Effect of Autocorrelation On Temporal Trends In Air-Temperature In A Northern Algeria And Links With Teleconnections Patterns


 This study investigates the effect of autocorrelation on temporal trends and step change on monthly, seasonal and annual temperatures of six meteorological stations of the North of Algeria from 1950 to 2016. Afterwards, links between the general atmospheric circulation, via six climate indices, and temperature are examined. Trends of temperature are analysed using six different versions of the Mann Kendall approach while the step change of the time series is computed using the original Pettitt test and the modified-Pettitt. Statistical tests have shown an increase in annual temperatures from 0.8 to 0.9°C since the 1980’s on the coastal regions and 90’s on the highlands. This warming most often exceeds 1°C on a seasonal scale, particularly in summer, while no significant trend is observed in winter. On a monthly scale, the increase in temperatures is marked between April and October. The analysis of relationships between six climate indices and average temperatures has shown that inter-annual temperature variability is most often associated with the East Atlantic oscillation for the entire study area. Winter temperatures are influenced by the Mediterranean oscillation as well as the North Atlantic oscillation. The East Atlantic oscillation is the dominant mode of circulation in spring and summer, while in autumn temperatures are strongly linked to West Mediterranean Oscillation. However, no significant correlations have been observed between temperatures and the Arctic Oscillation and El Nino southern oscillation.


Introduction
Global warming is one of the most sensitive issues of the 21 st century; it is considered by many as the most crucial issue facing humanity nowadays (Durand 2007). For several years, the scienti c community has been placing increasing importance on climate change, due to the changes observed over the last century. Indeed, since the mid-1970s the earth has experienced considerable climate variability characterized by a temperature increase of about 0.7°C (IPCC 2007(IPCC , 2014. This global warming has resulted in the appearance and persistence of some extreme events such as droughts and oods, which have affected water availability, agriculture and other socio-economic factors ( Air temperature plays a key role in understanding the complex climate system and it is one of the crucial variables affecting almost all environmental processes (Vancutsem et al. 2010). In addition, the identi cation of temperature trends is essential in the studies of climate change impacts on the hydrological cycle, environment and agriculture (Yu et al. 2019).
A slight increase in temperature can trigger hydrological droughts and oods (Dai 2011) and even air pollution (Zhang 2017). A better understanding of the temporal variability of temperatures is essential for the implementation of more effective climate change adaptation measures for sustainable water resources management and agriculture.
In Algeria, although much work has been devoted to the study of regional and local climate (Meddi 2009 Furthermore, the statistical tests of Mann-kendall (Mann 1945;Kendall 1976) and Pettitt (1979) are among the methods of analyzing the temporal variability of temperatures, most often used for trend analysis and step change detection, respectively. However, studies have shown that it is essential to take into account the effect of auto-correlation when analysing the temporal variability of a data set (von Storch 1995; Serinaldi  auto-correlation, such as the pre-whitening procedure (von Storch 1995), trend free prewhitening (TFPW) (Yue et al. 2002) and trend free prewhitening corrected unbiaised (Serinaldi and Kilsby, 2016). On the other hand, other methods suggest a correction of the variance of the Mann-Kendall test through the use of empirical formulas (Hamed and Ramachandra Rao 1998) and Monte Carlo simulations (Yue and Wang 2004). Due to the existence of long-term persistence within a climate datase which can also in uence the trend, another approach of the modi ed Mann kendall test has been proposed by Hamed (2008) and Kumar (2009). All these methods are increasingly used in recent studies that aim to show the effect of auto-correlation in the analysis of trends in a time series of climate data (e.g. Zamani (Toreti et al. 2010;Rios-Cornejo et al. 2015). It has to be noted that little is known about the links between the temperatures of Northern Algeria and the atmospheric circulation patterns (e.g. Zeroual et al. 2017). Relevant to this, the second objective of this work is to analyse the relations that might exist between temperature variability and climatic indices.
Furthermore, this study provides a complete analysis of the long-term temperature evolution, trend, break, persistence and their relationship with six modes of atmospheric circulation at different annual, seasonal and monthly time steps over the period 1950-2016, which is signi cant for climate prediction on a regional scale, in order to carry out necessary socio-economic impact studies for the implementation of plans supporting sustainable management of water resources and agriculture in Algeria. These  Our study was conducted on the northern part of Algeria located in northern Africa and on the southern shore of the Mediterranean ( g. 1), bounded to the east by Tunisia, to the west by Morocco and to the south by the Saharan Atlas. The climate that characterizes the study area is Mediterranean on the coast and semi-arid on the highlands. Landform is more accentuated in the East than in the west of Algeria, it is for this reason that the precipitation varies from approximately 700 mm to the East (Annaba) and less than 300 mm to the West (Oran). A rainfall gradient is also observed from North to South where the annual totals vary from approximately 600 mm (Algiers) to less than 200 mm (Djelfa).

