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

1. Introduction

Analysis of historical observations of climate variables indicates anthropogenic climate change [1,2,3]. Numerous studies at various spatial scales have considered the impact of projected changes in the climate system on hydrology and water resources [1,3,4,5,6]. Freshwater availability is the second most responsive environmental variable to climate change after air temperaturę [1,7,8,9,10]. The timeliness of temporal and spatial characteristics of precipitation and temperature, despite the many papers that have been written and analyses performed [11,12,13,14,15,16,17,18] especially in recent decades is increasing due to the documentation of theses of the impact of human activities on regional climatic factors [19,20,21]. Computational capabilities and complex analyses performed on long series of measurements are attracting interest [8,22,23] also as a result of the search for justification of theses of polarization of extreme phenomena [1,24,25,26,27,28,29,30]. The renewed interest in climatic factors including precipitation and temperature [13,31,32] is also due to accumulations of flood damage and periods of hydrological drought [33]. The study of changes in the amount of precipitation and temperature variability [5,9,14,34,35,36] depending on the observed period show changes in statistical trends and allow us to assess the form of the directions of these changes occurring [32,37,38,39]. The results of the evaluation of precipitation trends and temperature trends provide an opportunity to indicate the moment of trend change and determine the form of this trend. Given the importance of the variability of climate factors on water resources, an important research element is the use of statistical techniques [4] , [40] that allow the detection of changes and trends [2,38] in such characteristics as data of monthly precipitation totals and average monthly temperature, among others.

The Intergovernmental Panel on Climate Change (IPCC) predicts that increased greenhouse gas emissions following the industrialization of the world, due to large-scale burning of fossil fuels, human interference and land use change, will increase global temperatures [3,10,41,42]. The Intergovernmental Panel on Climate Change [43] noted that natural forces and human activities contribute significantly to changing climate patterns, i.e., increasing land and ocean surface temperatures, changing spatial and temporal precipitation patterns, increasing the frequency of extreme events, sea level rise and intensifying El Niño [3,4,31,34]. An average increase in global temperature of 0.6°C from 1901 to 2001 indicates that the Earth has been warming over the past many decades [43]. Changes in trends of precipitation and temperature can be observed and evaluated on the basis of anomalous values. The change in trends of various climatic variables can be quantified through the use of global circulation models (GCMs) or, for example, through the use of statistical models that recognize the timing of the trend change and its magnitude [8] . Polarization of climatic phenomena has been demonstrated [35], who found increasing trends mainly in flood climaxes in German river basins, which was attributed to climate change. In addition, seasonal analyses showed that these changes were greater in winter than in summer. Climate variability has a significant impact on flow values, which are sensitive to both precipitation and temperature. Many other studies have documented that human and economic activities can play an important role in reducing flow values directly related to precipitation. Many researchers have investigated the existence of a change point in the observed time series. Rapid changes, both in mean and variance, can be related to both climate (e.g., shifts in climate regimes, and anthropogenic effects (e.g., construction of dams and reservoir systems, changes in land use/land cover and agricultural practices, relocation of measurement points) [44,45]. Statistical analyses must be interpreted in conjunction with observed physical [3,7,46], social and economic phenomena [3,14,19,47]. Precipitation and temperature are very important climatic parameters because their changes can affect living conditions. Therefore, predicting temporal trends of precipitation and temperature is very useful in social and urban planning [48].

The world's rivers carry 40% of precipitation as runoff from the land and 95% of sediment to the oceans, linking the atmosphere, hydrosphere and biosphere [49,50]. Precipitation and temperature are among the main factors, along with evaporation, affecting the polarization of runoff. In this work, long-term sequences of precipitation and temperature are analyzed, which is also directly related to evaporation [51]. The analyses performed provide a basis for determining directions for the creation of decision-making tools to control extreme drought and flood events, as well as for assessing and preventing ecological damage in threatened regions [52].

