Multivariate Frequency Analysis Research Articles

Multivariate overall and dependence trend tests, applied to hydrology

Given climate change, trend detection is gaining increasing attention in the context of multivariate frequency analysis. In this paper, we propose new statistical tests for multivariate trend detection. The first one, a multivariate overall trend (MOT) test, is designed to detect trend in all components of the multivariate distribution (margins and dependence structure) whereas the second test is a multivariate dependence trend (MDT) test focusing on detecting trend in the dependence structure. A simulation study is used to evaluate the performance of the proposed tests. Results show that the proposed MOT test performs well when trend is present in margins, in the dependence structure and/or in both. Likewise, results of the proposed MDT test indicate a higher power when the trend is in the dependence structure. Moreover, an application to a real-world dataset is provided. Performing the proposed tests with the univariate tests provides a complete overview of trend detection.

A Multivariate Frequency Analysis Framework to Estimate the Return Period of Hurricane Events Using Event‐Based Copula

AbstractThis study proposed a framework to evaluate multivariate return periods of hurricanes using event‐based frequency analysis techniques. The applicability of the proposed framework was demonstrated using point‐based and spatial analyses on a recent catastrophic event, Hurricane Ian. Univariate, bivariate, and trivariate frequency analyses were performed by applying generalized extreme value distribution and copula on annual maximum series of flood volume, peak discharge, total rainfall depth, maximum wind speed, wave height and storm surge. As a result of point‐based analyses, return periods of Hurricane Ian was investigated by using our framework; univariate return periods were estimated from 39.2 to 60.2 years, bivariate from 824.1 to 1,592.6 years, and trivariate from 332.1 to 1,722.9 years for the Daytona‐St. Augustine Basin. In the Florida Bay‐Florida Keys Basin, univariate return periods were calculated from 7.5 to 32.9 years, bivariate from 36.5 to 114.9 years, and trivariate from 25.0 to 214.8 years. Using the spatial analyses, we were able to generate the return period map of Hurricane Ian across Florida. Based on bivariate frequency analyses, 18% of hydrologic unit code 8 (HUC8) basins had an average return period of more than 30 years. Sources of uncertainty, due to the scarcity of analysis data, stationarity assumption and impact of other weather systems such as strong frontal passages, were also discussed. Despite these limitations, our framework and results will be valuable in disaster response and recovery.

Dynamic Evolution and Copula-Based Multivariable Frequency Analysis of Meteorological Drought Considering the Spatiotemporal Variability in Northwestern China

Meteorological drought is a continuous spatiotemporal phenomenon that poses a serious threat to water resource security. Dynamic evolution and multivariable frequency analysis of meteorological drought are important for effective drought mitigation and risk management. Therefore, this study aims to analyze meteorological drought events in northwestern China between 1960 and 2018 based on the standardized precipitation evapotranspiration index (SPEI) through a three-dimensional identification method. This study investigates the meteorological drought dynamic evolution on different time and space scales and evaluates the frequency analysis considering the spatiotemporal variability based on Copula. The results show that SPEI presents an upward trend in Northwestern China. A trend towards increased humidity is observed in arid regions, contrasted by a trend towards aridification in semi-arid and semi-humid areas, indicating that the spatial distribution of drought in the study area tends towards homogenization. The possibility of high-intensity drought events occurring in the same area was relatively low, whereas low-intensity drought events were frequent. Additionally, this study analyzes the dynamic migration process of individual drought events from a three-dimensional perspective. Neglecting any one drought variable could significantly underestimate the occurring probability of severe drought events. Therefore, a multivariable frequency analysis considering the spatiotemporal variability plays a crucial role in the formulation of drought prevention and mitigation strategies, as well as drought forecasting.

Analysis of Hydrologic Drought Frequency Using Multivariate Copulas in Shaying River Basin

Droughts, considered one of the most dangerous and costly water cycle expressions, always occurs over a certain region, lasting several weeks or months, and involving multiple variables. In this work, a multivariate approach was used for the statistical characterization of hydrological droughts in Shaying River Basin with data from 1959–2008. The standard runoff index (SRI) and the run theory were employed to defined hydrological drought character variables (duration, severity, and intensity peak). Then, a multivariate joint probability analysis with four symmetric and corresponding asymmetric Archimedean Copulas was presented; and the multivariate frequency analysis with the joint return periods (Tand and Tor) were estimated. The results showed that the hydrological droughts have a severity of 4.79 and 5.09, and the drought intensity peak is of 1.35 and 1.50 in Zhoukou station and Luohe station, respectively; the rank correlation coefficients τ are more than 0.5, which means multivariate copulas can effectively describe the joint frequency distributions among multivariate variables. Drought risk shows a spatial variation: the downstream observed at Zhoukou station is characterized by a higher multivariate drought risk. In general, multivariate copulas provide a reliable method when constructing a comprehensive drought index and evaluating multivariate drought characteristics. Thus, this paper can provide useful indications for the multi-dimensional droughts’ risks assessment in Shaying River Basin.

