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DYNAMICS OF SEASONAL PATTERNS IN GEOCHEMICAL,
ISOTOPIC, AND METEOROLOGICAL RECORDS OF THE
ELBRUS REGION DERIVED FROM FUNCTIONAL DATA
Gleb A. Chernyakov1*, Valeria Vitelli2, Mikhail Y. Alexandrin1, Alexei M. Grachev1, Vladimir N. Mikhalenko1,
Anna V. Kozachek3, Olga N. Solomina1, Vladimir V. Matskovsky1
1Institute of Geography, Russian Academy of Sciences, 29 Staromonetniy lane, 119017, Moscow, Russia
2Department of Biostatistics, University of Oslo, Sognsvannsveien 9, 0372, Oslo, Norway
3Arctic and Antarctic Research Institute, 38 Bering st., 199397, St. Petersburg, Russia
*Corresponding author: firstname.lastname@example.org
Received: December 31th, 2019 / Accepted: August 9th, 2020 / Published: October 1st, 2020
A nonparametric clustering method, the Bagging Voronoi K-Medoid Alignment algorithm, which simultaneously
clusters and aligns spatially/temporally dependent curves, is applied to study various data series from the Elbrus region
(Central Caucasus). We used the algorithm to cluster annual curves obtained by smoothing of the following synchronous
data series: titanium concentrations in varved (annually laminated) bottom sediments of proglacial Lake Donguz-Orun; an
oxygen-18 isotope record in an ice core from Mt. Elbrus; temperature and precipitation observations with a monthly resolution
from Teberda and Terskol meteorological stations. The data of different types were clustered independently. Due to restrictions
concerned with the availability of meteorological data, we have fulfilled the clustering procedure separately for two periods:
1926–2010 and 1951–2010. The study is aimed to determine whether the instrumental period could be reasonably divided
(clustered) into several sub-periods using different climate and proxy time series; to examine the interpretability of the
resulting borders of the clusters (resulting time periods); to study typical patterns of intra-annual variations of the data series.
The results of clustering suggest that the precipitation and to a lesser degree titanium decadal-scale data may be reasonably
grouped, while the temperature and oxygen-18 series are too short to form meaningful clusters; the intercluster boundaries
show a notable degree of coherence between temperature and oxygen-18 data, and less between titanium and oxygen-18 as
well as for precipitation series; the annual curves for titanium and partially precipitation data reveal much more pronounced
intercluster variability than the annual patterns of temperature and oxygen-18 data.
KEY WORDS: Central Caucasus; paleoclimate archives; lake sediments; ice cores; clustering; functional data
CITATION: Gleb A. Chernyakov, Valeria Vitelli, Mikhail Y. Alexandrin, Alexei M. Grachev, Vladimir N. Mikhalenko, Anna V. Kozachek,
Olga N. Solomina, Vladimir V. Matskovsky (2020). Dynamics Of Seasonal Patterns In Geochemical, Isotopic, And Meteorological
Records Of The Elbrus Region Derived From Functional Data Clustering.
Geography, Environment, Sustainability.
ACKNOWLEDGEMENTS: The data analysis of the lake sediment core was supported by the RFBR project No. 17-05-01170 A;
the ice core data analysis was fulfilled with the support of RSF project No. 17-17-01270-П; the analysis of meteorological data
was performed in the framework of the State Assignment of Institute of Geography RAS No. 0148-2019-0004.
Conflict of interests: The authors reported no potential conflict of interest.
In the last decade, new paleoclimate archives were
obtained in the course of expeditionary work involving
the Institute of Geography of the Russian Academy of
Sciences. Among them are ice cores of the Western plateau
of Elbrus (Mikhalenko et al. 2015; Kozachek et al. 2017),
bottom sediments of the lakes Karakel, Donguz-Orun,
Khuko (Solomina et al. 2013; Alexandrin et al. 2018), etc.
The obtained cores were studied and dated by laboratory
methods; their elemental and isotopic compositions
were determined (Darin et al. 2015a; Darin et al. 2015b;
Kozachek et al. 2015). Until now, among the existing
statistical approaches, mostly the correlation-regression
and component analysis have been applied to study the
new data (Alexandrin et al. 2018). Among the applications
of cluster analysis to these data, only works on studying the
backward air mass trajectories and dust transfer are known
(Kutuzov et al. 2017; Khairedinova et al. 2017).