In assessing the polarization of climate phenomena, there are strong relationships between precipitation amounts and temperature [34,52,53]. An increase in global temperature can lead to an increase in water vapor in the atmosphere, which in turn can increase precipitation. In the case of lower temperatures, the opposite can occur, i.e. a decrease in precipitation. In extreme cases, such as prolonged droughts or floods, changes in temperature and precipitation can lead to catastrophic consequences, such as crop loss or property damage [1,45,54]. Proper monitoring of precipitation and temperature is crucial to understanding climate change and assessing the polarization of climate phenomena [55]. Long-term data series make it possible to identify trends in temperature and precipitation and to determine whether a particular location is experiencing polarization changes. In addition, analyzing the interrelationship between precipitation and temperature allows us to understand the magnitude of atmospheric phenomena, such as El Niño and La Niña cycles [56], which affect the global climate. An increase in temperature can affect the occurrence of intense precipitation, and a decrease in precipitation can lead to extreme heat waves [45]. It is also important to analyze precipitation amounts and temperature in the context of other climate phenomena, as interactions between these phenomena affect the ultimate effects of climate polarization [57]. Relationships between precipitation and temperature magnitudes are important for assessing the polarization of climate phenomena. Knowledge of these relationships is essential for understanding the impact of climate change on the environment, economy and society, as well as for taking action to reduce the negative effects of climate change.

2. Data accepted for analysis

The great interest in analyzing long precipitation sequences stems from the need to assess climate change and its impacts at all spatial scales. This demand has led to the initiation and support of many research and monitoring programs conducted by international organizations. In this context, the Global Precipitation Climatology Center (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 [31,58]. An equally important institution is 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, analyzing, updating and making available long-term precipitation and temperature sequences [59,60].

The goal of the aforementioned institutions is to meet users' requirements for the accuracy of the data and analysis results made available, as well as the timeliness and availability of the product. Among the various types of GPCC and NOAA products, gridded sets of long-term precipitation and temperature data, among others, are made available [58,59]. These 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. The data correspond to a spatial resolution of 0.5°x 0.5°, and are consistent in spatial and temporal extent. Products from both the GPCCC and NOAA are made available via the Internet.

NOAA (National Oceanic and Atmospheric Administration) and GPCC (Global Precipitation Climatology Center) are two institutions that provide the opportunity, based on the long-term data made available, to perform analysis and evaluation of the polarization of climatic phenomena, precipitation and temperatures. NOAA is engaged in research and monitoring of the atmosphere, oceans and climate around the world. 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. 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 continents. Assuming 509.9 10⁶ square kilometers of land area, 12.76% of the land area is included in the analysis. Table 1 shows the areas covered by the analysis.

GPCC data, rainfall totals by month from 1901 to 2010, calculated from grid data with a spatial resolution of 0.5°x 0.5° longitude and latitude, were converted to catchment areas. In this way, a sequence of monthly precipitation was obtained, which became the subject of the analyses presented in this article. GIS interpolation mechanisms were used in the spatial analysis of data preparation. The calculated values of the sequences were subjected to a simple statistical analysis, determining the basic statistics: the minimum and maximum value, the mean, the standard deviation of the sample and the value of the coefficient of variation. The data were analyzed in monthly cross sections as well as calendar years. Analyses of monthly precipitation covered the years 1901 to 2010.

NOAA monthly mean temperature data, from 1901 to 2010, calculated from grid data with a spatial resolution of 0.5°x 0.5° longitude and latitude, were converted to catchment areas. In this way, sequences of monthly average temperatures were obtained, which became the subject of the analyses presented in this article. GIS interpolation mechanisms were used in the spatial analysis of data preparation. The calculated values of the sequences were subjected to a simple statistical analysis, determining the basic statistics: the minimum and maximum value, the mean, the standard deviation of the sample and the value of the coefficient of variation. The data were analyzed in monthly as well as calendar year cross sections. Analyses of temperature data covered the years 1901 to 2010.

4. Polarization of precipitation and temperature phenomena

The occurrence of temporal variability and extreme events in air temperature and precipitation is usually assessed by analyzing a set of indicators that define variability and extreme conditions [39]. The common understanding of an extreme event is based on the assumption that a "normal" condition exists. In the context of the common understanding of an extreme event, polarization can be interpreted as a change in the nature of the occurrence of extremes, with extreme events - droughts, floods, extreme temperatures and precipitation - becoming more frequent, and the "normal" state becoming less frequent. Polarization of extreme events can result from a variety of factors, including human activities and climate change, which can affect precipitation cycles, the intensity of extreme events and temperature conditions. In this way, polarization can be seen as a process in which climate characteristics change and extreme events become more frequent, while the "normal" state becomes increasingly unstable. Polarization can have serious consequences for the functioning of ecosystems, the economy, and people's quality of life.

