Yamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70
https://doi.org/10.1186/s40645-023-00601-y

Progress in Earth andPlanetary Science

Open Access

METHODOLOGY

Validation of appropriate estimation criteria
for the number of components for separatinga polymodal grain‑size distribution
into lognormal distributions
Naofumi Yamaguchi1*   

Abstract
Polymodal particle size distributions are generally analyzed by separating them into lognormal distributions, but estimating the precise number of lognormal components required remains a considerable problem. In the present study,
appropriate evaluation criteria for the estimation of the number of components were examined by using artificial
data for which the true number of components was known. The characteristics of estimations of the number of components by four evaluation criteria, the mean square error (MSE), Akaike information criterion (AIC), Bayesian information criterion (BIC), and adjusted R-squared (ARS), were investigated. The results showed that the MSE and ARS wereless sensitive to the true number of components and tended to overestimate the number of components. By contrast, the AIC and BIC tended to underestimate the number of components, and their correct answer rates decreased
as the true number of components increased. The BIC tended to include the true number of components among its
higher ranked models. The present evaluation results suggest that the MSE, although frequently used, is not necessarily the most appropriate evaluation criterion, and that the AIC and ARS may be more appropriate criteria. Furthermore,checking whether the number of components estimated by the AIC or ARS is included among higher ranked BIC
models might prevent overestimation and thereby allow for more valid estimation of the number of components.
When the criteria were applied to grain-size distributions of lacustrine sediments, it was possible to estimate the number of components that reflected differences in grain-size distribution characteristics.
Keywords Polymodal grain-size distribution, Lognormal distribution fitting, Estimation criteria, Grain-size analysis
1 Introduction
The sediment grain-size distribution is fundamental
information for various types of sediments and is used inmany different fields because it reflects the origin of thesediments, the transport processes that acted on them,and the strength of the experienced transport forces.
Sediment grain-size characteristics have been used, for*Correspondence:
Naofumi Yamaguchi
naofumi.yamaguchi.sci@vc.ibaraki.ac.jp
1
Global and Local Environment Co‑creation Institute, Ibaraki University,
Ohu 1375, Itako, Ibaraki 311‑2402, Japan

example, to reconstruct past climate events and changesof depositional environments. The grain-size distributions of lake sediments have been used as an indicatorof hydrological conditions associated with climatic andenvironmental changes in the region where the lake islocated (e.g. Håkanson and Jansson 1983; Xiao et al.
2009; Dietze et al. 2014; Lu et al. 2018). Grain-size characteristics of aeolian sediments, including loess, havebeen regarded as a meaningful paleoclimatic proxy (e.g.
Vandenberghe et al. 1997; Sun et al. 2002; Sun 2004; Qinet al. 2005; Lim and Matsumoto 2006; Machalett et al.
2008; Antoine et al. 2009; Vandenberghe 2013; Lin et al.
2016; Schulte et al. 2018). Tephra grain-size distributions© The Author(s) 2023. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to theoriginal author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images orother third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory
regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.

Yamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70

have been used to estimate the characteristics of volcanic
eruptions and the dispersal of particles in the atmosphere (e.g. Rose and Durant 2009; Burden et al. 2011;Engwell and Eychenne 2016; Rossi et al. 2019; Miwa
et al. 2020). In addition, several methods have been proposed to estimate the transport pathways of modern surface clastic sediments based on spatial trend analyses of
their grain-size characteristics (McCave 1978; McLaren
1981; McLaren and Bowles 1985; Gao and Collins 1992;
Le Roux and Rojas 2007; Yamashita et al. 2018). Therefore, accurate methods of grain-size analysis and suitableinterpretation of grain-size distributions are an important topic in Earth science.
One major issue in the analysis of sediment grain-sizedata is how to deal with a complex distribution. Previous
studies have commonly assumed a unimodal distribution
and used representative statistical parameters such as
median diameter, sorting, skewness, and kurtosis to characterize the sediment grain size (Folk 1966). However,
the grain-size distribution of sediments in various natural environments is frequently complex and polymodal
rather than unimodal. In particular, with the advent oflaser grain-size analyzers, which reveal grain-size distributions over a wide size range at high resolution, it has
become clear that polymodal distributions are frequent
for various types of sediments. For example, lake sediments, which may have multiple sources and be affectedby multiple sedimentary processes, frequently havepolymodal grain-size distributions. Such distributions
complicate the analysis and interpretation of the representative parameters calculated by assuming unimodality. Aeolian deposits, including loess, also frequently havecomplicated polymodal grain-size distributions. Suchpolymodal grain-size distributions are thought to reflectmultiple sediment sources and sedimentary processes,
and each mode is thought to preserve environmental
information associated with them (Tanner 1964; Visher
1969; Middleton 1976; Ashley 1978). Therefore, it is not
appropriate to use just a few representative parameters
based on assumed unimodality to interpret polymodal
grain-size distributions. Appropriate analysis of suchcomplex grain-size distributions can potentially allow
each of the multiple origins and transport processes that
produced the sediment deposit to be reconstructed andto provide additional valuable information.
Several methods have been proposed for analyzing
complex polymodal distributions of grain size, including end-member mixing analysis (e.g. Weltje and Prins
2007; Parris et al. 2010; Dietze et al. 2012, 2013; Ijmker
et al. 2012; Yu et al. 2016) and discrete parametric curve
fitting. The latter method of separating grain-size distributions includes the use of prescribed distributionfunctions such as the Weibull distribution (Lim andPage 2 of 11

Matsumoto 2006; Park et al. 2014; Wu et al. 2020; Peng
et al. 2022a) and lognormal distributions as componentsof the polymodal distribution. In particular, a number ofprevious studies have used lognormal distributions (Qinet al. 2005; Xiao et al. 2009, 2012, 2013, 2015; Fettweis
et al. 2012; Wang et al. 2014; Gammon et al. 2017; Lu
et al. 2018; Miwa et al. 2020). The curve-fitting methodwith lognormal distributions is based on the assumptionthat various grain-size distributions can be represented
as a single lognormal distribution or a mixture of several: for example, a unimodal grain-size distribution withan asymmetrical or skewed shape can be regarded as a
mixture of several lognormal distributions (e.g. Tanner
1964; Ashley 1978). Thus, a given polymodal grain-size
distribution can be decomposed into several components
consisting of normal distributions on a logarithmic scale.
Methods based on parametric curve fitting, including
the lognormal distribution function fitting method, havethe advantage that they can be used for a single sample,
whereas end-member modeling analysis requires multiple samples. The lognormal distribution function fitting
method has been successfully used to obtain information
on past environmental changes from both lacustrine andaeolian sediments (Qin et al. 2005; Xiao et al. 2009, 2012,2013, 2015; Gammon et al. 2017; Lu et al. 2018).
The lognormal distribution function fitting method
involves two processes: (1) estimation of the parametersof the appropriate lognormal distributions (i.e. the meanand standard deviation of each, and their mixing proportions) and (2) determination of the number of components. Commercial data analysis software (e.g. Igor Pro,
PeakFitNagashima et al. 2004; Wang et al. 2014; Gammon et al. 2017; Miwa et al. 2020), original programs (e.g.
Sun et al. 2002; Sasaki and Kiyono 2003; Xiao et al. 2012,2013), and R platform packages (e.g. Buckland et al. 2021)
have all been used to estimate the lognormal distribution
parameters. In particular, several easy-to-use R packages
that have appeared in recent years have made parameter
estimation easier. By contrast, a method for determining
the number of components has not yet been establishedso this process remains arbitrary. The number of components has been estimated from the number of peaksin the grain-size distribution (e.g. Xiao et al. 2012), orby using the mean square error (MSE) as an evaluation
criterion (e.g. Xiao et al. 2013). When the MSE is used
as an evaluation criterion, the model for the number ofcomponents with the smallest MSE is adopted, or thatmodel is adopted having the smallest number of components with the MSE value below a certain thresholdvalue. Using the number of peaks in the grain-size distribution may lead to underestimation of the number ofcomponents because it can fail to find a small component
hidden by a larger adjacent component. In addition, itYamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70

