the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Grain-size distribution unmixing using the R package EMMAgeo

### Michael Dietze

The analysis of grain-size distributions has a long tradition in Quaternary Science and disciplines studying Earth surface and subsurface deposits. The decomposition of multi-modal grain-size distributions into inherent subpopulations, commonly termed end-member modelling analysis (EMMA), is increasingly recognised as a tool to infer the underlying sediment sources, transport and (post-)depositional processes. Most of the existing deterministic EMMA approaches are only able to deliver one out of many possible solutions, thereby shortcutting uncertainty in model parameters. Here, we provide user-friendly computational protocols that support deterministic as well as robust (i.e. explicitly accounting for incomplete knowledge about input parameters in a probabilistic approach) EMMA, in the free and open software framework of R.

In addition, and going beyond previous validation tests, we compare the
performance of available grain-size EMMA algorithms using four real-world
sediment types, covering a wide range of grain-size distribution shapes
(alluvial fan, dune, loess and floodplain deposits). These were randomly
mixed in the lab to produce a synthetic data set. Across all algorithms, the
original data set was modelled with mean *R*^{2} values of 0.868 to 0.995
and mean absolute deviation (MAD) values of 0.06 % vol to 0.34 % vol. The original
grain-size distribution shapes were modelled as end-member loadings with
mean *R*^{2} values of 0.89 to 0.99 and MAD of 0.04 % vol to 0.17 % vol. End-member scores reproduced the original mixing ratios in the
synthetic data set with mean *R*^{2} values of 0.68 to 0.93 and MAD
of 0.1 % vol to 1.6 % vol. Depending on the validation criteria, all models
provided reliable estimates of the input data, and each of the models
exhibits individual strengths and weaknesses. Only robust EMMA allowed uncertainties of the end-members to
be objectively estimated and expert knowledge to be included in the end-member definition. Yet, end-member interpretation should
carefully consider the geological and sedimentological meaningfulness in
terms of sediment sources, transport and deposition as well as
post-depositional alteration of grain sizes. EMMA might also be powerful in
other geoscientific contexts where the goal is to unmix sources and
processes from compositional data sets.

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## 1.1 Mixing of grain-size subpopulations in sedimentary deposits

Many studies in Quaternary Science aim to reconstruct past Earth surface dynamics using sedimentary proxies. Earth surface dynamics include a variety of processes that mix process-related components (Buccianti et al., 2006). Sediment from different sources can be transported and deposited by a multitude of sedimentological processes that have been linked to climate, vegetation, geological and geomorphological dynamics (Bartholdy et al., 2007; Folk and Ward, 1957; Macumber et al., 2018; Stuut et al., 2002; Tjallingii et al., 2008; Vandenberghe, 2013; Vandenberghe et al., 2004, 2018). During transport, grain-size subpopulations are affected by different transport energies, and, thus, distinct grain-size distributions are created upon deposition. Accordingly, it is possible to infer source areas, transport pathways and transport processes as well as the related sedimentary environment from measured grain-size distributions. This basic concept has been exploited for more than 60 years (Flemming, 2007; Folk and Ward, 1957; Hartmann, 2007; Visher, 1969). However, the approach is limited when sediments are transported by more than one process and become mixed during and after deposition (Bagnold and Barndorff-Nielsen, 1980; Vandenberghe et al., 2018).

The advent of fast, high-resolution grain-size measurements through
laser diffraction allows the assessment of grain-size distributions of large
sample sets in a short time and reveals the sediment mixing effects in
multiple modes or distinct shoulders in the grain-size distribution curves.
Although widely used, classic measures of bulk distributions such as sand,
silt and clay contents or mean grain size, *D*_{50}, sorting, skewness or
kurtosis (Folk and Ward, 1957) are non-informative in non-Gaussian,
multi-modal distributions and allow only a qualitative interpretation and
comparison of sedimentary processes that contributed to sediment formation.

To overcome these limitations and to improve process interpretation and attribution of associated drivers from sedimentary archives (Dietze et al., 2014), two ways have been proposed to decompose multi-modal grain-size distributions and to quantify the dominant grain-size subpopulation: parametric and non-parametric approaches. Among the former, commonly used curve fitting approaches describe a sediment sample as a combination of a finite number of parametric distribution functions such as (skewed) log-normal, log-hyperbolic or Weibull distributions (Bagnold and Barndorff-Nielsen, 1980; Gan and Scholz, 2017; Sun et al., 2002). However, curve fitting solutions are non-unique, and subpopulations might not be detected if a fixed number of functions are fitted to individual samples (Paterson and Heslop, 2015; Weltje and Prins, 2003), whereas other parametric approaches such as non- and semi-parametric mixture models (Hunter et al., 2011; Lindsay and Lesperance, 1995) are still very poorly explored in the field of grain-size distribution analyses.

Non-parametric end-member modelling or mixing analysis (EMMA) aims to
describe a whole data set as a combination of discrete subpopulations, based
on eigenspace analysis and compositional data constraints (Aitchison,
1986). A multidimensional grain-size data set **X** (i.e. *m* samples, each
represented by *n* grain-size classes) can be described as a linear combination
of transposed end-member loadings (**V**^{T}, representing individual
grain-size subpopulations), end-member scores (**M**, the relative contribution
of the end-member subpopulations to each sample) and an error matrix
**E** using
the function

Hence, it is possible to identify (using end-member loadings) and quantify (using end-member scores) sediment sources, transport and depositional regimes from mixed grain-size data sets. EMMA has been successfully applied to interpret and quantify past sedimentary processes from sediment deposits, beyond classical measures, for example in marine, lacustrine, aeolian, fluvial, alluvial and periglacial environments, across multiple spatial and temporal scales (Borchers et al., 2015; Dietze et al., 2013, 2016; Schillereff et al., 2016; Strauss et al., 2012; Toonen et al., 2015; Varga et al., 2019; Vriend and Prins, 2005; Wündsch et al., 2016).

