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Citation: Besseris, G. Databionic
Swarm Intelligence to Screen
Wastewater Recycling Quality with
Factorial and Hyper-Parameter
Non-Linear Orthogonal
Mini-Datasets. Water 2022,14, 1990.
https://doi.org/10.3390/w14131990
Academic Editor: Laura Bulgariu
Received: 27 May 2022
Accepted: 20 June 2022
Published: 21 June 2022
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water
Article
Databionic Swarm Intelligence to Screen Wastewater Recycling
Quality with Factorial and Hyper-Parameter Non-Linear
Orthogonal Mini-Datasets
George Besseris
Department of Mechanical Engineering, The University of West Attica, 12241 Egaleo, Greece; besseris@uniwa.gr
Abstract:
Electrodialysis (ED) may be designed to enhance wastewater recycling efficiency for
crop irrigation in areas where water distribution is otherwise inaccessible. ED process controls are
difficult to manage because the ED cells need to be custom-built to meet local requirements, and the
wastewater influx often has heterogeneous ionic properties. Besides the underlying complex chemical
phenomena, recycling screening is a challenge to engineering because the number of experimental
trials must be maintained low in order to be timely and cost-effective. A new data-centric approach is
presented that screens three water quality indices against four ED-process-controlling factors for a
wastewater recycling application in agricultural development. The implemented unsupervised solver
must: (1) be fine-tuned for optimal deployment and (2) screen the ED trials for effect potency. The
databionic swarm intelligence classifier is employed to cluster the L
9
(3
4
) OA mini-dataset of: (1) the
removed Na
+
content, (2) the sodium adsorption ratio (SAR) and (3) the soluble Na
+
percentage. From
an information viewpoint, the proviso for the factor profiler is that it should be apt to detect strength
and curvature effects against not-computable uncertainty. The strength hierarchy was analyzed
for the four ED-process-controlling factors: (1) the dilute flow, (2) the cathode flow, (3) the anode
flow and (4) the voltage rate. The new approach matches two sequences for similarities, according
to: (1) the classified cluster identification string and (2) the pre-defined OA factorial setting string.
Internal cluster validity is checked by the Dunn and Davies–Bouldin Indices, after completing a hyper-
parameter L
8
(4
1
2
2
) OA screening. The three selected hyper-parameters (distance measure, structure
type and position type) created negligible variability. The dilute flow was found to regulate the overall
ED-based separation performance. The results agree with other recent statistical/algorithmic studies
through external validation. In conclusion, statistical/algorithmic freeware (R-packages) may be
effective in resolving quality multi-indexed screening tasks of intricate non-linear mini-OA-datasets.
Keywords:
non-linear factorial screening; wastewater recycling; water quality index; electrodialysis;
databionic swarm intelligence; unsupervised cluster matching; dissimilarity validation
1. Introduction
Wastewater constitutes a precious resource, and its exigent exploitation has become an
international issue [
1
]. Therefore, worldwide sustainable development initiatives include
actionable goals that align with “the improved water quality through effluent treatment”
and the “improved water efficiency through the application of the 5R principles: reduce,
reuse, recover, recycle, replenish” [
2
]. There is a great demand for a variety of wastewater
treatment technologies that may not be limited to producing cleaner water only to serve
agricultural and industrial needs, but highly purified potable water as well [
3
–
5
]. Great
emphasis has been placed on making the most of wastewater stocks, which are intended
for agricultural irrigation purposes, especially in arid areas [
6
–
8
]. There is a universal
water scarcity that gradually places restrictions on the amount of available water that
may be distributed to irrigate farms; wastewater reuse techniques aspire to offer sustain-
able solutions [
9
]. However, wastewater separation processes are dictated by complex
Water 2022,14, 1990. https://doi.org/10.3390/w14131990 https://www.mdpi.com/journal/water
Water 2022,14, 1990 2 of 26
chemical phenomena. There are several engineering concerns that emerge from regulating
the removal of heavy metals from the influx stocks, while at the same time optimizing
nutrient recovery. A delicate balance is sought, such that irrigation water does not perturb
the groundwater quality status [
10
–
12
]. To solve the water crisis, numerous promising
membrane technologies have been developed [
13
]. Specifically, electrodialysis (ED) tech-
niques have become indispensable in industrial-level wastewater treatment projects around
the world [
14
,
15
]. Recycling polluted wastewater and using ED separation cells to pro-
duce irrigation water for crop growth could entail multifarious prospects [
16
]. In general,
to improve water treatment processes, it is crucial to conduct successful screening and
optimization studies on the operating controls by heeding to pragmatic conditions. A
water-quality-screening/optimization task, for a custom-made recycling ED process, is
bound to encounter the well-known drawbacks that arise from dealing with environmental
stochastic uncertainty in modelling water systems [
17
,
18
]. Modern machine learning tech-
niques have been recommended to assist in circumventing difficult theoretical aspects in
deciphering uncertainty through statistical data analysis [
19
–
22
]. Contemporary improve-
ment problem-solving tactics must embody design of experiments (DOE) techniques that
appeal to the fourth industrial revolution premise. A data-centric optimization approach
must account for elements of lean manufacturing, quality 4.0 and artificial intelligence [
23
].
It is profitable that the same mentality that is dominated by expediting innovation is applied
in water optimization projects [
24
]. A recommended pathway to achieve this is by adopting
robust engineering techniques that rely on the structured DOE framework [
25
–
27
]. Robust
DOE techniques and methods have embedded the ‘lean-and-green’ initiative, upon which
sustainable operations are established [
28
–
34
]; ‘lean-and-green’ operations minimize waste
and increase production uptime. Furthermore, green sampling is a principled innovative
tactic that makes the most of the gleaned information by encouraging: (1) the minimization
of the experimental trials, (2) the maximization of the investigated effects at a given research
effort, (3) the minimization of chemical material usage and costs and (4) the minimization
of environmental product impact [
35
,
36
]. Wastewater-recycling quality-improvement re-
search is susceptible to the green and sustainable chemistry stipulation that establishes
the necessity for green chemistry metrics to guide data-centric methods in order to reduce
waste and increase yield [37–41].
The work by Abou-Shady [
16
] has demonstrated that it is feasible to conduct lean-
and-green experiments on wastewater recycling by using Taguchi’s DOE methodology, for
robust product design, in a custom-made electrodialysis apparatus [
27
]. Taguchi methods
align to lean-and-green goals in several aspects. The main tactic is the drastic reduction in
trials by resorting to fractional factorial designs (FFDs) [
25
,
27
]. A second tactic consolidates
factorial screening and parameter optimization in a single phase even though they are
theoretically of different and distinct scope. A third tactic is that it minimizes the number
of factorial settings that are needed to profile and optimize the investigated characteristic
responses. A fourth conditional tactic is the designing of experiments without provisions
for trial run replications. Abou-Shady [
16
] showed that only nine pre-determined formu-
lations may be adequate to adjust four ED process factors to decontaminate wastewater
to become suitable for crop development in arid areas. The Taguchi-type optimization
scheme serves to collect concurrent information on factor strength along with the associ-
ated curvature effects. The characteristic guides are three regular water quality indices,
which are meaningful in agricultural engineering. From a scientific perspective, the Abou-
Shady [
16
] trials benefited from all mentioned tactics above. The trials were designed to
maximize factorial engagement for the allowable runs; the selected nine-run four-factor
non-linear orthogonal array (OA) scheme was saturated. Trial recipes were executed once
(unreplicated condition). Factorial screening and parameter optimization were conducted
in a single phase. The multi-response screening/optimization task was not carried out
because it is not available in the ordinary Taguchi methods. Finally, factorial strength
and curvature effects were pre-designed by selecting the proper experimental plan. A
second complication that had remained to be resolved was the quantification of the hidden
Water 2022,14, 1990 3 of 26
experimental uncertainty. It is a difficult subject and has become a theme of intense research
for a long time [
42
]. The problem of working with unreplicated schemes is not foreign to
agricultural research [
43
–
45
]. There have been some attempts to exploit the benefit that
unreplication offers in cost and time savings, but the data conversion process requires more
specialized approaches. Most commercial software packages provide a statistical analysis
for unreplicated–saturated FFD-collected datasets through one or more of the three options:
(1) the half-normal plot [
46
], (2) the Box–Meyer method [
47
] and (3) the Lenth method [
48
].