Cluster analysis is used to split a certain set of objects
into relatively homogeneous groups (clusters). In this work,
the clustering procedure was applied independently to
investigate several synchronous time series characterizing
the dynamics of the natural environment in the Central
Caucasus in the 20th century. As the result of this procedure,
each time series is divided into intervals corresponding to
dierent clusters. Thus, a time sequence of clusters or parts
of them appears in each data series.
In our work a nonparametric method of clustering
functional data, the Bagging Voronoi K-Medoid Alignment
(BVKMA) algorithm, was applied to analyze the data.
This clustering method and its application for studying
bottom sediments are described in detail in (Abramowicz
et al. 2017) and summarized in the Method section of our
paper. A specic feature of this approach is that splitting
annual data into clusters is based on the shape of intra-
annual variations of the parameter under investigation. In
that way, the analysis is aimed at answering the question
«which time periods are similar and which are dierent
in terms of their intra-annual variability». This clustering
method may be only applied to the data that have several
measurements for each period.
Moreover, the advantage of BVKMA compared to
previously developed methods of clustering functional
data is the ability of the method to deal jointly with two
eects, revealing by data, that can lead the clustering
procedure to a tendency of considering substantially
similar or interconnected data as completely dierent or
independent, what may be regarded as misclassication.
The rst issue is the eect of misalignment functional
data that is typically manifested in time lags. For instance,
if the functional data are represented as time-dependent
curves, one can notice time lags of peaks of some curves
compared to the others, despite a common shape and
reasons of the observed variability (see, e.g., Sangalli et al.
2010). The Alignment procedure, implemented in BVKMA,
is intended to prevent possible misclassication of the
curves during clustering due to their misalignment. In
climatic research, the misalignment may naturally occur in
annually repeated seasonal patterns.
The second eect that is important to account for, is a
possible time/spatial dependence of functional data. For
example, the neighboring annual curves, derived from
high resolution paleoclimatic records, may be regarded
as «dependent» because they are supposed to reect
common processes and eects characterizing the natural
environment of a given period. In BVKMA algorithm the
account for data dependency, expressed by a tendency to
attribute consecutive curves to the same cluster, is provided
by the usage of the Voronoi tessellation (Abramowicz et al.
2017). Also, the type of dependency (spatial or temporal)
does not impose restrictions on the applicability of the
method. Moreover, the spatial dependence observed for
the parameter of interest in annual layers of proxy data can
be transformed into temporal dependence on the base of
known dating of the proxy. In our study, this is the case for
lake sediment and ice core data.
The main purpose of our investigation is to assess
the degree of consistency of the resulting clusters for
geochemical, isotopic and meteorological data series and
to nd out essential or close time boundaries in dierent
series. This approach of combining various types of data in
order to reveal their implicit interrelations may be a useful
tool for creating paleoclimate reconstructions.
In many palaeoclimatic studies, researchers aim to nd
modern analogues for past climates, and thus reconstruct
specic parameters or palaeoenvironments for specic
time periods. The clustering method used in this study
previously was applied to cluster millennia-long time-
series, resulting in only one cluster covering the whole
instrumental period. The results of such an approach
could be hardly interpreted in terms of nding modern
analogues for past climates. Here for the rst time we apply
this method for time-series which are several decades
long and fully intersect with the instrumental period. The
purpose of this approach is (i) to determine whether an
instrumental period of usual length could be reasonably
divided (clustered) into several sub-periods using this
method and dierent climate and proxy time-series; (ii) to
study the shape of medoids (which represent intra-annual
variations of parameters) for dierent climate and proxy
time-series and their associations; and (iii) to examine the
resulting borders of the clusters, or resulting time periods,
in terms of their interpretability.
MATERIALS AND METHODS
The Greater Caucasus borders the Russian Plain from the
south. It is located in the temperate and subtropical zones
between the Black and Caspian Seas. Elbrus volcano (5642
m) – the highest peak of the Caucasus, supports extensive
modern glaciation. The climate in the region is dominated
by the westerlies. The continentality is increasing from the
west to east: the mean June temperature at the foothills
of Greater Caucasus is approximately +23–24 °C, while in
the east it is higher (25–29 °C): the annual precipitation,
on the contrary, decreases in the west-east direction from
4000 mm (Kodory valley) to 1000–1500 mm in the eastern
Caucasus (Gvozdetsky and Golubchikov 1987). Precipitation
maxima occur in July–September; the warmest month is
July, the coldest one is January.