has been generally pointed out that use of the MSE has
the statistical problem that the number of components
may be overestimated in evaluations based solely on theagreement of the curve fitting to the original data using
the MSE value (Bishop 2006). Although several otherevaluation criteria, including the Akaike information criterion (AIC; Akaike 1973) and the Bayesian information
criterion (BIC; Schwarz 1978), have seen generally wideuse for the estimation of the number of components,
they have not yet been applied to the separation of grainsize distributions, nor has their validity been examined.
In the present study, therefore, appropriate criteria forevaluating the number of components and their estimation characteristics when separating a grain-size distribution into lognormal distributions were investigated.
Separation tests were carried out on artificial grain-size
distributions for which the number of components andtheir parameters were known, and the estimation characteristics of each of four commonly used evaluation criteria were investigated to clarify the points that need tobe considered when they are used. In addition, the four
evaluation criteria were applied to an actual lake sediment sample, and the results were evaluated.

2 Method
2.1 Test procedure

The tests were carried out through the following
procedure.
(i) Given an artificial grain-size data at phi scale(= − ­log2(D/D0); D is particle size in milimeters andD0 is a reference diameter, equal to 1 mm) consisting of a mixture of normal distributions, for
an assumed number of components n of 1–9, theparameters of each component (mean, standard
deviation and mixing proportion) were estimated.
(ii) The values of the criteria to be validated wereobtained for each of the cases in (i).
The artificial data used in this procedure, the method
for estimating parameters by fitting, and the evaluation
criteria considered are described in detail below.

Page 3 of 11

standard deviation of each mixed normal distribution
was set to a uniform random number ranging from 0.2
to 0.8 phi. The mixing proportions were set by generating uniform random numbers that summed to 100% inincrements of 1%. The minimum value of a mixing proportion was set at 5%.
2.3 Separation into normal distributions

To decompose the artificial grain-size distributions (i.e.
mixed normal distributions at phi scale) and to estimate their parameters, the expectation–maximization(EM) algorithm (Dempster et al. 1977) was employed.
The EM algorithm is a frequently used iterative method
for finding (local) maximum likelihood estimates of
unknown parameters in statistical models (McLachlan and Krishnan 2007). The EM iteration alternates
between an expectation step and a maximization step:
the expectation step creates a function for the expectation of the log-likelihood using the current estimatesof the parameters, and the later maximization step
calculates parameters that maximize the expected loglikelihood found in the previous expectation step. Thisalgorithm yields the mixing proportions, means, andstandard deviations of the normal distributions thatcompose a given grain-size distribution. The analyses
were performed in R (version 4.1.2; R Core Team 2021),
using the package ‘mixR’ (version 0.2.0; Yu 2022) to run
the EM algorithm.
The quality of the fitting of the mixture of normal distributions to the artificial distributions was evaluatedby calculating the figure-of-merit (FOM) value (Penget al. 2022b) as follows:

FOM =

m
i=1 yi − yimi=1 yi× 100%

where yi is the volume percentage of the measured grain
size (i.e. the given artificial grain size in the present study)
yi , is the fitted grain-sizein the i-th grain-size interval, volume percentage in the i-th grain-size interval, and mis the number of grain-size intervals.

2.2 Artificial datasets used for testing

A total of five artificial datasets were generated, eachconsisting of a different number (ranging from 2 to 6) ofmixed normal distributions (components). Each dataset
consisted of 1000 cases of mixed normal distributionsof grain size at phi scale. For each case, the parametersof the normal distribution components used for the
mixing were chosen as follows: The mean value of one
component was fixed at 0 phi, and the distance between
the mean values of adjacent components was set to a
uniform random number ranging from 0.4 to 2 phi. The

2.4 Criteria for determining the number of components

A total of four evaluation criteria were tested: the meansquare error (MSE), which has been used in lake sediment studies (e.g. Xiao et al. 2013), and three otherevaluation criteria commonly used for model selection in various fields: the Akaike information criterion(AIC), the Bayesian information criterion (BIC), andthe adjusted R-squared (ARS). Each of these criteria
were calculated as follows:(2023) 10:70

m

MSE =

2
1 
yi − yimi=1

AIC = −2lnL + 2K
BIC = −2lnL + K lnm
m

i=1Page 4 of 11aRelative frequency (%)

Yamaguchi ﻿Progress in Earth and Planetary Science(yi −yi )2-0.1 phi
87.0%
0 phi
87.0%