Nevertheless, there seem to be disagreements in predictions among methods and software
packages [
49
]. Some prediction disparities may be attributed to deeper theoretical issues,
as in the case regarding the use of the half-normal plot [
50
]. Others may be identified in
unavoidably counting on auxiliary ‘pseudo-error’ estimations, or even in tentative prior
probabilities for the active factor group; speculative variance inflation factor estimations
may be part of the solver input. Irrespective of the hindrances and bad practices in ex-
ecuting DOE projects [
51
], DOE is a worthwhile empirical vehicle that drives to better
understanding large industrial-level processes because it is couched on simplicity [
52
,
53
].
The analysis process is a bit more complicated if a concurrent optimization is expected
that implicates several response variables [
54
]. Chemical qualimetrics have recognized the
contribution of multi-factorial screening for several characteristics as well as the influence
of the Taguchi methods to systematize laboratory trials in chemometrics [
55
,
56
]. Classical
analysis of variance (ANOVA) or general linear modeling (GLM) may not be readily de-
ployable techniques in unreplicated–saturated FFD schemes [
25
,
57
]. A prevailing notion
is to certainly assess predictions not only from the viewpoint of the statistical inferential
methods, but also to compare them to solutions that have been obtained from algorithmic
routines [58].
The unreplicated–saturated non-linear multi-response ED dataset of Abu-Shady [
16
]
was analyzed using descriptive and graphical Taguchi methods; a ‘one-response-at-a-time’
analysis is a rudimentary data reduction phase in aquametrics. There were further attempts
to add significance to the original experimental outcomes using: (1) a non-parametric multi-
factorial technique [
59
], (2) a combination method of a micro-clustered dimension-reduction
with rank learning [
60
] and (3) a combination method of unsupervised clustering with
entropic methods [
61
]. The proposed work implements a recently published algorithmic
classification method, the bionic swarm intelligence (DBSc) method [
62
], to structured multi-
dimensional mini datasets. DBSc has been developed to classify large multi-dimensional
datasets. However, the application of DBSc in this work is tested in the other end of the
spectrum, i.e., for screening a wastewater recycling ED process by classifying unreplicated–
saturated non-linear multifactorial multi-response FFD-sampled datasets. The new feature
is that no other post-processing is required besides the usual solution check through internal
validation. The advantage of adopting DBSc for clustering mini-data is multifaceted, and it
stems from the inner-workings of the algorithm itself: (1) the swarm intelligence searching
procedure, (2) optimized targeting in the absence of a definite global objective function,
(3) the upgraded non-parametric annealing driver and (4) the Nash equilibrium process
objective reachable by game theory and circumstantial symmetry. The specific innovative
intervention is that it is posited that the dominant factorial setting sequence(s) mirror
the clustered mini-dataset to a degree that it is internally measurable. The matching
of the two partitioned sequences, designed runs and clusters is monitored by internal
validity measures (Dunn Index [
63
], Davies–Bouldin Index [
64
] and Rand Index [
65
]).
Hyper-parameter screening for the fine-tuning of the DBSc routine is performed before
completing the wastewater recycling process screening. Hyper-parameter screening [
66
,
67
]
is a prophylactic step to ensure the optimal operation of the algorithm on the particular ED
process mini-dataset. To highlight the innovative aspects of this work, the concept’s main
features are summarized as follows:
1.
The approach attempts to nest the validated cluster solution to the individual factorial
recipe patterns.
Water 2022,14, 1990 4 of 26
2.
The unsupervised dimension reduction in the examined multiple aquametric indices
is independent of individual index performance goals.
3.
Self-organizing and emergent influences appear first from intermingling and labeling
multiple characteristic responses.
4.
Matching validated clusters to factorial recipe patterns circumvents the need to gauge
against a statistical reference law.
5.
The technique is operable on saturated–unreplicated multi-response multi-factorial
mini-datasets, in spite of standard aqua-qualimetrical treatments failing to return a
significance estimate.
6.
Easy-to-follow and trusted analysis steps are used because openly-sourced visual
tools and state-of-the-art artificial intelligence R-packages are deployed.
Extensive internal cluster validity comparisons are provided. Openly available DOE
and clustering tools are implemented on the freeware platform R for faster and more
reliable data-centric decision making [
68
–
70
]. The outcomes are discussed against those
results from previous published research.
2. Materials and Methods
2.1. Double Orthogonal Screening for Vital Factors and Solver Hyper-Parameters
A regular OA trial planner systematizes the progress of an accelerated product/process
screening/optimization study by compressing the total volume of experiments through the
balanced ‘tight-fitting’ of the scheduled factor setting combinations [
25
,
27
]. An OA sampler
is a prescribed matrix that standardizes the minimum number (n) of the formulated trial
recipes for a group of as many as minvestigated effects. The sampling tactics in this work
require two types of related screenings: (1) a saturated non-linear OA trial planner to
profile the hierarchical influence of the examined factors and (2) a mixed-structure OA
sampler to identify favorable hyper-parameter adjustments for the databionic algorithmic
solver [
62
]. The saturated non-linear OA trial planner is aimed to organize the experimental
runs in such a manner as to deliver simultaneous information on the potency status effect
and on the curvature tendencies for each of the investigated effects separately. Therefore,
an OA-based experimentation tactic speeds up the data-centric decision-making process
while demanding minimal research effort. During such a brief endeavor, the high-density
dataset configuration may also include a zero-replication regimen; it is the condition
of unreplication.
The non-linear minimal-data OA scheme for exemplifying the Taguchi L
9
(3
4
) matrix
test case is studied in its ‘full-utilization’ pattern, i.e., by exploiting the maximum input
assignment of four controlling factors [
26
,
27
]. This plan produces a total of nine data entries
per recorded characteristic response. Thus, the data requirements are maintained to a
minimum, because of the concurrent conditions of trial unreplication and factor saturation.
In its general representation with coded factor settings, the original Taguchi-type L
9
(3
4
)
OA, which is highlighted in this work, is shown in Table 1.
Table 1. The coded three-level four-factor Taguchi-type L9(34) OA.
Run # Factor A Factor B Factor C Factor D
11 1 1 1
21 2 2 2
31 3 3 3
42 1 2 3
52 2 3 1
62 3 1 2
73 1 3 2
83 2 1 3
93 3 2 1
Water 2022,14, 1990 5 of 26
On the other hand, effectively implementing an unsupervised solver on cluster-partitioning
tasks inherently requires fine-tuning feedback from its principal hyper-parameter gamut [
66
,
67
].
