One-step ahead forecasting of geophysical processes within a purely statistical framework

Abstract

The simplest way to forecast geophysical processes, an engineering problem with a widely recognised challenging character, is the so called “univariate time series forecasting” that can be implemented using stochastic or machine learning regression models within a purely statistical framework. Regression models are in general fast-implemented, in contrast to the computationally intensive Global Circulation Models, which constitute the most frequently used alternative for precipitation and temperature forecasting. For their simplicity and easy applicability, the former have been proposed as benchmarks for the latter by forecasting scientists. Herein, we assess the one-step ahead forecasting performance of 20 univariate time series forecasting methods, when applied to a large number of geophysical and simulated time series of 91 values. We use two real-world annual datasets, a dataset composed by 112 time series of precipitation and another composed by 185 time series of temperature, as well as their respective standardized datasets, to conduct several real-world experiments. We further conduct large-scale experiments using 12 simulated datasets. These datasets contain 24 000 time series in total, which are simulated using stochastic models from the families of Autoregressive Moving Average and Autoregressive Fractionally Integrated Moving Average. We use the first 50, 60, 70, 80 and 90 data points for model-fitting and model-validation and make predictions corresponding to the 51st, 61st, 71st, 81st and 91st respectively. The total number of forecasts produced herein is 2 177 520, among which 47 520 are obtained using the real-world datasets. The assessment is based on eight error metrics and accuracy statistics. The simulation experiments reveal the most and least accurate methods for long-term forecasting applications, also suggesting that the simple methods may be competitive in specific cases. Regarding the results of the real-world experiments using the original (standardized) time series, the minimum and maximum medians of the absolute errors are found to be 68 mm (0.55) and 189 mm (1.42) respectively for precipitation, and 0.23 °C (0.33) and 1.10 °C (1.46) respectively for temperature. Since there is an absence of relevant information in the literature, the numerical results obtained using the standardized real-world datasets could be used as rough benchmarks for the one-step ahead predictability of annual precipitation and temperature.

Full text

Papacharalampous et al. Geosci. Lett. (2018) 5:12
https://doi.org/10.1186/s40562-018-0111-1
RESEARCH LETTER
One-step ahead forecasting
of geophysical processes within a purely
statistical framework
Georgia Papacharalampous*, Hristos Tyralis and Demetris Koutsoyiannis
Abstract
The simplest way to forecast geophysical processes, an engineering problem with a widely recognized challenging
character, is the so-called “univariate time series forecasting” that can be implemented using stochastic or machine
learning regression models within a purely statistical framework. Regression models are in general fast-implemented,
in contrast to the computationally intensive Global Circulation Models, which constitute the most frequently used
alternative for precipitation and temperature forecasting. For their simplicity and easy applicability, the former have
been proposed as benchmarks for the latter by forecasting scientists. Herein, we assess the one-step ahead forecast-
ing performance of 20 univariate time series forecasting methods, when applied to a large number of geophysical
and simulated time series of 91 values. We use two real-world annual datasets, a dataset composed by 112 time series
of precipitation and another composed by 185 time series of temperature, as well as their respective standardized
datasets, to conduct several real-world experiments. We further conduct large-scale experiments using 12 simulated
datasets. These datasets contain 24,000 time series in total, which are simulated using stochastic models from the
families of AutoRegressive Moving Average and AutoRegressive Fractionally Integrated Moving Average. We use the
first 50, 60, 70, 80 and 90 data points for model-fitting and model-validation, and make predictions corresponding
to the 51st, 61st, 71st, 81st and 91st respectively. The total number of forecasts produced herein is 2,177,520, among
which 47,520 are obtained using the real-world datasets. The assessment is based on eight error metrics and accuracy
statistics. The simulation experiments reveal the most and least accurate methods for long-term forecasting applica-
tions, also suggesting that the simple methods may be competitive in specific cases. Regarding the results of the real-
world experiments using the original (standardized) time series, the minimum and maximum medians of the absolute
errors are found to be 68 mm (0.55) and 189 mm (1.42) respectively for precipitation, and 0.23 °C (0.33) and 1.10 °C
(1.46) respectively for temperature. Since there is an absence of relevant information in the literature, the numerical
results obtained using the standardized real-world datasets could be used as rough benchmarks for the one-step
ahead predictability of annual precipitation and temperature.
Keywords: ARFIMA, Benchmarking time series forecasts, Machine learning, Neural networks, Precipitation, Random
forests, Simple exponential smoothing, Support vector machines, Temperature, Univariate time series forecasting
© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License
(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,
provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license,
and indicate if changes were made.
Background
Forecasting geophysical variables in various time
scales and horizons is useful in technological applica-
tions (e.g. Giunta etal. 2015), but a difficult task as well.
Precipitation and temperature forecasting is mostly
based on deterministic models as the Global Circulation
Models (GCMs), which simulate the Earth’s atmosphere
using numerical equations; therefore, deviating from tra-
ditional time series forecasting, i.e. univariate time series
forecasting. is particular deviation has been ques-
tioned by forecasting scientists (Green and Armstrong
2007; Green etal. 2009; Fildes and Kourentzes 2011, see
also the comments in Keenlyside 2011; McSharry 2011).
Open Access
*Correspondence: papacharalampous.georgia@gmail.com
Department of Water Resources and Environmental Engineering, School
of Civil Engineering, National Technical University of Athens, Iroon
Polytechniou 5, 157 80 Zografou, Greece