## 1.2 Non-parametric grain-size unmixing approaches

Five approaches of non-parametric EMMA have been proposed: Weltje (1997) has developed a FORTRAN algorithm based on simplex expansion, which has been translated to a set of scripts for R (R Core Team, 2017) called RECA (R-based Endmember Composition Algorithm), including a fuzzy c-means clustering approach (Seidel and Hlawitschka, 2015). Available as MATLAB scripts, the algorithm by Dietze et al. (2012) has included eigenvector rotation, whereas Yu et al. (2015) have introduced a Bayesian EMMA (BEMMA) and Paterson and Heslop (2015) have used approaches from hyperspectral image processing (AnalySize). Based on the MATLAB algorithm by Dietze et al. (2012), Dietze and Dietze (2016) compiled a prototype R package (EMMAgeo v. 0.9.4).

Most EMMA approaches are deterministic (i.e. one single model solution
without any uncertainty estimates) and require the user to set a fixed
number of end-members *q* and further model parameters. In natural systems,
however, *q* is rarely known and, thus, often one of the reasons to employ EMMA.
Different approaches have been proposed to estimate *q*, such as the inflection
point in a *q*–*R*^{2} plot (Paterson and Heslop, 2015; Prins and
Weltje, 1999) or the iterative adjustment of model parameters such as the weight
transformation limit (Dietze et al., 2012), maximum convexity
error, number of iterations and weighting exponent (Weltje, 1997; Seidel and
Hlawitschka, 2015).

Previous studies of EMMA performance (Weltje and Prins, 2007; Seidel and Hlawitschka, 2015; Paterson and Heslop, 2015) either used measured data without information on the true loadings and scores or were based on ideally designed synthetic data. However, natural process end-members can overlap substantially and may have a varying or multi-modal grain-size distribution shape due to unstable transport conditions (e.g. gradual fining of aeolian dust with transport distance) and deposition (e.g. reworking by soil formation; Dietze et al., 2016; Vandenberghe et al., 2018).

Recently, van Hateren et al. (2018) compared the concepts and performances of AnalySize, RECA, BEMMA, EMMAgeo and a diffuse reflectance spectroscopy (DRS) unmixing approach (Heslop et al., 2007). They used numerically mixed real-world grain-size samples and compared the modelled end-member loadings with the real-world distributions and modelled scores with randomised mixing ratios, as suggested by Schulte et al. (2014). Van Hateren and others confirmed former studies and highlighted that geological background knowledge is crucial for end-member interpretation, but they also pointed to strong differences in model performance. However, the descriptions of van Hateren et al. (2018) are mainly based on verbal comparisons of graphic data representations, and the validation data are not available for future comparative studies.

Here, we introduce new operational modes and protocols for the comprehensive open-source R package EMMAgeo as a tool for quantifying process-related grain-size subpopulations in mixed sediments. We aim to clarify information provided by the reference documentation of the first version of the package (v. 0.9.4; Dietze and Dietze, 2016) and by Dietze et al. (2014), regarding parameter estimation and optimisation, and we add a new approach of uncertainty estimation of the end-member scores. We evaluate the performance and validity of EMMAgeo using a real-world grain-size data set with fully known end-member compositions and unbiased quantitative measures. For comparison, the same data set is modelled with other available grain-size end-member algorithms. An evaluation and validation of both process end-member distribution shapes and mixing ratios are provided. Finally, general constraints for the interpretation of end-members are discussed. The detailed Supplement shall help users to apply the EMMAgeo protocols and to reproduce the results, making use of the raw data published along with this study.

## 2.1 Background

EMMAgeo in its current version 0.9.6 (Dietze and Dietze, 2019) contains 22 functions (Table S1 in the Supplement), the example data set for this study and full documentation of these items. EMMAgeo provides a systematic chain of data pre-processing, parameter estimation and optimisation, the actual modelling and the inference of model uncertainties.

EMMAgeo is based on the EMMA MATLAB code by Dietze et al. (2012), which was slightly modified, i.e. vectorisation of looped
calculations to increase computation speed, a new coding of the scaling
procedure (Miesch, 1976) and additional measures of model
performance. Following Dietze et al. (2012), the core function
`EMMA()`

rescales the grain-size data matrix **X**
to constant row sums (i.e. *m* rows
of samples, *n* columns of grain-size classes). Then, a weight transformation
(Klovan and Imbrie, 1971) is performed using a weight constant *l* to
yield a weight matrix **W** that is not biased by variables with large standard
deviations (Weltje, 1997). The similarity matrix **A** is returned as the
minor product of **W**. From the similarity matrix, the eigenspace is
computed,
and eigenvalues (**L**) and their cumulative sums are calculated. Eigenvectors
are inferred and sorted by decreasing explained variance (**V**_{f}). These
eigenvectors are then rotated, by default using the Varimax rotation
(Dietze et al., 2012), in R using the package GPArotation
(Bernaards and Jennrich, 2005). Their order is inverted to yield
unscaled end-member loadings (**V**_{q}). Then, normalised end-member loadings
(**V**_{qn}) are computed by row-wise data normalisation of **V**_{q} or
are user-defined; i.e. a known **V**_{qn} can be used. A factor score matrix
(**M**_{q}) is calculated as a non-negative least squares estimate of **V**_{qn} and
**W**. Then, the data set can be described as a minor product of **M**_{q} and **V**^{T}
to yield the modelled weight matrix **W**_{m}. These values are
back-transformed and yield rescaled end-member loadings (**V**_{qsn}) and
quantitative scores (**M**_{qs}). A linear combination of **M**_{qs} and **V**^{T}
yields **X**^{′}, the modelled data set (see the mathematical formulation in Dietze
et al., 2012). Model evaluation measures are calculated by comparing **X** and
**X**^{′}: row-wise (sample) and column-wise (grain-size class) absolute model
deviation, data variance and root mean square errors (MAD_{m} and
MAD_{n}, ${R}_{m}^{\mathrm{2}}$ and ${R}_{n}^{\mathrm{2}}$ and
RMSE_{m} and RMSE_{n}), explained variance of each end-member
(*M*_{qsvar}) and total mean MAD_{t} and ${R}_{\mathrm{t}}^{\mathrm{2}}$ of the
model, as well as the number of overlapping end-member loadings (ol), defined as
one end-member having its main mode within the area of another end-member.