Data description
Six temperature series were selected to characterize the spatiotemporal variability of annual and monthly temperatures in northern Algeria. The choice of these meteorological weather stations is based on spatial representativeness and the availability of data. The work carried out by Taibi et al. (2015) highlighted a division of northern Algeria into 6 rainfall regions. For this purpose, we have selected a station by region ( g. 1), namely: the center (Algiers), the East (Annaba), the West (Oran), the eastern highlands (Constantine), the highlands West (Mascara) and the steppes (Djelfa). There are a large number of meteorological weather stations, however most temperature data are only available from the 1970s and 1980s. The monthly data for the six meteorological weather are available from 1950 to 2016 except for the stations of Djelfa and Mascara (Table 1). These data were collected from the National Meteorological O ce (NMO). . Wallace (1998, 2000) suggested that the NAO is the regional manifestation of the AO circulation mode. The AO index was calculated by Thompson and Wallace (2000) as the leading EOF of the monthly mean SLP over the northern hemisphere north of 20N.
The EA pattern as de ned by Wallace and Gutzler (1981) and Barnston and Livezey (1987) is considered as the second prominent mode of low frequency variability over the North Atlantic and appears as a leading mode in all months (NOAA 2018). The EA is structurally similar to the NAO and consists of a north-south dipole of centers of anomalies covering the North Atlantic from east to west. The EA centres of anomaly are shifted southeast to the approximate nodal lines of the NAO model. It is for this reason that the EA is often interpreted as a NAO shifted towards the south. However, the lower-latitude center contains a strong subtropical link in association with modulations in the subtropical ridge intensity and location. This subtropical link makes the EA pattern distinct from its NAO counterpart (NOAA 2018).
The MO designates the difference normalized pressure at the 500hPa geopotential level between Algiers and Cairo (Conté et al., 1989). The MO results from the opposite behavior of barometric, thermal and pluviometric variability between the the western and eastern Mediterranean ENSO pattern is represented by the south oscillation index (SOI) which is de ned as the pressure difference between Tahiti and Darwin. Tahiti represents the  high subtropical pressures of the eastern Paci c, and Darwin represents the low equatorial pressures of the northern Indian Ocean and Indonesia. Martin-Vide and Lopez-Bustins (2006) proposed a new regional telecommunication index de ned from synoptic data from the western Mediterranean basin and its surroundings and which they called the Western Mediterranean Oscillation (WeMO). The WeMO index indicates the pressure difference between the regions of the north of the Italian peninsula and the south-west of the Iberian Peninsula.

