Quantile Regression Based Methods for Investigating Rainfall Trends Associated with Flooding and Drought Conditions
Conducting trend analysis of climatic variables is one of the key steps in many climate change impact studies where trend is often checked against aggregated variables. However, there is also a strong need to investigate the trend of the data in different regimes – examples include high flow versus low flow, and heavy precipitation versus prolonged dry period. For this matter, quantile regression (QR) based methods are preferred as they can reveal the temporal dependencies of the variable in question for not only the mean value, but also its quantiles. As such, the tendencies revealed by the QR methods are more informative and helpful in studies where different mitigation methods need to be considered at different severity levels.In this paper, we demonstrate the use of several quantile regressions methods to analyse the long-term trend of rainfall records in two climatically different regions: The Dee River catchment in the United Kingdom, for which daily rainfall data of 1970–2004 are available; and the Beijing Metropolitan Area in China for which monthly rainfall data from 1950 to 2012 are available. Two quantiles are used to represent heavy rainfall condition (0.98 quantile) and severe dry condition (0.02 quantile). The trends of these two quantiles are then estimated using linear quantile regression before being spatially interpolated to demonstrate their spatial distribution (for Dee river only). The method is also compared with traditional indices such as SPI. The results show that the quantile regression method can reveal patterns for both extremely wet and dry conditions of the areas. The clear difference between trends at the chosen quantiles manifests the utility of QR in this context.
KeywordsClimate change Precipitation Quantile regression Trend analysis
In recent decades, climate change has had an increasing impact on water resources, agricultural activities and the environment (Shi and Xu 2008). Increased variability of the magnitude and frequency of precipitation and temperature are among the major impacts of climate change (Dinpashoh et al. 2014). It is projected that globally, by the 2050s, annual average runoff will increase by 10–40% at high latitudes and in some wet tropical regions; on the other hand, there will be a decrease of 10–30% in some dry areas at mid-latitudes and in the dry tropics (IPCC 2007). For water resources planning and management, it is often preferable to use trend analysis of climatic variables such as precipitation, temperature and river flows, which has been widely reported in many studies.
There have been various studies on the trend of extreme precipitation events in different regions of the world, such as Donat et al. (2013); You et al. (2011); Zhai et al. (2005); Powell and Keim (2015). Conversely, extreme events due to the lack of precipitation, e.g., droughts can also have a considerable impact on social economic development and the environment. Unlike extreme precipitation events whose impact is often readily perceived as severe flooding, the onset of droughts is dependent on many different factors. It also takes much longer for the impact of droughts to be fully appreciated than that of heavy precipitation events.
Meteorological drought originates from a lack of precipitation over a long time period that is often appended by remarkably high winds, high temperature, high solar radiation and low humidity which cause increased evapotranspiration and is categorised by the dryness degree compared to a normal condition (Utah Division of Water Resources 2007).
Agricultural drought originates from the deficit of evapotranspiration and rainfall over an extended time period, which leads to prolonged periods of low soil moisture that would affect agriculture productivity and cause failure of ecosystem rehabilitation (Lei et al. 2015).
Hydrological drought appears when a deficit of precipitation perseveres for a long period, leading to an overall shortage of water supply in different forms such as streamflow, reservoir storage. etc. (Van Loon and Van Lanen 2012). Additionally, anthropogenic activities have a considerable influence on hydrological droughts.
Socioeconomic drought links the demand and supply of some economic goods with elements of hydrological, meteorological and agricultural drought (Shiferaw et al. 2014).
Drought can be characterised by three quantities: the areal extent, the severity of the occurrence and the duration (Tsakiris et al. 2013) which are often represented by the indices derived from chosen climatic observations. These indices can assist the policy makers and economists in the planning and management of water resources and decision-making process. In the literature, it is no surprise that there is over 100 drought indicators reflecting the complexity of the subject matter (Lloyd-Hughes 2014), to name just a few, the standardized precipitation index (SPI; McKee et al. 1993); the Palmer drought severity index (Palmer 1965); the vegetation drought response index (Brown et al. 2008); the multivariate standardized drought index (Hao and Aghakouchak 2013); the surface water supply index (Shafer and Dezman 1982) and the drought severity index (Mu et al. 2013). A comprehensive review on the drought indices has been given by Tsakiris et al. (2007) and Stagge et al. (2017).
It is worth noting that although the multi-factor based indices can describe drought events more accurately, other indices using only the precipitation data such as SPI, have also been widely adopted in many studies thanks to the high availability of precipitation data, especially when studying future climate where other factors, such as vegetation are often unavailable or need to be further derived.
As to the methods used for trend analysis, ordinary linear regression is among the first choices of many researchers. This method is often accompanied by other non-parametric methods, such as Mann-Kendall test (Mann 1945; Kendall 1975) helping to further confirm the statistical significance of the trends detected, e.g., Martinez et al. (2012) and Song et al. (2014). In many cases, although the trend indicated by the fitted regression line may not be statistically significant, its gradient is used nonetheless as a rough indicator.
