An R function for the estimation of trend signifcance under the scaling hypothesis- application in PET parametric annual time series

Abstract

We present an R function for testing the significant trend of time series. Te function calculates trend significance using a modified Mann-Kendall test, which takes into account the well-known physical behavior of the Hurst-Kolmogorov dynamics. Te function is tested at 10 stations in Greece, with approximately 50 years of PET data with the use of a recent parametric approach. A significant downward trend was detected at two stations. Te R software is now suitable for extensive use in several fields of the scientific community, allowing a physical consistent of a trend analysis.

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Open Water Journal
Volume 4
Issue 1 Open Water Science and Soware Article 6
2017
An R function for the estimation of trend
signicance under the scaling hypothesis-
application in PET parametric annual time series
Aristoteles Tegos
National Technical University of Athens, tegosaris@yahoo.gr
Hristos Tyralis
montchrister@gmail.com
Demetris Koutsoyiannis
National Technical University of Athens, dk@itia.ntua.gr
Khaled Hamed
hamedkhaled@hotmail.com
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Tegos, Aristoteles; Tyralis, Hristos; Koutsoyiannis, Demetris; and Hamed, Khaled (2017) "An R function for the estimation of trend
signi=cance under the scaling hypothesis- application in PET parametric annual time series," Open Water Journal: Vol. 4 : Iss. 1 , Article
6.
Available at: h>p://scholarsarchive.byu.edu/openwater/vol4/iss1/6

An R function for the estimation of trend signicance
under the scaling hypothesis- application in PET
parametric annual time series
Soware Introduction
Aristoteles Tegos1*, Hristos Tyralis2, Demetris Koutsoyiannis3, Khaled H.
Hamed4
1,2,3Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical
University of Athens, Iroon Polytechniou 5, 15780 Zografou, Greece
4Department of Irrigation and Hydraulics, Faculty of Engineering, Cairo University
*Corresponding Author: tegosaris@yahoo.gr
ABSTRACT
We present an R function for testing the signicant trend of time series. e function calculates trend signicance using
a modied Mann-Kendall test, which takes into account the well-known physical behavior of the Hurst-Kolmogorov
dynamics. e function is tested at 10 stations in Greece, with approximately 50 years of PET data with the use of a
recent parametric approach. A signicant downward trend was detected at two stations. e R soware is now suit-
able for extensive use in several elds of the scientic community, allowing a physical consistent of a trend analysis.
Keywords
Hurst; Potential evapotranspiration; Parametric model; R soware; Trend analysis
1.0 Introduction
Trend estimation in hydro-climatic time series has
focused the attention of the scientic community (Sen,
2013). Many studies have examined the trend of precip-
itation, streamow, groundwater regime, temperature,
potential evapotranspiration both at annual and seasonal
scales (Markonis et al. 2016, Stevens et al. 2016, Panda et
al. 2012, Arora et al. 2005, Kumar et al. 2010). Specically,
trend estimation in potential or actual evapotranspiration
pay the attention of the researchers (Gocic and Trajkovic,
2014, Mo et al. 2015, Tabari et al., 2011). Generally, the
trend results are mixed across dierent climatic regions,
as Tabari et al (2011) found a positive trend for 70% of 20
Iranian meteorological stations during the period 1996-
2005, but Gocic and Trajkovic (2014) calculated a signif-
icant increasing downward trend in 70% of 12 Serbian
meteorological stations (study period 1980-2010). Finally,
Mo et al. 2015, by investigating the areal evapotranspira-
tion in China for the period 1981-2010 with remote sens-
ing data, observed an increasing trend from the 1980s to
the mid-1990s, followed by a decreasing trend. For the
examination of physical variability, the Mann-Kendall
under the independence assumption has been pro-