Meteorological data at the high elevation of the
Caucasus are quite scarce. In this paper, we used the
data from Terskol station located in the area where
our other proxies (ice core and lake sediments) are
situated and Teberda station with longer meteorological
records. Shahgedanova et al. (2014) noticed positive
trends in summer temperature and precipitation of the
accumulation period (October–April) recorded at the high-
elevation Terskol and Klukhorsky Pereval stations in the
period between 1987 and 2010.The glaciers are however
retreating since the early 20th century and the retreat rate
The study area and the locations of the proxy records
and meteorological stations used as the sources of data are
marked on the map below (Fig. 1).
The following data were used in the work.
a) Data on the elemental composition of the core of
the annually laminated bottom sediments of Lake Donguz-
Orun. The top core used (160 mm) contains annual layers
formed during the period 1922–2010 (Alexandrin et al.
2018). Among the chemical elements present in the
sample, the terrigenous element titanium (Ti) was selected
for cluster analysis, because variations in its content
correlate most strongly with the series of meteorological
observations in the region (Alexandrin et al. 2018).
b) The vertical prole of the oxygen isotope content
(δ18O) in the ice cores of the Western plateau of Elbrus
(depth – up to 182 m; dated part – from 1774 to 2013; see
(Preunkert et al. 2019; Kutuzov et al. 2019)).
c) Monthly data on average air temperature from
observations at the Teberda (since 1926) and Terskol (since
1951) weather stations.
d) Monthly data on precipitation totals from
observations at the above mentioned meteorological
stations for the same periods.
For data (a) and (b), the vertical proles were converted
to a time distribution based on the known depth–age
Gleb A. Chernyakov, Valeria Vitelli et al. DYNAMICS OF SEASONAL PATTERNS IN GEOCHEMICAL ...
GEOGRAPHY, ENVIRONMENT, SUSTAINABILITY 2020/03
Clustering of annual curves was carried out separately
for the following two periods.
1) 1926–2010 – the maximum period of time provided
simultaneously by geochemical, isotopic and
meteorological data. The Teberda weather station was
selected as providing the longest series of observations in
2) 1951–2010 – the period of observations at the Terskol
weather station and the simultaneous availability of lake
sediment and ice core data. The weather station Terskol
was selected as the closest to Lake Donguz-Orun and Mt.
In accordance with the designations introduced, below
we will indicate the data type with a letter (a – d), and the
study period with a number (1 or 2). For example, (a1) will
denote the titanium data for 1926–2010.
To study the data, we have applied a recently developed
nonparametric method of clustering functional data,
the Bagging Voronoi K-Medoid Alignment, which
simultaneously clusters and aligns by phase the data
elements (annual curves), using the information about the
dependence (sequence) of these curves (Abramowicz et
al. 2017). The method is a generalization of the previous
Bagging Voronoi Clustering (Secchi et al. 2013), which does
not handle misalignment of the data. All computations and
analysis of the data are performed in the R programming
language (R Core Team 2020).
Preprocessing. From the time series representing our
raw data, the associated functional form was reconstructed
via a smoothing procedure. In order to do that, a series of
each parameter (Ti, δ18O, temperature, precipitation) was
divided into sub-series of observations for individual years.
The annual data were centered with respect to their mean
value. Without loss of information the yearly time scale was
converted to a reference one by uniformly distributing the
time instances on the interval [0, 1] (such that for each year
the rst time instance is associated to 0 and the last one to 1).
Next, the centered annual data were normalized with respect
to the maximum absolute value of the whole time series.
Finally, after all previously described normalizations and
transformations, a continuous function was reconstructed
from each annual series by smoothing via a sum of the rst
few Fourier harmonics. Typically, we used from 5 to 9 Fourier
basis functions depending on the stability or oscillations of
the initial data. Thus, a series of annual curves was obtained
for each parameter. This allowed us to apply the BVKMA
algorithm designed for clustering functional data.
Let us set out at a qualitative level the main stages of the
BVKMA algorithm, following (Abramowicz et al. 2017), and
the procedure for tuning its input parameters.