15

10

1.8 phi
13.0%
1.9 phi
13.0%

5

0
-2

ARS = 1 − m−3n−12mi=1 (yi −y)

3 Results
The FOM for the optimal fitting curve obtained by the
EM algorithm for each mixed normal distribution case
was less than 2.9%. The parameters of the normal distribution components were not always estimated correctly
even when the number of true components was given,and errors in parameter estimation were more frequent
the higher the true number of mixture components in theartificial grain-size distribution (Fig. 1, Additional file 1:
Fig. S1). Note that the accuracy of the parameter estimation depended on the degree of overlap of the adjacentcomponents in the artificial data. Details on this point
are described in Additional file 2.
The rate at which the number of components estimated as optimal matched the true number of components (Nt) (i.e. correct answer rate) differed among thecriteria and depended on the value of Nt for the given
grain-size distribution (Fig. 2). For all criteria, the correct answer rate tended to decrease as Nt increased(Fig. 2). When Nt was relatively small, the AIC and
BIC tended to have higher correct answer rates thanthe other criteria, whereas when Nt was larger, the
ARS and MSE tended to have higher correct answer
rates than the AIC and BIC. When Nt was two, the BIC
had the highest correct answer rate (89.6%), followed

2

4

6

Grain size (phi)m−11.6 phi-0.1 phi
45.0%
28.9%
1.7 phi
0 phi
28.7%
28.0%
1.0 phi
35.6%
1.0 phi
20.0%bRelative frequency (%)

where L is the maximized value of the likelihood function, K is the number of degrees of freedom, y is themean of yi, and n is the number of components. In thepresent study, the number of degrees of freedom was
3n−1. Note that for the criteria MSE, AIC, and BIC, thepreferred model is the one for which the value of the criterion is smaller, whereas for ARS, the preferred modelis the one for which the value of the criterion is larger.
MSE is simply the goodness of fit of the estimated mixture of fitting curves to the original data. Therefore, MSE
tends to select excessively complex models, as describedabove (Bishop 2006). By contrast, the other three criteria
are proposed to select the appropriate model by penalizing the increase in the number of components and thecomplexity of the model.

0

15

10

5

3.5 phi
6.8%
3.6 phi
7.0%

0
-2

0

2

4

6

Grain size (phi)True component
Separated component
Mixture of components estimatedFig. 1 Examples of the artificial grain-size distributions
and separation results. Examples of the artificial grain-size
distributions used in the present study and separation results
when the true numbers of components were a two and b four. The
blue solid lines show the true normal distribution components.
The red and green dashed lines show the components separated
by the EM algorithm and their mixing curves, respectively. The
values shown above each component are the mean grain sizes
and the mixing proportions of the true and separated components(blue and red, respectively)by the AIC (81.3%), ARS (69.6%), and MSE (62.9%).
However, compared to the other criteria, the BIC
showed a greater decrease in the correct answer rate
as Nt increased, and its correct answer rate was lowest
when Nt was larger than four (44.9% for Nt = 4; 30.7%
for Nt = 5; 22.7% for Nt = 6). By contrast, the MSE did
not decrease as rapidly as Nt increased, and its correct answer rate was second highest among the criteria when Nt was 5 or more (49.0% for Nt = 5; 41.4% for
Nt = 6). The correct answer rate of AIC was high for
Nt counts up to 4 (75.1% for Nt = 3; 59.6% for Nt = 4),
and it was comparable to those of the MSE for Nt of
5 and above (46.2% for Nt = 5; 37.5% for Nt = 6). The
ARS showed a similar trend to the MSE, but its correct
answer rate was higher than that of the MSE regardlessYamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70

60
40

Rate (%)

80

100

number of components was 6, the top four ranked models contained the true number of components in 95% of
the cases (Fig. 4e).

