Assessing Polarisation of Climate Phenomena Based on Long-Term Precipitation and Temperature Sequences

1. Introduction

Analyses of historical observations of climate variables clearly indicate the anthropogenic causes of climate change [1,2,3]. Numerous studies at various spatial and temporal scales have thoroughly explored the impact of projected changes in the climate system on hydrology and water resources [1,3,4,5,6]. After air temperature, fresh water is proving to be the second most reactive environmental variable to climate change [1,7,8,9,10]. Despite the numerous existing studies and analyses of the temporal and spatial characteristics of precipitation and temperature, the timeliness of these studies is becoming increasingly appreciated [11,12,13,14,15,16,17,18], especially in the context of documenting the impact of human activities on regional climate factors [19,20,21]. Increasing computational capabilities and comprehensive analyses based on long measurement series [8,22,23] are attracting attention, especially in the context of the search for the justification of theses on the polarisation of extreme climatic events [1,24,25,26,27,28,29,30].

The polarisation of climatic elements is a phenomenon in which certain climatic elements or features tend to separate into two extreme or opposite categories [31,32]. This means that over time, certain climatic features become more clearly concentrated in the two opposite extremes, which can lead to increased variability and diversity in atmospheric conditions. Polarisation can manifest itself as a clear separation between extremes of temperature, precipitation or other climatic parameters, with implications for long-term patterns and trends in the climate of an area. The property of bimodality emphasises the concentration of features in two opposite extremes [8,33,34,35]. The polarisation of precipitation events can be defined as a process in which the frequency and intensity of extreme precipitation events, such as rainstorms, hail, storms or floods, increases. Simultaneously, the "normal" precipitation condition is becoming less frequent. In the case of temperature phenomena, polarisation can be defined as a process in which the frequency and intensity of extreme temperatures, such as heat waves or extreme cold, increases. Again, the "normal" state of temperatures is becoming increasingly rare. Both of these phenomena are part of a broader process of climate change that affects atmospheric circulation and processes, leading to increased weather variability and a greater risk of extreme weather. Polarisation can be interpreted as a change in the nature of the occurrence of extreme events, where extreme events, specifically extreme temperatures and precipitation, are becoming more frequent, while "normal" conditions are becoming less frequent.

The increased interest in climatic factors such as precipitation and temperature [13,36,37] is also due to the consequences of floods and periods of hydrological drought [38]. Studies of the variability of precipitation and temperature over different periods [5,9,14,39,40,41] have made it possible to analyse statistical trends and assess the signal of these changes [37,42,43,44]. The results of these analyses provide information about the moments of trend change and the form of them. In the context of the importance of the variability of climatic factors, it is important to apply statistical techniques [4,45] that allow the detection of changes and trends [2,43], in among other sources, data relating to monthly precipitation totals and monthly average temperatures.

Climate change predicted by the IPCC (Intergovernmental Panel on Climate Change) occurs as a result of increases in greenhouse gas emissions, which are mainly caused by the industrialisation of the world, the burning of fossil fuels and changes in land use [3,10,46,47]. According to the Intergovernmental Panel on Climate Change [48], both natural forces and human activities contribute to changes in climate patterns, including increases in land and ocean surface temperatures, changes in spatial and temporal precipitation patterns, increases in the frequency of extreme events, rising sea levels and increased El Niño [3,4,36,39]. It is clearly emphasised that human influence significantly outweighs the action of natural forces. The warming of the Earth is evident in data that show a 0.6°C increase in global temperature from 1901 to 2001 [48]. To see changes in precipitation and temperature trends, anomalous values and statistical models are used to measure the timing of the trend change and its magnitude. The change in trends of various climate variables can be quantified using global atmospheric circulation models (GCMs) or, for example, using statistical models that identify the moment of change of the trend and its magnitude [8].

Many studies have shown that climatic variability has a significant impact on river flow values, which are sensitive to both precipitation and temperature. The polarisation of climatic phenomena has been demonstrated by researchers [40], who found increasing trends mainly in flood magnitude in German river catchments, which was attributed to climate change. In addition, seasonal analyses have shown that these changes were greater in winter than in summer. Rapid changes in both mean and variance can be attributed to both climate (e.g., changes in climate regimes) and anthropogenic effects (e.g., construction of dams and reservoir systems, changes in land use and vegetation cover, changes in the location of measurement points) [49,50]. Statistical analyses must be interpreted in the context of observed physical [3,7,51], social and economic phenomena [3,14,19,52]. Therefore, predicting temporal trends in precipitation and temperature is very useful in social and urban planning [53].