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Traditional time series forecasting can be performed
using several classes of regression models, as reviewed
in De Gooijer and Hyndman (2006), while the two major
classes are stochastic and machine learning. Regression
models are in general fast-implemented in contrast to
their computationally intensive alternative in precipita-
tion and temperature forecasting, i.e. the GCMs. For
their simplicity and easy applicability, the former have
been proposed as benchmarks for the latter by Green
etal. (2009).
Recognizing the necessity of introducing traditional
forecasting methods in temperature and precipitation
forecasting, Armstrong and Fildes (2006) have recom-
mended a relevant issue in one of the Journals special-
ized in forecasting. Since then and despite the fact that
considerable parts of books in hydrology are devoted to
such methods (Sivakumar 2017, pp 63–145; Remesan
and Mathew 2015, pp 71–110), there has not been a sys-
tematic approach to the subject. However, studies adopt-
ing statistical forecasting approaches in geoscience are
sporadically published in a variety of Journals. Within a
statistical framework, Tyralis and Koutsoyiannis (2014,
2017) use Bayesian techniques for probabilistic climate
forecasts under the established assumption of long-
range dependence of the observed time series. In the
latter study information from GCMs is used to improve
the performance of the time series forecasting methods.
Moreover, Table 1 presents some examples of studies
using univariate time series forecasting approaches that
do not utilize exogenous predictorvariablesto forecast
precipitation or temperature variables, and streamflow or
river discharge variables. e former can be considered
as climatic or meteorological variables depending on the
time scale of interest, while the latter can be considered
as the results of precipitation (and other) variables and
are more frequently modelled by describing this depend-
ence using either deterministic or statistical methods.
Such statistical approaches to modelling hydrological
variables can be found in Chen et al. (2015), Gholami
etal. (2015) and Taormina and Chau (2015).
In a somehow different direction, Papacharalampous
etal. (2017c) conduct a multiple-case study, i.e. a syn-
thesis of 50 single-case studies, by using monthly pre-
cipitation and temperature time series of various lengths
observed in Greece. Some important points regard-
ing the comparison of univariate time series forecast-
ing methods and additional concerns introduced when
implementing the machine learning ones (hyperparam-
eter optimization and lagged variable selection) in one-
and multi-step ahead forecasting are illustrated in the
latter study. Nevertheless, only large-scale forecast-pro-
ducing studies could provide empirical solutions to sev-
eral problems appearing in the field of (geophysical) time
series forecasting. Such studies are rare in the literature.
Beyond geoscience, Makridakis and Hibon (2000) use a
real-world dataset composed by 3003 time series, mainly
originating from the business, industry, macroeconomic
and microeconomic sectors, to assess the one- and multi-
step ahead forecasting accuracy of 24 univariate time
series forecasting methods. In geoscience, on the other
hand, there are only four recent studies, all companions
of the present, to be subsequently discussed.
Papacharalampous etal. (2017a) compare 11 stochastic
and nine machine learning univariate time series fore-
casting methods in multi-step ahead forecasting of geo-
physical processes and (empirically) prove that stochastic
and machine learning methods can perform equally well.
e comparisons are conducted using 24,000 simulated
time series of 110 values, 24,000 simulated time series of
310 values and 92 mean monthly time series of stream-
flow with varying lengths, as well as 18 metrics. ese
20 methods are also found to collectively compose a
representative sample set, i.e. exhibiting a variety of
forecasting performances with respect to the different
metrics. Alongside with this study, Papacharalampous
et al. (2017b) investigate the error evolution in multi-
step ahead forecasting when adopting this specific set
of methods. e tests are performed on 6000 simulated
time series of 150 values, 6000 simulated time series of
350 values and the streamflow dataset used in Papacha-
ralampous etal. (2017a). Some different behaviours are
revealed within these experiments, also suggesting the
fact that one- and multi-step ahead forecasting are dif-
ferent problems to be examined for the same methods.
Moreover, Tyralis and Papacharalampous (2017) focus
on random forests, a well-known machine learning algo-
rithm, with the aim to improve its one-step ahead fore-
casting performance by conducting experiments on
16,000 simulated and 135 annual temperature time series
of 101 values. Finally, Papacharalampous et al. (2018)
investigate the multi-step ahead predictability of monthly
precipitation and temperature by applying seven auto-
matic univariate time series forecasting methods to a
sample of 1552 monthly precipitation and 985 monthly
temperature time series of 480 values.
Herein, we examine the fundamental problem of one-
step ahead forecasting, also complementing the results
of the four above-mentioned studies. In more detail, we
expand the former of these studies by exploring the one-
step ahead forecasting properties of its methods, when
applied to geophysical time series. Emphasis is put on the
examination of two real-world datasets, a precipitation
dataset and a temperature dataset, together containing
297 annual time series of 91 values. ese datasets are
examined in both their original and standardized forms.
We further perform experiments using 24,000 simulated

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
time series of 91 values. ese experiments complement
the real-world ones by allowing the examination of a
large variety of process behaviours, while they are also
controlled to some extent, facilitating generalizations
and increasing the understanding on the examined prob-
lem. e number of forecasts produced using these real-
world and simulated datasets are 47,520 and 2,130,000,
respectively, i.e. the largest among its companion studies.
Our aim is twofold, to provide generalized results regard-
ing one-step ahead forecasting within a purely statisti-
cal framework [justified, for example, in Hyndman and
Athanasopoulos (2013)] in geoscience and hopefully to
establish the results obtained by the examination of the
standardized real-world datasets as rough benchmarks
Table 1 Examples of univariate time series forecasting in geoscience
s/n Study Process Number of origi-
nal time series Forecast time scale Forecast horizon(s)
[step(s) ahead] Univariate time series forecasting
method(s)
1 Hong (2008) Precipitation 9 Hourly 1 (1) Support vector machines
(2) Hybrid model, i.e. a combination
of recurrent neural networks and
support vector machines
2 Chau and Wu (2010) 2 Daily 1, 2, 3 (1) Neural networks
(2) Hybrid model, i.e. a combination
of neural networks and support
vector machines
3 Htike and Khalifa
(2010)1 Monthly, biannually,
quarterly, yearly 1 Neural networks
4 Wu et al. (2010) 4 Monthly, daily 1, 2, 3 (1) Linear regression
(2) k-nearest neighbours
(3) Neural networks
(4) Hybrid model, i.e. a combination
of neural networks
5 Narayanan et al.
(2013)6 Yearly 21 × 3 (months) AutoRegressive Integrated Moving
Average (ARIMA)
6 Wang et al. (2013) 1 Monthly 12 Seasonal AutoRegressive Integrated
Moving Average (SARIMA)
7 Babu and Reddy
(2012)Temperature 1 Yearly 10 (1) ARIMA
(2) Wavelet-based ARIMA
8 Chawsheen and
Broom (2017)1 Monthly 121 SARIMA
9 Lambrakis et al. (2000)Streamflow
or river
discharge
1 Daily 1 (1) Farmer’s model
(2) Neural networks
10 Ballini et al. (2001) 1 Monthly 1, 3, 6, 12 (1) AutoRegressive Moving Average
(ARMA)
(2) Neural networks
(3) Neurofuzzy networks
11 Yu et al. (2004) 2 Daily 1 (1) Support vector machines coupled
with an evolutionary algorithm
(2) Standard chaos technique
(3) Naïve
(4) Inverse approach
(5) ARIMA
12 Komorník et al. (2006) 7 Monthly 1, 3, 6, 12 (1) Threshold AutoRegressive (AR)
with aggregation operators
(2) Logistic smooth transition AR
(3) Self-exciting threshold AR
(4) Naïve
13 Yu and Liong (2007) 2 Daily 1 (1) Support vector machines coupled
with decomposition
(2) Standard chaos technique
(3) Naïve
(4) Inverse approach
(5) ARIMA
14 Koutsoyiannis et al.
(2008)1 × 12 (months) Yearly 1 (1) Stochastic
(2) Analogue method
(3) Neural networks
15 Wang et al. (2015) 3 Monthly 12 SARIMA