## 2.2 Theory of operational modes and protocols

A deterministic and a robust operational EMMA mode can be run by a function
and two protocols, respectively. First, EMMA can be performed with a
user-based decision on all parameters, which is comparable to existing
algorithms. This *deterministic EMMA* is mainly useful for exploratory studies, such as
investigating the effect of different numbers of end-members *q*, weight
transformation limits *l* or factor rotation types. The function call
`$\text{EM\_det}\mathrm{<}\mathrm{-}\text{EMMA}\mathrm{(}X\mathrm{=}X$, `

with
*q*=4, plot = TRUE),`X`

being the data set and `q`

the number of suggested end-members, returns the
final end-member loadings and scores, the modelled data set and several
quality criteria. Additional function arguments can be provided, such as `l`

,
other factor rotation methods, predefined unscaled end-member loadings,
the grain-size class limits of the input data set or a series of plot
arguments in standard R language.

The second and third protocol of *robust EMMA* account for the real mixing conditions
being generally unknown. In such cases, it is reasonable to evaluate
different model realisations within meaningful parameter ranges; i.e. `q`

and `l`

can
be varied to infer the range of uncertainty associated with the set of model
scenarios in a probabilistic framework. The central goal of robust EMMA is
to set the boundary conditions for these two parameters `q`

and `l`

, which allows
all resulting scenarios to be modelled to identify emergent robust end-members
and to describe these by statistical measures. There are two ways to run
robust EMMA (Fig. 1). An extended protocol is suitable for more exploratory
studies, in which parameter settings can be explored and manipulated in detail
(Fig. 1a). A compact protocol allows calculation of all important input and
output parameters in five steps (Fig. 1b). Both protocols follow the same
workflow but with different requirements and possibilities to interact.

The *extended protocol of robust EMMA* (Fig. 1a) starts with defining *l*_{min}, a lower limit for `l`

(step 1).
By default, *l*_{min} is set to zero (according to Miesch, 1976). The upper
limit *l*_{max} (step 2) represents the maximum possible value that still
allows eigenspace calculation and is either found by testing the possibility
of eigenspace computation for a sequence of possible `l`

values (function
`test.l()`

) or by iteratively approximating the highest possible `l`

value
(`function test.l.max()`

). When `l`

approaches *l*_{max}, EMMA generates
increasingly unreliable output (e.g. negative loadings), which is why
*l*_{max} should be set to a lower value, for example, the default value
95 % of *l*_{max}. Based on *l*_{min} and *l*_{max}, a sequence of likely
values for `l`

is created (step 3). The number of these values (here *n*=20)
should balance sufficient `q`

scenarios and reasonable computational time.

The range of the number of end-members *q* is then modelled for each element of
this sequence of `l`

. Step 4 sets *q*_{min} by testing how much of the data
variance can be explained with a given *q* prior to eigenspace rotation. Dietze et
al. (2012) suggested a minimum *R*^{2} of 0.95. A reasonable
estimate of the highest meaningful value *q*_{max} (step 5) can be the
first local maximum of total mean ${R}_{\mathrm{t}}^{\mathrm{2}}$ after all
end-members were modelled. In EMMA (Dietze et al. 2012),
${R}_{\mathrm{t}}^{\mathrm{2}}$ rises as more end-members are included until, after
a local maximum, it drops again, which is related to the forced constant-sum
constraints (Paterson and Heslop, 2015). Alternative criteria can be a fixed
upper threshold of ${R}_{\mathrm{t}}^{\mathrm{2}}$ or a fixed user-defined value for
*q*_{max} (step 5). Note that this approach differs from the way that other
models identify *q*. While Weltje (1997) and Prins and Weltje
(1999) use the inflection point of the *q*–*R*^{2} relationship to
set one fixed *q*, robust EMMA provides a range of ** q** that include this inflection
point. In step 6, the ranges of

`q`

and `l`

are combined to a parameter matrix `P`

,
used to model all likely end-member scenarios. In `P`

,

*q*_{min}must be at least 2,

*q*_{max}must be at least as high as the corresponding

*q*_{min}and there must be no NA (not available) values (see Supplement).

End-member loadings from different model parameter settings tend to cluster
at similar main mode positions, which Dietze et al. (2012, 2014) used to
manually identify robust end-members. To identify these mode clusters within
EMMAgeo (step 7), `EMMA()`

is evaluated for each combination of ** q** between

*q*

_{min}and

*q*

_{max}for each element of

**. Step 8 generates the limits around the mode clusters of the robust end-member loadings. The limits are a two-column matrix with the lower and upper limit class for each robust end-member. With these class limits, all robust end-member loadings can be extracted (step 9), and their class-wise means and standard deviations are calculated.**

*l*With the mean robust loadings, i.e. the unweighted mean of all similarly
likely loadings of step 9, it is possible to optimise the model with respect
to different quality criteria by changing `l`

to yield an optimal EM solution
(*l*_{opt}, step 10). The default quality criterion is
${R}_{\mathrm{t}}^{\mathrm{2}}$. Other possible criteria are thresholds in mean
sample- and class-wise *R*^{2} and ** E** and the number of
overlapping end-members. With the uncertainty ranges of robust loadings and

*l*

_{opt}(or any other user-defined

*l*values), it is possible to quantify the uncertainties of end-member scores using Monte Carlo simulations (step 11). The simulations generate many sets of unscaled end-member loadings, by default 100 times

`q`

. `EMMA()`

is performed with each subset of
loadings, and the scores are extracted. Their overall scatter is described by
the sample-wise standard deviation. The Monte Carlo approach cannot
propagate a specific **to the scores calculation because loadings are randomly sampled with no information about the initial**

*l**l*with which they have been created. Hence, the Monte Carlo approach only delivers an estimate of the standard deviation of the scores (default) or asymmetric confidence intervals, whereas the mean derives from the optimal EM model.