Multivariate Frequency Analysis for Streamflow Drought Having Different Time Resolution Using Archimedean Copula Functions

This study explored return periods of the streamflow deficit characteristics by threshold levels with different time resolutions (daily, weekly, and monthly) using univariate and Copula functions, reflecting hydrological regimes like rapid recoveries from drought and anthropogenic dam operation by monthly basis in South Korea. As a result, the daily-based threshold had a large variability in severities and durations. This is because threshold levels include a large variability of daily streamflow with a small time interval. Joint return periods of duration under extreme droughts were similar to those from univariate frequency analysis. The severities in these extreme droughts were also around the average level, but the duration was relatively long, leading to an increase in the joint return period. This result helps to effectively manage water resource preparing for the assessment on the drought risk with a long duration.

Quantifying uncertainty in multivariate quantile estimation of hydrometeorological extremes via copula: A comparison between bootstrapping and Markov chain Monte Carlo

AbstractThe performance of uncertainty estimation methods, namely bootstrapping and Markov chain Monte Carlo (MCMC), in univariate frequency analysis of hydrometeorological extremes has been well tested in the literature. However, the two methods have not been thoroughly compared for multivariate frequency analysis of such events. In this study, we compare the performance of bootstrapping and MCMC in estimating the uncertainty of bivariate quantiles of extremes as defined by the return period quantiles of hydrologic drought duration and severity, and concurrent meteorological drought and heat wave. Using a copula framework, we analyse the accuracy and size of confidence intervals of the bivariate quantiles, and bias in point estimates of them. We also investigate the performance of the two methods in estimating the uncertainty of univariate quantiles of the marginal distributions of the resulting bivariate copulas. This is to evaluate if any advantage of one method over the other is consistent, whether in estimating the univariate or bivariate quantiles. We conduct this study with synthetic datasets of various sample sizes and predefined distributions derived from a set of empirical data. The results show MCMC to be superior when estimating the uncertainty of bivariate quantiles where the sample size is small (~50). Where the sample size is large (~100 and ~200), the results show bootstrapping to be the better option for estimating uncertainties of bivariate quantiles. For estimating uncertainties of univariate quantiles, bootstrapping is performing better under all investigated sample sizes. Results and conclusions in this study will be beneficial for hydrometeorological risk assessment, hydrologic infrastructure design, and water resources assessment.

A copula-based drought assessment framework considering global simulation models

Study regionSão Paulo state – Brazil. Study focusCompound events, such as droughts and heat waves, may have severe impacts on human activities. Traditionally, they are characterized considering a univariate perspective. However, this approach may not be the most adequate to characterize such hazards as they often result from a combination of variables interacting in space and time. Alternatively, several studies adopt the multivariate frequency analysis as it allows the consideration of concurrent drivers and their dependencies. Nevertheless, few of them evaluated this methodology in a climate change context. In view of this, this study aims to compare the uni and multivariate approaches to characterize extreme drought events considering both historical and future scenarios, using the severe water crisis experienced in the southeast region of Brazil in 2014–2015 as a study case. New hydrological insights for the regionThe univariate approach can substantially underestimate the risk associated with extreme events. For future scenarios, differences between the two methodologies reached 90% of the estimated return period. Significant increasing trends were found only for temperature. Both approaches indicated that drought events will be more common and intense in the future. However, the univariate framework may misspecificate the associated risks, as it not account for the expected warming condition that may trigger or exacerbate extreme drought events.