Nt

Nt ±1
MSE
AIC
BIC
ARS

20
0

Page 5 of 11

2

3

4

5

6

Nt
Fig. 2 Correct answer rates for each criterion relative to the true
number of components. Rates at which the true numberof components Nt (filled symbols) or the true number of components
within Nt ± 1 (white symbols) were estimated by each of the criteria
in relation to Nt

of Nt, and its correct answer rate was highest for truecomponent counts of 4 and above (62.3% for Nt = 4;
58.8% for Nt = 5; 47.6% for Nt = 6). The trend in the rate
at which the difference between the estimated numberof components was within Nt ± 1 also differed among
criteria and depended on Nt. For the BIC, that rate
decreased markedly when Nt was 4 or above, whereas
for the MSE and ARS, it remained approximately constant at around 81% and 88%, respectively, regardless of
Nt. The rates for the AIC were roughly comparable tothose for the ARS, but the AIC rate was highest when
Nt was 4.
The difference between the number of components
estimated as optimal (Ne) and Nt (Ne minus Nt) tended todiffer among the criteria (Fig. 3). The MSE and ARS often
overestimated the number of components regardless ofNt, whereas the AIC and BIC often underestimated thenumber of components, especially as Nt increased. The
BIC more often estimated the number of components to
be one fewer than the true number of components when
Nt was 4 or more. When the AIC was used to estimate
the number of components for data with low Nt, it rarely
overestimated the number of components by more thanfive.
Among the nine models (i.e. with an assumed numberof components of 1–9), the frequency with which themodel with the true number of components was selectedin the higher ranks of the criterion also tended to vary
among the criteria (Fig. 4). In the present study, the percentage of cases with the true component number was
highest for the BIC, exceeding 95%. Even when the true

4 Discussion
From the FOM values, it can be assumed that the parameter estimation in the present study resulted in good fitting. However, the parameter estimation of the normal
distributions was not always correct, even when the true
number of components was given (Fig. 1; Additional
file 1: Fig. S1). In the case of estimation using actual natural grain-size distributions, the parameters of the truecomponents are not known. For this reason, the discussion here is based on the values of the evaluation criteria obtained for each case, irrespective of the correctnessof the parameter estimates. Note that the accuracy ofparameter estimation and the correct answer rate foreach criterion decreased with the degree of overlap ofadjacent components, while it did not affect the characteristics of each criterion, discussed below. Caution
should be paid with regard to the accuracy of the estimation of parameters and the number of components forcases with significant overlap of adjacent components,
but this effect of the degree of overlap does not need tobe taken into account in the comparison of the characteristics of each evaluation criterion in the following discussion. Details are described in Additional file 2.
The present results suggest that it is important to estimate the number of components by taking into accountthe estimation characteristics of each criterion: the AIC
and BIC can be calculated with high accuracy when theactual number of components is small, but they tend tobe slightly underestimated as the true number of components increases (Figs. 2 and 3). For this reason, caution
should be exercised in their use when the grain size distribution is complex and may consist of more than fivenormal distributions. In contrast, the present results
suggest that MSE and ARS are less sensitive to the valueof the true number of components than AIC and BIC(Fig. 2); therefore, they may be effectively used for complex grain-size distributions. It should be noted, however,that the MSE and ARS may be overestimated.
The results of the present validation using artificial datasuggested that the AIC or ARS would be preferable for
estimating the number of components by using a single
criterion for natural grain-size data where the true number of components is unknown. The present results indicate that the correct answer rate of the MSE, which has
been used in previous studies of sediment grain size, is
not higher than the rates of the other criteria; thus, it is
not necessarily the most suitable criterion for estimating
the correct number of components (Fig. 2). The correct
answer rate of the ARS, which similarly tends to be less