The world's rivers play a key role in linking the atmosphere, hydrosphere and biosphere by transporting 40% of precipitation runoff and 95% of sediment to the oceans [54,55]. The shaping of river flow depends on many factors, such as precipitation, temperature, evaporation, soil conditions, landforms and land use changes [56]. With the available long-term precipitation and temperature data, it is very important to conduct long-term analyses of this data in order to understand the polarisation of climate phenomena and its potential consequences [57]. In addition, such analyses provide valuable information that can be used to develop decision-making tools for the purposes of effectively managing extreme events, such as droughts and floods, and to mitigate ecological damage in regions particularly vulnerable to these hazards [57].

When assessing climate polarisation, a process in which certain climatic features become concentrated in two opposing categories, there are strong links between precipitation and temperature [39,57,58]. An increase in global temperatures can lead to an increase in the amount of water vapour in the atmosphere. More water vapour in the atmosphere increases the potential for cloud formation and precipitation, especially in areas with higher temperatures where there is access to more thermal energy As a result, more water vapour may in turn increase precipitation, meaning that areas with higher temperatures may experience more precipitation. Conversely, lower temperatures can result in reduced precipitation. In extreme cases, such as prolonged droughts or floods, fluctuations in temperature and precipitation can have catastrophic consequences, including the failure of crops and damage to property [1,50,59].

Long-term data series make it possible to identify temperature and precipitation trends and help to determine whether a region is experiencing changes in polarity. Furthermore, the analysis of precipitation-temperature interactions provides a better link between important atmospheric phenomena, such as El Niño and La Niña cycles [60,61], which affect global climate patterns. An example of this is the fact that rising temperatures can affect the occurrence of intense precipitation, while reduced precipitation can contribute to extreme heat waves [50]. It is import to consider the amount of precipitation and temperature along with other climate phenomena as their interaction can affect the overall effect of climate polarisation [62]. Understanding magnitudes and the relationship between precipitation and temperature is crucial for assessing the polarisation of climate phenomena and its consequences for the environment, economy and society. This knowledge is essential with regard to taking preventive measures in order to mitigate the negative effects of climate change.

The main topic of this study is the analysis of long-term sequences of precipitation and temperature data in terms of the possibility of introducing the concept of polarisation of climate extremes. Section one reviews the literature and presents the background of the variability of extreme climate events that provide the basis for analysing the polarisation phenomenon. Section two presents the temporal and spatial extent of the data that was used in the analysis. Section three characterizes the phenomenon of the polarisation of extreme climate events in the areas of precipitation and temperature, and discusses the impact of polarisation on important consequences for ecosystems, economies and infrastructure. The fourth section presents the approaches used in constructing a measure of polarisation and presents a proposal for the introduction of a static measure based on a stationary time series and a dynamic time series based on trends. The fifth and sixth sections present known statistical techniques in trend analysis and trend change point recognition. The seventh section discusses and evaluates the results of the performed analyses. The eighth section presents conclusions.

2. Preparation of Data for Analysis

The great interest in analysing long precipitation sequences stems from the need to assess climate change and its impacts on all spatial and temporal scales. This demand has led to the initiation and support of many research and monitoring programs conducted by international organisations. In this context, the Global Precipitation Climatology Centre (GPCC) was established on behalf of the World Meteorological Organization (WMO) in 1989. This institution is supported and operated by the Deutscher Wetterdienst (DWD, Germany's National Meteorological Service) as a German contribution to the World Climate Research Program (WCRP). The main objective of the GPCC is the global analysis of monthly precipitation on the Earth's surface based on "in situ" precipitation station data [36,63]. An equally important institution is the NOAA (National Oceanic and Atmospheric Administration) the US government agency for the study of the atmosphere, oceans and climate. It conducts research and monitoring of climate change, including collecting, processing, analysing, updating and making long-term precipitation and temperature sequences available [64,65].

The goal of the aforementioned institutions is to meet user requirements for the accuracy of the data and analysis results made available, as well as the timelines and availability of the product. Among the various types of GPCC and NOAA products, gridded sets of long-term precipitation and temperature data are made available [63,64]. These items of data are not made available in real time. This paper relies on grid data of monthly precipitation totals from the GPCC products made available and grid data of monthly mean temperatures from NOAA products. This data corresponds to a spatial resolution of 0.5°x 0.5°, are consistent in spatial and temporal extent and cover the years 1901 to 2010. Products from both the GPCCC and NOAA are made available via the Internet.