Autocorrelation
In any trend or step analysis exercise, it is also important to evaluate the existence of serial correlation. In such cases, the presence of serial correlation in the time series can impact considerably the outcome of trend analysis. Positive autocorrelation can arti cially induce trend in a time series (e.g. von Storch 1995; Kulkarni and von Storch 1995), while negative autocorrelation can weaken the trend if it is in fact present (Yue et al, 2002). Such statistical features have serious implications for trend and step analyses. The time series of temperature data require to be veri ed for autocorrelation before applying a trend tests and step analysis. The autocorrelation of a time series X_t means that this latter is linked with its own previous (t-K) or following (t + K) values (Cunderlik and Burn 2004). This dependence of time series is measured by the coe cients of the autocorrelation at different lag-time (k). The hypothesis of serial independence is checked by t test statistic (Mondal et al. 2012). The null hypothesis that ρ ̂ = 0 is tested against the alternative hypothesis that ρ ̂ ≠ 0. If we reject the null hypothesis, at a signi cance level of α. In the case of this work, the autocorrelation coe cient was calculated for a shift k=1 (Lag-1) at a signi cance level of 5%.

Mann-kendall test and sen's slope
The Mann-Kendall (MK) test (Mann 1945 andKendall 1975) allows to detect the existence of a signi cant global trend within a time series data as well as the direction of the trend. Trends are assessed by estimating the Sen slope (Sen 1968) associated with the Mann-Kendall test. It is a non-parametric method based on the median slope. The latter being less sensitive to outliers than traditional regression methods, it allows a more reliable assessment of the trend (Eichner et al. 2003).

Modi ed Mann-kendall (MMK)
In order to eliminate the effect of autocorrelation on temporal trends in the time series, several authors have modi ed the Mann-kendall test according to different approaches. As part of our work, six statistical modi ed tests based on three different approaches were used.

a) Approaches based on prewhitening
The three approaches used are those proposed by von Storch (1995) Yue et al. (2002) and Serinaldi and Kilsby (2016). Von storch (1995) developed the prewhitening method (MMK-PW) and suggest using it when the series has a lag-1 autocorrelation otherwise the original Mann-kendall test is su cient. Afterwards, Yue et al. (2002) proposed the TFPW method to also addressed the autocorrelation in time series data. However, Serinaldi and kilsby (2016) have shown that the trend-free-pre-whitening approach (TFPW) does not considered the variance in ation of the uncorrelated residues and is not able to provide a pre-whitened of the time series used for the modi ed Mann-Kendall test. To that effect, it suggested the corrected and unbiased trend-free-pre-whitening (TFPWcu) approach (Serinaldi and kilsby 2016).

b) Approaches based on variance correction
The modi cation of the variance of the Mann-Kendall test is an approach proposed by Hamed and Rao (1998) (MMKH) and Yue and Wang (2004) The variance correction is done by empirical formulas for the MMKH test, while the MMKY test uses the Monte-Carlo simulations to remove the effect of the auto-correlation in the MK test.

c) Long term persistance (LTP)
Some studies have shown that long-term persistence (LTP), also known as the Hurst phenomenon, can underestimate the importance of the trend in climate variability (Koutsoyiannis 2003;Koutsoyiannis and Montanari 2007;Hamed 2008). To this end, Hamed (2008) modi ed the MK test to consider this phenomenon in the time series. This method was used as part of this work to verify the long-term persistence of temperatures in Northern Algeria. For more information, the article of Kumar (2009) illustrates in detail the different stages of this method.

Pettitt test
The Pettitt test (Pettitt 1979) is a non-parametric test used to verify the stationarity of a series of data. The null hypothesis of this test means that the series is stationary while the alternative hypothesis means that the data series is not stationary and that a break point of mean exists from a given date.
2.4.5. Modi ed Pettitt test using TFPW-cu (1979). This test is applied when one wishes to detect a breaking point within a series of autocorrelated data. The mathematical equations and the sequential steps of the framework are explained in detail in Serinaldy et klisby (2016) and Achour et al. (2020).