Consequently, an L
8
(4
1×
2
2
) OA was re-arranged to probe the stability of the implemented
databionic swarm intelligence analyzer on explaining the particular L
9
(3
4
) OA mini-
dataset [
26
,
27
]. The proposed L
8
(4
1×
2
2
) OA scheme, which mixes three categorical
hyper-parameter variables, each available in nominal scaling, is shown in Table 2; the
hyper-parameter settings are listed in coded form.
Table 2.
The coded mixed-level Taguchi-type L
8
(4
1×
2
2
) OA for algorithmic hyper-parameter screening.
Run # Hyper-Parameter A Hyper-Parameter B Hyper-Parameter C
11 1 1
21 2 2
32 1 2
42 2 1
53 2 1
63 1 2
74 2 2
84 1 1
Nevertheless, there is an intrinsic relationship between the two distinct OA-based
screenings. The controlling fractional factorial L
9
(3
4
) OA generated the multi-response
mini-dataset. The unsupervised clustering was firstly applied on the OA mini-dataset, and
then the cluster-labelling results from the hyper-parameter (mixed OA) screening were
obtained. Finally, the algorithmic stability of the clustered data predictions was internally
checked on the structured and pre-labeled factorial (columns) vectors of the generically
coded L
9
(3
4
) OA scheme. This stability needs to be individually checked across all m
controlling factors. Thus, the overall concept is naive and easy to comprehend. The examined
controlling factors are labeled as:
Xj
for 1
≤
j
≤
m(m
N), and their respective factor settings
are correspondingly coded as x
ij
(x
ij
N) for 1
≤
i
≤
n(n
N) and 1
≤
j
≤
m. Particularly,
a non-linear OA scheme is pre-formed with k
j
levels for the jth factor (1
≤
j
≤
m) and
3
≤
k
j≤
K
j
(K
j
N), as shown in Table 3. Hence, the non-linear OA matrix is an input ordered
array that produces an output matrix {r
ic
} with
Rc
multiple characteristic responses, with
1≤i≤nand 1 ≤c≤L (L N); each cth matrix column is a single response vector.
Table 3. A general arrangement of an OA matrix, and its cluster-labelled output.
Controlling Factors (Input) Multiple Characteristics (Output) Labelled Clusters
run # X1X2...Xm
1x11 x12 ...x1m
2x21 x22 ...x2m
. . . .
. . . . . . .
. . . .
nxn1xn2...xnm
→
run # R1R2...RL
1r11 r12 ...r1L
2r21 r22 ...r2L
. . . .
. . . . . . .
. . . .
nrn1rn2...rnL
→
run # Id
1l1
2l2
. .
. .
. .
nln
The mini-dataset is converted to a cluster vector,
Id
, which is labelled into a total
number of Z clusters with cluster members:
li
| 1
≤li≤
Z (Z
N) and 1
≤
i
≤
n. Since the
Xj
vectors in an OA provide probable ‘pre-clustering’ information through the locations of
their pre-defined factor settings, they may be directly tested individually for accuracy in
comparison to the final cluster identification vector
Id
. The hyper-parameters that favor
accuracy amelioration are retained to permit the prediction of the final solution.
2.2. The Databionic Swarm Intelligence Classifier
The orthogonal multi-dimensional L
9
(3
4
) OA mini-dataset is morphed into a single
cluster identification vector by employing the databionic swarm intelligence classifier
(DBSc) [
62
]. The DBSc data conversion engine is benefited from several attractive built-in
Water 2022,14, 1990 6 of 26
processor features, which may also promote the effective cluster member fingerprinting of
high-density mini-datasets. Strictly speaking, the DBSc is geared toward cluster analysis
for large high-dimensional datasets. Notably, the DBSc-extracted information is assembled
from three conspicuous nature-inspired agents such as: (1) emergence, (2) self-organization
and (3) swarm intelligence. Theoretically, DBSc may also be applicable, in a broader sense,
to non-linear OA-apportioned multi-response mini-datasets. Two particularly intriguing
traits of the DBSc data-conversion engine render its adoption highly desirable to OA-
generated multi-response multifactorial mini-datasets: (1) the swarm intelligence searching
procedure, which seeks optimized target states in the absence of an established global
objective function and (2) the upgraded non-parametric annealing driver, which is astutely
impelled toward a Nash equilibrium, and hence is guided by solid game theory and
symmetry considerations. DBSc has been designated for this study, instead of other more
familiar clustering methods, because it is the only unsupervised classifier that manages
to fingerprint previously uncharacterized data. Briefly, the DBSc classifier comprises
three sequential co-processors: (1) a parameter-free projection engine, (2) a parameter-free
high-dimensional data visualization facility and (3) a zero-sensitivity cluster membership
identifier. However, it is reiterated that the important caveat in this intended application is
attributed to the inherent ‘data smallness’. Unequivocally, the classifier visualization phase
may not be responsive to the generated ‘small-and-dense’ OA dataset. The topographic
map, which is an essential component of the DBSc solver, may be restricted in providing
enough databot details that visually ascertain the aptness standing of the final cluster
formations. Irrespective of the extent of the usability of the topographic map in clustering
OA data, the DBSc engine advances to recommend a dendrogram and a final identification
prediction of the cluster memberships. Probing the stability of the DBSc routine becomes
advisable under such previously unexplored conditions. Therefore, three hyper-parameters
reasonably emerged in congruence to modulating the cluster prediction accuracy. The
three DBSc hyper-parameters were: (1) the type of the distance measure, (2) the structure
type of clusters and (3) the position matrix preference. Since this is a first examination on
the subject, the types of distance measures were limited to a group of ordinary candidate
models. They were mainly chosen because the computing of their non-metric multi-
dimensional scaling values was available by popular freeware modules in data science. For
demonstrating purposes, the well-known Shepard–Kruskal approach was employed to
furnish information on dissimilarity trends with the Shepard diagram [
71
] and to assist
in differentiating between two nominated cluster numbers; their Kruskal stresses [
72
]
were also computed and compared. With a lack of prior studies, it is worthwhile to start
exploring how five of the most widely implemented continuous distance measure models
impact the OA mini-data dissimilarity estimations: (1) the Euclidean distance, (2) the
Maximum/Chebyshev distance, (3) the Manhattan distance, (4) the Canberra distance
and (5) the Minkowski distance. After a pre-screening, an attempt was made to shorten
the initial list of distance measure models. The four proposed distance measure models
for the hyper-parameter screening were (Table 4): (1) the Euclidean distance, (2) the
Maximum/Chebyshev distance, (3) the Manhattan distance and (4) the Canberra distance.
Moreover, ‘compact’ and ‘connected’ structure types of clusters were altered in running
the DBSc module [
73
]. Finally, the position options were set as either ‘Projected Points’
(automatic clustering projection from the Pswarm algorithm) or as ‘Best Matches’ (involving
the ‘GeneralizedUmatrix’ module).
Water 2022,14, 1990 7 of 26
Table 4. The Taguchi-type L8(41×22) OA arrangement for algorithmic hyper-parameter screening.