The NOAA (National Oceanic and Atmospheric Administration) and the GPCC (Global Precipitation Climatology Centre) are two institutions that provide the opportunity, based on the available long-term data, to perform analysis and evaluation of the polarisation of climatic phenomena. The NOAA is engaged in the research and monitoring of the atmosphere, oceans and climate around the world. The NOAA collects data from various sources such as satellites, ocean buoys, aircraft and weather stations to assess the state of the climate and predict changes. The GPCC is involved in assessing precipitation around the world. The GPCC uses a variety of data sources, such as weather stations, radar and satellites, to develop global precipitation maps. The two institutions use their knowledge and data to assess changes in climate phenomena, including precipitation and temperatures around the world, and predict how they will change in the future. They are also working with other climate institutions to increase understanding of the climate and to help make decisions about climate change. This paper examines global trends in monthly precipitation totals and monthly mean temperatures from the area of 377 river basins distributed over all of the continents. Assuming 509.9 10⁶ square kilometres of land area on the earth, the analysis covers 12.76% of the total land area of the globe. Table 1 shows the areas covered by the analysis. The regions in the text are identified by code $WMO\_REG=n$ where $n=1,...,6$ (Table 1, Figure 1).

GPCC precipitation data and NOAA temperature data by month from 1901 to 2010, calculated from grid data with a spatial resolution of 0.5°x 0.5° ° latitude and longitude, were converted to catchment areas. In this way, a sequence of monthly data was obtained, which became the subject of the analyses presented in this article. GIS interpolation mechanisms were used in the spatial analysis of the data preparation. The monthly precipitation amount and the monthly temperature value were calculated as a weighted average of the corresponding features of the grid surface elements that completely covered the catchment area, taking into account the surface area of these elements.

The area of one of the smallest catchments in this study: SKJERN A (Europe, Denmark) is, according to calculations, 1090 km². An area of 0.5°x 0.5° at the latitude of Denmark covers an area of 3100 km². The location of this catchment area, for calculations of precipitation or temperature, requires the consideration of five neighbouring areas of 0.5°x 0.5°. With even such a small catchment area, precipitation and temperature characteristics are averaged. The calculated values of the sequences were subjected to a simple statistical analysis, determining the basic statistics: the minimum and maximum value, the mean and, the standard deviation of the sample. The data was analysed in monthly as well as calendar year cross sections. Analyses of temperature data covered the years 1901 to 2010.

4. The polarisation of precipitation and temperature phenomena

The occurrence of temporal variability and extreme events in air temperature and precipitation is usually assessed by analysing a set of indicators that define variability and extreme conditions. The common understanding of an extreme event is based on the assumption that a "normal" condition exists [44]. In the context of the common understanding of an extreme event, polarisation can be interpreted as a change in the nature of the occurrence of extremes, with extreme events – extreme temperatures and precipitation – becoming more frequent and the "normal" state becoming less frequent. The polarisation of extreme events can result from a variety of factors, including human activity and climate change, which can affect precipitation cycles, the intensity of extreme events and temperature conditions. In this way, polarisation can be interpreted as the evolution of climatic features, resulting in an increase in the frequency of extreme events while destabilising the 'typical' state. Polarisation can have serious consequences for the functioning of ecosystems, the economy and the quality of human life.

Analysis of the polarisation of precipitation and temperature is essential because of their key role in global energy and water cycles and their impact on climate change. Accurate knowledge of precipitation and temperature is particularly important for assessing the amount of available freshwater and managing water resources, which is essential for reducing the risk of floods and droughts. In addition, there is growing scientific evidence that human activities are influencing climate change, contributing to shorter periods of intense precipitation and longer dry periods of high temperature and low precipitation. The polarisation of extreme events, such as floods and droughts, is increasingly evident and can be attributed to the unevenness and intensity of human activity. Investigating the polarisation factors associated with monthly precipitation and monthly average temperatures can significantly influence the development of water management strategies and reduce the risk of extreme climatic events.