4

6

2.8%

2

Ne ‒ Nt

4

6

32.7%

0
0

2

N e ‒ Nt

4

6

−2

0

2

4

6

0

0
−4

4

3.4%

2

4

0

2

Ne ‒ Nt

7.7%

−4

4

30.0%

−2

Ne ‒ Nt

0

2

Ne ‒ Nt

4−20

Ne ‒ Nt

2

11.8%

50.7%

−2

0

Ne ‒ Nt

2

4

−4

4.1%

−2

0

Ne ‒ N t

2

5.9%

71.4%

0
−2

−4

9.5%

65.2%

0

30.2%

0

100

−2

6

2

0

100
4

100

0.2%

2

N e ‒ Nt0

Ne ‒ Nt

51.7%

0

0

0

−2

3.0%

30.7%

0

Ne ‒ Nt100

2

Ne ‒ Nt

−2

0

100
0

−4

44.3%

100

3.6%

4

0
−2

6

2

8.1%

32.3%

0
4

100

6.8%

2

N e ‒ Nt0

Ne ‒ Nt

100

9.3%

15.6%

0

0

−2

0

6

Overestimated
41.4%

100

4

Underestimated
17.2%

0

2

Ne ‒ Nt

100

16.9%

0

Overestimated
39.5%

100

−2

−2

0

Ne ‒ Nt

13.6%

100

6

Underestimated
11.5%

2

4

27.6%

0

4

100

1.8%

2

Ne ‒ Nt

0

0

0

0

Overestimated
41.2%

100

Underestimated
6.2%

Nt = 6100

Overestimated
40.9%

Nt = 5100

Underestimated
2.6%

Nt = 4100

100Overestimated
36.9%

100

MSE

Nt = 30

Relative frequency [%]

AIC
BIC

Relative frequency [%]Relative frequency [%]ARS

Relative frequency [%]Underestimated
0.2%

Page 6 of 11

−4

−2

0

Ne ‒ N t
21.0%

2

31.4%

0

Nt = 2(2023) 10:70

100

Yamaguchi ﻿Progress in Earth and Planetary Science

−4

−2

0

2

4

Ne ‒ Nt

−4

−2

0

2

Ne ‒ N t

Fig. 3 Histograms of the difference between the estimated and the true number of components. Histograms of the difference
between the number of components estimated as optimal (Ne) and the true number of components (Nt) (Ne−Nt) for each criterion and each truecomponent number. Dark gray bars indicate cases when the true number of components was estimated (Ne = Nt). The red lines are the medianvaluesdependent on the value of the true number of components, is higher in all component numbers from 2 to 6.
Although the correct answer rate of the AIC is lower thanthat of the ARS when the true number of components islarge, the AIC successfully estimates the true component
number ± 1 at a higher rate overall; this finding suggeststhat use of the AIC is less likely to result in large estimation failures. Use of the AIC would be also preferred
when analysing less complex grain-size distributions.
Furthermore, more appropriate component number
estimation might be achieved by taking into account theestimation characteristics of each criterion and using
ones with complementary characteristics. For example,
for the BIC, the top four models contained the correct

number of components with a probability of more than
95% (Fig. 4). Considering this feature of the BIC, it should
be possible to avoid the overestimation problem of, forexample, the AIC or ARS, by checking whether the number of components they selected was among the higher
ranked models of the BIC.

5 Application to natural grain‑size data:an example
5.1 Sample and method of analysis

The four criteria tested in the present study were applied
to the grain-size distribution of sediment samples from
Lake Kitaura (Fig. 5). Lake Kitaura is a freshwater lake incentral Japan, with an area of 35.2 ­km2, a maximum depth(2023) 10:70

Page 7 of 11

Nt = 3

a

b

1

2

3

4

5

6

7

8

9

0

20

20

40

40

60

60

80

80

100

Nt = 20

Cases with true number ofcomponents (%)

100

Yamaguchi ﻿Progress in Earth and Planetary Science

1

2

3

4

5

6

7

8

9

5
6
Rank

7

8

9

Rank

Nt = 5

c

d

1

2

3

4

5
Rank

6

7

8

9

0

20

20

40

40

60

60

80

80

100

Nt = 40

Cases with true number ofcomponents (%)

100

Rank

1

2

3

4

e

20

40

60

80

MSE
AIC
BIC
ARS

0

Cases with true number ofcomponents (%)

100

Nt = 6

1

2

3

4

5

6

7

8

9

Rank

Fig. 4 Percentage of cases in which the true component number was included above the rank. Percentage of cases in which the true component
number was included above the rank for datasets with a–e 2–6 true components. The horizontal dashed line indicates 95%

of 7 m (except where it has been artificially deepened by
dredging), and a mean depth of 4 m (Fig. 5). Samples ofbottom surface sediment were collected with an Ekman–
Birge grab sampler (15 cm × 15 cm) from two sites close
to the center of the lake, KT2104-06 and KT2104-07(Fig. 5c), in April 2021, where the water depth was 6.5
and 5.9 m, respectively. Sites KT2104-06 and KT2104-07
were 916 and 424 m, respectively, from the nearest shore.
Approximately 2 g of each sample was placed in a beakerand pre-treated with 50 mL of 10% H­ 2O2 to remove
organic matter. Then, each sample was rinsed with purified water and dispersed with 30 mL of 5.5 g/L (­ NaPO3)6.
The grain-size distributions were measured with a laser
grain-size analyzer (SALD-2300, Shimadzu Corporation,
Kyoto, Japan).
The resulting grain-size distributions of the two samples were polymodal and exhibited their highest values
at around 6.0 phi (Fig. 5d). The two distributions sharedYamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70