One of the major drawbacks of this method is that the trend it manifests is often expressed as the mean value of climatic variables that are conditioned on time. As such, it is difficult to gain further necessary insights as to how the events associated with more extreme values vary with time. For example, water managers are usually more concerned with the trends of severe storms or extreme dry spells than those of the ‘mean’ conditions. To a certain degree, such problem can be mitigated by stratifying the data into different categories, e.g. high flows versus low flows; however, one must realise that doing so would effectively reduce the sample size and hence sacrifice the information of data variability.
The quantile regression (QR) method (Koenker and Bassett 1978; Koenker 2005) extends the ordinary linear regression to explain how the quantiles of response variables are conditioned on the input variables, which offers a new window through which different regimes of the response variables can be examined in detail. Clearly, there is a need for identifying trends of climatic variables in different quantity regimes as mitigation measures would be more effective with such refined information.
Since quantiles are often a convenient measurement of the data departing from its mean and thus offer an association with the rarity of those values, it follows naturally to use the QR method for revealing the trend of ‘extreme’ events as indicated by different quantiles. A more vigorous approach of linking QR with extreme value distribution can be referred to Cai and Reeve (2013).
In this paper, we demonstrate a study of using the QR based methods to identify the rainfall trends in two remarkably different climate regions: the Dee river catchment in the UK and the Beijing Metropolitan Area (BMA) in China. Focus is set on the trend of both wet (indicated by high quantiles) and dry conditions (indicated by low quantiles) as they are of great value as far as flood risk management and water resources management are concerned. We make use of a higher quantile 0.98 to represent wet conditions – where severe flooding may occur; and a lower quantile 0.02 for dry conditions where prolonged droughts may be induced. We further show that the spatial distribution of such trends can also help produce a coherent, refined spatial structure for the use of flood risk and water management purposes.
The rest of this paper is organised as follows: the two study areas are introduced in section 2 followed by the discussion of the methodology. A detailed discussion on the result is given in section 4, and finally, several concluding points are drawn in section 5.
2 Study Area
The annual rainfall over the Dee catchment ranges from 650 mm in the downstream area in the east to 1200 mm in the upstream in the west (BADC 2015). Daily rainfall records from 13 rain gauges over a period of 35 years (1970–2004) are used in this study. Figure 1a shows the catchment elevation and the locations of the rain gauges.
In contrast, the metropolitan area of Beijing is far larger (16,410 km2) yet with a similar layout of topography with its west and north (68% in area) having elevations between 1000 and 1500 m, while the central and south-east parts are just 20–60 m above the sea level. The monsoon-driven humid continental climate in this region is rather opposite to the Dee catchment, having cold and dry winters and hot humid summers. Over the last two decades, the area has suffered both very dry winters/springs and wet summers which caused severe shortages in the water supply as well as local flooding in the urban areas. Annual rainfall recorded at the 45 rain gauges (Fig. 1b) over the period of 1960–2012 were obtained alongside the monthly areal average precipitation over the entire region for the same period. Monthly values at the rain gauges, however, were not available due to licence restriction.
3.1 Quantile Regression
The Quantile Regression (QR) (Koenker and Bassett 1978; Koenker 2005) is a statistical technique that was initially introduced for regression analysis in econometrics as an alternative and possibly better tool to the ordinary least square method (OLS). It has since been gradually applied in many other disciplines. The method has received considerable attention in many statistical literatures, but less so in areas related to water resources analysis and environmental studies (Tareghian and Rasmussen 2013). An overview of some of the reported studies of the using of quantile regression in water and environmental studies is given by Abbas and Xuan (2019).
The QR method is a powerful extension to ordinary linear regression in a way that the quantiles instead of the mean of the given response variables are conditioned on independent variables with many additional benefits as discussed in Koenker (2005).
The suitable optimisation algorithms are outlined in Koenker (2005) and the outcome is the linear quantile regression model. Although the linear form of QR is very common, parametric models that are non-linear in parameters (i.e. models in which the form of the non-linear regression equation is explicitly specified by the modeller) can also be estimated (Cannon 2011).
The linear form of such relationship (Eq. 1) can then be used to describe the magnitude (in terms of its gradient) of the trend. It should be noted that proposing a linear tendency of the response variable quantiles on the input variable (time) renders the process parametric.
The QR method is not limited to its linear form only; where necessary, it can be conveniently extended to nonlinear case as well as non-parameteric; however, to serve our purpose of identifying a general trend in a time series, it is convenient and beneficial to use its linear form. While non-parametric quantile regression is not necessarily the best choice for representing trends, it can clearly give a better fit in many cases and indeed needs to be explored in further studies.