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
for the one-step ahead predictability of annual precipita-
tion and temperature. e establishment of forecasting
benchmarks is meaningful, especially for the one-step
ahead attempts, as the latter constitute the most simple
ones and their accuracy can be quantified using a single
metric, i.e. the absolute error.
Data and methods
We use the datasets briefly described in Tables2 and
3. e PrecDat and TempDat datasets are annual and
originate from two larger monthly datasets, available in
Peterson and Vose (1997) and Lawrimore et al. (2011)
respectively. e sample period is from 1910 to 2000, so
the following two conditions are simultaneously met: (1)
there are no missing values, (2) the number of stations
around the globe is the largest possible. We note that
for sample periods extending after 2000 the number of
retained stations would decrease rapidly. Figure1 pre-
sents the maps of the retained stations. e precipitation
ones create a sufficiently dense network in the United
States of America and in Scandinavia, while the retained
temperature stations in the United States of America, in
Japan and in a part of South Korea. As it is apparent from
Table 2, the StandPrecDat and StandTempDat datasets
simply contain the standardized time series of PrecDat
and TempDat respectively.
Figure 1 also presents the histograms of the Hurst
parameter maximum likelihood estimates (Tyralis and
Koutsoyiannis 2011) of the formed real-world time series.
ese estimates are of importance within this study for
two reasons: (1) we implement a univariate time series
forecasting method (see later on in this section) that
takes advantage of this information under the established
assumption of long-range dependence, (2) we standard-
ize the original real-world time series using the mean
and standard deviation maximum likelihood estimates
(estimated simultaneously with the Hurst parameter) of
the Hurst–Kolmogorov process. e standard deviation
estimates would be considerably different if we mod-
elled the time series using independent normal variables
(Tyralis and Koutsoyiannis 2011). For consistency pur-
poses with respect to the real-world datasets of the pre-
sent study (but also to approximate the typical length of
annual geophysical time series), the simulated time series
are of 91 values as well. ey originate from the families
of AutoRegressive Moving Average (ARMA(p,q)) and
AutoRegressive Fractionally Integrated Moving Aver-
age (ARFIMA(p,d,q)), the definitions of which can eas-
ily be found in the literature, for example in Wei (2006),
pp 6–65, 489–494. e simulations are performed with
mean 0 and standard deviation of 1. Hereafter, to spec-
ify a used R algorithm, we state its name accompanied
Table 2 Datasets of this study (part 1): real-world datasets
s/n Abbreviated name Process Type Primal dataset R algorithm Number of time series
1 PrecDat Precipitation Original Peterson and Vose (1997) 112
2 TempDat Temperature Lawrimore et al. (2011) 185
3 StandPrecDat Precipitation Standardized PrecDat mleHK {HKprocess} 112
4 StandTempDat Temperature TempDat 185
Table 3 Datasets of this study (part 2): simulated datasets
s/n Abbreviated name Process Parameter(s) R algorithm Number of time series
5 SimDat_1 AR(1) φ1 = 0.7 arima.sim {stats} 2000
6 SimDat_2 AR(1) φ1 = −0.7
7 SimDat_3 AR(2) φ1 = 0.7, φ2 = 0.2
8 SimDat_4 MA(1) θ1 = 0.7
9 SimDat_5 MA(1) θ1 = −0.7
10 SimDat_6 ARMA(1,1) φ1 = 0.7, θ1 = 0.7
11 SimDat_7 ARMA(1,1) φ1 = −0.7, θ1 = −0.7
12 SimDat_8 ARFIMA(0,0.30,0) fracdiff.sim {fracdiff}
13 SimDat_9 ARFIMA(1,0.30,0) φ1 = 0.7
14 SimDat_10 ARFIMA(0,0.30,1) θ1 = −0.7
15 SimDat_11 ARFIMA(1,0.30,1) φ1 = 0.7, θ1 = −0.7
16 SimDat_12 ARFIMA(2,0.30,2) φ1 = 0.7, φ2 = 0.2, θ1 = −0.7, θ2 = −0.2

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Fig. 1 Precipitation and temperature data from Peterson and Vose (1997) and Lawrimore et al. (2011) respectively