67Open Water
posed as a standard statistical measure for ethe valuation
and quantication of trends (Ahn and Palmer, 2015).
Furthermore, dierent Mann–Kendall statistical
methodologies have been developed and proposed,
namely the Mann–Kendall under the Markovian behav-
ior assumption aer trend-free pre-whitening, the
Mann–Kendall with complete autocorrelation struc-
ture and the Mann–Kendall under the long-term per-
sistence assumption (Kumar et al. 2009). e latter
test, proposed by Hamed (2008), oers a consistent
framework to consider the Hurst phenomenon, which
is observed in many climatological and hydrological
processes, resulting in the increase of physical variabil-
ity (Koutsoyiannis 2003; Koutsoyiannis and Montanari
2007). Hurst coecient was rst introduced by engineer
Harold Hurst during the design of the Aswan reservoir
(Sutclie et al. 2016) and plays a signicant role in the
hydrological variability (O’Connell et al, 2016). Its pres-
ence in large measured hydrometeorological samples is
ubiquitous (Iliopoulou et al. 2016) Comparative analy-
sis of dierent trend model shows signicant dierences
in the totally results (Hamed 2008, Kumar et al. 2009)
and thus a physical consistent framework is needed.
is study presents an R function embedded in an
automatic and user-friendly environment fol-
lowing modern views of water resources mod-
eling tools (Guo et al. 2016, Turner and Ganelli
2016). e package implements the modied
Hamed’s (2008) framework and the procedure is
tested in annual parametric PET time series from
10 sites in Greece, which cover the period 1950-2000.
2. Materials and methods
2.1 e parametric-PET model
e parametric model was rst introduced by
Koutsoyiannis and Xanthopoulos in 1998, mainly as
a framework to ll and extend PET time series and
included the calibration of the parametric model in
a Penman-Monteith time series. Later, Tegos et al.
(2009, 2013, 2015) implemented the model at point
and regional scale in Greece territory and com-
pared the results with Hargraves and Oudin models.
Recently, an extended and comparative analysis in
dierent climatic regimes were made which included
the development of the model in the well-known
CIMIS network (California, U.S.A) as well in European
stations (Tegos et al. 2015). e results of the imple-
mentation were quite satisfactory and the frame-
work allowed consistent monthly and annual PET
estimation at point and especially in regional scale.
Another key conclusion was the better agreement
with Penman-Monteith measured samples against
other world-recognized radiation-based models such
as Hargreaves, Oudin, Jensen and McGuiness. e
most recently application was the daily and monthly
implementation of the model for the PET mapping
in an irrigated plain of Greece (Malamos et al., 2015).
e mathematical expression of the paramet-
ric model for every time step is the following:
where, PET potential evapotranspiration (mm)
Ra (KJm-2) is the extraterrestrial radiation, a
(KgKj-1), b (Kgm-2) and c (C-1) are the calibrated
parameters, while T (C) is the mean air temperature.
e extraterrestrial radiation Ra, for each
day of the year and for dierent latitudes is esti-
mated from the solar constant, the solar decli-
nation and the time of the year by the formula:
where Ra (MJm-2d-1) extraterrestrial radiation, Gsc
solar constant = 0.0820 MJm-2min-1, dr inverse rel-
ative distance Earth-Sun, ωs (rad) sunset hour
angle, φ latitude (rad), δ solar declination (rad).
2.2 Mann-Kendall test under the scaling hypothesis
e Mann-Kendall test under the scaling hypothesis
consists of three consecutive hypothesis tests, namely O
(Original MK test), H (Hurst Parameter test) and M. e
mathematic background and framework are presented
from Hamed (2008). Let H0i denote the null hypothesis
aRa-b
1-cT
(1)
PET =
Gsc dr [ωssin(φ)sin(δ)+cos(φ)cos(d)sin(ωs)](2)
24(60)
π
Ra =