This article analyses long-term sequences of precipitation and temperature to assess the polarisation of climate phenomena. The term "polarisation" in terms of precipitation or temperature is used in climate science to describe the process by which some areas of the planet become more extreme in terms of temperature and/or precipitation. This can lead to significant changes in the local environment, including changes in the distribution of plant and animal species, changes in weather patterns and changes in sea level [66]. Long-term sequences of precipitation and temperature are an essential tool for studying climate polarisation. These sequences provide a detailed record of an area's climate over many years, allowing trends and patterns in the data to be identified. By analysing these sequences, it is possible to identify areas where the climate is becoming increasingly polarized and track changes over time. One of the key advantages of using long-term precipitation and temperature sequences is that they provide a high degree of accuracy and precision. The time series are created using advanced measurement techniques, such as data from satellites and weather stations, and are subject to rigorous quality control measures [27,63,67]. As a result, they provide a reliable record of climate data in a given area, enabling precise observations and measurements.

The impact of polarisation can be observed in various areas of life on earth: in ecosystems, the economy and infrastructure, in changes in the size and location of water resources. The impact of polarisation on ecosystems in areas of extreme temperature and precipitation changes affects biodiversity and ecosystem functioning [66]. The polarisation of climatic factors can lead to changes in the distribution of plant and animal species [68,69], which can have negative consequences for entire ecosystems. The consequences of polarisation for agriculture can affect crops and food production [17,41]. In some regions, climate polarisation may lead to a reduction in the amount of available water resources, which in turn affects agricultural production and the health of the industry. The consequences of polarisation for infrastructure can affect roads, bridges, buildings, water and sewage networks and water management systems. In extreme cases, severe damage or destruction can occur. Due to the complex and dynamic nature of the climate [70,71], polarisation is variable and difficult to predict. It is also difficult to predict exactly what the consequences of polarisation will be in a given region. In addition, climate variability means that some regions may experience polarisation in different years, complicating the process of predicting and monitoring changes. The interaction between polarisation and other climate phenomena can trigger certain climatic events, such as hurricanes, tornadoes or droughts. In this way, one phenomenon can amplify or weaken another, complicating the process of understanding the scale of climate change. The influence of anthropogenic factors on the polarisation of climate factors can occur through greenhouse gas emissions [17,47,57,62], among other factors.

Polarisation of climate events refers to changes in the intensity and frequency of influential weather events. NOAA and the GPCC monitor these changes [63,72] to determine the factors influencing the polarisation of climate phenomena, their effects, and ways to reduce the negative consequences associated with these events. Assessing temperature variability is crucial to understanding climate change and its impact on ecosystems and people [73]. All of this information is essential for developing strategies and actions to manage change and to adapt and minimise its negative impacts. NOAA and GPCC are key sources of information for scientists, policymakers and other stakeholders who make decisions about climate change.

5. The concept of polarisation measure

The concept of measuring climate polarisation is based on measuring the degree of diversity and variability of climate elements in a given area. This measure is used to determine the level of polarisation of climate elements in the region of interest. Polarisation refers to the extremes of parameters such as temperature, precipitation, humidity, etc. occurring in a specific area. A high level of polarisation indicates significant differences between maximum and minimum values, which can lead to an increased risk of extreme weather events. The measure of polarisation can be determined from various climatic parameters, such as temperature, precipitation, humidity and atmospheric pressure. This can be calculated on different spatial and temporal scales, for example, for the entire country over the course of a year, for a specific region over the course of a month, or for individual weather stations over the course of a day. Depending on the purpose of the study and the availability of data, the measure of polarisation can be determined for different combinations of parameters. It is worth noting that a high level of polarisation does not necessarily mean unfavourable climatic conditions. For example, some regions have significant temperature differences between summer and winter, which can have a positive impact on seasonal tourism. However, in other cases, high polarisation can lead to serious problems, such as droughts, floods or storms. Therefore, a measure of polarisation can be useful in identifying areas that require special attention when planning climate change adaptation activities.

Different approaches are possible for constructing measures of polarisation in climate phenomena, depending on the type and scale of the data under analysis. One of the basic methods is to use probabilistic techniques based on classical theories of one- or multi-dimensional random variables with distributions built using Copula functions [53,74,75]. Other, simpler methods are based on statistical characteristics [4,10,33,45]. A simple concept is to construct indicators measuring the degree of extremity that a phenomenon can take, for example, by measuring the dispersion relative to the long-term average. An example of such an indicator is the coefficient of variation. Another way to create a polarisation indicator is to define it as the degree of extremity compared to the mean value relative to the variability measured by the dispersion measure [12,76,77]. Such indicators can be based on classical measures, such as variance, standard deviation, mean deviation and the coefficient of variation, or on positional measures, such as range, quartile deviation and the coefficient of variation [33].