The *compact protocol of robust EMMA* (Fig. 1b) combines steps of the extended protocol and automates the
identification of plausible grain-size class limits for robust end-member
extraction. After data input checks, the ranges of ** l** (step 1) and

**(step 2) are determined. These boundary conditions are used to evaluate multiple EMMA scenarios (step 3). Cluster limits can be identified by a kernel density estimate for all available grain-size modes (step 4) that are used to define robust end-members. Kernel density estimates are curves that depict the continuous empiric distribution of data, in this case grain-size mode classes, by sliding a window (the kernel) over the data and counting the number of values within it for each sliding step. The window size (or kernel bandwidth) is the parameter controlling the shape of the resulting curve. Here, a default kernel bandwidth of 1 % of the number of grain-size classes of the input data set is used. To identify mode cluster limits, the density curve needs to be cut off at a given threshold. Above that threshold, the limits bracketing the modes can be derived. By default, the cut-off threshold is defined as the 0.7 quantile of the density values. These empirical default values were found to be appropriate during extensive tests with synthetic data sets during package development. However, they are not universal and may be adjusted when needed. With all modelled end-member loadings (from step 3) and the class limits (from step 4), the robust end-member loadings and scores can be extracted (step 5; Fig. 1).**

*q*## 3.1 Example data set

Sediment outcrops of four depositional environments were sampled near the
city of Dresden, Germany (Fig. 2). These represent natural sedimentological
end-members (EM_{nat}) of a floodplain section (EM_{nat}1, with main
grain-size mode at 19 µm) of an Elbe River tributary, a loess deposit
(EM_{nat}2, mode at 36 µm) of the Ostrau profile described by
Meszner et al. (2013), a sandur dune (EM_{nat}3, mode at 330 µm)
and a Weichselian alluvial fan (EM_{nat}4, mode at 406 µm; Fig. 2).
These natural environments were selected to cover a broad range
of transport regimes, grain-size distribution shapes, degrees of mode
overlapping (EM_{nat}4 and EM_{nat}3) and number of modes (EM_{nat}1).
Note that these samples are potentially composed of multiple grain-size
populations themselves and are far from “ideal” for synthetic unmixing. We
decisively chose this approach not only to compare the performance of
different EMMA methods but also to explore drawbacks in the modelling
procedure under such conditions.

Three parallel samples (0.3–2.0 g) per outcrop were chemically treated with
10 % NaCl, 15 % H_{2}O_{2} and 1.25 mL Na_{4}P_{2}O_{7}, each
for 48 h, and measured with a Beckman Coulter LS 13 320 Laser
Diffraction Particle Size Analyzer at RWTH Aachen, delivering 116 classes
(0.04–2000 µm). Between 7 and 16 aliquots per sample were investigated
in triplicates. Grain-size distributions were derived applying the Mie
theory with the following parameters: fluid refraction index: 1.33; sample refraction index: 1.55;
imaginary refraction index: 0.1. The resulting median grain-size distributions (Fig. 2)
were manually mixed in the lab to generate 100 samples with randomly
assigned individual contributions. Within this data set **X**_{nat}, 50 samples
contained all four EM_{nat}, 25 were prepared without EM_{nat}4 material
and a further 25 were prepared without EM_{nat}1. Hence, in contrast to
other studies, we fully know the grain-size distributions of the underlying
natural process end-members and their mixing ratios, which allows a detailed
evaluation of performance and comparison of all available EMMA algorithms.

## 3.2 Model performance of different EM analyses

The example data set **X**_{nat} was decomposed with deterministic EMMA of
EMMAgeo using `q=4`

and `l=0`

. Robust EMMA was run with both the extended and
the compact protocol. To be as conservative and as unbiased as possible, both
protocols were executed with the default parameterisations as suggested
above and were only modified when results obviously prompted manual parameter
adjustments.

To run the FORTRAN-based approach by Weltje (1997), provided by Jan-Berendt Stuut
(personal communication, 2017), the grain-size classes of **X**_{nat} needed to be
aggregated; i.e. the resolution decreased by a factor of 2. For consistent comparisons
with the other approaches, the resulting end-member loadings were
interpolated back to the initial grain-size resolutions (see Supplement). The down-sampling and subsequent up-sampling of all EM_{nat}
values resulted in negligible artefacts with an average *R*^{2}>0.999. The modelled data set **X**^{′} was computed by combining
loadings and scores according to Eq. (1), excluding the error matrix **E**.

Running the collection of the five RECA R scripts (Seidel and
Hlawitschka, 2015) required manual installation of the additional package
compositions (Van den Boogaart et al., 2014), e1071
(Meyer et al., 2017) and nnls (Mullen and van Stokkum,
2012), loading all scripts and manual screen input of the model parameters.
RECA needs to be run completely to the end until consequences of parameter
changes can be inspected. The decision on *q* is based on a
*q*–${R}_{n}^{\mathrm{2}}$ plot (e.g. using the inflection point). Here, RECA
was run with four end-members, a maximum convexity error of −6, confirmation
of the first start model, 100 iterations and a weighting exponent of 1, as
suggested by Seidel and Hlawitschka (2015).

AnalySize by Paterson and Heslop (2015) provides a MATLAB GUI, in
which *q* is set manually. The numeric MATLAB output, end-member loadings and
scores, was imported to R using the package R.matlab (Bengtsson,
2018).

Bayesian EMMA (BEMMA) in MATLAB (Yu et al., 2015) does not allow
a predefined *q* to be specified. With repeated model runs, the number of output
end-members changed unsystematically between two and four. Depending on *q*, the
shape and mode positions of the unmixed distributions fluctuated, which
prevented the output from different model runs from being grouped. Hence, we did not
include BEMMA in this comparison.

## 3.3 Evaluation and validation criteria

The performance of all approaches was evaluated in two steps. First, we
compared the original data set **X**_{nat} and the modelled data sets **X**^{′} using
(i) coefficients of determination (mean total ${R}_{\mathrm{t}}^{\mathrm{2}}$,
sample-wise ${R}_{m}^{\mathrm{2}}$, class-wise ${R}_{n}^{\mathrm{2}}$)
and (ii) the absolute differences between **X**_{nat} and **X**^{′} (total MAD_{t},
sample-wise MAD_{m}, class-wise MAD_{n}). This comparison resembles the
classical evaluation step when the true natural end-member composition is
unknown.