Page 8 of 11

Fig. 5 Study site location and grain-size distributions of the two samples. a Location and b bathymetry of Lake Kitaura, and c locationsof the sampling sites (KT2104-06 and -07) and d grain-size distributions of the sediment samples from those two sites. The depth isobath interval
is 0.5 m. Bathymetric data are from the Geospatial Information Authority of Japan (2018)some peaks and inflexion points; for example, both distributions had peaks at around 6.0 and 7.6 phi and an inflexion point at around 8.6 phi (Fig. 5d). The sample fromsite KT2104-07, which was closer to the shore and at ashallower water depth, also had a slope inflexion point ataround 4.3 phi, whereas the sample from site KT2104-06
lacked this feature.
The grain-size data were separated into normal distributions at phi scale, and the values of the four criteria
were obtained in the same manner as in the validation on
the artificial data described above (Sects. 2.1, 2.3 and 2.4).
5.2 Estimating the number of components using the four
evaluation criteria

For the sample from site KT2104-06, which was further
away from the shore, different numbers of components
were estimated by each of the four criteria (Fig. 6a–d):in the MSE-, AIC-, BIC-, and ARS-based estimations,the optimal number of components was 9, 4, 3, and 6
respectively. Both the AIC-preferred model of 4 components and ARS-preferred model of 6 components wereamong the top four BIC models (Fig. 6c). In contrast, theMSE-preferred model of 9 components was not included
among the higher ranked BIC models; this result suggeststhat the MSE-preferred model may be an overestimation.
Different numbers of components were also estimatedby the four criteria for the sample from site KT2104-07,
which was closer to shore (Fig. 6f–i): in the MSE-, AIC-,
BIC-, and ARS-based estimations, 8, 5, 4, and 8 components, respectively, were found to be optimal. The MSEpreferred and ARS-preferred model of 8 components
was not included among the higher ranked BIC models (Fig. 6h); this result suggests overestimation. Whenthe grain-size distributions of the samples from sites
KT2104-06 and -07 were separated into normal distributions at phi scale with 4 and 5 components, respectively,

Yamaguchi ﻿Progress in Earth and Planetary Science

KT2104-06
a

32800

0.30

AIC
32200

0.20
2

3 4 5 6 7 8
Number of components

1

5

9

3 4 2

Volume (%)

2 3 4 5 6 7 8
Number of components

9

KT2104-07f

5.9 phi
47.3%

e

6

7.5 phi
38.3%

58.7 phi
10.5%

43

9.4 phi
3.9%

2
1
0
0

1

2

3

4

5

6

7

8

9

10

11

12

Grain size (phi)1

2 3 4 5 6 7 8
Number of components

9

1

9

3 1
1

i

2 3 4 5 6 7 8
Number of components

9

1

4
1 2 3

2 3 4 5 6 7 8
Number of components

944.1 phi
6.7%

3
2

0
0

1

2

3

4

5

6

7

8

9

10

11

12

Grain size (phi)

0.95

35600

5

7.6 phi
26.9%
8.7 phi
11.4%
9.4 phi
3.3%

5

1

ARS
0.97

BIC
36000

5
2 4

4 5 2 1 3

0.99h

2 3 4 5 6 7 8
Number of components

35400

4 5 2 1 3

5.8 phi
51.7%

j

6

AIC
35800

MSE
0.05
0.10

36200

0.15

g

Volume (%)