Further methods of building indicators are based on the idea of unevenness [78,79], which can also be a measure of polarisation. One such indicator is kurtosis [80], which is influenced by the intensity of extreme values, so it measures what is happening in the "tails" of the distribution. Other types of indicators are measures that have their genesis in economics and econometrics. One such example is the Gini index, also called the Gini coefficient [77,81,82,83,84], built using the Lorentz curve, which describes the degree of concentration of a one-dimensional distribution of a random variable with non-negative values [82]. The Gini index ranges from a minimum value of zero, when all values are equal, to a theoretical maximum of one in an infinite population in which every element except one has a magnitude of zero. In the context of climate change, the Gini index can be used to measure inequality in exposure to the effects of climate change, such as droughts, floods or sea level rises [62]. Higher values of the Gini index indicate greater inequality in the occurrence of these phenomena. Several other examples of measures can be cited, including the following:

• The concentration ratio [85] determines the degree of concentration of values at one end of the distribution and is similar to the Gini coefficient. However, it should be noted that the Gini coefficient may be less useful in analysing asymmetric distributions, which means that other indicators such as the concentration ratio or Lorenz curve should be considered in such cases.
• The GMD (Gini Mean Difference) index [86] is an inequality measure used in statistical and econometric analysis to measure polarisation or inequality in a sample distribution. Unlike the Gini coefficient, which measures unevenness, the GMD index enables the analysis of unevenness in the distribution of any variable, such as income, age, weight, height, precipitation and temperature. The GMD index ranges from zero to one, where zero indicates complete evenness in the distribution and one indicates the concentration of all values in one class. The higher the GMD index value, the greater the unevenness in the variable distribution.
• The Theil index [84,87] is a measure of inequality in the distribution of quantitative variables, based on the idea of information entropy, taking into account differences between groups of values in the distribution, similar to the Gini coefficient, but with a greater emphasis on extreme values.
• The Lorenz indicator [78,82] is an inequality indicator in a distribution, which is based on the Lorenz curve. It is often used to measure income inequality but can also be used to measure inequality in other quantitative variables, including climate change studies. Higher values of the Lorenz curve indicate greater inequality in the occurrence of climate change effects such as droughts, floods, or sea level rises, meaning that some regions or social groups are more vulnerable to the effects of climate change than others.
• The Atkinson index [77] is a measure of inequality in the distribution of quantitative variables, which is based on the idea of absolute deviations. It takes into account the differences between groups of values in the distribution; it is similar to the Gini index, but focuses more on average values than extreme values.
• Range relates to values calculated as max-min usually refering to the difference between the maximum and minimum values of a given variable in a specific period of time. In the case of assessing the polarisation of precipitation and temperature, max-min can be used as a measure of the amplitude of these variables in a given period.

In this study, two measures are adopted to evaluate the phenomenon of polarisation, the first $P_1$ is built based on a stationary time series, the second $P_2$ is based on calculated trends.

The first measure is defined as follows:

$$P_1=\frac{max-min}{\sigma }$$

where:

$(max-min)$- the amplitude of change, range, the evaluated changes or ranges are taken over a selected period of time (110 years), the difference between the maximum and minimum value of a characteristic - is a measure that characterizes the empirical area of variation of the studied characteristic;

$\sigma$ - standard deviation.

The amplitude of change values can be useful in identifying extreme climate conditions, such as periods of drought or heat waves, as well as periods of intense precipitation or extremely low temperatures. However, max-min as a single measure may be limited in its usefulness because it does not take into account other factors such as the length and intensity of the period, as well as other variables that affect weather conditions. Due to the fact that it ignores all data except for two extreme values, it does not provide information about the diversity of individual feature values in the population. Therefore, along with max-min values, other measures such as mean values, standard deviations, or cumulative indices are typically used to obtain a more comprehensive and diverse view of climate variability. For example, max-min values can be calculated for individual months, seasons or years, and then compared with mean values, standard deviations, or other measures to better describe climate variability over time and space.