Second, knowing which natural end-members have been mixed to create the
example data set **X**_{nat}, we compare (i) *R*^{2} and MAD for
EM_{nat} distributions and loadings (${R}_{n\mathrm{1}}^{\mathrm{2}}$ to
${R}_{n\mathrm{4}}^{\mathrm{2}}$ and MAD_{n1} to MAD_{n4}), (ii) *R*^{2} and MAD for mixing ratios and scores
(${R}_{m\mathrm{1}}^{\mathrm{2}}$ to ${R}_{m\mathrm{4}}^{\mathrm{2}}$ and MAD_{m1} to
MAD_{m4}) and (iii) the absolute deviations of the mode positions of
EM_{nat} distributions and loadings. For comparisons of
EM_{nat} distributions with modelled loadings, all results were truncated
to grain-size classes of EM_{nat} higher than 0.1 % vol and rescaled
to 100 %. There are two reasons for this: first, due to the generally
narrow grain-size distributions, EM_{nat} contained many grain-size
classes of only zeros, which biases the resulting measures
(Ciemer et al., 2018). This bias is severe: correlating, for
example, two sequences of random values (e.g. 3.1, 5.2, 4.0 and 9.2, 8.3,
3.5) typically yields a very low correlation coefficient (e.g. $r=-\mathrm{0.065}$). However, padding these sequences with zeros strongly increases
the correlation coefficient (e.g. *r*=0.87, including five zeros). Second,
it is known that EMMA (Dietze et al., 2012), but also other
approaches, causes spurious secondary modes directly below the mode positions
of other end-members (Paterson and Heslop, 2015). The spurious modes
are obviously not related to the underlying sedimentation regime and are not
intended to be interpreted genetically (Dietze et al.,
2014). As they would also bias the model comparison measures, we excluded
them from model evaluation.

## 4.1 Evaluation of model performance

### 4.1.1 Deterministic EMMA

Figure 3 shows the default graphical output after the EMMA algorithm has
modelled the data set with four end-members. Panels a and b depict
*R*^{2} values (squared Pearson correlation coefficients) organised
by grain-size class and sample. Overall, the data set was reproduced with
a mean ${R}_{\mathrm{t}}^{\mathrm{2}}$ of 0.969 and MAD_{t} of 0.2 % vol (Table 1).
Panels c and d show modelled end-member loadings and scores.
End-member loadings (EM1-4) had modes at 16, 40, 310 and 450 µm.
Spurious secondary modes of less than 2.5 % vol are visible below
primary modes of other end-members. Apart from the multi-modal EM4, all
end-members have a log-normal shape. Figure 3a shows
that grain-size class-wise ${R}_{n}^{\mathrm{2}}$ decreases between 946 and
1830 and 117 and 177 µm, both grain-size class intervals
that contribute less than 0.9 % vol to **X** (Fig. 2). Mean sample- and
class-wise absolute deviations are shown in Table 1.

The scores of EM1 to EM4 accounted for 20 %, 20 %, 31 % and 29 % of the
variance of **X**^{′}, respectively. Sample-wise ${R}_{m}^{\mathrm{2}}$ is 0.98 on
average (Table 1). Four outliers had ${R}_{m}^{\mathrm{2}}<\mathrm{0.95}$
(samples 16, 57, 64, 75; Fig. 3b). However, neither removing these
samples nor changing the rotation type from Varimax to Quartimax or the oblique rotation Promax improved the modelling of loadings
and scores (not shown).

### 4.1.2 Robust EMMA – extended and compact protocol

In the extended protocol, an *l*_{min} of zero was used according to Miesch (1976),
and *l*_{max} was set to 0.37, i.e. 95 % of the modelled absolute
*l*_{max} of 0.39 (see methods in Sect. 3). However, when using this value, negative
loadings occurred. Therefore, the value was set to 80 %, yielding a more
realistic and valid *l*_{max} of 0.31. With 20 values between
*l*_{min} and *l*_{max}, *q*_{min} varied between 2 and 3
(Fig. 4a),
and *q*_{max} showed a trend of decreasing ${\mathit{R}}_{\mathrm{t}}^{\mathrm{2}}$ with
increasing *l* (Fig. 4c, d), until even NA values were produced for some parameter
combinations (blue graph, Fig. 4b). Accordingly, after a local optimum at
*q*_{max} between 4 and 9 (open circles, Fig. 4b), adding more end-members
leads to numerical instabilities.

Figure 5a shows all 223 end-member loadings from 96 EMMA runs that agree
with the parameter space of `P`

. Note that the protocol can be run with user-defined grain-size units (function argument classunits) or simply the raw
grain-size class numbers (default, and used in the following sections for
simplicity). Several end-members have main mode position clusters between
grain-size class numbers 63 and 66, 74 and 77, 94 and 97 and 99 and 102 (orange
polygons). These class limits were used to model the robust end-member
loadings, excluding the negative loadings that were modelled due to *l*_{max} values
that are too high. A fifth cluster at classes 71–73 exists (not marked), although
the distribution of this end-member is rather broad and overlaps with the
distribution shapes of the two other clusters in this range. It was rejected
as a robust end-member (see below).

The resulting robust EM3 and EM4 loadings show high class-wise standard
deviations (SDs) around the mode positions (Fig. 6a). EM1 has a continuously
narrow uncertainty envelope (i.e. mean±1 SD), and EM2 shows the
largest and most variable envelope. Mean class-wise SDs range from 0.06 (EM4)
to 0.38 % vol (EM2). The main mode positions of the robust loadings are
identical with those of deterministic EMMA; only the EM2 mode is one class
off. Using the mean robust loadings, *l*_{opt} was 0.0163 when maximising
${R}_{\mathrm{t}}^{\mathrm{2}}$. Based on this, mean robust scores were modelled
(Fig. 6b) with an average SD of 9.9 % vol, 7.8 % vol, 11.3 % vol and 9.5 % vol for EM1 to
EM4.