Copula‐Based Bivariate Return Period Analysis and Its Implication to Hydrological Design Event

AbstractIn univariate frequency analysis, the return period of an event has a one‐to‐one correspondence with its characteristic value, and the response of the hydraulic structure to hydrological load expressed by the hydrological event is monotonic. Thus, the design criteria of the hydraulic structure can be equivalently represented by the return period of the hydrological event, and consequently, design event‐based design parameters evaluated have been widely used in practical engineering. However, the monotonic correspondence between the return period of the hydrological event and the response of the hydraulic structure does not exist in the multivariate context, and hydrological load with a larger joint return period does not always produce a more unsafe response. Misunderstandings of concepts of return periods of hydrological event, and estimation of hydrological design events usually take place in multivariate frequency analysis. This study theoretically derives the relations between different types of joint return periods, joint return period and its marginal return periods, the occurrence of bivariate extreme events and their return periods, and then the theoretical framework is tested. Results from the case contribute to the understanding of bivariate return periods of hydrological event, and the results demonstrate that design criteria cannot be equivalently represented by joint return periods of hydrological load, and design parameters of the hydraulic structure should not be determined by multivariate hydrological design events.

Estimating the probability of compound floods in estuarine regions

Abstract. The quantification of flood risk in estuarine regions relies on accurate estimation of flood probability, which is often challenging due to the rareness of hazardous flood events and their multi-causal (or “compound”) nature. Failure to consider the compounding nature of estuarine floods can lead to significant underestimation of flood risk in these regions. This study provides a comparative review of alternative approaches for estuarine flood estimation – namely, traditional univariate flood frequency analysis applied to both observed historical data and simulated data, as well as multivariate frequency analysis applied to flood events. Three specific implementations of the above approaches are evaluated on a case study – the estuarine portion of Swan River in Western Australia – highlighting the advantages and disadvantages of each approach. The theoretical understanding of the three approaches, combined with findings from the case study, enable the generation of guidance on method selection for estuarine flood probability estimation, recognizing issues such as data availability, the complexity of the application/analysis process, the location of interest within the estuarine region, the computational demands, and whether or not future conditions need to be assessed.

Drought Frequency Analysis Based on the Development of a Two-Variate Standardized Index (Rainfall-Runoff)

Drought is one of the most drastic events, which has imposed irreparable damages on human societies and may occur in any climate regime. To define drought, given its properties of multidimensionality and randomity, one cannot rely on a single variable/index (e.g., precipitation, soil moisture, and runoff). Accordingly, implementing a novel approach, this study investigated drought events in two basins with different climatic regimes, using multivariate frequency analyses of drought duration, severity, and severity peak, based on developing a Two-variate Standardized Index (TSI). The index was developed based on the concept of copula, by applying rainfall-runoff data (1974–2019) and comparing them with two popular drought indices, the Standardized Precipitation Index (SPI) and Standardized Stream Flow Index (SSFI), in terms of derived drought characteristics. The results show that TSI determined more severe drought conditions with fewer return periods than SPI and SSFI in a specific drought event. This implies that the disadvantages of SPI and SSFI might not be found in TSI. The developed index can be employed by policymakers and planners to protect water resources from drought.

Multivariate Frequency Analysis of Annual Maxima Suspended Sediment Concentrations and Floods in the Jinsha River, China

AbstractStreamflow plays a critical role in sediment transport. Traditional frequency analysis of annual maximal (AM) suspended sediment concentrations (SSCs) is usually based on univariate distrib...

Wind-electricity-heat Correlation and Potential Uncertainty Analysis Based on Copula Function

In the economic dispatch and optimal operation of power system, it is necessary and effective to consider the correlation among heat load, power load and wind power output to formulate a reasonable dispatch plan or to evaluate the reliability of the system. Therefore, based on the theory of Copula correlation analysis, a multivariate Copula analysis toolbox (MvCAT) is proposed to infer Copula parameters and estimate potential uncertainties. Firstly, the mixed evolution Markov Chain Monte Carlo (MCMC) method in Bayes framework calculates the posterior distribution of Copula parameters, evaluate their uncertainties relative to fitting, and then select the appropriate Copula function by goodness-of-fit test. Finally, a the typical daily data of a province as samples for analysis, proposed method solves the limitation that the local optimization method often falls into the local minimum, and quantitative evaluation of the correlation between the response variables and the uncertainty associated with the length of the recorded data is essential for multivariate frequency analysis.

Two Ways to Quantify Korean Drought Frequency: Partial Duration Series and Bivariate Exponential Distribution, and Application to Climate Change

Studies using drought index to examine return levels of drought can be classified into two approaches: univariate frequency analysis using annual series extracted from drought index time series and multivariate frequency analysis that simultaneously reflects various characteristics of drought. In the case of drought analysis, it is important to properly consider the duration, so, in this study, univariate frequency analysis is performed using the partial duration series. In addition, a bivariate frequency analysis is performed using a relatively simple bivariate exponential distribution to give a more realistic return level to major drought events in the past while reflecting the correlation between drought severities and durations. The drought severity–duration–frequency curves using each of the two frequency analyses are derived, and these curves are used to examine how the drought phenomenon currently in progress is evolving. From this, the advantages and disadvantages of the two approaches, as well as the points to be aware of in application, are discussed. Finally, using the two approaches to the proposed drought frequency analysis, the behavior of Korea’s future extreme droughts is investigated under the conditions of various future climate change scenarios.