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
by the name of the R package, denoted with {}. All algo-
rithms are used with predefined values, unless specified
differently.
We implement the forecasting methods described in
Tables4, 5 and 6. e latter constitute an adapted repro-
duction from Papacharalampous et al. (2017a). Naïve
is the last observation benchmark, while random walk
(RW) is a commonly used variation of Naïve (Hyndman
and Athanasopoulos 2013). Regarding the AutoRegres-
sive Integrated Moving Average (ARIMA) methods, the
ARIMA_f and auto_ARIMA_f forecasting models use the
same algorithm with the ARIMA_s and auto_ARIMA_s
simulation models respectively, although the innovations
are set to be zero in the former ones. e latter applies
to the auto_ARFIMA method as well, which is com-
monly used for modelling processes that are assumed to
exhibit long-range dependence. ese five methods esti-
mate the involved parameters using the maximum likeli-
hood method. BATS, ETS and SES stand for Box-Cox
transformation, ARMA errors, Trend and Seasonal com-
ponents (De Livera etal. 2011); Error, Trend and Season-
ality (or ExponenTial Smoothing); and Simple Exponential
Smoothing respectively. Further information about the
latter models can be found in Hyndman and Athanaso-
poulos (2013), while eta is introduced in Assimakopou-
los and Nikolopoulos (2000). All the stochastic methods
use procedures like those presented in Hyndman and
Khandakar (2008). e machine learning methods, on the
other hand, are based on a somehow different algorith-
mic approach. is fact is easily perceivable through the
alongside examination of Tables4, 5 and 6.
e assessment of the one-step ahead forecasting per-
formance is based on the error metrics and accuracy sta-
tistics of Table7.
We conduct the experiments described in Tables8 and
9. We use each dataset in five experiments; every time
examining different parts of the time series according
to Table9. While the application of the stochastic meth-
ods does not require a validation set (since all the model
parameters are estimated using other procedures, such as
the maximum likelihood estimation), the same does not
apply to the application of the machine learning methods
(except NN_3). For each of the latter, we fit the candidate
models defined in Table5 to the fitting set, i.e. the first
33, 40, 47, 53 or 60 values, and subsequently use them to
make predictions corresponding to the validation set, i.e.
the next 17, 20, 23, 27 or 30 values respectively. Finally,
we decide on the “optimal” model, i.e. the one exhibiting
the smallest root mean square error on the validation set.
We fit this model to the first 50, 60, 70, 80 or 90 values
and make predictions corresponding to the 51st, 61st,
71st, 81st or 91st value respectively.
e only assumption of our methodological approach
concerns the application of the auto_ARFIMA method
within the real-world experiments and is that the annual
precipitation and temperature variables can be suf-
ficiently modelled by the normal distribution. is
assumption is rather reasonable (implied by the Central
Limit eorem; Koutsoyiannis 2008, chapter 2.5.6) and
could hardly harm the results. In general, such funda-
mental assumptions are preferable to the introduction of
extra parameters, e.g. to using the Box-Cox transforma-
tion to normalize the data. e rest of the methods are
non-parametric and, thus, not affected by the possible
non-normality. To take advantage of some well-known
theoretical properties, in the SE_1i–SE_7i simulation
experiments the ARIMA_f and ARIMA_s methods are
given the same AutoRegressive (AR)and Moving Aver-
age (MA)orders used in the respective simulation pro-
cess, while d is set 0. ese two methods, as well as the
simple, auto_ARIMA_f, auto_ARIMA_s and auto_
ARFIMA methods serve as reference points within our
Table 4 Univariate time series forecasting methods of this study (part 1): stochastic methods
s/n Abbreviated name Category R algorithm(s) Implementation notes
1 Naïve Simple
2RW rwf {forecast} drift = TRUE
3 ARIMA_f AutoRegressive Integrated Moving
Average (ARIMA) Arima {forecast}, forecast {forecast} Arima {forecast}: include.mean = TRUE,
include.drift = FALSE, method = ” ML”
4 ARIMA_s Arima {forecast}, simulate {stats}
5 auto_ARIMA_f auto.arima {forecast}, forecast {forecast}
6 auto_ARIMA_s auto.arima {forecast}, simulate {stats}
7 auto_ARFIMA AutoRegressive Fractionally Integrated
Moving Average (ARFIMA) arfima {forecast}, forecast {forecast} arfima {forecast}: estim = ”mle”
8BAT S State space bats {forecast}, forecast {forecast}
9 ETS_s ets {forecast}, simulate {stats}
10 SES Exponential smoothing ses {forecast}
11 Theta thetaf {forecast}

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Table 5 Univariate time series forecasting methods of this study (part 2): machine learning methods
s/n Abbreviated name Category Model structure information R algorithm(s) Implementation notes
Hyperparameter optimized
using grid search (grid values) Lagged variable selection proce-
dure (see Table 6)
12 NN_1 Neural networks Single hidden layer multilayer
perceptron CasesSeries {rminer}, fit {rminer},
lforecast {rminer}, nnet {nnet} Number of hidden nodes (0, 1,
…, 15) 1
13 NN_2 2
14 NN_3 nnetar {forecast} 3
15 RF_1 Random forests Breiman’s random forests algo-
rithm with 500 grown trees CasesSeries {rminer}, fit {rminer},
lforecast {rminer}, randomForest
{randomForest}
Number of variables randomly
sampled as candidates at each
split (1, …, 5)
1
16 RF_2 2
17 RF_3 3
18 SVM_1 Support vector machines Radial basis kernel “Gaussian”
function, C = 1, epsilon = 0.1 CasesSeries {rminer}, fit {rminer},
lforecast {rminer}, ksvm {kernlab} Sigma inverse kernel width (2n,
n = − 8, − 7, …, 6) 1
19 SVM_2 2
20 SVM_3 3