With the compact protocol, the same parameter space (*l*_{min}, *l*_{max},
*q*_{min} and *q*_{max}) was calculated as with the extended protocol. Robust
end-member definition is supported by the plot output of the function
`robust.EM()`

. Figure 5b shows five clusters with mode positions at 13–16,
27–33, 38–47, 250–320 and 390–500 µm (i.e. class numbers 63–66,
71–74, 75–77, 95–97 and 100–102). The colour scheme reveals that the
cluster at 27–33 µm (grey bar, Fig. 5b) systematically occurs when
EMMA was run with three end-members (red curves, Fig. 5a). Clusters at 13–20
and 38–47 µm emerge especially when four end-members were included in
a model run (green curves). A similar case exists for the two coarse
end-members, at 250–320 and 390–500 µm. Hence, models with a value
for `q`

that is too small systematically merge distinct grain-size distributions
into spurious, broad curves. Values for `q`

that are too high instead caused
spikey loadings (blueish, pink curves, Fig. 5b).

Defining the limits by the automatic kernel density estimate approach
suggested only three out of four natural end-members as robust ones,
combining all loadings around class 100 (Fig. 5b, black line). Setting the
kernel bandwidth arbitrarily to 0.5 would allow separation of the two
overlapping modes around EM_{nat}2 while missing EM_{nat}1 and
misinterpreting the cluster around the two coarsest end-members (not shown).
Thus, for strongly overlapping mode clusters, automatic class limit
detection was not appropriate. Hence, we set the mode limits similar to the
extended protocol to class numbers 63–66, 75–77, 94–97 and 99–102,
changing the definition of EM2 by just one grain-size class (extended
protocol: 74–77) to better exclude the cluster at 27–33 µm (red
curves, Fig. 5b) and to assess varying robust end-member definitions.

The resulting end-members are shown in Fig. 6b. They are similar to the plotted output of the deterministic version (Fig. 3) but extended by uncertainty polygons, the different representation of scores and slightly different mode positions, grain-size class-wise ${R}_{n}^{\mathrm{2}}$ (0.93) and sample-wise ${R}_{m}^{\mathrm{2}}$ (0.98). Mean end-member contributions to the variance of the data set (20 %, 19 %, 29 % and 32 %) are almost identical to the deterministic version.

### 4.1.3 Comparison to other end-member unmixing algorithms

The full benchmark reveals that all approaches successfully model the data sets. The output of RECA shows difficulties in reaching the minimum convexity error of −6 with the initial 100 iterations, but by increasing the value to 200 iterations the issue was solved.

The average ${R}_{\mathrm{t}}^{\mathrm{2}}$ values were higher than 0.868 in all
cases, up to 0.995 (Table 1). Sample-wise ${R}_{m}^{\mathrm{2}}$ values were
always higher than the grain-size class-wise ${R}_{n}^{\mathrm{2}}$ values.
Deterministic EMMA yielded slightly better results than the two robust EMMA
protocols, which in turn were very similar. The lowest (highest) and highest
(lowest) ${R}_{\mathrm{t}}^{\mathrm{2}}$ (MAD_{t}) values are related to RECA and
AnalySize, respectively.

The main absolute deviations of **X**^{′} from **X**_{nat} are associated with
grain-size classes between 100 and 1000 µm, regardless of the model
(Fig. 7). Except for AnalySize, all approaches show systematic
underestimation of these grain-size classes of up to −2.5 % vol per
class. Vice versa, finer grain-size classes between 1 and 100 µm are
slightly overestimated by ca. 1 % vol per class. The effects of the
applied sample mixing scheme of *X*_{nat} are clearly visible in all model
results (Fig. 7). Samples 51 to 75 (without coarse EM_{nat}4) show an
overestimation of coarse and underestimation of fine classes. Samples 76 to
100 (without fine EM_{nat}1) show the opposite picture. AnalySize yielded
the overall best unmixing, with average deviations of ca. ±1 % vol.

## 4.2 Validation against known data set composition

The above criteria quantify how well the approaches modelled the data set
(Eq. 1), whereas their ability to reproduce the true “mixed ingredients”
is addressed here. The *R*^{2} values between loadings and input
EM_{nat} grain-size distributions (Table 2a) were on average between 0.4
and 0.99 and, thus, systematically larger than *R*^{2} values
linking scores and mixing ratios (0.77 to >0.99; Table 2b). Both
EMMAgeo and AnalySize performed less well in modelling one out of three
EM_{nat} distributions (EM1 for EMMAgeo and EM4 for AnalySize). The MAD
was below 0.8 % vol for all models and end-members, except for EM4
scores (AnalySize).

A graphical comparison of the grain-size class-wise deviations of input
end-member distributions and modelled loadings (Fig. 8) shows that all
EMMAgeo-based models underestimate the main mode grain-size classes (i.e.
curves are below the 1:1 line). This is the result of the emergence of
spurious modes that shift class-wise percentages (up to −3.2 % vol) from
the main modes to classes around the spurious modes (up to 1.3 % vol)
that actually contain no grains (vertically aligned points at zero
*x* values). The other EMMA approaches also show mismatches between natural
end-members and modelled loadings. Especially the alluvial fan EM_{nat}4
is affected, most severely in AnalySize. Percent volume (% vol) shifts due to spurious
secondary modes also occur for the algorithm of Weltje (1997) and
RECA. Overall, the latter approach yields the most accurate representations
of the input distributions.

Concerning the reproduction of the initial mixing ratios (Fig. 9, Table 2b),
variability among the models is higher, and all approaches show some
unsystematic over- and underestimation, especially for EM in samples in which real mixing ratios were zero (vertical point clusters along the 0 %
*x* axis; Fig. 9). Except for RECA and AnalySize, the opposite effect is also
visible: the models suggest zero contribution from end-members that are
actually present in a sample with up to ca. 20 % (horizontal points along
the 0 % *y* axis; Fig. 9).

The modal grain-size classes of the four EM_{nat} were modelled with
different levels of success (Fig. 8, legends). The main modes of the coarse
end-members were detected with only one or two grain-size classes' difference, whereas finer end-members differed by up to three classes. Modal
classes of EM_{nat}2 and EM_{nat}3 were correctly depicted by EMMA of
Weltje (1997), RECA and AnalySize. Most models yielded a value of
EM_{nat}1 that is slightly too coarse, deviating by one or two grain-size
classes. EM_{nat}4 caused the largest scatter among the models.

