## 1. Introduction

As a consequence of the increasing interest in the change of natural resources available due to climate change, many studies properly invest their efforts to address the management of such resources. Among others, water management plays a key role since water scarcity will be one of the main issues to be addressed by human beings [

1,

2], especially because of its subsequent effects on, the agricultural sector. Thus, many studies consider it of prominent importance to detect changes in temporal trends of drought, particularly at the regional scale. Drought is the result of a complex interaction among precipitation deficiencies, excessive evapotranspiration and the demands of water use. Whereas aridity is permanent and occurs only in low rainfall–high evapotranspiration areas, drought is a temporary feature that occurs mostly in all climatic zones. The definition of drought varies according to five types: meteorological, hydrological, agricultural, socioeconomic, or groundwater. This depends on the duration of the reduction in the amount of precipitation and the contributing variables that affect evapotranspiration, such as temperature, humidity, and wind. A complete review of drought definitions and concepts is given in [

3].

In fact, the Earth warming intensifies the global hydrological cycle by increasing the globally averaged precipitation, evaporation, and runoff. In this framework, the Standardized Precipitation Evapotranspiration Index (SPEI) [

4] is widely used since it combines the sensitivity of the Palmer Drought Severity Index (PDSI) [

5] to the changes in evapotranspiration with the multi-temporal feature of the Standardized Precipitation Index (SPI) [

6]. The possibility of being scale-free, i.e., calculated for a variety of time scales, allows SPEI to monitor both short-term water supplies and long-term water resources and, consequently, to be proper for either agricultural or hydrological drought. Recall that agricultural drought is mostly linked to soil moisture conditions that respond to precipitation anomalies on a relatively short scale, whereas hydrological drought regards groundwater, stream-flow, and reservoir storage that reflect the long-term precipitation anomalies.

Within this context, the trend analysis is an important tile to start the risk management associated with these type of events and the choice of the statistical methods to assess trend is one of the causes that determine differences in the results of studies devoted to this subject [

7,

8,

9].

In general, the trend analysis of a variable includes both the assessment of direction and magnitude. Trend direction reveals whether there is an increasing (positive) or decreasing (negative) dependence of the variable on-time, while trend magnitude expresses the measure of this variation in terms of relative change per unit of time. The Mann–Kendall (M-K) statistical test has been frequently used to assess the significance of trends in hydro-meteorological time series because it requires fewer assumptions than the parametric tests [

10]. Many studies apply M-K test to assess the SPEI trend either locally or globally. For instance, Vicente-Serrano et al. [

11] used M-K test to assess trends in the seasonal SPI and SPEI time series representative of different regions obtained through principal component analysis in Bolivia; Li et al. [

12] combined M-K trend test of SPEI time series and an analysis of the average variation of dry episodes duration in a region of South Tibet; Liu et al. [

13] used the analysis of SPEI trend for classifying the risk of the crop yield; and Tan et al. [

14] used M-K test for Malaysia drought trend assessment where droughts are identified by means of SPI. About SPI, which is straightforward comparable to SPEI being built in a very similar methodological framework, several studies have used a parametric approach to evaluate the change in time of droughts. Among others, Moreira et al. [

7] applied a log-linear model to assess significant change in the fixed classes of risk; Krysanova et al. [

15] applied both linear and general linear models to account for drought change in the Elba basin; and Bazrafshan [

16] proposed a trend drought assessment of Iran based on both the nonparametric and parametric test using either SPI or SPEI.

Another approach for the analysis of drought trend is to investigate the statistical properties of drought. In particular, once SPEI has been fixed as drought measure, it is possible to calculate the duration, severity, and intensity of droughts throughout the time series [

17]. This computation relies on the assumption that a drought event is considered to occur at a time when the value of SPEI is continuously negative and ends when SPEI becomes positive. Then, the return period of duration, severity, and intensity properties gives a tendency of drought expectation for the future. For instance, Mishra and Singh [

18] used the annual values of drought severity for frequency analysis and fit a distribution to analyze the associated risk of droughts using the exceedances’ probability, while Mortuza et al. [

19] used a copula approach to identify the joint probability distribution of drought duration and severity, then computed the return period of drought according to different thresholds. Finally, Onyutha [

20] introduced a modified version of the extreme value distribution used for frequency analysis of drought extreme events, which takes into account the non-stationarity of the data.

The scientific question posed in this paper regards how to evaluate the trend of SPEI in a specific drought risk class among moderate, severe and extreme. The typical methodological framework for trend analysis in hydrology is to combine M-K for significance and

$\beta $-Sen method for direction and magnitude. The M-K is a powerful tool that can also be extended to the detection of a trend in non-normally distributed time series. For instance, Yue et al. [

21] examined the power of the M-K test, i.e., the probability of rejecting a null hypothesis when it is false for a few commonly used distributions in hydrology: the extreme value distributions, the Pearson type 3, and the lognormal. Even though the advantages of such a nonparametric test, for example, the assumption of normality or of variance homogeneity are not needed, there are several shortcomings that can influence the result of the M-K and need to be addressed: (i) the serial correlation in the time series [

10,

22,

23]; and (ii) the dependence of test power on the magnitude of trend, sample size, and the amount of variation within a time series [

21].