The adopted measure, which is the ratio of the max-min values to the standard deviation $\sigma$, can be used as a measure to evaluate the polarisation of precipitation and temperature. The $max-min$ values refer to the amplitude of changes of a given variable over a period of time, while the standard deviation σ refers to the degree of variability of these values around their mean. By applying this measure, we can see how large the amplitude of changes is in relation to the variability around the mean. Values of this measure greater than 1 suggest that the amplitude of changes is greater than the variability around the mean. If the variability is relatively low compared to the amplitude, it may indicate the occurrence of periods of extreme climatic conditions, such as periods of drought or heatwaves, as well as periods of intense rainfall or extremely low temperature. However, the use of a single measure may be limited because it does not take into account other factors affecting climate variability. Therefore, it is worth using it together with other measures that show the character of climate variability. Note that the values of this measure for precipitation and temperature will always take a non-negative value.

The second form of adopted measures could be dynamic indicators showing the variability of the indicator over a longer period of time. The simplest indicators are based on the idea of a trend coefficient. Trend is a long-term feature referring to the general direction of changes over time in the value of a variable. Trend can indicate an increase, decrease, or stabilization of the variable over time. In time series analysis, the trend coefficient is one of the basic elements that help to forecast future values of the variable. The trend can be analysed for different time scales, from short-term cycles to long-term trends [2]. Trend analysis is applied in many fields, such as economics, finance, meteorology, earth sciences, medicine, sociology, and marketing. In scientific research, trend analysis is often used to study changes in long-term time series, such as climate change, demographic changes, or evolution of species.

The second measure is defined as:

$$P_2=\frac{trend(max-min)}{trend\left(\sigma \right)}$$

where

$trend(max-min)$ is the trend of the amplitude of changes, range or difference between the maximum and minimum value of the variable - it is a measure characterizing the empirical range of variability of the studied feature;

$trend\left(\sigma \right)$ is the standard deviation to be evaluated are taken over a selected period of time (110 years).

The above measure can be interpreted as the ratio of the trend in variability of the amplitude to the trend in variability of the standard deviation. The trend in variability of the amplitude refers to the direction and speed of changes in the maximum and minimum values of a given variable over time, while the trend in variability of the standard deviation refers to the direction and speed of changes in the variability of the values of a given variable around their mean over time. Values of this measure greater than 1 suggest that the trend in the increase of amplitude changes is stronger than the trend in the increase of variability around the mean. This means that changes in the maximum and minimum values of a given variable are increasing faster than the variability around their mean, which may indicate the occurrence of extreme weather conditions. In the proposed measure, the trend of the ratio of the indicators was not calculated, but the trend of the numerator and denominator was separately maintained. Both measures are used to assess trends in data, but they differ in how they relate to data variability.

The measure: $trend\left(\frac{max-min}{\sigma }\right)$ was not used because max-min changes are normalised by the standard deviation, meaning that high data changes in one direction can be offset by changes in the other direction.

The measure $\frac{trend(max-min)}{trend\left(\sigma \right)}$, by contrast, uses two indicators of data variability: trend amplitude and trend standard deviation. The use of amplitude in this formula means that more emphasis is placed on the extreme values of the data, rather than its overall variability. Therefore, when one is interested in more extreme values in the data, the measure adopted may be a better choice.

Note that the values of this measure for precipitation and temperature can be negative as well as positive. In the case of negative values, this means that the trends of polarisation factors (amplitude and variability) will be opposite, in the case of a positive value, the trends of polarisation factors will be consistent.

6. Detecting a change point in the trend

Various methods can be applied to determine change points in a time series [88], [8,89,90,91]. In this analysis, the non-parametric Pettitt change point test (PCPT) [92] was used to detect the occurrence of changes. The Pettitt test (PCPT) is a non-parametric test for detecting sudden changes in a time sequence. It is used to detect the turning point where a sudden change, known as a "jump", occurs in the time series. The Pettitt test (PCPT ) involves comparing the sum of ranks of two subsets of data, which are divided by a threshold value, to determine whether there is a statistically significant change in the time sequence.

This test can be applied to analyse data with any distribution, and the test result is not dependent on the data having a normal distribution. The result of the Pettitt test (PCPT ) is a test statistic value, which is compared with the critical value for the level of significance to determine whether the null hypothesis of no sudden changes in the time sequence can be rejected. The rank is given after sorting and is dependent on the variable that gives the order of the records in the set.