For the sake of clarity, let us describe an input dataset
as a rectangular array of numbers (matrix), organized as
follows1. Each row of the array contains the sequential
values of the parameter of interest, belonging to a particular
year (annual curve). We will call any row of the array and
the data contained in it as a site. For instance, in the case
of the temperature data, the ith row contains the values
of temperature during the ith year of the studied period,
obtained by smoothing of 12 monthly observations at a
meteorological station. Thanks to the Fourier smoothing, for
each type of data we have increased the number of annual
values up to 50. Therefore, each of our datasets contains 50
columns. The number of rows N in our datasets is either 85
or 60, depending on the number of years in the analyzed
time period, starting either from 1926 or from 1951.
The execution of the BVKMA proceeds in two phases –
bootstrap phase and aggregation phase.
Bootstrap phase. This three-step procedure is being
applied to the same input data array a specied number
of times B. The individual replications of this procedure are
independent and their results are being saved.
Step 1. Generation of a random Voronoi tessellation. At this
step the data array is randomly divided into a given number n of
Fig. 1. The study area. The map of the Elbrus region (Central Caucasus) with the marked locations of data sources: Lake
Donguz-Orun, Mt. Elbrus, the weather stations Terskol and Teberda. Space image from Google Earth
1Note that for each parameter (Ti, δ18O, temperature, precipitation), each meteorological station (Teberda, Terskol), and each study
period (1926–2010, 1951–2010) a separate dataset is formed and analyzed independently.
sub-arrays (Voronoi cells). Each cell is a set of several consecutive
sites. The cells can vary in size, since being formed randomly.
This step demands an input parameter L=N/n – the expected
number of sites within a Voronoi cell. Varying L, we change the
measure of supposed dependence of annual data as preliminary
information given to the algorithm.
Step 2. Identication of local representatives (medoids) for each
cell of the tessellation. First, the Alignment procedure is applied
to annual curves (sites) of each cell. Then medoids are chosen as
the curves in each cell which are the most similar to all the other
aligned curves in the same cell. The similarity of the curves is
determined by a metric (see below). As a result of this procedure,
for each Voronoi cell a new 1-dimensional array (an additional
annual curve, also called local representative) is created, which
summarizes the information carried by all sites of the cell. So,
having completed the second step, we have a set of Voronoi
cells and a set of their local representatives – one for each cell.
Step 3. Clustering of the local representatives, formed at the
previous step. The number K of clusters is assigned a priori, before
executing the algorithm, and the clustering algorithm used on
the local representatives is the K-medoid algorithm. All clusters
are being labeled, and the label of each cluster refers also to all
local representatives forming it. Next, all sites of each Voronoi
cell get the same cluster label as the one that was assigned to
the local representative of this cell. Thus, after completion of
this step, the entire initial data array will be divided into clusters.
In other words, for each site it will be indicated which one of K
clusters it belongs to.
Aggregation phase. Since the Bootstrap phase is repeated
B times, and every time a Voronoi tessellation is created
randomly, the resulting cluster distributions are expected to be
dierent. Thus, for each site (year) a frequency distribution of
cluster assignments along the B replicates is provided. At the
Aggregation phase these frequencies are calculated for each
site, and the cluster label, which was encountered more often
than others, is nally assigned to a site. As the result of this
majority vote procedure the nal partitioning of the data array
into clusters is formed.
Parameter selection. The above mentioned variables B, L,
and K are the input parameters of the BVKMA algorithm.
In all runs of the algorithm we kept the number of bootstrap
replications B equal to 1000. This value turned out to be sucient
to provide the robustness of the results.
The expected length of a Voronoi cell L was varied signicantly
in order to encompass all possible numbers of Voronoi cells n. The
inevitable restriction imposed by the algorithm on n is K+1≤n≤N,
expressing the fact that the number of Voronoi cells should be
greater than the number of clusters, but cannot exceed the
number of sites in the dataset. Hence, using the equality L=N/n,
one can easily derive the restrictions on it: 1≤L≤N/(K+1). Thus,
for each number of clusters K we executed the algorithm with
various possible values of the expected length of a Voronoi cell L.
To determine the most adequate value of L we used the average
entropy estimator implemented in BVKMA. The mean entropy
Ē is the measure of the misclassication of the data during
clustering. Therefore, the optimal value of L is the one, providing
the minimum of Ē.