BIC

1

3 4 5

ARS
0.92
0.96
0.88

3 4 5
1 2
1

31600

d

1 2

1.00

9

32800

2 3 4 5 6 7 8
Number of components

0.93

MSE

0.10
0.00

1

32200
31600

Page 9 of 11b

5 2 4 31c

0.00

(2023) 10:70

Separated component
Mixture of components estimated1

2 3 4 5 6 7 8
Number of components

9Fig. 6 Values of each criterion for the two lake sediment samples and examples of component separation. Values of a MSE, b AIC, c BIC, and d ARS
for 1–9 assumed components, and e component separation of the sample from KT2104-06. Values of f MSE, g AIC, h BIC, and i ARS for 1–9 assumed
components, and j component separation of the sample from KT2104-07. The boldface numbers above component number show the rankingof those models

based on the AIC criterion results given above, normal
distribution components with shared mean values at 5.8–
5.9, 7.5–7.6, 8.7, and 9.4 phi were estimated (Fig. 6e andj). Of these shared components, those at 5.8–5.9, 7.5–7.6,
and 8.7 phi were generally consistent with the peaks andslope inflexion points visually observed in the grain-size
distributions (Fig. 5d). A component with a mean valueof 4.1 phi, which was estimated only for the sample fromsite KT2104-07, corresponds to the slope inflexion pointat around 4.3 phi visually observed in the grain size distribution (Fig. 6j). Although one advantage of the methodof separation into lognormal distributions for finding thenumber of component is that it can be applied to a single sample (Peng et al. 2022b), as in the present example,

Yamaguchi ﻿Progress in Earth and Planetary Science(2023) 10:70

the number of components can be estimated with greater
validity and geological meaning by analyzing samples
from the same vicinity and comparing the results.

6 Conclusions
To investigate appropriate estimation criteria for thenumber of components when separating a grain-size
distribution into lognormal distributions, four estimation criteria were evaluated by using artificial grain-sizedata where the number of components was known. The
criteria mean square error (MSE), Akaike information
criterion (AIC), Bayesian information criterion (BIC),
and adjusted R-squared (ARS) were evaluated, and theestimation characteristics of each were investigated. The
results showed that estimation characteristics of thenumber of components differed among the criteria: theMSE and ARS were less affected by the true number ofcomponents and tended to overestimate the number ofcomponents. In comparison, the AIC and BIC showed atendency to underestimate the number of components,
with the correct answer rate decreasing as the true number of components increased. The BIC tended to include
the true number of components among its higher ranked
models. These evaluation results suggest that the frequently used criterion MSE is not necessarily the mostappropriate evaluation criterion and that the AIC and
ARS may be more appropriate. Furthermore, by checking
whether the number of components estimated by the AICor ARS is included among the higher ranked BIC models,the possibility of overestimation can be avoided, and theestimation of the number of components may be more
valid. As an example, the four evaluation criteria wereapplied to the estimation of the number of components
in grain-size distributions of bottom surface sedimentsfrom Lake Kitaura. The number of components obtainedwith the AIC appeared to reflect observed differences in
the characteristics of the grain-size distributions.
Abbreviations
AIC	Akaike information criterion
ARS	Adjusted R-squared
BIC	Bayesian information criterion
EM algorithm	Expectation–maximization algorithm
FOM	Figure-of-merit
MSE	Mean square error

Supplementary Information
The online version contains supplementary material available at https://​doi.​
org/​10.​1186/​s40645-​023-​00601-y.
Additional file 1: Fig. S1. Relationship between the mean, mixing
proportion and standard deviation of the true normal distribution component s obtained during artificial dataset prepara tion versus those of
the estimated distribution components.

Page 10 of 11

Additional file 2. Detailed descriptions of the effects of the degree ofoverlap of adjacent components in the artificial grain size data in thepresent test on the estimation of the distribution parameters and thenumber of components.
Acknowledgements
I thank two anonymous reviewers for their helpful comments, which
improved this paper.
Author contributions
NY designed the study, analyzed the data, and wrote the manuscript. The
author read and approved the final manuscript.
Funding
This work was partly supported by a Grant-in-Aid for Scientific Research (C)(No. 21K03677) from the Japan Society for the Promotion of Science to NY.
Availability of data and material
The data analyzed during the current study are available from the corresponding author on reasonable request.

Declarations
Competing interests
The author declares that he has no competing interest.
Received: 29 May 2023 Accepted: 9 December 2023

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