## 5.1 Operational modes of EMMAgeo

The functionality of EMMA has improved significantly since the introduction of the MATLAB algorithm of Dietze et al. (2012). Not only an increase in computation speed, which was already 1 to 3 orders of magnitude faster than for other algorithms (Paterson and Heslop, 2015), but also many new and detailed ways to explore end-members (with deterministic EMMA) and to estimate and describe associated uncertainties of all end-member components (with robust EMMA) were implemented. The plot output of both EMMA modes is a comprehensive visualisation of all relevant information. It allows direct process interpretation in terms of plausibility of loadings and scores, model performance and identification of outliers.

Both EMMA modes, deterministic and robust, result in consistently similar
outputs. Deviations of individual modes of robust loadings from known
EM_{nat} distributions by one or two grain-size classes are within the
model uncertainty of robust EMMA. Therefore, a key step is the definition of
robust end-members by setting the grain-size class limits that bracket
robust, parameter-independent main modes, which overcomes the problem of
relying on statistical measures like the inflection point of a
*q*–*R*^{2} graph (van Hateren et al., 2018). The
workflow of robust EMMA offers ways to explore the ability of different
kernel bandwidths and density thresholds, but in complicated cases, like the
one provided in this study, expert knowledge-based limit definition might be
the most practicable option. Hence, each data set should be considered
individually, and deviations from common patterns may be significant in their
own right (see discussion by van Hateren et al., 2018).

## 5.2 Performance test and validation

Unmixing quality is very high regardless of the model used, suggesting that all approaches in this benchmark are able to reproduce the input grain-size data set with unmixed end-member subpopulations. There is no model with an outstanding performance. Model deviations of <1 % (especially for grain-size classes with >0.1 % vol) are low in the light of uncertainties related to process interpretation (see below).

The validation against known input end-member composition showed that all
EMMA approaches are equally applicable. When comparing end-member loadings
with the EM_{nat} distributions, *R*^{2} mainly represents
shifts in mode positions, whereas MAD reacts to both shifts in the modes of
individual grain-size distributions and differences in the volume
percentages per class. Yet, each algorithm has certain strengths depending
on the specific dimension of the investigation: if the main goal is to
identify the most likely *q* that builds an empirical data set, robust EMMA
provides a set of tools that go beyond classical approaches (e.g. the
inflection point of the *q*–*R*^{2} plot) – allowing the inclusion
of expert knowledge in the quantification and interpretation of grain-size
subpopulations.

If the correct grain-size distribution shape of underlying process end-members is targeted, RECA of Seidel and Hlawitschka (2015) and EMMA by Weltje (1997) are most suitable from our benchmark study (Table 2a). RECA had problems with reaching the convexity error threshold, which could result from our data set with largely overlapping natural process end-members.

When quantifying the contribution of end-members to a given sample, robust EMMA, EMMA according to Weltje (1997) and AnalySize performed best (Table 2b). Robustly estimated scores using EMMAgeo reproduced original mixing proportions very well and in a range comparable to the other available end-member algorithms. However, as all approaches and earlier EMMA evaluations showed, very low and high scores (<20 % and >80 %) of one end-member might be under- or overestimated within the compositional mixture (McGee et al., 2013). Hence, extremely high (e.g. 100 %) contributions of one end-member to a sample should not be interpreted as complete absence of the other end-members but rather as the dominance of this one end-member (and vice versa).

If uncertainty estimates for both loadings and scores are considered important, then only robust EMMA is suitable. The inclusion of uncertainties for loadings and scores is a key precondition for propagating model results to further data analysis, for example to interpret grain-size end-members as proxies for sediment sources (loadings) in environmental archives as they evolve with time (scores). As van Hateren et al. (2018) emphasise, changes in the model results will inevitably result in diverging interpretations of the assumed sedimentary processes. Also, the interpretations of the scores in their spatial (samples across a landscape) or temporal (samples downcore) context will be affected. Thus, it is extremely important to provide some estimate of the inherent uncertainty in both the proxy definition and in the sample domain. So far only robust EMMA can deliver such information. Yet, necessary parameter estimates and diverging start conditions evidently exist in the other models too.

If the distribution shape of an inherent natural grain-size end-member is known, EMMAgeo allows quantification of its contribution to the data set by including it as unscaled loadings in both deterministic and robust EMMA or by assigning the known main mode class limits when selecting robust end-members (step 4; Fig. 1b). Finally, if free and open-source software is a criterion – which is increasingly the case for journals and funding agencies (David et al., 2016; Munafò et al., 2017) – RECA and EMMAgeo remain the only options.

## 5.3 Comparison with other benchmark studies

In previous benchmark studies, EMMAgeo performed less well, which
Paterson and Heslop (2015) attributed to the implementation of the
non-negativity and sum-to-one constraints. van Hateren et
al. (2018) pointed to the secondary modes as cause of the deviations of
scores from the mixing ratios. We cannot confirm the poor performance of
EMMAgeo in our study, as it is not fully clear how van
Hateren et al. (2018) determined the EMMAgeo loading curves, which they
evaluate graphically. They note that in EMMAgeo the *q* is not set by the
inflection point of the *q*–*R*^{2} relationship, but robust EMMA
would lead to one *q*, and not a sequence of 2 to 5, as discussed in their study.
Additionally, it is unclear which realisation from within the robust EMMA
uncertainty range was evaluated by van Hateren et al. (2018). Accordingly,
detailed introduction of the EMMA protocols is
essential to avoid future misinterpretations.

Yet, the occurrence of artificial secondary modes below the main modes of the end-members is more pronounced in EMMAgeo compared to other unmixing algorithms. The inherent compositional data constraints lead to an intimate linkage of the distribution shape of one end-member with the distribution shapes of other loadings. However, when excluding hardly interpretable secondary modes from global measures of model quality, the performance of EMMA is well within the range of other available algorithms. As repeatedly noted in articles applying EMMAgeo (Dietze et al., 2012, 2014) but also highlighted for other approaches in the benchmark study of Paterson and Heslop (2015), secondary modes are model artefacts and should not be interpreted genetically.