Open Water 68
of each test and let H1i denote the alternative hypoth-
esis, where i = O, H, M denotes the step of the Mann-
Kendall test under the scaling hypothesis. We dene:
• H0O: No trend under the independence
assumption
• H1O: Signicant trend under the indepen-
dence assumption.
• H0H: No signicant LTP.
• H1H: Signicant LTP.
• H0M: No trend under LTP assumption.
• H1M: Signicant trend exists under LTP
assumption.
• en the three steps of the test are summa-
rized by the following sequences
• {H0O}: No trend.
• {H1O}: Possible signicant trend. Proceed to
step H.
• {H1O, H0H}: Signicant trend exists.
• {H1O, H1H}: Possible LTP eect. Proceed to
step M.
• {H1O, H1H, H0M}: No trend.
• {H1O, H1H, H1M}: Signicant trend exists.
Hurst coecient can be dened by a simple pow-
er-law relationship of its standard deviation:
σκ=κH-1σ
where σ ≡ σ(1) and H is the entropy production
in logarithmic time (Koutsoyiannis 2011), and the
parameter ranges between 0 and 1. For values H
> 0.5, the process exhibits long-term persistence,
while for H < 0.5 the process is anti-persistent.
For the test implementation, we used the R func-
tion MannKendallLTP from the HKprocess R pack-
age (Tyralis, 2015). e R function computes the
p-value in each step of the test. If the p-value is
higher than a predened signicance level α (e.g.
α = 0.05), then we cannot reject H0. A p-value less
than or equal to α gives evidence that H1 is true.
2.3 Study area and procedures
Ten meteorological stations (National
Meteorological Services of Greece) well-distrib-
uted over Greece were used. Table 1 presents the list
of the meteorological stations used in our study.
Stations φ(o) z (m)
Heraklion 35.20 39
Ioannina 39.42 484
Kavala 40.54 63
Kerkyra 39.37 2
Kozani 40.18 626
Larissa 39.39 74
Lemnos 39.54 17
Methoni 36.50 34
Skyros 38.54 5
Tripoli 37.32 663
Based on our previous study (Tegos et al. 2013) the
parametric model was calibrated and tested in monthly
time step for the period 1968-1989. For the purposes of
this study, monthly air temperature data for the period
1950-2000 were collected and the parametric model was
applied to the total length. Finally, every monthly time
series was aggregated into annual step with the use of
the HYDROGNOMON soware (Kozanis et al. 2010).
3. Results
Table 2 presents the results of our analysis. In seven
out of the ten stations tested, no trends were found
under the independence assumption. e estimate of
the Hurst parameter for annual PET time series var-
ies in the range from 0.43 to 0.76. Out of the three
stations that had signicant trends under the inde-
pendence assumption, only two stations (Ioannina,
Limnos) showed a signicant downward trend.
In Figure 1, we present the PET at Ioannina. In Table 2,
we observe a signicant trend under the independence
assumption. is assumption is valid. At Kerkyra (see
Figure 2) we do not observe any signicant trend. At
Larissa (Figure 3), we nd a signicant trend under the
independence assumption; however, this trend is not
signicant under the long-term dependence assump-
tion. Finally, we observe a signicant trend under a valid
independence assumption at Limnos (see Figure 4).
Table 1. Meteorological stations with their latitude (φο) and
elevation (z).