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
approach. In particular, ARIMA_f, auto_ARIMA_f and
auto_ARFIMA are theoretically expected to be the most
accurate within our simulation experiments [for an expla-
nation see Papacharalampous etal. (2017a), chapter2],
while BATS is also expected to perform well in these
experiments, since it comprises an ARMA model. In
summary, the experiments are controlled to some extent,
while their components (datasets, methods and metrics)
are selected to provide a multifaceted approach to the
problem of one-step ahead forecasting in geoscience.
Results and discussion
In this section, we summarize the basic quantitative and
qualitative information gained from the experiments
of the present study, while the total amount is available
in the Additional files 1, 2, 3, 4, 5, 6 and 7. We further
Table 6 Lagged variable selection procedures adopted for the machine learning methods of Table 5
s/n Time lags R algorithm
1 The corresponding to an estimated value for the AutoCorrelation Function (ACF) acf {stats}
2 The corresponding to a statistical important estimated value for the ACF. If there is no statistical important estimated value for the
ACF, the corresponding to the largest estimated value acf {stats}
3 According to nnetar {forecast}, i.e. the time lags 1, …, n, where n is the number of AutoRegressive (AR) parameters that are fitted to
the time series data ar {stats}
Table 7 Error metrics and accuracy statistics of this study
s/n Abbreviated name Full name Category Values Optimum value
1EError Error metrics (−∞, +∞) 0
2 AE Absolute error [0, +∞) 0
3 PE Percentage error (−∞, +∞) 0
4 APE Absolute percentage error [0, +∞) 0
5 MdoAE Median of the absolute errors Accuracy statistics [0, +∞) 0
6 MdoAPE Median of the absolute percentage errors [0, +∞) 0
7 LRC Linear regression coefficient (−∞, +∞) 1
8 R2 Coefficient of determination [0, 1] 1
Table 8 Experiments of this study
The symbol i can take the values stated in Table 9
s/n Abbreviated name Category Dataset (see Table 3) Forecasting methods (see Tables 4 and 5)Metrics (see Table 7)
1 RWE_1iReal-world PrecDat 1, 2, 7–20 1–8
2 RWE_2iTempDat
3 RWE_3iStandPrecDat 1, 2, 7–20 1, 2, 5, 7, 8
4 RWE_4iStandTempDat
5 SE_1iSimulation SimDat_1 1–6, 8–20 1, 2, 5, 7, 8
6 SE_2iSimDat_2
7 SE_3iSimDat_3
8 SE_4iSimDat_4
9 SE_5iSimDat_5
10 SE_6iSimDat_6
11 SE_7iSimDat_7
12 SE_8iSimDat_8 1, 2, 7–20
13 SE_9iSimDat_9
14 SE_10iSimDat_10
15 SE_11iSimDat_11
16 SE_12iSimDat_12

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
discuss the findings and explicate their contribution in
light of the literature.
Experiments using the precipitation datasets
For the experiments using the PrecDat dataset, the mini-
mum AE value is 0 (practically) and the maximum around
1,750mm (for forecasts produced by the simple forecast-
ing methods, i.e. Naïve and RW), while the respective
values for the APE error metric are 0 (practically) and
1.64 (for a forecast produced by NN_1). e MdoAE
and MdoAPE values are summarized in Tables10 and 11
respectively. e minimum MdoAE is 68mm, while the
maximum is 189mm. ese two values are in the same
order of magnitude as the smallest and average standard
deviation estimates of the time series respectively. e
minimum MdoAPE value is 0.09 and the maximum 0.22,
while the respective LRC values are 0.73 and 1.18. e
best LRC value (1.00) is measured within RWE_1c for
the simple forecasting methods, while the best R2 value
(0.84) is measured within RWE_1d for BATS. e worst
LRC and R2 values are 0.73 for RF_2 within RWE_1d and
0.54 for NN_1 within RWE_1a respectively.
In Fig.2 we present a graphical summary of the experi-
ments using the PrecDat dataset. e values in the
three upper heatmaps are scaled in the row direction
and the darker the colour within a specific row the bet-
ter the forecasts. In fact, heatmaps are used in this study
instead of conventional tables, since they allow the easy
extract of qualitative information. e relative perfor-
mance of the forecasting methods differs to some degree
across the various RWE_1i experiments, with ETS_s and
NN_1 being the worst performing in terms of MdoAE
and MdoAPE, followed by the simple methods. On the
other hand, in terms of LRC Naïve and RW exhibit rather
the best overall performance. In the downer heatmap
of Fig.2 we zoom into the RWE_1b experiment. By its
examination we observe that all the implemented fore-
casting methods can perform well or bad, depending on
the individual case. is fact is also apparent in the side-
by-side boxplots of Fig.2. Furthermore, we observe that
for one specific time series the AE values measured are
very high for all the forecasts apart from those produced
by the simple forecasting methods.
Regarding the experiments using the StandPrec-
Dat dataset, the minimum AE value is 0 and the maxi-
mum around 10. e MdoAE values are summarized in
Table12. e minimum MdoAE is 0.55, while the maxi-
mum is 1.42. ese two values are 45% smaller and 42%
larger than 1 (standard deviation of the standardized
time series) respectively. Since there is an absence of rel-
evant information in the literature, these values could
be used as rough benchmarks for the predictability of
annual precipitation. Most preferably, a representative
sample set of univariate time series forecasting meth-
ods could be implemented at least for benchmarking
purposes alongside with any other forecasting attempt.
Moreover, the minimum and maximum LRC values are
−0.25 and 0.25 respectively, the former measured for
ETS_s and the latter for RW. Finally, the minimum R2
value is 0 (practically), while the maximum is 0.09, meas-
ured within SE_3a for ETS_s. In addition to this numeri-
cal information, Fig. 3 presents a brief comparison
between the experiments using the PrecDat and Stand-
PrecDat datasets. As illustrated in this figure, the relative
performance of the forecasting methods with respect to
AE and MdoAE in the experiments using the latter data-
set is mostly similar to the one in the experiments using
Table 9 Part of the time series used within each experi-
ment according to the i value
s/n iData points of each time series
used for the model-fitting (required
for all models) and model-validation
(required for the machine learning
models)
Data points of each
time series used
for model-testing
1 a 1, 2, 3, …, 50 51
2 b 1, 2, 3, …, 60 61
3 c 1, 2, 3, …, 70 71
4 d 1, 2, 3, …, 80 81
5 e 1, 2, 3, …, 90 91
Table 10 Minimum, maximum and mean values of the
MdoAE within the experiments using the PrecDat dataset
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum (mm) Maximum (mm) Mean (mm)
RWE_1a 111 (RF_1) 172 (NN_1) 135
RWE_1b 68 (SVM_1) 146 (ETS_s) 91
RWE_1c 91 (SVM_3) 171 (ETS_s) 119
RWE_1d 143 (BATS) 189 (RF_2) 162
RWE_1e 98 (Theta) 150 (NN_1) 122
Table 11 Minimum, maximum and mean values of the
MdoAPE within the experiments using the PrecDat dataset
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
RWE_1a 0.12 (RF_1) 0.21 (RW ) 0.16
RWE_1b 0.09 (SVM_1) 0.18 (ETS_s) 0.12
RWE_1c 0.12 (SVM_3) 0.21 (NN_1) 0.15
RWE_1d 0.15 (BATS) 0.22 (NN_1) 0.17
RWE_1e 0.12 ( Theta) 0.18 (NN_1) 0.15