Analysis of the polarization of precipitation and temperature is essential because of their key role in the 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 fresh water and managing water resources, which is essential for reducing the risk of floods and droughts. In addition, there is a growing body of scientific evidence confirming that human activities are influencing climate change, contributing to shorter periods of high-intensity precipitation and longer periods of high temperature and low precipitation. The polarization of extreme phenomena, such as floods and droughts, is increasingly evident and can be attributed to the unevenness and intensity of human activities. Therefore, studying polarization factors related to monthly precipitation and average monthly temperatures is key to understanding climate change at the regional level and developing strategies to manage water resources and reduce flood and drought risks.

This article analyzes long-term sequences of precipitation and temperature to assess the polarization of climate phenomena. The term "polarization" in terms of precipitation or temperature is used in climate science to describe the process by which certain areas of the planet become more extreme in terms of temperature 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 [61]. Long-term sequences of precipitation and temperature are an essential tool for studying polarization in climate. These sequences provide a detailed record of an area's climate over many years, allowing the identification of trends and patterns in the data. By analyzing these sequences, it is possible to identify areas where the climate is becoming more 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 [27,58,62]. As a result, they provide a highly reliable record of climate data in a given area, enabling precise observations and measurements.

The impact of polarization can be observed in various areas of life on Earth: in ecosystems, economy, and infrastructure, in changes in the size and location of water resources. The impact of polarization on ecosystems in areas of extreme temperature and precipitation changes affects biodiversity and ecosystem functioning [61]. Polarization of climate factors can lead to changes in the distribution of plant and animal species [63,64], which can have negative consequences for entire ecosystems. The consequences of polarization for agriculture can affect crops and food production [17,36]. In some regions, climate polarization can lead to a decrease in the amount of available water resources, which in turn will affect agricultural production and the condition of industry. The consequences of polarization for infrastructure can affect roads, bridges, buildings, water and sewerage networks, and water management systems. In extreme cases, serious damage or destruction may occur. The variability and difficulties in predicting polarization - due to the complex and dynamic nature of climate [65], it is difficult to predict exactly what the consequences of polarization will be in a given region. In addition, climate variability means that some regions may experience polarization in different years, complicating the process of predicting and monitoring changes. The interaction between polarization and other climate phenomena can affect other climate phenomena such as hurricanes, tornadoes, or droughts. In this way, one phenomenon can strengthen or weaken others, complicating the process of understanding the scale of climate change. The impact of anthropogenic factors on the polarization of climate factors, among other things, can occur through the emission of greenhouse gases [17,42,52,57]. Anthropogenic climate change, or changes resulting from human activity, leave a lasting mark on physical and biological systems around the world. Such changes include glacier melting, decline in some animal populations, changes in bird migration, and premature flowering of plants [66]. Analyzing the relationship between human activity and polarization can help understand and mitigate the negative effects of climate change.

The polarization of climate phenomena refers to changes in the intensity and frequency of extreme weather events, such as droughts, floods, hurricanes, and hailstorms. NOAA and GPCC monitor these changes [58,67] to determine the factors that influence the polarization of climate phenomena, their effects, and how to reduce the negative consequences associated with these events. Assessing temperature variability is crucial for understanding climate change and its impact on ecosystems and humans. All of this information is essential for developing strategies and actions to manage changes, adapt, and minimize their negative effects. NOAA and GPCC are a key source of information for scientists, policymakers, and other stakeholders who make decisions about climate change.