Modified Maximum Pseudo Likelihood Method of Copula Parameter Estimation for Skewed Hydrometeorological Data

For multivariate frequency analysis of hydrometeorological data, the copula model is commonly used to construct joint probability distribution due to its flexibility and simplicity. The Maximum Pseudo-Likelihood (MPL) method is one of the most widely used methods for fitting a copula model. The MPL method was derived from the Weibull plotting position formula assuming a uniform distribution. Because extreme hydrometeorological data are often positively skewed, capacity of the MPL method may not be fully utilized. This study proposes the modified MPL (MMPL) method to improve the MPL method by taking into consideration the skewness of the data. In the MMPL method, the Weibull plotting position formula in the original MPL method is replaced with the formulas which can consider the skewness of the data. The Monte-Carlo simulation has been performed under various conditions in order to assess the performance of the proposed method with the Gumbel copula model. The proposed MMPL method provides more precise parameter estimates than does the MPL method for positively skewed hydrometeorological data based on the simulation results. The MMPL method would be a better alternative for fitting the copula model to the skewed data sets. Additionally, applications of the MMPL methods were performed on the two weather stations (Seosan and Yeongwol) in South Korea.

Copula-Based Multivariate Frequency Analysis of the 2012–2018 Drought in Northeast Brazil

The 2012–2018 drought was such an extreme event in the drought-prone area of Northeast Brazil that it triggered a discussion about proactive drought management. This paper aims at understanding the causes and consequences of this event and analyzes its frequency. A consecutive sequence of sea surface temperature anomalies in the Pacific and Atlantic Oceans, at both the decadal and interannual scales, led to this severe and persistent drought. Drought duration and severity were analyzed using run theory at the hydrographic region scale as decision-makers understand impact analysis better at this scale. Copula functions were used to properly model drought joint characteristics as they presented different marginal distributions and an asymmetric behavior. The 2012–2018 drought in Ceará State had the highest mean bivariate return period ever recorded, estimated at 240 years. Considering drought duration and severity simultaneously at the level of the hydrographic regions improves risk assessment. This result advances our understanding of exceptional events. In this sense, the present work proposes the use of this analysis as a tool for proactive drought planning.

Assessing coincidence probability for extreme precipitation events in the Jinsha River basin

Global analyses show that climate change has already increased the frequency and strength of extreme weather events. The observed and remote-sensed precipitation productions have been deeply studied considering the stationary and non-stationary characteristics of extreme series. However, as these studies largely addressed the evolution of precipitation extremes at a single station, the consistency risk of extremes at multiple stations within close days has been little researched. In this paper, a multivariate frequency analysis method is proposed to reveal the exceeding and coincidence probabilities of the most dependent extreme precipitation series. First, a generalized extreme distribution, which has proved better at describing the tail behavior of extreme series, is developed to simulate the marginal distribution of rainfall extremes. Then, a mixed von Mises distribution is introduced to fit the occurrence time interval of the extreme events. Taking advantage of its flexibility in choosing arbitrary marginals, a copula is constructed to implement a joint probability analysis of extreme precipitation events at multiple stations. The upper reaches of the Yangtze River, where frequent heavy rains are the main source of the regional floods, were selected as a typical research area. Five hundred twenty-eight copula models were finally constructed by arbitrarily choosing two precipitation extreme series among the total 33 meteorological stations. The results revealed the risk of extreme rainfall occurrence once in 2, 5, 10, 20, 50, or 100 years, within a time interval of 1–90 days. The research has great significance for understanding complicated extreme events and is helpful for the disaster emergency response and management.