Here, we propose a trend test for the different classes of drought risk based on the Poisson process and we compare its results with those of the M-K test. In fact, when a truncation level is imposed, the exceedances of a climatological time series can be modeled as a point process. In the framework of the time series of SPEI values calculated at a specific time scale and for a given month, we can consider the inter-arrival times—years in this case—between drought episodes as independent and identically distributed. Consequently, the point process becomes a counting process (among many, see Par. 4.2 of [

24]) and the natural distribution used to describe such a counting process is the Poisson distribution. The latter is entirely characterized by a unique parameter: the intensity or rate of the process, which can be interpreted as the mean number of drought episodes per unit of time (years). Furthermore, if the arrival rate does not remain constant through time, then it follows a Non-Homogeneous Poisson Process (NHPP). This framework is particularly useful because, as demonstrated by Park et al. [

25]: “drought expressions by a statistical index such as SPI can be distorted by stationary assumption and cautious approach is needed when deciding drought level considering climate changes”. We consider the use of a special case of the NHPP: the power law process defined in [

26]. The power-law functional form used to model the dependence of the process intensity on time suggests the use of the

${\chi}^{2}$ test for time-trend analysis. This approach is commonly used in risk evaluation, for example Ho [

27] used it to evaluate volcanic eruptions time-trend or in lifetime analysis. Nevertheless, applications for SPI in this theoretical framework already exist, for example Achcar et al. [

28] used NHPP to identify change points in time series of droughts in Brazil. The main advantage of this approach is that the trend is completely determined by the exponent of the power-law [

27].

Moreover, we present the computer code used for the computation of SPEI time series at a global scale as well as that for the trend analysis. The part of code regarding the SPEI computation starts from that proposed by Beguería et al. [

29] and it is based similarly on the precipitation and evapotranspiration time series of the CRU dataset [

30]. Our code [

31] is openly accessible from the link

https://zenodo.org/badge/latestdoi/194230602, and differs from that released by Beguería [

32]: (1) because it is based on the

raster [

33] instead of

ncdf4 R package; and (2) because it is possible to set the time window for the reference period to be used in the parameter estimation phase. The SPEI dataset resulting from our computation is comparable in principle with the SPEI global database released by Beguería et al. [

29]; however, the comparison is not feasible since the code released by these authors does not allow setting the reference period and, moreover, it seems to contain a bug in the part regarding the expansion of potential evapotranspiration from the mm/day scale to mm/month.

This paper is organized as follows.

Section 2 contains a brief description both of the dataset and the programming code used to calculate the SPEI; in particular,

Section 2.3 introduces our methodological proposal for testing the significance of SPEI trend and its nonparametric counterpart, the Mann–Kendall test; and

Section 3 presents a comparison of the two methods in terms of differences for 3, 6, 12, and 24 monthly temporal scales either in the form of global maps or box plots. Finally,

Section 4 concludes the paper with some discussion. In this study, we used R software [

34], and

SPEI [

35] and

modifiedmk [

36] R packages for SPEI calculation and M-K test computation, respectively.

## 4. Discussion

Trend analysis of climatic extremes plays a key role in understanding climate change. However, complex phenomena such as drought have several features that imply different impacts. An increase in moderate droughts, in fact, has a different impact on natural resources compared to an increase in the extreme classes, even at short time scale, i.e., SPEI-3. To satisfy such an operative request, we introduced a testing approach that allows the user know whether there is an increased risk of drought events or not and, furthermore, to classify the risk in terms of drought severity. The trend testing approach that we suggest in this paper is a methodological procedure that includes a common trend test, such as Mann–Kendall, but is not limited to one. In fact, the use of SPEI in the analysis of droughts suggests the possibility of developing a parametric test based on the Poisson process because the definition of thresholds accounting for different levels of severity is intrinsically connected to the standardization process used for the SPEI computation, and their definition allows treating the corresponding exceedances as a counting process.

To demonstrate our intuition, we adopted a known special case of Non-Homogeneous Poisson Process that is based on the power-law and we performed the trend analysis of SPEI time series from 1901 to 2018 at the 3-, 6-, 12- and 24-month scales for each month. Then, we compared the results obtained with this method to those from the classical Mann–Kendall test at the global spatial scale.

The results of the comparison between the two approaches show that the M-K identifies a significantly smaller number of the cases where there is a positive trend of drought episodes, especially for hydrological droughts. In addition, for the months from October to April, it emerges that M-K identifies more cases with a negative trend for meteorological and agricultural droughts than NHPP (SPEI-3 and SPEI-6). Furthermore, the areas under the risk of drought increasing are considerably larger if the NHPP approach is used instead of M-K to test the trend. Furthermore, this difference between the two approaches is more significant at two-time scales: for winter and spring months, and at long-timescale (SPEI-24). Both time scales are crucial for water availability and management. For this reason, a comparison of the results obtained from the application of two trend statistical approaches, i.e., M-K and NHPP, and their combination, suggested that, even though the M-K test is one of the most used and robust statistical trend methods, it cannot explain, alone, the significance of all these characteristics.

To summarize, when indices such as SPEI are used to define and monitor drought’s occurrence and evolution, the adoption of a counting process approach for testing the trend allows the possibility of distinguishing the results among different classes of risk. This is particularly useful for risk management, such as the management of water resources.

The proposed methodology applies to all drought indices where the phenomenon is addressed by any type of threshold that allows a sequence of drought episodes throughout the observed period. Thus, this method could be also applied, for example, to the SPI (Standardized Precipitation Index) [

6,

48], the PDSI (Palmer Drought Severity index) [

49], the EDI (Effective Drought Index) [

50], the Deciles [

51], the CMI (Crop Moisture Index) [

52] and the SWSI (Surface Water Supply Index) [

53].