69Open Water
4. Discussions and conclusions
We present an R function that implements the Mann-
Kendall test under the long-term persistence hypothe-
sis. e test applied and tested in annual time series of
PET estimated from a recent parametric approach. e
parametric model estimation allows the consistent esti-
mation of the PET with minimal data requirements and
it’s useful for climatic studies when crucial hydrome-
terological data are missing (wind velocity, relative
humidity, extraterrestrial radiation). e results of
our preliminary case study analysis show that in seven
cases, no signicant trend was detected under the inde-
pendence assumption. In one case, no signicant trend
was detected under the long-term persistence assump-
tion, while the trend was signicant under the inde-
pendence assumption. In the remaining two cases, we
found a signicant downward trend under both the
independence and the long-term persistence assump-
tions. In summary, an R function is ready and user-
friendly for use in other eld of water resources studies.
Acknowledgement
e authors wish to kindly acknowledge one anon-
ymous reviewer for his/her constructive suggestions
which improved earlier version of this manuscript.
Appendix A Supplementary material
Supplementary data and code for reproducing the
analysis of this paper as well as additional Figures,
associated with the present study but not included
here for brevity, are available as supporting material in
Appendix A.
Soware Availability
Name of soware: MannKendallLTP R function in
the HKprocess R package
Developers: Hristos Tyralis
Contact: Hristos Tyralis, Athens 10443, contact:
montchrister@gmail.com
Year rst available: 2016
Required soware: R (≥ 3.2.3)
Cost: Free. e function is public available in
the R package at https://CRAN.R-project.org/
package=HKprocess
Table 2. Summary results of the application of the Mann-Kendall modied test to the PET data. e Hurst parameter was estimated
using the maximum likelihood estimator (Tyralis and Koutsoyiannis 2011). e trend identication is performed for a predened level
α = 0.05 in each step.
Staons Hurst pa-
rameter
esmate
Mann-Kendall
2-sided p-value
(Step O)
Signicance
of H, 2-sided
p-value (Step
H)
Mann-Kend-
all LTP 2-sided
p-value (Step M)
Trend idenca-
on
Heraklion 0.67 0.31 {H0O}, no trend
Ioannina 0.58 0.05 0.27 {H1O,H0H}, trend
exists
Kavala 0.76 0.63 {H0O}, no trend
Kerkyra 0.71 0.90 {H0O}, no trend
Kozani 0.63 0.31 {H0O}, no trend
Larissa 0.76 0.04 0.00 0.42 {H1O,H1H,HOM} no
trend
Lemnos 0.74 0.00 0.26 {H1O,H0H}, trend
exists
Methoni 0.69 0.06 {H0O}, no trend
Skyros 0.46 0.40 {H0O}, no trend
Tripoli 0.43 0.46 {H0O}, no trend

Open Water 70
Figure 1. Annual PET at Ioannina
Figure 2. Annual PET at Kerkyra
Figure 3. Annual PET at Larissa