Run # Distance Measure Structure Type Position Type
1Euclidean Connected Best Matches
2Euclidean Compact Projected Points
3Maximum Connected Projected Points
4Maximum Compact Best Matches
5Manhattan Compact Best Matches
6Manhattan Connected Projected Points
7Canberra Compact Projected Points
8Canberra Connected Best Matches
2.3. The Water Quality Case Study
To elucidate databionic double-screening on a crucial data-centric application, a pub-
lished wastewater treatment optimization study was revisited [
16
]. The great importance
of the selected case study rests on several key aspects: (1) a newly designed electrodial-
ysis apparatus was launched, (2) the intended purpose of the research was to improve
the recycling process of wastewater stocks in order to supply arid/semi-arid areas with
agricultural irrigation water, (3) the ED cell dialyzed a great variety of polluted water
resources from different locations and origins, (4) the wastewater properties could not be
assumed homogeneous, owing to the underlying complex chemical systems, (5) a struc-
tured non-linear orthogonal mini-dataset was planned and gathered, (6) a triplet of water
quality indices were to be concurrently screened/optimized, (7) the Taguchi-type OA sam-
pler generated a one-time dataset to satisfy the synchronous screening and optimization
tasks, (8) the optimized water quality indices were to achieve agricultural conformity
levels comparable to those necessitated to crop nourishment, yield and safety, (9) the con-
straints of trial unreplication and factor saturation were present in the experimental plan,
(10) the recommended Taguchi-type SNR data conversion was not applicable due to the
unreplication constraint, (11) the multi-response dataset was in percentage or ratio form,
i.e., dimensionless and bounded on both ends of the scale, (12) the remaining unexplainable
experimental error was not feasible to be recovered from standard analysis methods and
(13) the overall unique nature of the application.
The trial-planning scheme was a four-factor three-level nine-run L
9
(3
4
) OA, and it
was executed once. The implemented OA sampler was fully exploited and was thus
saturated with the following four controlling factors: (1) A: the dilute flow; (2) B: the
cathode flow; (3) C: the anode flow; and (4) D: the voltage rate [
16
]. The dataset consisted
of the responses of the three water quality indices: (1) RS: the removed sodium content (%);
(2) SAR: the sodium adsorption ratio; and (3) SSP: the soluble sodium percentage (%). There
are previous comments on the investigated water quality indices and their pivotal value in
the cultivation management of productive crops [
60
]. The dataset is described in ref. [
16
];
the entire input–output arrangement is in conjunction to Tables 3and 4, respectively.
2.4. The Methodological Outline
The proposed methodology may be recapitulated as follows:
(1)
Determine the relevant water quality characteristics that represent the wastewater
recycling efficiency performance with respect to the specific agricultural application.
(2)
Select a group of ED process controlling factors.
(3)
Determine the minimum group of factor settings that covers the operational range
and allows for the detection of potential characteristic non-linearities.
(4)
Program lean trials by implementing the proper OA-sampling scheme that accommo-
dates non-linear trend detection.
(5) Execute the prescribed OA runs (step 4), and gather the multi-characteristic mini-dataset.
(6)
Pre-screen each characteristic response using visual information from the boxplot
(original/notched) [
74
], the adjusted boxplot [
75
], the violin plot [
76
] and the bean
plot [77].
Water 2022,14, 1990 8 of 26
(7) Determine the number of candidate (mini-dataset) clusters by pre-screening the available
distance measures using the Shepard plot trends and the Kruskal stress estimations.
(8)
Shortlist the distance measures from step 7 if there is evidence that their dissimilarity
estimations are not as competitive as the alternative candidate models.
(9)
Apply OA screening on the algorithmic hyper-parameters of the databionic swarm
intelligence analyzer [62,73].
(10)
Assess the best partitioning performance of the clustered mini-dataset against the
original controlling factorial OA pre-labeled settings (individual testing of the
Xj
vectors and the final cluster identification vector Id).
(11) Obtain the cluster dendrogram and the solver-labelled clusters of the mini-OA-dataset.
(12) Evaluate cluster similarity using the Rand Index [
65
] across different hyper-parameter
screenings; the proximity of the clustering results to the controlling factorial OA
pre-settings is facilitated by the estimation of, respectively, the Dunn Index [
63
] and
the Davies–Bouldin Index [64].
(13)
Provide a hyper-parametrized hierarchy for the profiled factorial landscape.
2.5. The Computational Aids
All required computational work was conducted on the free statistical analysis soft-
ware platform R (v. 4.1.3) [
69
]. The non-linear L
9
(3
4
) OA array was constructed using the
module ‘param.design()’ from the R-package ‘DoE.base’ (v. 1.2). Similarly, the L
8
(4
2×
2
2
)
OA was prepared to setup and carry out the hyper-parameter screening process; it is an
assessment of the DBSc-based cluster membership accuracy against a pre-selected group
of distance measures, the DBSc projection procedures and the structure type of clusters.
The module ‘boxplot()’ was used from the R-package ‘graphics’ (v. 4.1.3) to obtain regular
and notched visuals for the summary statistics of the three water quality characteristics.
The module ‘adjbox()’ in the R-package ‘robustbase’ (v. 0.93-9) provided the corresponding
adjusted boxplot depictions of the three responses. To prepare the customized violin and
bean plots, the corresponding R-packages ‘vioplot()’ (v. 0.3.7) and ‘beanplot()’ (v. 1.2) were
employed. Moreover, to assemble the recommended assortment of Shepard graphs in
order to pre-assess the available options for the distance measures (‘euclidean’, ‘maximum’,
‘manhattan’, ‘canberra’ or ‘Minkowski’), the modules ‘isoMDS()’ and ‘Shepard()’ were uti-
lized from the R-package ‘MASS’ (v. 7.3-55); two k-dimensional cluster configurations were
obtained from their generated (non-metric multi-dimensionally scaled) distance matrices,
along with their respective computed Kruskal stresses. The self-organized clustering of the
three water-quality-index mini-datasets was accomplished through the implementation
of the swarm intelligence machinery, the R-package ‘DatabionicSwarm’ (v. 1.1.5) [
73
].
Furthermore, it furnished the ‘ClusteringAccuracy()’ and the ’DelaunayClassificatioError()’
modules in order to estimate the normalized aggregation of all fingerprinted true-positive
data points in addition to finding the first k-nearest Delaunay neighbors in the input space,
and subsequently in order to evaluate them in the output space. The dendrogram visu-
alization and the labelled cluster list was generated from the module ‘DBSclustering()’.
To coordinate the swarm-based databot projections, which were also required in half of
the hyper-parameter screening runs, the module ‘Pswarm()’ was executed to complete the
2D reduction (polar) mapping. The parallel distance matrix computation using multiple
threads was facilitated by the R-package ‘parallelDist’ (v. 0.2.6). The R-package ‘Gen-
eralizedUmatrix’ (v. 1.2.2) supplied the ‘Bestmatches’ positions as the alternative to the
‘ProjectedPoints’, which were estimated by intermediate distance matrix processing using
the ‘GeneratePswarmVisualization()’ module. The validation status of the cluster partition-
ing performance of the water quality index mini-dataset against the structured factorial
OA formulations was conducted by estimating the Rand Index, the Dunn Index and the
Davies–Bouldin Index, applying the modules accordingly: (1) ‘rand.index()’ (R-package
‘fossil’ (v. 0.4.0)), (2) ‘dunn()’ (R-package ‘clValid’ (v. 0.7)) and ‘index.DB()’ (R-package
‘clusterSim’ (v. 0.49-2)).
