not affected by the marginals they only depend upon the copula , and may provide a hint of the dependence structure at play. Here, the left panels concern the Observations Obs , the middle panels concern the Control data Ctrl , and the right panels concern the RCP4.
The results of the Kendall and Spearman tests mentioned above find a graphical visualization in Fig. The Bottom panels of Fig. Similar conclusions can be drawn considering the plots in Fig. In particular, the fact that the pseudo-observations are to different extents aligned along the main diagonal is in agreement with the previously claimed different degree of concordance viz. The shapes of the isolines of the Empirical Copula of interest are in agreement with the conclusions previously drawn.
Also reported are the sample sizes in parentheses. The Bottom panel shows the isolines of the Empirical Copulas, averaged over all the N R randomizations, for the levels 0. The ability of the linked climate-hydrology models to introduce changes in the probability distributions of the relevant variables over a temporal domain of interest , both at a univariate and at a multivariate level, can be investigated via suitable non-parametric statistical tests: clearly, these studies may provide important information about the features and the behavior of the models like, e.
As a matter of fact, considering the temperature data shown in Fig. However, such an analysis would not have been possible without the preliminary indication of the break-point year provided by the Change-Point test, which may help in spotting and identifying possible climate changes.
The vertical labels indicate possible change-point years. In particular, the results shown in Fig. In turn, the climate-hydrology model is unable to reproduce the copula of the observed data over the control period, while it replicates the one of the Control data over the future temporal horizon: this may represent a flaw of the models. In this work, a thorough investigation of the behavior and the performances of linked climate-hydrology models is carried out. On the one hand, innovative analysis paradigms, involving the probability distributions of the data, are exploited.
Suitable non-parametric statistical tests are used to check a number of features and assumptions of utmost importance in climate-hydrology studies, especially considering the investigation of climate changes. In particular, the following issues are illustrated and discussed:. The analyses proposed exploit recent and innovative statistical tests: used in combination with traditional validation techniques, they may help in improving the assessment of climate-hydrology models, and provide important information concerning the behavior and the flaws of such models.
Most importantly, the methodological approach outlined here is appropriate also in contexts different from climate-hydrology studies, in order to evaluate the performance of any model of interest: in particular, methods to check a model per se are sketched out, investigating whether its outcomes are statistically consistent. Invaluable discussions and suggestions by F. All authors analysed the results and reviewed the manuscript.
Electronic supplementary material. Supplementary information accompanies this paper at Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Gianfausto Salvadori, Email: ti. Carlo De Michele, Email: ti. National Center for Biotechnology Information , U. Sci Rep.
Published online Sep Author information Article notes Copyright and License information Disclaimer. Corresponding author. Received May 18; Accepted Jul Abstract One of the ultimate goals of climate studies is to provide projections of future scenarios: for this purpose, sophisticated models are conceived, involving lots of parameters calibrated via observed data. Open in a separate window. Figure 1. Drought time series The hydrological response of the Po river basin to climate inputs is investigated through the analysis of the droughts measured at the five river sections mentioned above.
Univariate analysis A first fundamental step of the investigation procedure consists in considering the univariate distribution functions of the variables at play. Multivariate analysis: dependence structures In case the independence among the variables at play is statistically rejected, it is then important to check whether the model of interest is able to reproduce the dependence structures i.
Treatment of repeated observations A final important issue concerns the particular structure of some data bases, which may adversely affect both the univariate and the multivariate analyses. Results and Discussion In this section, the results of several univariate and multivariate analyses are presented and discussed, involving both climate and hydrological variables. The outcomes shown later may answer at least three important questions.
An overview of hydrologic models
Figure 2. Figure 3. Figure 4. Figure 5. Multivariate analysis: dependence structures A visual inspection of the available measurements is traditionally carried out via the scatter-plots of the data: these are shown in Fig. Figure 6. Figure 7. Change-Point and Copula-equality tests The ability of the linked climate-hydrology models to introduce changes in the probability distributions of the relevant variables over a temporal domain of interest , both at a univariate and at a multivariate level, can be investigated via suitable non-parametric statistical tests: clearly, these studies may provide important information about the features and the behavior of the models like, e.
Figure 8. Figure 9. Summary and Conclusions In this work, a thorough investigation of the behavior and the performances of linked climate-hydrology models is carried out. In particular, the following issues are illustrated and discussed: whether the model is able to reproduce the distributions of the data over which it has been calibrated, both univariate and multivariate via homogeneity and Copula-equality tests ; to what extent the model is able to reproduce or alter the degree of association between different non-independent variables via Kendall and Spearman tests ; whether the model is able to introduce change-points in the probability laws of interest via Change-Point tests , which represent the fingerprints of possible climate changes; possible failures of the statistical tools adopted.
Modeling Hydrologic Change: Statistical Methods
Electronic supplementary material Supplementary Material 1. Acknowledgements Invaluable discussions and suggestions by F. Author Contributions R. Notes Competing Interests The authors declare that they have no competing interests. Electronic supplementary material Supplementary information accompanies this paper at Contributor Information Gianfausto Salvadori, Email: ti.
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