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Fig. 2 Results in brief of the experiments using the PrecDat dataset

Page 11 of 19
Papacharalampous et al. Geosci. Lett. (2018) 5:12
the former dataset. Nevertheless, the LRC (and R2) val-
ues are far worse when using the standardized datasets.
In fact, standardization results to processes with different
predictability with respect to the original.
Experiments using the temperature datasets
In Fig.4 we present a graphical summary of the experi-
ments using the TempDat dataset. For these experiments
the minimum AE value is 0 and the maximum around
43 °C (for a forecast produced by NN_2), while the
respective APE values are 0 and 9.64 (for a forecast pro-
duced by ETS_s). e MdoAE and MdoAPE values are
summarized in Tables13 and 14 respectively. e mini-
mum MdoAE is 0.23°C, while the maximum is 1.10°C.
ese two values are in the same order of magnitude as
the smallest and largest standard deviation estimates of
the temperature time series respectively. e respective
values for MdoAPE are 0.02 and 0.08. e minimum LRC
value is 0.95 and the maximum is 1.02; all the LRC val-
ues are close to the optimum. Finally, the minimum R2
value is 0.78, measured for NN_2 within RWE_2b, while
all the rest R2 values are higher than 0.97 with maximum
1 (practically), measured for the auto_ARFIMA method
within RWE_2b. In summary, the relative performance
of the forecasting methods varies across the different
experiments conducted using the TempDat dataset. e
auto_ARFIMA, BATS, SES, eta and NN_3 seem to be
well performing in terms of MdoAE and MdoAPE when
applied to these temperature time series compared to
the overall picture, while the simple methods are far the
best in terms of MdoAE within the RWE_2d experiment.
ETS_s and NN_1 are the worst performing within all the
experiments apart from RWE_2c, in which the simple
methods exhibit the worst performance. Finally, by com-
paring the numerical results of the experiments using the
PrecDat and TempDat dataset, we observe the fact that
temperature is more predictable than precipitation.
Regarding the experiments using the StandTempDat,
the minimum AE value is 0 and the maximum around
18.91. e MdoAE values are summarized in Table15.
e minimum MdoAE value is 0.33, while the maximum
is 1.46. ese two values are 67% smaller and 46% larger
than 1 (standard deviation of the standardized time
series) respectively and could be used as rough bench-
marks for the predictability of annual temperature (for
an explanation see the subsection entitled “Experiments
using the precipitation datasets”). e minimum LRC
value is 0.04 and the maximum is 0.76, the former meas-
ured for SVM_1 and the latter for RW. Finally, the mini-
mum R2 value is 0.03, while the maximum is 0.48. e
latter value is measured for Naïve in RWE_4a. Figure5
facilitates a comparison between the experiments using
the TempDat and StandTempDat datasets. Here as well,
we observe that the relative performance of the fore-
casting methods with respect to AE and MdoAE in the
experiments using the standardized precipitation time
series mostly does not vary from the respective relative
performance when using the original temperature time
series. We further note that the LRC (and R2) values are
worse when using the standardized temperature dataset,
while they are better for the latter than for the standard-
ized precipitation dataset.
Experiments using the simulated datasets
e subsequently reported information constitutes the
provided empirical solution to the problem of one-step
ahead forecasting in geoscience. Nonetheless, this solution
is rather qualitative than quantitative (although the results
are also stated quantitatively), since the respective experi-
ments use unscaled data that could be assumed as real-
world data in a standardized form (such as StandPrecDat
and StandTempDat) with different predictability than the
original (for example, see the subsections entitled “Experi-
ments using the precipitation datasets” and “Experiments
using the temperature datasets”). In fact, the experiments
using standardized precipitation and temperature can
facilitate a connection between the experiments using the
same data in their original form and the experiments using
the simulated datasets. A graphical summary of the latter
experiments is available in Fig.6.
e generalized findings of the present study are the
following:
(1) e E values are approximately symmetric around 0
(mean value of the simulations).
(2) e results may vary significantly across the simula-
tion experiments using different simulated datasets
and across the different time series within a specific
experiment depending on the forecasting method.
(3) Consequently, the relative performance of the fore-
casting methods may also vary significantly across
the simulation experiments using different simulated
datasets.
Table 12 Minimum, maximum and mean values of the
MdoAE within the experiments using the StandPrecDat
dataset
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
RWE_3a 0.70 (RF_1) 1.22 (NN_1) 0.92
RWE_3b 0.55 (SVM_2) 0.95 (ETS_s) 0.69
RWE_3c 0.72 (BATS) 1.42 (NN_1) 0.86
RWE_3d 0.99 (Theta) 1.42 (ETS_s) 1.14
RWE_3e 0.69 (Theta) 1.07 (ETS_s) 0.89

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Fig. 3 Comparison in brief between the experiments using the PrecDat and StandPrecDat datasets

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Fig. 4 Results in brief of the experiments using the TempDat dataset