Water 2022,14, 1990 9 of 26
3. Results
3.1. Graphical Pre-Screening of the Three Water Quality Indices
The data pre-screening starts with the portrayals of the responses of the three water
quality indices with respect to the three adaptations of boxplots: (1) the original, (2) the
notched and (3) the adjusted. It can be immediately noticed from Figure 1that the original
and the adjusted versions of the boxplot do not agree with the removed sodium content (RS)
dataset. The lone outlier point appears to be situated on opposite sides in the two graphs.
Inspecting the adjusted boxplot suggests that the RS dataset is rather asymmetrical, and it
may be worth exploring further with other techniques. This is focal because impending
data non-normality may render outlier detection arbitrary. The original and adjusted
boxplots for the sodium adsorption ratio (SAR) and the soluble sodium percentage (SSP),
when contrasted, exhibit matching behavior in both cases. Moreover, from Figure 1, it
is also evident that the (original boxplot) interquartile ranges for all three water quality
indices coincide with the 95% confidence interval for the medians (notched boxplots). This
is perceived as a substantial variation in locating a robust central tendency for all three
examined water quality indices, and it is usually attributed to the small sample limitation.
In Figure 2, the original boxplot, the violin plot and the bean plot provide additional
comparative information for the response dispositions. The violin and the bean plots for
the RS dataset seem to affirm the hint for the presence of an asymmetrical spread in the
original/adjusted boxplots; the lower values of the data distribution may perhaps uphold a
wider tail formation. Individual outliers are not expected to be detectible in the violin plots.
The adjusted boxplot and the bean plot agree that the outlier point is on the lower end of
the distribution. Multimodality for this RS dataset may not be ruled out according to its
profile in the corresponding bean plot. In comparison, the estimated kernel density trends
in the violin plot of the SAR dataset reveal a skewness that is barely visible in the original
boxplot. It is, rather, credited to a higher concentration of datapoints below the mean value
in the bean plot. Finally, the respective violin and bean plots for the SSP dataset depict a
symmetric spread around the location point, in agreement to the two boxplots; the nature
of the distribution tails [78] still remains not transparent.
Water 2022, 14, x FOR PEER REVIEW 10 of 26
Figure 1. Original, notched and adjusted boxplots for the three water quality indices: (1) RS (first
row), (2) SAR (second row) and (3) SSP (third row).
Figure 2. Boxplots, violin plots and bean plots for the three water quality indices: (1) RS (first row),
(2) SAR (second row) and (3) SSP (third row).
3.2. Graphical Pre-Screening of the Candidate Distance Measures
Choosing a suitable distance measure for the Databionic Swarm classifier is simpli-
fied by a naïve exploration of the group of the nominated distance measures in two cluster
number settings of relevance. Since the multi-response dataset was programmed to de-
liver information at three preset adjustments, for each probed factor, the dominant fac-
tor(s) are netted in three-cluster configurations if they are indeed non-linear, or otherwise
in two-cluster reduced form if they behave linearly. Therefore, cluster number settings of
2 and 3 were decided to be tested. Moreover, the candidate distance measures were indic-
atively selected: (1) the Euclidean distance, (2) the Maximum/Chebyshev distance, (3) the
Manhattan distance, (4) the Canberra distance and (5) the Minkowski distance. For the
Minkowski distance measures, the model parameter p was tested at values 2 and 4,
Figure 1.
Original, notched and adjusted boxplots for the three water quality indices: (1) RS (
first row
),
(2) SAR (second row) and (3) SSP (third row).
Water 2022,14, 1990 10 of 26
Water 2022, 14, x FOR PEER REVIEW 10 of 26
Figure 1. Original, notched and adjusted boxplots for the three water quality indices: (1) RS (first
row), (2) SAR (second row) and (3) SSP (third row).
Figure 2. Boxplots, violin plots and bean plots for the three water quality indices: (1) RS (first row),
(2) SAR (second row) and (3) SSP (third row).
3.2. Graphical Pre-Screening of the Candidate Distance Measures
Choosing a suitable distance measure for the Databionic Swarm classifier is simpli-
fied by a naïve exploration of the group of the nominated distance measures in two cluster
number settings of relevance. Since the multi-response dataset was programmed to de-
liver information at three preset adjustments, for each probed factor, the dominant fac-
tor(s) are netted in three-cluster configurations if they are indeed non-linear, or otherwise
in two-cluster reduced form if they behave linearly. Therefore, cluster number settings of
2 and 3 were decided to be tested. Moreover, the candidate distance measures were indic-
atively selected: (1) the Euclidean distance, (2) the Maximum/Chebyshev distance, (3) the
Manhattan distance, (4) the Canberra distance and (5) the Minkowski distance. For the
Minkowski distance measures, the model parameter p was tested at values 2 and 4,
Figure 2.
Boxplots, violin plots and bean plots for the three water quality indices: (1) RS (
first row
),
(2) SAR (second row) and (3) SSP (third row).
3.2. Graphical Pre-Screening of the Candidate Distance Measures
Choosing a suitable distance measure for the Databionic Swarm classifier is simplified
by a naïve exploration of the group of the nominated distance measures in two cluster
number settings of relevance. Since the multi-response dataset was programmed to deliver
information at three preset adjustments, for each probed factor, the dominant factor(s)
are netted in three-cluster configurations if they are indeed non-linear, or otherwise in
two-cluster reduced form if they behave linearly. Therefore, cluster number settings of
2 and 3 were decided to be tested. Moreover, the candidate distance measures were in-
dicatively selected: (1) the Euclidean distance, (2) the Maximum/Chebyshev distance,
(3) the Manhattan distance, (4) the Canberra distance and (5) the Minkowski distance. For
the Minkowski distance measures, the model parameter pwas tested at values 2 and 4,
respectively. The measure parameter pset at a value of two coincides with the Euclidean
distance measure definition. Figure 3provides visual differences for the tendencies of the
configuration distances against their similarities for distance measures and both cluster
numbers using the illuminating Shepard graphs. It is immediately obvious that the Eu-
clidean distance performs more optimally than the rest of the distance models, and it does
not discriminate between cluster numbers. As expected, the same behavior is observed for
the Minkowski model set at p= 2. However, for p= 4, the similarity-tracking tendencies
dramatically deteriorate. A general observation is that the measure-based similarity effi-
ciency favors the three-cluster option. The same conclusion is reached by computing the
corresponding Kruskal stresses for the 12 plots in Figure 3. In Table 5, it becomes evident
that the Euclidean distance measure (Minkowski model at p= 2) drastically minimizes the
Kruskal stresses. Stepping down from a three-cluster to two-cluster model, it is accompa-
nied by a 35% increase in the magnitude of its Kruskal stress prediction. The three-cluster
partitioning consistently minimizes the Kruskal stress estimations in all examined distance
measure models. There is a significant difference in the Kruskal stress estimations using
the Minkowski model set at p= 2 and p= 4.
Water 2022,14, 1990 11 of 26
Table 5. Kruskal stress pre-screening for the distance measures at two cluster number settings.