3.2 Significance Test of the Linear Trend
A warranted question related to any trend analysis is whether the trend is statistically significant. For quantile regression, several bootstrap methods are developed to test the significance of the fit. Discussion of this topic goes beyond the scope of this paper. The analysis in this study is conducted using the R-package ‘quantreg’ (Koenker et al. 2016) which includes both the fitting and the significance test methods.
3.3 Choice of the Quantiles
3.4 Standardised Precipitation Index (SPI)
Classification of SPI values (McKee et al. 1993)
The SPI index is an extensively utilised technique that permits harmonic comparison across both space and time and has the flexibility to evaluate a deficit of rainfall over a defined accumulated time period. It also provides a sign of probable occurrence of a drought with increasingly negative values representing a severe drought condition (Lloyd-Hughes and Saunders 2002). SPI has been employed in many analyses based on long-term precipitation data (e.g. Nalbantis and Tsakiris 2009).
The relative flexibility, comparability and simplicity of computation of the SPI have resulted in an authorisation by the World Meteorological Organisation as the indicator for monitoring meteorological drought (Hayes et al. 2011). Despite these advantages of the SPI, there are identified disadvantages. The utilisation of the rainfall dataset alone does not consider the evaporative demand, which may lead to underestimated drought severity in seasons or areas with high levels of evapotranspiration. Furthermore, the selection of a suitable theoretical probability distribution for the precipitation data is still under study (e.g. Stagge et al. 2015) and the fitting of an appropriate probability distribution function with a high percentage of zeros is challenging.
The SPI can be computed for any given periods (3, 6, 9, 12, 24 or 48 months). In our study, the monthly rainfall amounts are fitted with a gamma probability density function using an R package ‘precintcon’ (Povoa et al. 2016) to produce the annual SPI values at each rain gauges, then the QR method was applied to this series to investigate its variation over time.
3.5 Extreme Precipitation Indices
Total precipitation on very wet day (daily precipitation >95th percentile) R95PTOT;
Total precipitation on extreme wet day (daily precipitation >99th percentile) R99PTOT;
Days with heavy precipitation (daily precipitation >10 mm) R10MM;
Days with very heavy precipitation (daily precipitation >20 mm) R20MM.
An R package ‘climdex. Pcic’ (Bronaugh 2015) is used to produce the above extreme indices at each rain gauges, then the QR method was applied to this series to investigate its variation over time.
4 Results and Discussion
For the Dee catchment, daily rainfall records from the 13 rain gauges were aggregated into monthly and yearly datasets from which the linear trend of 0.98 and 0.02 quantiles are produced at each rain gauges, before being interpolated over the catchment using the Inverse Distance Weighted (IDW) method.
It is interesting to see how SPI compares with the QR trends in terms of indicting dry and wet conditions. The trend line of the 0.98 quantile from the QR analysis indicates how the amount of rainfall at a given frequency (return period) varies with time (increase or decrease). SPI, however, is a measurement against the entire series assuming no change in the distribution. A single SPI value is like a quantile of rainfall to some extent (as it measures the distance from the mean). Therefore, in general, the value of high SPI should somehow resemble the trend of the 0.98 quantile. This has been supported by the similarity of the pattern shown in both Figs. 3a and 7.
It is clear from Fig. 6 that the use of the overall trend of the annual SPI is not very helpful to indicate the trend of wet and dry conditions, as even with a slightly increasing trend of all stations, severe dry conditions get intensified during the end part of the period. This again shows that the QR regression can better capture those trends.
The QR based trend analysis shows more detailed information regarding more extreme conditions (wet and dry). This is particularly useful for water managers who are more concerned with these values rather than the average one.
The involvement of quantiles brings an extra benefit of bridging trend analysis with frequency, which implies a great potential of its use in studying climate change impact on engineering design without being constrained by assumptions of data stationarity.
It helps better to understand the climate change impact. As already shown in the Beijing case, a decreasing trend in summer rainfall may still be accompanied with increasing severe storms in the same season.
Not only can the QR method capture the pattern detected by other indices such as SPI, but it can also conveniently reveal the temporal trend of different values of concern.
Salam A. Abbas has been supported by the scholarship provided by the Higher Committee for Education Development in Iraq; Yunqing Xuan has been partly supported by the Royal Academy of Engineering’s UK-China Urban Flooding Research Programme (Grant: UUFRIP\10021), which are both gratefully acknowledged. The authors are grateful for the data provided by the Natural Resource Wales, the Centre for Ecology and Hydrology UK and the British Atmospheric Data Centre. We also thank the editor and the anonymous reviewers for their valuable advices and comments helping to improve the paper.
Compliance with Ethical Standards
Conflict of Interest
The authors declare no conflict of interests. This paper is based on the study previously presented in the 10th World Congress of EWRA 2017 with the same title but has since been substantially extended for the invited submission for journal publication.
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