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
(4) On the contrary, the relative performance of the fore-
casting methods is slightly affected by the length of
the time series for the experiments of the present
study. e same has been found to mostly apply to
the multi-step ahead forecasting performance of the
same methods in Papacharalampous etal. (2017a) for
two other time series lengths.
(5) Some forecasting methods are more accurate than
others. e best-performing methods are ARIMA_f,
auto_ARIMA_f, auto_ARFIMA, BATS, SES and
eta. is good performance of the former four
methods when applied to ARMA and ARFIMA
processes is expected from theory, while the eta
forecasting method has also performed well in the
M3-Competition (Makridakis and Hibon 2000) and
is expected to have a similar performance with SES
(Hyndman and Billah 2003). e five above-men-
tioned forecasting methods are all stochastic.
(6) All the machine learning methods except for NN_1
(mostly NN_3 and SVM_3) are comparable to the
best-performing methods, as it has also been found
to apply in the experiments of Papacharalampous
et al. (2017a, b). Likewise, in Tyralis and Papacha-
ralampous (2017), random forests are competitive
with the ARFIMA and eta benchmarks.
(7) e simple methods are competitive in specific simu-
lation experiments, as suggested for specific cases in
Cheng et al. (2017), Makridakis and Hibon (2000)
and Papacharalampous etal. (2017a) as well. Never-
theless, they also stand out because of their bad per-
formance in other simulation experiments.
(8) Most of the far outliers are produced by neural net-
works.
e minimum AE value for the forecasts is 0 (prac-
tically) and the maximum around 155 (produced by
NN_2). e MdoAE values are summarized in Tables16
and 17. Especially, the latter is useful in supporting
Observations (5–7). e minimum MdoAE is 0.65, while
the maximum is 2.91. ese two values are 35% smaller
and 191% larger than 1 (standard deviation of the simula-
tions) respectively. Furthermore, in spite of Observation
(4), the MdoAE values may decrease on the level of the
second or even the first decimal, when moving from the
simulation experiments using time series of 51 values to
those of 91 values, with the NN_1 forecasting method
exhibiting the largest improvement. e minimum LRC
value is −0.88 and the maximum is 0.94, both measured
for RW, while the minimum and maximum values pro-
duced by Naïve differ in the second and third decimal
respectively. is range holds a complete interpretation
of the observed within the real-world experiments varia-
tions in the performance of the simple methods in terms
of LRC from extremely good to extremely bad (with
respect to the overall picture). Finally, the minimum R2
value is 0 (practically), measured for ETS_s within several
experiments, while the maximum is 0.84 within SE_9b
for Naïve.
Conclusions
e simulation experiments reveal the most and least
accurate methods for long-term one-step ahead fore-
casting applications, also suggesting that the simple
methods may be competitive in specific cases. Further-
more, the relative performance of the forecasting meth-
ods is slightly affected by the time series length for the
Table 13 Minimum, maximum and mean values of the
MdoAE within the experiments using the TempDat dataset
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum (°C) Maximum (°C) Mean (°C)
RWE_2a 0.42 (NN_3) 0.72 (NN_1) 0.51
RWE_2b 0.23 (Theta) 0.54 (NN_1) 0.32
RWE_2c 0.38 (BATS) 0.66 (RW) 0.47
RWE_2d 0.78 (RW) 1.10 (NN_3) 1.01
RWE_2e 0.38 (Theta) 0.62 (ETS_s) 0.46
Table 14 Minimum, maximum and mean values of the
MdoAPE within the experiments using the TempDat data-
set
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
RWE_2a 0.04 (Theta) 0.06 (NN_1) 0.04
RWE_2b 0.02 (auto_ARFIMA) 0.05 (ETS_s) 0.03
RWE_2c 0.03 (SVM_1) 0.06 (RW) 0.04
RWE_2d 0.07 (Naïve) 0.08 (NN_1) 0.08
RWE_2e 0.03 (RF_1) 0.05 (NN_1) 0.04
Table 15 Minimum, maximum and mean values of the
MdoAE within the experiments using the StandTempDat
dataset
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
RWE_4a 0.61 (BATS) 0.93 (ETS_s) 0.71
RWE_4b 0.33 (Theta) 0.73 (NN_1) 0.47
RWE_4c 0.56 (SES) 0.96 (ETS_s) 0.69
RWE_4d 1.20 (NN_1) 1.46 (Theta) 1.36
RWE_4e 0.48 (Theta) 0.82 (ETS_s) 0.61

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
simulation experiments of this study (using time series
of 51, 61, 71, 81, 91 values), while it strongly depends on
the process. Also importantly, the experiments using the
original real-world time series result to minimum and
maximum medians of the absolute errors of 68 and 189
mm for precipitation, and 0.23 and 1.10°C for tempera-
ture respectively. Additionally, the experiments using
the standardized real-world time series suggest that the
Fig. 5 Comparison in brief between the experiments using the TempDat and StandTempDat datasets

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Fig. 6 Results in brief of the experiments using the simulated datasets