Distance Measure Cluster Number Kruskal Stress Value
Euclidean 2 7.35 ×10−14
Euclidean 3 5.44 ×10−14
Maximum 2 2.28
Maximum 3 0.72
Manhattan 2 1.96
Manhattan 3 5.73 ×10−3
Canberra 2 8.67 ×10−3
Canberra 3 7.31 ×10−3
Minkowski (p= 2) 2 7.35 ×10−14
Minkowski (p= 2) 3 5.44 ×10−14
Minkowski (p= 4) 2 6.75 ×10−3
Minkowski (p= 4) 3 2.18 ×10−3
Water 2022, 14, x FOR PEER REVIEW 11 of 26
respectively. The measure parameter p set at a value of two coincides with the Euclidean
distance measure definition. Figure 3 provides visual differences for the tendencies of the
configuration distances against their similarities for distance measures and both cluster
numbers using the illuminating Shepard graphs. It is immediately obvious that the Eu-
clidean distance performs more optimally than the rest of the distance models, and it does
not discriminate between cluster numbers. As expected, the same behavior is observed
for the Minkowski model set at p = 2. However, for p = 4, the similarity-tracking tendencies
dramatically deteriorate. A general observation is that the measure-based similarity effi-
ciency favors the three-cluster option. The same conclusion is reached by computing the
corresponding Kruskal stresses for the 12 plots in Figure 3. In Table 5, it becomes evident
that the Euclidean distance measure (Minkowski model at p = 2) drastically minimizes the
Kruskal stresses. Stepping down from a three-cluster to two-cluster model, it is accompa-
nied by a 35% increase in the magnitude of its Kruskal stress prediction. The three-cluster
partitioning consistently minimizes the Kruskal stress estimations in all examined dis-
tance measure models. There is a significant difference in the Kruskal stress estimations
using the Minkowski model set at p = 2 and p = 4
(A)
(B)
Figure 3. Cont.
Water 2022,14, 1990 12 of 26
Water 2022, 14, x FOR PEER REVIEW 12 of 26
(C)
(D)
(E)
Figure 3. Cont.
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(F)
(G)
(H)
Figure 3. Cont.
Water 2022,14, 1990 14 of 26
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(I)
(J)
(K)
Figure 3. Cont.
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Water 2022, 14, x FOR PEER REVIEW 15 of 26
(L)
Figure 3. Shepard graphs of the three water quality index datasets with respect to the five measure
types at the two cluster number settings, k: (1) Euclidean distance ((A): k = 2, (B): k = 3), (2) Maximum
distance ((C): k = 2, (D): k = 3), (3) Manhattan distance ((E): k = 2, (F): k = 3), (4) Canberra distance
((G): k = 2, (H): k = 3), (5) Minkowski distance for p = 2 (I): k = 2, (J): k = 3) and (6) Minkowski distance
for p = 4 ((K): k = 2, (L): k = 3).
Table 5. Kruskal stress pre-screening for the distance measures at two cluster number settings.
DISTANCE MEASURE
CLUSTER NUMBER
KRUSKAL STRESS VALUE
EUCLIDEAN
2
7.35×10−14
EUCLIDEAN
3
5.44×10−14
MAXIMUM
2
2.28
MAXIMUM
3
0.72
MANHATTAN
2
1.96
MANHATTAN
3
5.73×10−3
CANBERRA
2
8.67×10−3
CANBERRA
3
7.31×10−3
MINKOWSKI (p = 2)
2
7.35×10−14
MINKOWSKI (p = 2)
3
5.44×10−14
MINKOWSKI (p = 4)
2
6.75×10−3
MINKOWSKI (p = 4)
3
2.18×10−3
3.3. Unsupervised Multi-Response Screening Using the Databionic Classifiier
Based on the pre-screening information of Sections 3.1 and 3.2, it was decided to ex-
amine in more depth the behavior of the four distance measures: (1) the Euclidean dis-
tance, (2) the Maximum/Chebyshev distance, (3) the Manhattan distance and (4) the Can-
berra distance in the unsupervised multi-response screening output using the Databionic
classifier. A three-cluster-partition setup was maintained constant in all trial runs. To in-
vestigate the influence of the three proposed hyper-parameters on the grouping accuracy
of the DBSc, the L8(41*22) OA arrangement (Table 4) was conducted. The eight resulting
dendrograms are shown in Figure 4. The screening process generated grouping patterns
that follow different pairing pathways for different hyper-parameter combinations. How-
ever, from Table 6, out of the eight executed recipes, only three outcomes are distinct. Six
outcomes produced the same membership groupings (runs #3–#8). The outcomes from
runs #1 and #2 differentiate between them in four membership identifications. The
Figure 3.
Shepard graphs of the three water quality index datasets with respect to the five measure
types at the two cluster number settings, k: (1) Euclidean distance ((
A
): k = 2, (
B
): k = 3), (2) Maximum
distance ((
C
): k = 2, (
D
): k = 3), (3) Manhattan distance ((
E
): k = 2, (
F
): k = 3), (4) Canberra distance
((
G
): k = 2, (
H
): k = 3), (5) Minkowski distance for p=2(
I
): k = 2, (
J
): k = 3) and (6) Minkowski distance
for p= 4 ((K):k=2,(L): k = 3).
3.3. Unsupervised Multi-Response Screening Using the Databionic Classifiier
Based on the pre-screening information of Sections 3.1 and 3.2, it was decided to exam-
ine in more depth the behavior of the four distance measures: (1) the Euclidean distance,
(2) the Maximum/Chebyshev distance, (3) the Manhattan distance and (4) the Canberra
distance in the unsupervised multi-response screening output using the Databionic classi-
fier. A three-cluster-partition setup was maintained constant in all trial runs. To investigate
the influence of the three proposed hyper-parameters on the grouping accuracy of the
DBSc, the L
8
(4
1×
2
2
) OA arrangement (Table 4) was conducted. The eight resulting den-
drograms are shown in Figure 4. The screening process generated grouping patterns that
follow different pairing pathways for different hyper-parameter combinations. However,
from Table 6, out of the eight executed recipes, only three outcomes are distinct. Six out-
comes produced the same membership groupings (runs #3–#8). The outcomes from runs
#1 and #2 differentiate between them in four membership identifications. The sequence
outcome from run #2 does not agree in only one position with respect to the rest of the runs
(runs #3–#8).
Table 6. Cluster membership identifications from screening the three DBSc hyper-parameters.
Cluster Member Identification
Hyper-Parameter Screening Run # Run #1 Run #2 Run #3 Run #4 Run #5 Run #6 Run #7 Run #8 Run #9
11 1 2 1 3 1 1 1 1
2112121333
31 1 2 1 2 1 3 1 3
4112121313
51 1 2 1 2 1 3 1 3
6112121313
71 1 2 1 2 1 3 1 3
8112121313
Water 2022,14, 1990 16 of 26
Water 2022, 14, x FOR PEER REVIEW 16 of 26
sequence outcome from run #2 does not agree in only one position with respect to the rest
of the runs (runs #3–#8).
(A)
(B)
(C)
Figure 4. Cont.
Water 2022,14, 1990 17 of 26
Water 2022, 14, x FOR PEER REVIEW 17 of 26
(D)
(E)
(F)
Figure 4. Cont.
Water 2022,14, 1990 18 of 26
Water 2022, 14, x FOR PEER REVIEW 18 of 26
(G)
(H)
Figure 4. The eight dendrogram outputs from the hyper-parameter OA screening (Table 4). (A)
Run #1, (B) Run #2, (C) Run #3, (D) Run #4, (E) Run #5, (F) Run #6, (G) Run #7 and (H) Run #8.
Table 6. Cluster membership identifications from screening the three DBSc hyper-parameters.