It is predicted that global warming will lead to an increase in the frequency of extreme weather events [1]. Although there is no direct evidence linking the increased frequency of extreme events to climate change, the observed positive trend indicates an increased likelihood of droughts and flash floods. Furthermore, changes in land use such as urbanization, reduction of natural retention, and improper water management can strongly influence the number of extreme events. The need to adapt water management to factors influencing the formation of water resources and the generation of hazards requires a comprehensive study of the current state and forecasts regarding the number, intensity, and frequency of extreme events. Extreme events such as droughts and floods can be characterized by their severity, duration, intensity, time between occurrences, and other direct and indirect parameters. Such a complex multiparameter characterization of these extreme events prompts modelers to simplify the description of the phenomenon and focus on one dominant parameter that is significant for the task or to use multidimensional methods, commonly used for example in frequency analysis (FFA, DFA) and risk analysis (FRA, DRA) [9,11,35,57,68].

5. The concept of polarization measure

The concept of measuring climate polarization is based on measuring the degree of diversity and variability of climate elements in a particular area. This measure is used to determine the level of polarization of climate elements in a given region. Polarization refers to extreme values of a parameter such as temperature, precipitation, humidity, etc., occurring in a specific area. A high level of polarization indicates significant differences between maximum and minimum values, which can lead to increased risk of extreme weather events. The measure of polarization can be determined based on different climatic parameters such as temperature, precipitation, humidity, atmospheric pressure, etc. It 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 meteorological stations over the course of a day. Depending on the research purpose and data availability, the measure of polarization can be determined for different combinations of parameters. It is worth noting that a high level of polarization does not necessarily mean unfavorable climatic conditions. For example, in some regions, there are significant differences in temperature between summer and winter, which can have a positive impact on seasonal tourism. However, in other cases, high polarization can lead to serious problems such as droughts, floods, or storms. Therefore, the measure of polarization can be useful in identifying areas that require special attention when planning actions related to climate change adaptation.

Different approaches are possible for constructing measures of polarization in climate phenomena, depending on the type and scale of the analyzed data. 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 [48,69,70]. Other, simpler methods are based on statistical characteristics [4,10,40,71]. 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 polarization indicator is to define it as the degree of extremity compared to the mean value, relative to variability measured by the dispersion measure [12,72,73]. Such indicators can be based on classical measures, such as variance, standard deviation, mean deviation, coefficient of variation, or on positional measures, such as range, quartile deviation, coefficient of variation [71].

Another method of constructing indicators is based on the idea of inequality [74,75], which can also be a measure of polarization. One such indicator is kurtosis, which is influenced by the intensity of extreme values and measures what happens in the "tails" of the distribution. Other types of indicators have their origins in economics and econometrics. An example is the Gini coefficient [73,76,77,78,79], which is constructed using the Lorenz curve, which describes the degree of concentration of a one-dimensional random variable with non-negative values [77]. The Gini coefficient ranges from a minimum value of zero when all values are equal, to a theoretical maximum of one in an infinite population where all but one element has a value of zero. In the context of climate change, the Gini coefficient can be used to measure inequality in exposure to climate change impacts such as droughts, floods, or sea level rise [57]. Higher Gini coefficient values indicate greater inequality in the occurrence of these phenomena.

Further methods of building indicators are based on the idea of unevenness [74,75], which can also be a measure of polarization. One such indicator is kurtosis, 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. An example is the Gini index [73,76,77,78,79] built using the Lorentz curve, which describes the degree of concentration of a one-dimensional distribution of a random variable with non-negative values [77]. 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 rise [57]. Higher values of the Gini index indicate greater inequality in the occurrence of these phenomena. Several other examples of measures can be cited, such as:

• The concentration ratio [80], which determines the degree of concentration of values at one end of the distribution, similar to the Gini coefficient. However, it should be noted that the Gini coefficient may be less useful in analyzing 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 [81] is an inequality measure used in statistical and econometric analysis to measure polarization or inequality in a sample distribution. Unlike the Gini coefficient, which measures unevenness, the GMD index allows for the analysis of unevenness in the distribution of any variable, such as income, age, weight, height, precipitation, or temperature. The GMD index ranges from zero to one, where zero indicates complete evenness in the distribution, and one indicates concentration of all values in one class. The higher the GMD index value, the greater the unevenness in the variable distribution.
• The Theil index [79,82] 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.
• Lorenz indicator [74,77] - 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 rise, meaning that some regions or social groups are more vulnerable to the effects of climate change than others.
• Atkinson index [73] - 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, similar to the Gini index, but focuses more on average values than extreme values.
• Range - values calculated as max-min usually refer to the difference between the maximum and minimum values of a given variable in a given period of time. In the case of assessing the polarization 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 polarization, the first $P_1$ is built based on static statistics, the second $P_2$ is based on calculated trends.