However, to test the impact of artificial secondary modes on model
performance, we modelled the EM_{nat} data set with user-defined
end-members. We manually set the unscaled loadings outside the known primary
end-member modes to zero and used these updated loadings for the modelling
process (see Supplement for R code). Although the resulting
end-member loadings are now free of secondary modes, the mixing ratios are
only marginally better modelled (−1 % to 4 % deviation). Thus, such a
truncation may help in tuning the shape of the modelled end-members but
cannot improve deviations of the scores from mixing ratios. Still, the
uncertainty ranges of the robust scores included the expected EM_{nat}
mixing ratios (66.5 % of the samples are within the modelled 1 standard
deviation range).

## 5.4 Constraints on end-member interpretation

Going beyond classical measures of grain-size properties, EMMA is well suited to quantify sedimentary processes from mixed sediment sequences in space and time. However, interpretation of grain-size end-members requires expert knowledge about the investigated sedimentary system. Hence, when applying EMMA to any set of grain-size data, the interpretability of the resulting end-members needs to be checked. For this, both end-member components should be considered: the shape and position of the main modes of the loadings and the spatio-temporal or stratigraphic context of the scores. For example, the effectiveness of a process in sorting sediment could be interpreted in the classical sense from the shape of the end-member loadings (excluding artificial modes), with broader peaks being more poorly sorted than narrow peaks (Friedman, 1961).

As any other statistical method, EMMA is a tool, and interpretation of grain-size end-members relies on contextual knowledge. There may be processes that contribute to the overall sediment composition and that are not size-selective or sort sediment of various grain-size classes in a typical way. For example, event-triggered turbidity currents in lakes caused problems in attributing a single sedimentary process to end-members in the study by Dietze et al. (2014) because the typical fining-upwards trend was also reflected by several end-members that contributed to samples of “normal” deposition.

Closely related is the constraint of stationarity in processes, which implies that through space and time each transport process must create an identical grain-size distribution. For example, fining of aeolian material from one distinct source area with downwind transport distance (Pye, 1995) might rather be explored by a gradual approach, e.g. by running EMMA in a moving window over a data set to detect shifts in stationarity.

Post-depositional processes that change grain-sizes, e.g. due to permafrost conditions or soil formation, could strongly disturb the original grain-size characteristics. In the worst case, a lacustrine sediment archive composed of different aeolian and fluvial sediment end-members (Dietze et al., 2013) can be affected by ongoing cryogenic and active-layer dynamics in a way that all modelled end-members were overlapping and peaking in similar grain-size classes – “erasing” primary signals related to sediment deposition. If post-depositional activity overprints the original depositional processes, EMMA can detect them as single end-members and would allow quantification of the intensity of the overprint, e.g. soil formation (Dietze et al., 2016) or weathering (Sun et al., 2002; Xiao et al., 2012).

Sediments affected by the processes mentioned above can affect end-member modelling in manifold ways. For example, EMMA could result in rather low explained variances, and the modes of affected end-member loadings would become broader and/or may even be better represented by additional but nevertheless spurious end-members. In the worst case, modes of end-member loadings overlap strongly or cannot be unmixed at all.

EMMAgeo allows the characterisation of multi-modal grain-size distributions by end-member subpopulations. New protocols allow a quick analysis, including modelling of associated uncertainties for both end-member loadings and scores. Using four known natural end-members, which represent typical sediment types found in terrestrial systems, the performance of EMMAgeo in unmixing the correct end-member distribution shapes and mixing ratios is within the same order as the performance of other available end-member modelling algorithms, which all perform very well. Hence, all of these algorithms are powerful tools for characterisation of different sediment source, transport, depositional and even post-depositional processes. In comparison to other algorithms, EMMAgeo is the only available open-source grain-size unmixing approach that includes uncertainty estimates. An inherent strength of the fully free R package is a large flexibility for users to modify the parameter settings and workflows with the new protocols, reproduce results and continue data evaluation.

Once genetically interpretable grain-size end-members are derived, their loadings can be described by classical descriptive measures (Folk and Ward, 1957; Blott and Pye, 2001). This allows a statistically robust determination and comparison of mean, sorting and shape measures across sites and data sets by describing and quantifying processes that sort sediment better or poorer than other processes.

Many future applications in the fields of Quaternary Science, sedimentology, geology, geomorphology and hydrology could gain new insights from applying EMMAgeo to compositional data sets that represent mixtures. In contrast to classical linear decomposition methods such as principle component analysis, EMMA has the potential to quantify (and not just qualify) different sources or processes of modern and past sedimentary environments that contribute to a sample set, including associated model uncertainties.

The Supplement contains the example data set, end-member measurement data, mixing ratios and output of the other approaches included in the comparison. The R package EMMAgeo in its latest release version 0.9.6 (Dietze and Dietze, 2019; https://doi.org/10.5880/GFZ.4.6.2019.002) is available on the Comprehensive R Archive Network (R Core Team, 2017) and on GitHub. Please report any bugs and improvements to the maintainers of the package.

The supplement related to this article is available online at: https://doi.org/10.5194/egqsj-68-29-2019-supplement.

ED and MD improved the original EMMA algorithm, workflows and auxiliary functionalities. ED compiled the operational modes of EMMA and MD established the EMMAgeo package. Both authors wrote the paper.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Connecting disciplines – Quaternary archives and geomorphological processes in a changing environment”. It is a result of the First Central European Conference on Geomorphology and Quaternary Sciences, Gießen, Germany, 23–27 September 2018.

Thomas Hösel and Claudia Ziener prepared the example data set. Philip Schulte performed grain-size analysis using the Laser particle sizer at RWTH Aachen. Jan-Berend Stuut provided the data from the FORTRAN code of Weltje (1997), and Mitch D'Arcy provided language editing. Kai Hartmann and Andreas Borchers supported the initial development of EMMA and Kirsten Elger the DOI and landing page coordination. Many users of former versions of the MATLAB and R scripts greatly helped to improve EMMAgeo.

The article processing charges for this open-access publication were covered by a Research Centre of the Helmholtz Association.

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- How to cite
- Abstract
- Introduction
- The R package EMMAgeo
- Practical application: material and methods
- Results: the different model performances
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- References
- Supplement

- How to cite
- Abstract
- Introduction
- The R package EMMAgeo
- Practical application: material and methods
- Results: the different model performances
- Discussion
- Conclusions
- Code and data availability
- Author contributions
- Competing interests
- Special issue statement
- Acknowledgements
- Financial support
- References
- Supplement