Cluster Member Identification
Hyper-Parameter
Screening Run #
Run
#1
Run #2
Run #3
Run #4
Run #5
Run #6
Run #7
Run #8
Run #9
1
1
1
2
1
3
1
1
1
1
2
1
1
2
1
2
1
3
3
3
3
1
1
2
1
2
1
3
1
3
4
1
1
2
1
2
1
3
1
3
5
1
1
2
1
2
1
3
1
3
6
1
1
2
1
2
1
3
1
3
7
1
1
2
1
2
1
3
1
3
8
1
1
2
1
2
1
3
1
3
Using the DBSc’s ‘ClusteringAccuracy()’ function, the clustering result sequence was
contrasted against each of the four factorial setting sequences as prescribed in L9(34) OA.
Figure 4.
The eight dendrogram outputs from the hyper-parameter OA screening (Table 4). (
A
) Run
#1, (B) Run #2, (C) Run #3, (D) Run #4, (E) Run #5, (F) Run #6, (G) Run #7 and (H) Run #8.
Using the DBSc’s ‘ClusteringAccuracy()’ function, the clustering result sequence was
contrasted against each of the four factorial setting sequences as prescribed in L
9
(3
4
) OA. A
radar graph (Figure 5) summarizes all this information, from which there is no clear indica-
tion that some factor prevails over the others in affecting cluster membership predictions
for a particular hyper-parameter combination. It is noted that the radar graph encom-
passes synchronous information from both experimental designs, i.e., from
L9(34) OA
and
L8(41×22) OA
schemes. Similarly, a radar graph (Figure 6) was prepared where the Rand
Index evaluation of the eight hyper-parameter screening clustering solutions are laid out
for contrasting against the L
9
(3
4
) OA factorial setting sequences. Again, it is not conclusive
that one particular factor dominates the rest of the group. However, a bar chart (Figure 7)
of the Dunn Index performance across the eight hyper-parameter combinations reveals
that run #2 maximized internal consistency (compact clusters using Euclidean distance
measure with projected points). Cluster validity for all runs varied from
0.659 (run #2)
to 0.553 (runs #3–8). The only factorial (pre-set) partition pattern that may standout in
comparison to the three unsupervised cluster predictions is identified to factor A (dilute
Water 2022,14, 1990 19 of 26
flow), as the performance of the remaining three factors is close to a Dunn Index value
of zero for all hyper-parameter screening runs (Figure 7). It may be inferred that factor
A may simultaneously influence all three water quality indices. Irrespective of adopting
a Euclidean or non-Euclidean distance measure (Figure 8), the Delaunay classification
error is more comparable to the cluster predictions, and it is generally minimized when
the factorial partitioning for factor A is imposed on the mini-OA-dataset. Hence, it is
implied that a factorial classification may be feasible when the setting sequence of factor
A in the OA-planner nearly mimics the cluster solution, which is derived from the unsu-
pervised outcome; factor A adequately reflects back a visualization of the most accurate
two-dimensional projection of the three-response data system. Along these lines, what are
equally convincing are the two radar graphs for the Davies–Bouldin Index performances
across all eight run outcomes (Figure 9). The Davies–Bouldin Index has been prepared for
both centrotype cases: (1) medoids (Figure 9A) and (2) centroids (Figure 9B). The result is
similar for both versions: factor A appears to provide an approximate membership distri-
bution that tightly encircles the unsupervised cluster identification solution. Factor A is
capable of noticeably minimizing the Davies–Bouldin Index, and its OA setting groupings
may pose as sequence imitations. The remaining factorial performances are much more
distant to describing the cluster solution. The Dunn Index and the two versions of the
Davies–Bouldin Index (centrotype cases: medoids and centroids) estimations of the cluster
membership identification results are accumulated in Table 7. The Lenth test (function
‘LenthPlot’ in R-package BsMD (30 April 2020)) was used to complete the hyper-parameter
screening using the three internal validity responses (BDIc, BDIm and DI). In the Lenth
plots, the coded hyper-parameters are A: distance measure type; B: structure type; and
C: position type. It becomes apparent that none of the internal checks of cluster validity
promote a specific hyper-parameter setting in any of the three Lenth plots of Figure 10. This
means that hyper-parameter settings carry no critical statistical significance, and they may
be adjusted with respect to practicality. Therefore, the Euclidean distance measure is se-
lected due to its efficiency, as is demonstrated in Table 5. The simpler structure and position
types are also preferred. Hence, the two other hyper-setting selections that are assigned in
the final solution are: the ‘Compact’ structure and the ‘Projected Points’ position (Table 4).
The final clustering solution is identified for the sequence generated from run #2 (Table 6).
Consequently, the overall factorial screening prediction suggests the predominance of factor
A (dilute flow), as it is now clearly construed from Figures 7and 9. The final outcome
agrees with: (1) a non-parametric statistical method prediction [
59
], (2) a combination of
semi-unsupervised (silhouette method)/statistical method with confirmation data [
60
] and
(3) a combination of unsupervised (affinity propagation clustering)/information-entropic
methods with confirmation data [61].
Table 7.
Validity checks of the cluster identification results (Table 6) with Dunn Index (DI) and
Davies–Bouldin Index (BDIc: centroids, and BDIm: medoids).
Run # BDIc BDIm DI
10.416 0.721 0.605
20.732 0.839 0.659
30.416 0.721 0.474
40.766 0.912 0.544
50.766 0.912 0.574
60.416 0.653 0.531
70.766 0.912 1.022
80.766 0.912 1.022
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Water 2022, 14, x FOR PEER REVIEW 20 of 26
Figure 5. Radar graph of the clustering accuracy, estimated from the self-organized databionic
swarm intelligence algorithm.
Figure 6. Radar graph of the Rand Index for the hyper-parameter screening clustering solutions.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
A B C D
0
0.2
0.4
0.6
0.8
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
A B C D
Figure 5.
Radar graph of the clustering accuracy, estimated from the self-organized databionic swarm
intelligence algorithm.
Water 2022, 14, x FOR PEER REVIEW 20 of 26
Figure 5. Radar graph of the clustering accuracy, estimated from the self-organized databionic
swarm intelligence algorithm.
Figure 6. Radar graph of the Rand Index for the hyper-parameter screening clustering solutions.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
A B C D
0
0.2
0.4
0.6
0.8
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
A B C D
Figure 6. Radar graph of the Rand Index for the hyper-parameter screening clustering solutions.
Water 2022,14, 1990 21 of 26
Water 2022, 14, x FOR PEER REVIEW 21 of 26
Figure 7. Bar chart for the Dunn Index performance (factorial presetting vs. cluster partitioning).
Figure 8. Radar graph for DCE comparison of Euclidean and non-Euclidean distance measure per-
formance.
0.605
0.659
0.553
0.409
0.076 0.061 0.066
DBSC RUN#1 DBSC RUN#2 DBSC
RUNS#3-8
FACTOR A FACTOR B FACTOR C FACTOR D
0
1
2
3
4Cls
A
BC
D
EUCLIDEAN NON-EUCLIDEAN
Figure 7. Bar chart for the Dunn Index performance (factorial presetting vs. cluster partitioning).
Water 2022, 14, x FOR PEER REVIEW 21 of 26
Figure 7. Bar chart for the Dunn Index performance (factorial presetting vs. cluster partitioning).
Figure 8. Radar graph for DCE comparison of Euclidean and non-Euclidean distance measure per-
formance.
0.605
0.659
0.553
0.409
0.076 0.061 0.066
DBSC RUN#1 DBSC RUN#2 DBSC
RUNS#3-8
FACTOR A FACTOR B FACTOR C FACTOR D
0
1
2
3
4Cls
A
BC
D
EUCLIDEAN NON-EUCLIDEAN
Figure 8.