We restricted the number of clusters K to be equal to 2, 3
or 4. We have not enlarged this number because the amount
of sites (years) is relatively small (maximum 85). To tune K, we
applied another built-in estimator of the BVKMA algorithm – the
so-called λ-criterion (for more details, see (Abramowicz et al.
2017; Sangalli et al. 2010)). Again, the optimal value of K is the
one, providing the minimum of λ.
Thus, the way to nd the optimal values of the parameters
for each dataset was the following. First, for each K the optimal
value of L was determined with the help of the entropy criterion.
After that, we found the optimal number of clusters K, using the
λ-criterion, among the cases of optimal values of L.
The optimal values of the parameters and corresponding
values of the statistical indicators, resulting from our analysis, are
presented in Table 1.
In addition to the numeric input parameters discussed
above, for running BVKMA one has to set a metric to quantify
the similarity between annual curves, and a family of warping
functions necessary for the Alignment procedure. Our choice of
these two functional parameters is the same as in (Abramowicz
et al. 2017). Namely, we used the normalized L2-based distance as
a metric, and the group of positive slope ane transformations
as a family of warping functions. More details and denitions can
be found in (Abramowicz et al. 2017; Vantini 2012).
RESULTS AND DISCUSSION
As a result of applying the BVKMA algorithm to dierent
types of data in the dierent studied cases, we have obtained
either 2 or 3 clusters. Typically, the resulting cluster assignment
led to one cluster less than the prescribed number K. This means
that in the nal year-by-year cluster assignment by majority vote
one of the clusters never comes out as the modal one.
The results of applying the algorithm are depicted in
Fig. 2: the cluster distributions over time (left side), and the
corresponding medoids of each cluster (right side). The medoids
represent intra-annual variability of the data, thus the left end
of each medoid corresponds to the beginning of the year, and
the right end – to the end of the year. They may be shifted in
the direction of the abscissa due to the Alignment procedure.
The Alignment is essential in the process of clustering, thought
it has no physical signicance for representation of the
resulting medoids. In fact, the medoid of the cluster is the most
representative curve in the cluster transformed in abscissa as
a result of the Alignment (shifted and stretched/compressed).
Nevertheless, the overall shape of the curve, subjected to such
transformation, is preserved.
Gleb A. Chernyakov, Valeria Vitelli et al. DYNAMICS OF SEASONAL PATTERNS IN GEOCHEMICAL ...
Case K L Ē λ
(a1) 3 20 0.71 0.48
(a2) 4 12 0.68 0.37
(b1) 3 10 0.61 0.93
(b2) 3 15 0.55 0.86
(c1) 4 12 0.61 0.81
(c2) 4 10 0.59 0.81
(d1) 2 15 0.60 0.45
(d2) 3 8 0.58 0.67
Table 1. The cases of optimal parameter selection and their numerical characteristics
GEOGRAPHY, ENVIRONMENT, SUSTAINABILITY 2020/03
Fig. 2. The results of clustering of smoothed annual curves (left), and the corresponding medoids (right) for two study
periods: 1926–2010 (1) and 1951–2010 (2). The clustered data are: titanium content in the bottom sediment core from
Lake Donguz-Orun (a); oxygen-18 isotope content in the ice core from Elbrus (b); monthly average temperature at
Teberda and Terskol weather stations (c); monthly sum of precipitation at the same weather stations (d). The colors of
the medoids match the colors used in the diagrams of cluster distribution over time for each type of data
The data on titanium concentrations in the Donguz-
Orun sediment core show the most prominent inter-
cluster diversity (Fig. 2 (a1, a2, right)). In the original
study by Alexandrin et al. (2018) titanium was related to
precipitation, having signicant correlation (r = 0.44) with
annual precipitation measured at Teberda weather station.
However, clustering fullled for the two parameters
showed dierent results (Fig. 2 (a1, a2, d1, d2, left)). It
might be related to mild correlation strength, but also to
dierent factors driving intra-annual patterns of variability
of two parameters. Titanium is sometimes claimed to
mimic terrigenous runo, and thus reecting precipitation.
