Applied Multivariate Statistics in Geohydrology and Related Sciences

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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.

A distributional multivariate approach for assessing performance of climate-hydrology models

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.


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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.

Bibliographic Information

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.

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.

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Figure 6. Figure 7.

Session 1 Applied Multivariate Statistics - Part 1: Lecture

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.

Bibliographic Information

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.

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.

References 1. An overview of CMIP5 and the experiment design. Bulletin of the American Meteorological Society. WMO Bulletin. Flato, G.

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Evaluation of climate models. In Stocker, T. Kirchner JW. Getting the right answers for the right reasons: Linking measurements, analyses, and models to advance the science of hydrology. Water Resour. Local control on precipitation in a fully coupled climate-hydrology model. Scientific Reports. Taylor K. Summarizing multiple aspects of model performance in a single diagram. Journal of Geophysical Research: Atmospheres.

Evaluation of daily precipitation characteristics in the clm and their sensitivity to parameterizations. Meteorologische Zeitschrift. On the need for bias correction of regional climate change projections of temperature and precipitation. Geophysical Research Letters. About this Textbook It has been evident from many years of research work in the geohydrologic sciences that a summary of relevant past work, present work, and needed future work in multivariate statistics with geohydrologic applications is not only desirable, but is necessary.

Show all. Charles E. Pages Correlation Brown, Dr.

Factor Analysis Brown, Dr. Canonical Correlation Brown, Dr. Multiple Regression Brown, Dr. Multivariate Analysis of Variance Brown, Dr. Multivariate Analysis of Covariance Brown, Dr. Principal Components Brown, Dr. Intensive stream water monitoring provided data to calculate P loads from the ha farm watershed for all runoff events during a two-year pre-treatment period and a four-year post-treatment period.


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  • Statistical control for inter-annual climatic variability was provided by matched P loads from a nearby ha forested watershed, and by several event flow variables measured at the farm. A sophisticated multivariate analysis of covariance ANCOVA provided estimates of both seasonal and overall load reductions. Statistical power and the minimum detectable treatment effect MDTE were also calculated.

    Changes in farm management practices and physical infrastructure clearly produced decreases in event P losses measurable at the small watershed scale. Editorial Board. Permissions information. Subscription Questions. Manuscript Invoice Payment. Skip to main content.

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