Radar graph for DCE comparison of Euclidean and non-Euclidean distance measure performance.
Water 2022,14, 1990 22 of 26
Water 2022, 14, x FOR PEER REVIEW 22 of 26
(A)
(B)
Figure 9. Radar graphs of the Davies–Bouldin Index for the centrotype cases: (A) medoids and (B)
centroids.
Table 7. Validity checks of the cluster identification results (Table 6) with Dunn Index (DI) and
Davies–Bouldin Index (BDIc: centroids, and BDIm: medoids).
RUN #
BDIc
BDIm
DI
1
0.416
0.721
0.605
2
0.732
0.839
0.659
3
0.416
0.721
0.474
4
0.766
0.912
0.544
5
0.766
0.912
0.574
6
0.416
0.653
0.531
7
0.766
0.912
1.022
8
0.766
0.912
1.022
0
2
4
6
8
10
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
Cls A B C D
0
2
4
6
8
Run#1
Run#2
Run#3
Run#4
Run#5
Run#6
Run#7
Run#8
Cls A B C D
Figure 9.
Radar graphs of the Davies–Bouldin Index for the centrotype cases: (
A
) medoids and
(B) centroids.
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Figure 10: Lenth method results for the direct factorial screening of the BDIc, BDIm and DI estima-
tions of Table 7 (function ‘LenthPlot’ in R-package ‘BsMD’ (30 April 2020)).
4. Conclusions
Improving the efficiency of wastewater separation processes is a very crucial subject,
especially when the applied technology is geared toward supplying large water volumes
for irrigating crops in arid regions. Remote locations may benefit from certain advantages
that the electrodialysis technique can offer. To regulate the efficiency of the wastewater
ED treatment process to a level at which the filtered water product is suitable for agricul-
tural usage, an empirical screening of the ED process, based on pertinent water quality
characteristics, is pivotal. No single theory can describe the complex phenomena that gov-
ern a custom-made ED separation process. This is because the ED design variables may
be unique to the particular application. In addition, the wastewater resources are antici-
pated to be of diverse origins with unpredictable non-homogeneous chemical properties.
Consequently, there is an urgency from an engineering perspective to facilitate quick and
economic ED process profiling. Diagnosing palpably strong factors eases the concurrent
adjustment of several operational quality characteristics to an optimal level. A newly pub-
lished artificial intelligence method, the databionic swarm intelligence classifier, was at-
tempted to be adapted to deal with a constrained (small) aquametric dataset. The non-
parametric self-organized search for an ‘objective-free’ Nash equilibrium, which is tracked
by emerging databots, may be doable in classifying the multi-response output from high-
density OA trials. Implementing fractional factorial designs downsizes the sampling re-
quirements to a mini-dataset. In the case study, the resulting ‘small-and-dense’ ED-based
dataset carries information for three key water quality indices that are associated with
crop production performance. The four ED-controlling factors were screened for effective-
ness as well as for curvature trends at the same time. The experimental sampling plan was
organized according to an unreplicated–saturated Taguchi-type L9(34) orthogonal array.
The databionic swarm intelligence classifier received hyper-parameter fine-tuning before
the machine learning capabilities of the unsupervised profiler were assessed on predicting
factorial strength and tendencies. The three controlling hyper-parameters were: (1) the
distance measure time, (2) the structure type and (3) the position type. The algorithmic
screening was conducted by programming an unreplicated L8(41*22) OA sampler. In the
proposed implementation, the databionic swarm intelligence analyzer was found to be
Figure 10.
Lenth method results for the direct factorial screening of the BDIc, BDIm and DI estimations
of Table 7(function ‘LenthPlot’ in R-package ‘BsMD’ (30 April 2020)).
4. Conclusions
Improving the efficiency of wastewater separation processes is a very crucial subject,
especially when the applied technology is geared toward supplying large water volumes
for irrigating crops in arid regions. Remote locations may benefit from certain advantages
that the electrodialysis technique can offer. To regulate the efficiency of the wastewater
ED treatment process to a level at which the filtered water product is suitable for agricul-
tural usage, an empirical screening of the ED process, based on pertinent water quality
characteristics, is pivotal. No single theory can describe the complex phenomena that
govern a custom-made ED separation process. This is because the ED design variables
may be unique to the particular application. In addition, the wastewater resources are
anticipated to be of diverse origins with unpredictable non-homogeneous chemical prop-
erties. Consequently, there is an urgency from an engineering perspective to facilitate
quick and economic ED process profiling. Diagnosing palpably strong factors eases the
concurrent adjustment of several operational quality characteristics to an optimal level. A
newly published artificial intelligence method, the databionic swarm intelligence classifier,
was attempted to be adapted to deal with a constrained (small) aquametric dataset. The
non-parametric self-organized search for an ‘objective-free’ Nash equilibrium, which is
tracked by emerging databots, may be doable in classifying the multi-response output from
high-density OA trials. Implementing fractional factorial designs downsizes the sampling
requirements to a mini-dataset. In the case study, the resulting ‘small-and-dense’ ED-based
dataset carries information for three key water quality indices that are associated with crop
production performance. The four ED-controlling factors were screened for effectiveness
as well as for curvature trends at the same time. The experimental sampling plan was
organized according to an unreplicated–saturated Taguchi-type L
9
(3
4
) orthogonal array.
The databionic swarm intelligence classifier received hyper-parameter fine-tuning before
the machine learning capabilities of the unsupervised profiler were assessed on predicting
factorial strength and tendencies. The three controlling hyper-parameters were: (1) the
distance measure time, (2) the structure type and (3) the position type. The algorithmic
screening was conducted by programming an unreplicated L
8
(4
1×
2
2
) OA sampler. In
the proposed implementation, the databionic swarm intelligence analyzer was found to be
impervious to any hyper-parameter tuning. Despite being destined to handle large high-
dimensional data problems, the databionic swarm intelligence solver may be also useful for
extracting information from structured mini-datasets. The novel introduction in this work
is the validity assessment of cluster similarity, as evidenced from the sequence matching
of the unsupervised cluster formation strings to each of the pre-defined factorial setting
strings dictated by the implemented OA scheme. The internal validity measures, such as
Water 2022,14, 1990 24 of 26
the Dunn Index and the Davies–Bouldin Index, were evaluated. Estimations of the data
cluster similarities were supplemented by also using the Rand Index. The algorithmically
recommended Delauney classification error furnished estimations of cluster similarities
while focusing on how well-visualized the reduction was in the three-dimensional input
space to the designated two-dimensional projection. The databionic swarm intelligence
solver identified the wastewater diluted flow to be the only dominant variable that regu-
lated the water quality performance in the ED cell. Therefore, the new approach resolves
the effect for each of the studied ED process controlling factors by using them as a magnify-
ing glass, and the transparent information is directly gleaned from the tightness of the grip
of the individual factorial trial recipe output as it encircles the validated cluster solution.
It is stressed that this is achieved in a novel fashion by self-organizing the multiple aqua-
metric indices into cluster members against no predefined characteristic-based objectives.
Future work can seek unsupervised factorial screenings for larger numbers of investi-
gated water quality indices as well as larger numbers and variable types of the nominated
controlling factors.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Not applicable.
Conflicts of Interest: The author declares no conflict of interest.
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