Pearson coe cient correlation
To explore the links between average temperatures and climate indices the Pearson correlation coe cient is used. This coe cient has already been used in numerous studies to demonstrate the in uence of general atmospheric circulation on climate variability (eg. Plewa

Autocorrelation analysis
Before analyzing the trends in the temperature time series, we investigate the autocorrelation as suggested by Hamed and Rao (1998) and Serinaldi and Kilsby (2016). The autocorrelation is checked at the different annual, seasonal and monthly time steps. Fig.2 represents the auto-correlogram for each station, the autocorrelation coe cient was determined for a lag-1. The temperature series is autocorrelated to the signi cance threshold of 5% when its value is above (below) that of the upper (lower) limit. The results highlight the presence of an auto-correlation for all the annual average temperature series. At the seasonal scale, the autocorrelation is observed in autumn, spring and summer at the stations of Algiers, Annaba, Oran and Constantine, while it is observed in spring and summer at Mascara and only in autumn at Djelfa station. Winter temperature series are not auto-correlated for all stations.
On a monthly scale, the presence of autocorrelation in the data series differs from one station to another ( g.2). The monthly temperatures for April, June, July, and August are autocorrelated at Algiers station, while at Annaba and Constantine stations, autocorrelation is observed only in August and October. It is at Oran station that autocorrelation is strongly present since it is detected within the monthly series going from March to August. The monthly data series for March, June and August are also autocorrelated at the Mascara station, contrary to the Djelfa station which is characterized by the absence of autocorrelation on a monthly scale. The results also show that the autocorrelation detected within the different data series is positive, which according to Hamed and Rao (1995) which would further increase the probability of detecting a trend then it does not exist (Serinaldi and Kilsby 2016). It therefore becomes essential to eliminate autocorrelation to analyze temperature trends, by applying the different methods illustrated in section (2.4).

Temperature trend analysis
The Mann-kendall test is most often used to analyze trends in a time series data. However, the presence of autocorrelation may in uence the results obtained. For this purpose, we analyzed the trend of the mean temperature series using the original Mann Kendall test as well as six other MMK tests developed using different approaches in order to eliminate the effect of autocorrelation. It is a question of comparing for each station the results of the Z obtained by the seven statistical tests with the different time steps annual, seasonal, and monthly, only for the auto-correlated series. When the series is not auto-correlated, only the original Mann-Kendall test is applied. A trend is said to be signi cant when Z is greater (or less) than 1.96 (-1.96). The sen slope is also calculated to measure the magnitude of warming (or to quantify the variation in temperatures by decades).
In Algiers station, the analysis of the trend in annual temperatures series by the MK test highlights a signi cant positive trend (Z = 3.99) (table 2). This time series is autocorrelated and eliminating the effect of autocorrelation using the different approaches of MMK also showed the presence of a positively respectively (see Table xx). The impact of global warming in the summer months translates into a dramatic rise in temperatures of around 0.3°C/decade. The month of October recorded a signi cant positive trend much more important than the other months which is manifested by a signi cant rise in temperatures of around 0.33°C/decade.    1986,1994,1991,1995,1993,1985,1982 and1984. The P-value is generally less than 0.0001 except for March (p-value= 0.02) and September (p-value=0.03). After using TFPW cu approach, the temperature series of Mars, April, May, June, July and August were tested again for shift using Pettitt's test.
The results showed that K T values are decreased ( At Mascara station, the Analysis of temperature time series using original Pettitt test shown that the annual temperature has change point in the year 1996 with an increase of 0.9°C (p-value=0.00), while for seasonal time scales, spring, summer and autumn data series have re ected a shift respectively in the year 1996 (+ 1.3 °C), 1997 (+1.6 °C) and 2000 (+0.9 °C) and having respectively a p-value of 0.001, 0.002 and 0.04 (table3). Winter season has not detected any change point at 5% significance level. After corrected and unbiased trend free prewhitening (TFPWcu), the annual, spring and summer temperature series were tested again for shift using Pettitt's test. The results showed that the p-value is increased beyond the threshold of the signi cance of 5% for annual (Pvalue =0.1) and summer temperature (P-value=0.06). For the spring season, the change point date has been brought backward of four years (1992) compared with the original Pettitt's test.
At Djelfa station, the Analysis of temperature time series using original Pettitt test shown that the annual temperature has change point in the year 1993 with an increase of 0.8°C (p-value=0.00), while for seasonal time scales, spring, summer and autumn data series have re ected a shift respectively in the year any change point at 5% significance level. After using TFPWcu approach, the results show that the P-value calculated for annual and autumn auto-correlated temperature series is close to the critical value (0.05) that's why the detected change point has been eliminated. Analysis of monthly temperatures series using original Pettitt test shows no break in time series of January, February, May, June, September, November and December, while a change point is observed in March (+1.2°c), April (+1.6°c), July (+1,3°c), August (+1,4°c), and October (+1.9°c) respectively at the date 1999,1982,1992,1985 and 1998. It is noted that at the monthly scale, all of the temperature series are not auto-correlated.