The PCPT has been widely used to detect changes in observed climatic and hydrological time series [8,16,93,94]. The Pettitt test is also applicable to investigate an unknown change point by considering a sequence of random variables ${(X}_1,X_2,...,X_T)$, which have a change point at $\tau$. As a result $(X_1,X_2,...,X_{\tau })$ has a common distribution $F_1(·)$, but $(X_{\tau +1},X_{\tau +2},...,X_T)$ have a different distribution $F_2(·)$, where $F_1(·)\ne F_2(·)$. The null hypothesis $H_0$: no change (but $\tau =T$); is tested against the alternative hypothesis $H_1$: change (lub $1\le \tau <T$); using the non-parametric statistic $K_T=max\left|U_{t,T}\right|=max(K_{T+},K_{T-})$ where:

$$U_{t,T}=\sum _{i=1}^t\sum _{j=t+1}^{T}sgn(X_t-X_j)$$

$$sgn\left(X_t-X_j\right)=\left[\begin{array}{cc}1& (X_t-X_j)>0\\ 0& (X_t-X_j)=0\\ -1& (X_t-X_j)<0\end{array}\right]$$

$K_{T+}=max{U}_{t,T}$ for the downward shift and $K_{T-}=-min{U}_{t,T}$ or the upward shift [91]. The confidence level associated with $K_{T+}$ lub $K_{T-}$ is approximately determined by:

$$\rho =\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-6{K_T}^2}{T^3+T^2}\right)$$

When $p$ is smaller than the specified confidence level (for example, in this study, 0.95 was adopted), the null hypothesis is rejected. The approximate probability of significance for the change point is defined as:

$$P=1-\rho$$

It is obvious that in the case of a significant change point, the series is segmented at the change point into two sub-time series.

The main aim of this study is to investigate the existence of change points in the time series of monthly sum precipitation and monthly average temperature characteristics. For time series showing a significant change point, the trend test is applied to partial series, and if the change point is not significant, the trend test is applied to the entire time series [8].

7. Trend test

To examine the trend in a given time series, the Mann-Kendall test (MKT) can be applied. Originally, this test was used by Mann [95], and later Kendall in 1975 derived the distribution of the test statistic [91]. This test is independent of the type of distribution and we do not need to adopt any specific form of the data distribution function [96]. This test was widely recommended by the World Meteorological Organization for public applications, and has been used in many scientific studies to assess trends in water resource data [2,8,91]. Therefore, the MKT has been recognised as an excellent tool for trend detection by other scholars in similar applications. It should also be noted that the MKT test considers only the relative values of all elements in the series $X=\{x_1,x_2,\dots ,x_n\}$. for analysis. The test statistic for the MKT test is given by:

$$S=\sum _{i=1}^{n-1}\sum _{j=i+1}^{n}sgn(x_j-x_i)$$

$$sgn\left(x_j-x_i\right)=\left[\begin{array}{cc}1& (x_j-x_i)>0\\ 0& (x_j-x_i)=0\\ -1& (x_j-x_i)<0\end{array}\right]$$

where $x_j,x_i$ are the consecutive values of the data, and n is the number of elements in the data. Under the null hypothesis of no trend, and assuming that the data are independent and identically distributed with a zero mean and a variance $V\left(S\right)$ denoted by ${\sigma }^2$, which is calculated as ($t_i$ - the repeated values in the analyzed sequence):

$${V\left(S\right)=\sigma }^2=\frac{1}{18}\left(n·\left(n-1\right)·\left(2n+5\right)-{\sum }_{i=1}^m{t}_i·(t_i-1)·({2t}_i+5)\right)$$

To test a hypothesis, the standard normal distribution is used, denoted as the test statistic $Z$, which is defined as follows for the trend test:

$$Z=\left[\begin{array}{cc}\frac{S-1}{\sigma }& S>0\\ 0& S=0\\ \frac{S+1}{\sigma }& S<0\end{array}\right]$$

In the two-sided test for trend, the null hypothesis is represented as: $H_0$: there is no trend in the dataset, which will be rejected if the calculated $Z$ statistic is greater than the critical value of this statistic obtained from the standard normal distribution table corresponding to the previously established level of significance. A positive value of $Z$ indicates an increasing trend, and a negative value indicates a decreasing trend.

The magnitude of the trend is estimated using a non-parametric slope estimator based on the median proposed by Sen [97] and extended by Hirsch [98]. The slope estimator is given by:

$$\beta =Median\left[\frac{x_j-x_k}{j-k}\right]\mathrm{for}\mathrm{all}k<j$$

Where $1<k<j<n$, and $\beta$ is treated as the median of all possible pairs of combinations for the entire dataset.