71Open Water
References
Ahn, K. H., & Palmer, R. N. (2015). Trend and Variability in
Observed Hydrological Extremes in the United States. Journal
of Hydrologic Engineering, 21(2), 04015061.
Arora, Manohar, N. K. Goel, and Pratap Singh. “Evaluation of
temperature trends over India.”Hydrological sciences journal
50.1 (2005).
Gocic, Milan, and Slavisa Trajkovic. “Analysis of trends in refer-
ence evapotranspiration data in a humid climate.”Hydrologi-
cal Sciences Journal 59.1 (2014): 165-180.
Guo, D., Westra, S., & Maier, H. R. (2016). An R package for
modelling actual, potential and reference evapotranspiration.
Environmental Modelling & Soware, 78, 216-224
Hamed KH (2008) Trend detection in hydrologic data: e
Mann-Kendall trend test under the scaling hypothesis. Journal
of Hydrology 349(3-4):350-363.
Iliopoulou, T., Papalexiou, S. M., Markonis, Y., & Koutsoyiannis,
D. (2016). Revisiting long-range dependence in annual precip-
itation. Journal of Hydrology.
Kozanis, S., Christodes, A., Mamassis, N., Efstratiadis, A. and
Koutsoyiannis, D., 2010, May. Hydrognomon–open source
soware for the analysis of hydrological data. In EGU General
Assembly Conference Abstracts (Vol. 12, p. 12419).
Koutsoyiannis, D., 2003. Climate change, the Hurst phenom-
enon, and hydrological statistics. Hydrological Sciences-
Journal-des Sciences Hydrologiques 48 (1).
Koutsoyiannis, D., Montanari, A., 2007. Statistical analysis of
hydroclimatic time series: uncertainty and insights. Water
Resources Research 43, W05429.
Koutsoyiannis D. 2011. Hurst–Kolmogorov dynamics as a
result of extremal entropy production. Physica A: Statistical
Mechanics and its Applications, 390(8): 1424-1432
Koutsoyiannis, D., and  Xanthopoulos. “Engineering hydrolo-
gy.”Edition 3 (1999): 418.
Kumar, S., Merwade, V., Kam, J., & urner, K. (2009).
Streamow trends in Indiana: eects of long term persistence,
precipitation and subsurface drains. Journal of Hydrology,
374(1), 171-183.
Markonis, Y., Batelis, S. C., Dimakos, Y., Moschou, E., &
Koutsoyiannis, D. (2016). Temporal and spatial variability of
rainfall over Greece. eoretical and Applied Climatology,
1-16.
Mo, X., Liu, S., Lin, Z., Wang, S., & Hu, S. (2015). Trends in land
surface evapotranspiration across China with remotely sensed
NDVI and climatological data for 1981–2010.Hydrological
Sciences Journal, 60(12), 2163-2177.
O’Connell, P. E., Koutsoyiannis, D., Lins, H. F., Markonis, Y.,
Montanari, A., & Cohn, T. (2016). e scientic legacy of
Harold Edwin Hurst (1880–1978). Hydrological Sciences
Journal, 1-20.
Panda, D. K., A. Mishra, and A. Kumar. “Quantication
of trends in groundwater levels of Gujarat in western
India.”Hydrological Sciences Journal57.7 (2012): 1325-1336.
Stevens, Andrew J., Derek Clarke, and Robert J. Nicholls. “Trends
in reported ooding in the UK: 1884–2013.”Hydrological
Sciences Journal 61.1 (2016): 50-63.
Sutclie, J., Hurst, S., Awadallah, A. G., Brown, E., & Hamed,
K. (2016). Harold Edwin Hurst: the Nile and Egypt, past and
future. Hydrological Sciences Journal, 1-14.
Tabari, H., Maro, S., Aeini, A., Talaee, P. H., & Mohammadi,
K. (2011). Trend analysis of reference evapotranspiration in
the western half of Iran. Agricultural and Forest Meteorology,
151(2), 128-136.
Tegos, A., Efstratiadis, A., & Koutsoyiannis, D. (2013). A para-
metric model for potential evapotranspiration estimation
based on a simplied formulation of the Penman-Monteith
equation. Evapotranspiration-an overview. InTech, Rijeka,
Croatia, 143-165.
Tegos, A., Malamos, N., & Koutsoyiannis, D. (2015). A parsimo-
nious regional parametric evapotranspiration model based on
a simplication of the Penman–Monteith formula. Journal of
Hydrology, 524, 708-717.
Tegos, A., Efstratiadis, A., Malamos, N., Mamassis, N., &
Koutsoyiannis, D. (2015). Evaluation of a parametric approach
for estimating potential evapotranspiration across dierent
climates. Agriculture and Agricultural Science Procedia, 4,
2-9.
Tegos, A., N. Mamassis, and D. Koutsoyiannis. “Estimation
of potential evapotranspiration with minimal data depen-
dence.”EGU General Assembly Conference Abstracts. Vol. 11.
2009.
Turner, S. W. D., & Galelli, S. (2016). Water supply sensitivity
to climate change: An R package for implementing reser-
voir storage analysis in global and regional impact studies.
Environmental Modelling & Soware, 76, 13-19.
Tyralis H, Koutsoyiannis D (2011) Simultaneous estimation of
the parameters of the Hurst-Kolmogorov stochastic process.
Stochastic Environmental Research and Risk Assessment
25(1):21-33.
Tyralis H. (2016). HKprocess: Hurst-Kolmogorov process.
R package version 0.0-2. https://CRAN.R-project.org/
package=HKprocess