Links between climate indices and temperature
At the annual scale, the Pearson coe cient highlights a strong statistically signi cant positive correlation between the EA index and all of the annual temperature series ( Table 4)    Following these outcomes, it seems that the EA index affects the variability of temperatures during the different months of the year, however it is during the winter (December and February) and spring (March, April and May months) that it has the greatest in uence. According to NOAA (2018), the positive phase EA is associated with above-average surface temperatures in Europe (Mediterranean) during all the months, which explains the rise in temperatures observed in spring and summer North of Algeria.
In December and January, the MO index has a great in uence on the temperatures of the study area. Dünkeloh and Jacobeit (2003)   In spring and summer, the temperature increase is mainly related to the positive phase of EA at the different scales (annual and monthly). In autumn, it is clear that the temperature increase is associated with the negative phase of the WMO and particularly during the month of October which is characterized by a signi cant temperature increase where strong correlations are observed with this index for all stations. EA is the second mode also in uencing temperatures in autumn.

Conclusion
The objective of this work was to analyse the variability of mean temperatures in northern Algeria and its relationship with the general atmospheric circulation through six rainfall stations during the period 1950-2016. Recent studies have shown that it is necessary to remove the effect of autocorrelation when analysing trends or breaks in a series of autocorrelated hydro-climatic data using different approaches to identify trends and nd their signi cance in an acceptable manner.
To do this, a comparison between several statistical tests, using different approaches, was carried out on all temperature series and at different time scales. It appears that all these approaches do not systematically eliminate the trend observed in an autocorrelated series. However, for some temperature series, the PW and TFPW-cu preblanking methods were found to be the most effective in eliminating the autocorrelation effect. However, the elimination of the persistence effect by the LTP test rendered the trends insigni cant. Our objective is not to highlight the best suitable test for analysing trends or breaks, but to use more than one approach to identify trends and nd their signi cance in an acceptable way for consideration in planning action and decision making and for coping with global warming.
The coastal region of Algeria is marked by a warming of 0.8 to 0.9°C observed since the 1980s. This warming most often exceeds 1°C on a seasonal scale, particularly in summer, while no signi cant trend is observed in winter. On the highlands, the warming is most noticeable on a monthly scale between April and October, exceeding 1°C and reaching 2°C in October. These ndings show that global warming has also affected the Algerian interior as far as more than 100 km from the Mediterranean Sea.
The interannual variability of temperatures in northern Algeria is in uenced by the EA index. This circulation pattern seems to affect every month of the year.
Its positive phase is responsible for the temperature increase. On the other hand, the absence of warming during the winter months seems to be associated with the activity of the NAO and the MO, known for their in uence, particularly in January, by a drop in temperatures in the Mediterranean region. The WMO is very active in autumn and seems to explain the observed warming, especially in October.
Finally, this work has shown that the observed large-scale warming affects the different regions of Algeria. The impact of this warming has already been felt in the past through a long period of drought characterized by a signi cant rainfall de cit which has strongly affected the availability of water resources for drinking water supply and agriculture in Algeria. However, taking into account the climate projections indicated by the IPCC in 2014, this warming is likely to increase further and could reach 4°C by the end of the century, if no mitigation measures are taken, which will worsen the current situation. For this reason, adaptation measures are needed by carrying out socio-economic and environmental impact studies.

Declarations
Funding (Not applicable)