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
minimum and maximum medians of the absolute errors
are 0.55 and 1.42 for precipitation, and 0.33 and 1.46
for temperature respectively. ese latter numerical
results could be used as a rough upper boundary for the
one-step ahead predictability of annual precipitation and
temperature.
We subsequently state the limitations of this study and
some future directions. e provided empirical solution
to the problem of one-step ahead forecasting in geosci-
ence is rather qualitative than quantitative, while the
experiments using standardized precipitation and tem-
perature data have offered rough benchmarks only. In
the future more real-world data could be used to develop
improved benchmarks for assessing the respective pre-
dictabilities. It would be of interest to further investigate
how these predictabilities depend on the location from
which the data originate. In this case, more stations span-
ning around the globe would be required. Moreover, a
direct and large-scale comparison, set on a common base
(if this is feasible), between deterministic and statistical
approaches to forecasting geophysical processes would be
useful and interesting. Another limitation of this study is
related to the adopted modelling approach, i.e. the data-
driven one, according to which the selection of the model
does not depend on the properties of the examined pro-
cess and, therefore, the latter are mostly not investigated.
Furthermore, the improvement of the performance of the
machine learning models requires extensive comparisons
between different procedures of hyperparameter opti-
mization and lagged variable selection. Finally, future
research could focus on the examination of the respective
predictabilities, when using exogenous predictor vari-
ables as well, while a definitely worth-stated future direc-
tion is related to the adoption of probabilistic forecasting
methods, instead of the point forecasting ones.
Abbreviations
ACF: AutoCorrelation Function; AR: AutoRegressive; ARFIMA: AutoRegressive
Fractionally Integrated Moving Average; ARIMA: AutoRegressive Integrated
Moving Average; ARMA: AutoRegressive Moving Average; MA: Moving Aver-
age; SARIMA: Seasonal AutoRegressive Integrated Moving Average.
Authors’ contributions
GP and HT worked on the analyses equally and to all of their aspects. GP,
HT and DK discussed the results and contributed in the writing of the main
manuscript. All authors read and approved the final manuscript.
Additional files
Additional file 1. Exploration of the datasets.
Additional file 2. Experiments using the PrecDat dataset.
Additional file 3. Experiments using the TempDat dataset.
Additional file 4. Experiments using the StandPrecDat dataset.
Additional file 5. Experiments using the StandTempDat dataset.
Additional file 6. Experiments using the simulated datasets, part 1.
Additional file 7. Experiments using the simulated datasets, part 2.
Table 16 Minimum, maximum and mean values of the
MdoAE within the simulation experiments
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
SE_1i0.68 (ARIMA_f | SE_1a) 1.05 (NN_1 | SE_1a) 0.80
SE_2i0.67 (ARIMA_f | SE_2c) 1.82 (RW | SE_2e) 0.95
SE_3i 0.65 (ARIMA_f | SE_3c) 1.04 (NN_1 | SE_3a) 0.81
SE_4i0.67 (ARIMA_f | SE_4c) 1.21 (ETS_s | SE_4a) 0.84
SE_5i0.66 (ARIMA_f | SE_5e) 1.48 (RW | SE_5c) 0.90
SE_6i0.68 (ARIMA_f | SE_6b) 1.20 (ETS_s | SE_6d) 0.89
SE_7i0.66 (auto_ARIMA_f | SE_7d) 2.91 (RW | SE_7e) 1.22
SE_8i0.67 (auto_ARFIMA | SE_8c) 1.02 (NN_1 | SE_8a) 0.77
SE_9i0.67 (auto_ARFIMA | SE_9d) 1.05 (NN_1 | SE_9b) 0.80
SE_10i0.67 (auto_ARFIMA | SE_10e) 1.22 (RW | SE_10e) 0.83
SE_11i0.68 (Theta | SE_11e) 1.10 (NN_1 | SE_11a) 0.77
SE_12i0.69 (auto_ARFIMA | SE_12b) 1.06 (NN_1 | SE_12a) 0.78
Table 17 Minimum, maximum and mean values of the
MdoAE for each method within the simulation experi-
ments
The minimum of the minimum values and the maximum of the maximum
values are in italic
Minimum Maximum Mean
Naïve 0.68 (SE_3c) 2.88 (SE_7a) 1.12
RW 0.69 (SE_9c) 2.91 (SE_7e) 1.13
ARIMA_f 0.65 (SE_3c) 0.72 (SE_7a) 0.69
ARIMA_s 0.91 (SE_2a) 1.04 (SE_3a) 0.96
auto_ARIMA_f 0.66 (SE_7d) 0.75 (SE_6c) 0.70
auto_ARIMA_s 0.91 (SE_4c) 1.02 (SE_3d) 0.97
auto_ARFIMA 0.67 (SE_10e) 0.73 (SE_10d) 0.69
BAT S 0.67 (SE_3c) 0.76 (SE_6c) 0.71
ETS_s 0.93 (SE_3d) 2.11 (SE_7e) 1.14
SES 0.66 (SE_3c) 1.52 (SE_7e) 0.83
Theta 0.66 (SE_3c) 1.57 (SE_7a) 0.84
NN_1 0.90 (SE_7e) 1.16 (SE_7a) 1.01
NN_2 0.72 (SE_8c) 0.89 (SE_5b) 0.79
NN_3 0.69 (SE_8c) 0.84 (SE_6c) 0.74
RF_1 0.71 (SE_8c) 1.08 (SE_6a) 0.82
RF_2 0.72 (SE_8c) 1.04 (SE_6c) 0.83
RF_3 0.72 (SE_3c) 0.98 (SE_6c) 0.80
SVM_1 0.71 (SE_8e) 1.23 (SE_7a) 0.86
SVM_2 0.68 (SE_8c) 1.01 (SE_7a) 0.81
SVM_3 0.68 (SE_8c) 0.92 (SE_6c) 0.76

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Papacharalampous et al. Geosci. Lett. (2018) 5:12
Acknowledgements
We thank the Editor Bellie Sivakumar and two anonymous reviewers, whose
comments have substantially improved the quality of this paper.
The analyses and visualizations have been performed in R Programming
Language (R Core Team 2017) by using the contributed R packages forecast
(Hyndman and Khandakar 2008, Hyndman et al. 2017), fracdiff (Fraley et al.
2012), gdata (Warnes et al. 2017), ggplot2 (Wickham 2016), HKprocess (Tyralis
2016), kernlab (Karatzoglou et al. 2004), knitr (Xie 2014, 2015, 2017), nnet
(Venables and Ripley 2002), randomForest (Liaw and Wiener 2002), readr
(Wickham et al. 2017) and rminer (Cortez 2010, 2016).
We acknowledge the Asia Oceania Geoscience Society (AOGS) for provid-
ing the publication cost. A preliminary research by Papacharalampous et al.
(2017d) was presented in the 14th AOGS Annual Meeting.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
This is a fully reproducible research paper; all the codes and data, as well as
their outcome results, are available in the Additional files (Papacharalampous
and Tyralis 2018). The sources of the real-world datasets are Lawrimore et al.
(2011) and Peterson and Vose (1997).
Funding
This research has not received any funding.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in pub-
lished maps and institutional affiliations.
Received: 31 August 2017 Accepted: 17 March 2018
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