The first measure is defined as follows:

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

$(max-min)$- the amplitude of changes, range, the difference between the maximum and minimum value of a variable, is a measure characterizing the empirical range of variability of the analyzed feature, $\sigma$ - standard deviation.

$Max-min$ 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. Since 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 understand 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 polarization 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 temperatures. 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 showing 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 can 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. The trend is a long-term feature, the general direction of changes over time in the value of a variable. The trend can indicate an increase, decrease, or stabilization of the variable over time. In time series analysis, the trend is one of the basic elements that help to forecast future values of the variable. The trend can be analyzed at 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, and $trend\left(\sigma \right)$ is the standard deviation.

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 $trend\left(\right(max-min)/\sigma )$ evaluates the trend magnitude by the variability of the data - the amplitude normalized by the standard deviation. $Max-min$ changes are normalized by the standard deviation, which means that high changes in one direction can be balanced by changes in the opposite direction.

On the other hand, the measure $\left(trend\right(max-min\left)\right)/\left(trend\right(\sigma \left)\right)$ uses two indicators of data variability: the trend of the amplitude and the trend of the standard deviation. The use of the amplitude in this formula means that extreme values of the data are emphasized more than their overall variability. Therefore, if we are more interested in extreme values in the data, the adopted measure may be a better choice. However, if we want to obtain a more balanced assessment of the trend, taking into account the overall variability of the data, the measure $trend\left(\right(max-min)/\sigma )$ may be more appropriate. It should be noted that the values of this measure for precipitation and temperature can be negative and positive. In the case of negative values, this means that the trends of polarization factors (amplitude and variability) will be opposite, in the case of a positive value, the trends of polarization 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 [83], [8,84,85,86]. In this analysis, the non-parametric Pettitt change point test [87] was used to detect the occurrence of changes. The Pettitt test 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 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 analyze data with any distribution, and the test result is not dependent on the assumption of normality of the data. The result of the Pettitt test 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 Pettitt test has been widely used to detect changes in observed climatic and hydrological time series [8,16,88,89]. 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 τ. As a result, (X_1,X_2,...,X_τ) has a common distribution F_1 (·), but (X_(τ+1),X_(τ+2),...,X_T) has the same distribution as F_2 (·), where F_1 (·)≠F_2 (·). The null hypothesis H_0: no change (or τ=T); is tested against the alternative hypothesis H_1: change (or 1≤ τ <T); using the non-parametric statistic K_T = max|U_(t,T) |=max (K_(T+) ,K_(T-)) where:

The Pettitt test has been widely used to detect changes in observed climatic and hydrological time series [8,16,88,89]. 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 $(X_{\tau +1}, X_{\tau +2},..., X_T)$ has the same distribution as $F_2(·)$, 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:

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

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

$K_{T+} = max{U}_{t,T}$ for the downward shift and $K_{T-} = -min{U}_{t,T}$ or the upward shift [86]. 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 will be applied to partial series, and if the change point is not significant, the trend test will be applied to the entire time series [8].

7. Trend test

To examine the trend in a given time series, the Mann-Kendall test can be applied. Originally, this test was used by Mann [90], and later Kendall in 1975 derived the distribution of the test statistic [86]. This test is independent of the type of distribution and we do not need to assume any specific form of the data distribution function [91]. 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 [2,8,86]. Therefore, the Mann-Kendall (MK) test has been recognized as an excellent tool for trend detection by other scholars in similar applications. It should also be noted that the MK 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 MK test is given by:

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

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 $S$ denoted by ${\sigma }^2$, which is calculated as ($t_i-$ - the repeated values in the analyzed sequence):

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

To test hypotheses, the standard normal distribution, denoted as the test statistic $Z$, is used, 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 [92] and extended by Hirsch [93]. 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.