Non-stationary frequency analysis of annual extreme rainfall volume and intensity using Archimedean copulas: A case study in eastern China

Owing to hydrological change and human activities, increasing attention is being paid to the evolution of dependence structure of hydrological characteristics, especially for non-stationary multivariate frequency analysis. In this study, we performed a non-stationary frequency analysis of annual extreme rainfall of four study regions in eastern coastal China. Two indexes separately accounting for rainfall volume and intensity were derived from daily precipitation data recorded by four pairs of gauges. In particular, based on the compatibility of data sets examined by the copula equality test, index series sharing similar dependence structures were merged to enrich data bases for copula modelling. A moving time window of 30 years was applied to the index series for non-stationarity analysis. An investigation of dependence evolution was conducted by detecting trends and change-points in the correlation coefficient series. Time-dependent Archimedean copulas and Generalized Extreme Value (GEV) distribution were employed to model the joint and marginal distributions, respectively. Kendall’s return period was employed to compute the design values of the indexes. Results showed an intensifying tendency of extreme rainfall volume and/or intensity in three of the study regions and uneven spatial and temporal distribution of precipitation. Finally, the occurrence risk of annual extreme rainfall corresponding to different joint return periods (JRPs) was analyzed based on the observed index series. The study highlights incorporating non-stationarity in multivariate hydrological analysis and merging compatible data sets to improve copula inference.

A Bivariate Frequency Analysis of Extreme Wave Heights and Periods Using a Copula Function in South Korea

Kim, Y.-T.; Park, J.-H.; Choi, B.-H.; Kim, D.H., and Kwon, H.-H., 2018. A Bivariate Frequency Analysis of Extreme Wave Heights and Periods Using a Copula Function in South Korea. In: Shim, J.-S.; Chun, I., and Lim, H.S. (eds.), Proceedings from the International Coastal Symposium (ICS) 2018 (Busan, Republic of Korea). Journal of Coastal Research, Special Issue No. 85, pp. 566–570. Coconut Creek (Florida), ISSN 0749-0208.Sea storms are generally described by a set of variables such as wave height, wave period and wave direction, and these random variables are typically treated as independent of one another. Moreover, univariate wave frequency analysis has been applied to estimate design wave heights corresponding to various return levels (e.g. 20, 50 and 200-year) using significant wave heights under stationary conditions. However, it has been acknowledged that these variables are often correlated with each other, and such dependence structure needs to be considered in the estimation of extreme quantiles. More specifically, a joint estimation of quantiles for different combinations of these variables such as wave heights and wave periods is required to reliably assess optimal design of coastal (or offshore) structures. Over the last several decades, accelerated sea level rise (SLR) and its impact on coastal areas have been reported in many parts of the world. Estimation of extreme quantiles of SLR as a nonstationary process play a crucial role in assessing these impacts. In these contexts, a multivariate frequency model using a copula function approach is introduced to describe sea storm risk, which is mainly characterized by wave heights and periods. The proposed multivariate frequency analysis offers several advantages over widely used univariate stationary frequency analysis including uncertainty estimation, improved representation of inter-dependency and significant improvement of compound risk estimation.

Multivariate frequency analysis of urban rainfall characteristics using three-dimensional copulas

Urban runoff is a major cause of urban flooding and is difficult to monitor in the long term. In contrast, long term continuous rainfall data are generally available for any given region. As a result, it has become customary to use design rainfall depth as a proxy for runoff in urban hydrological analyses, with an assumption of the same frequency for runoff and rainfall. However, this approach has lack of overall coordination and cannot fully reflect the variability of rainfall characteristics. To address this issue, this study presents a three-dimensional copula-based multivariate frequency analysis of rainfall characteristics based on a long term (1961-2012) rainfall data from Guangzhou, China. Firstly, continuous rainfall data were divided into individual rainfall events using the rainfall intensity method. Then the characteristic variables of rainfall (design rainfall depth, DRD; total rainfall depth, TRD; peak rainfall depth, PRD) were sampled using the annual maximum method. Finally, a copula method was used to develop the multivariate joint probability distribution and the conditional probability distribution of rainfall characteristics. The results showed that the copula-based method is easy to implement and can better reflect urban rainstorm characteristics. It can serve a scientific reference for urban flood control and drainage planning.

Multivariate Frequency Analysis of Meteorological Drought Using Copula

The multivariate frequency analysis of droughts for Agartala (India) was carried out in the present study. The meteorological drought was modelled using Standardised Precipitation Index(SPI) at the time scale of 1, 3, 6 and 12 months. Three droughts variables i.e., duration, severity, interval were determined for SPI at the time scale of 1, 3, 6 and 12 months. For the construction of bivariate and trivariate joint distributions Archimedean and metaelliptical copulas were used. Upper tail dependence test was also carried out. The best copula was selected based on minimum value Akaike’s information criteria (AIC)) and Schwarz information criterion(SIC). The drought risk was estimated using joint probabilities and return period for the study area.