However, precipitation during the cold season may
generate runo only in spring during snow melt, hence
the intra-winter distribution of precipitation would not be
related to the spring peak of runo, but the total amount of
precipitation in the winter will matter. Hence, we underline
that the results of the BVKMA clustering algorithm
should be always interpreted keeping in mind possible
disagreement in intra-annual variability of interconnected
On the contrary, the temperature and especially
oxygen-18 records reveal similar and stable seasonal
patterns (Fig. 2 (b1, b2, c1, c2, right)), and therefore the
dierences among clusters are less signicant for these
types of data. Temperature is known to have larger
correlation distance than precipitation. In this regard, we
cannot nd a realistic explanation for dierent clustering
results for temperature measured on two weather stations
having similar results for precipitation. Hence we interpret
these results as follows. The instrumental period may be not
long enough to obtain reasonable clusters for a parameter
with stable intra-annual variability, such as temperature.
Moreover, some inter-cluster time boundaries occur closely
in timing for some series of data (Fig. 2): in the 1940s (b1,
c1, left) and in the late 1960s (b2, c2, left).
We also nd very close inter-cluster boundaries in the
titanium and oxygen-18 data (Fig. 2 (a1, b2, left)).
The precipitation data series of meteorological stations
Teberda and Terskol have a similar structure of cluster
distributions over time: a large cluster encompassing
most part of the studied period followed by a small cluster
attributed to the latter period (Fig. 2 (d1, d2, left)).
Intra-annual patterns revealed by the respective
medoids (Fig. 2 (d1, d2, right)) also have a similarity: for
both weather stations we can observe maximum values
of precipitation in the middle of the season (summer)
for the rst period (green) and two local maxima (spring
and autumn) with reduction in summer for the second
period (magenta). These results show that, in contrast to
very stable intra-annual variations of temperature, those
of precipitation are variable enough to be reasonably
clustered into several periods. The consistency of these
periods for two remote weather stations may indicate that
the results of the clustering catch common underlying
forcing of changed precipitation seasonality in 2000s.
Originally, the BVKMA algorithm was applied for a 6000-
year long varved sedimentary sequence (Abramowicz
et al. 2017). It had proven to be suitable for registering
centennial to millennial scale variations in the distribution
of the seasonal values of the selected parameters, thus
providing important paleoclimatic implications. In this
study, we apply the BVKMA algorithm for signicantly
shorter sequences (60 and 85 years long). The climatic
variations (temperature and precipitation) as well are their
proxies (sedimentary Ti-values and ice core δ18O) at such a
short time scale were obviously incomparably smaller than
those for the half Holocene time span.
The two-three clusters provided by the algorithm tend
to represent minor uctuations – especially clear with the
curves of temperature and δ18O. A certain incoherence
of the Ti-values can be attributed to the uncertainty of
distinguishing the annual layers in varved sediments (done
with the use of geochemical markers rather than direct
visual observation in the case of Lake Donguz-Orun).
On the time scale of centuries to millennia the physical
basis for cluster analysis of the paleoclimatic data is
much more robust. Application of the BVKMA algorithm
for shorter sequences provides a necessary basis for its
application for the longer ones that are expected for lake
sediments, ice core data and possibly other sources of
paleoclimatic information in the Caucasus.
The seasonal patterns of four types of proxy and
meteorological data series from the Elbrus region (Ti
concentrations, δ18O, temperature, and precipitation) are
derived by applying the clustering algorithm Bagging
Voronoi K-Medoid Alignment, separately for two periods:
1926–2010 and 1951–2010.
The time dynamics of clusters and the corresponding
cluster medoids are obtained.
The seasonal patterns of oxygen-18 and temperature
data occurred to be relatively similar and unchangeable.
A notable degree of consistency of the resulting
clusters and their time boundaries for oxygen-18 and
temperature data is gured out. This result is encouraging
for future attempts to dene modern analogues of past
climates using this clustering technique.
A less pronounced, but still the observable consistency
of the cluster distributions is found for Ti and O-18 data,
as well as for precipitation data of meteorological stations
Teberda and Terskol.
Our results demonstrate that for parameters with
relatively stable intra-annual patterns of variability (like
temperature) the usual length of instrumental period may
be not enough for its reasonable division into sub-periods.
Highly variable parameters (like precipitation), on the
contrary, may be reasonably clustered even inside several
decades of data.
Also, we underline that the results of the BVKMA
clustering algorithm should be always interpreted keeping
in mind possible disagreement in intra-annual variability
of interconnected parameters, as we demonstrated in
titanium data from Donguz-Orun sediment core and
precipitation from Teberda weather station.
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