Decades of Miscomputation in Genomic Clades and Distances

Abstract

Hardly a week seems to go by without encountering a new genetics study that contains a diagram of specimen genetic similarities and clades. For these diagrams, biologists have long relied on university-based and/or commercial computational packages which are not only prone to pilot errors but also contain “analysis” methods which should never be used for genetic distance or clustering. Not that all the software is poor – it appears there is a mixture of good and bad in each package. The troublesome methods, however, have enjoyed acceptable use for so long that serious errors are published on a frequent basis. What follows is a list of concerns that will hopefully be useful to authors and reviewers alike. The report concludes with a graph-theoretical alternative to the current status quo in genomics.

Full text

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
DOI:10.5121/ijcsa.2022.12401 1
DECADES OF MISCOMPUTATION IN
GENOMIC CLADES AND DISTANCES
Richard B. Frost
Frost Concepts, Vista CA, USA
ABSTRACT
Hardly a week seems to go by without encountering a new genetics study that contains a diagram of
specimen genetic similarities and clades. For these diagrams, biologists have long relied on university-
based and/or commercial computational packages which are not only prone to pilot errors but also contain
“analysis” methods which should never be used for genetic distance or clustering. Not that all the software
is poor – it appears there is a mixture of good and bad in each package. The troublesome methods,
however, have enjoyed acceptable use for so long that serious errors are published on a frequent basis.
What follows is a list of concerns that will hopefully be useful to authors and reviewers alike. The report
concludes with a graph-theoretical alternative to the current status quo in genomics.
KEYWORDS
Bayesian clustering, Graph partitioning, Missing values, Pair joining, Pseudo-metrics.
1. ITEMS OF CONCERN
1.1. Use of pseudo-metrics
A portion of the genomic literature utilizes pseudo-metrics to compute similarity or dissimilarity
among genetic profiles. But to be used as a distance, the values are not valid for comparison
unless the measure is a qualified metric [1]. This matter was contested 38 years ago by
Felsenstein [2] who insisted biologists were only making adjacency comparisons in nodes of
topological ancestry trees. However, the construction of those trees involves all-to-all
comparisons of dissimilarities[3].
The well-known 1979 coefficient of Nei & Li [4] (eqns. 8 and 26) is an example of a pseudo-
metric:
󰇛󰇜  
󰇛󰇜
Consider the 8 specimens with 8 random 0,1 markers listed in Table 1. In the spatial domain Nei
& Li's 1979 coefficient produces 2 infinite distances due to zero denominators. Ignoring these,
the remaining subset violates the metric triangle axiom 16 times out of 270. In the marker
frequency domain (Nei's intended use) the results are equally poor with the  measure
producing 49 triangle axiom errors out of 336 tests plus 1 zero distance value. A list of non-
metric measures offered as “distances” in a selection of commonly used software packages is
given in Supplemental Table S1. It is also worth noting that only one biostatistical software
package is careful with the term “metric” and refers to all genetic “distances” as dissimilarities
[5].

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Table 1.
Part 1: Eight example specimens with randomly generated marker values.
Part 2: Marker value frequencies.
Part 3: Jaccard metric spatial distances, scaled to integers so that values represent number of
marker mismatches. In Part 3, nearest neighbours have the smallest value in any row or column.
For example, row #6 indicates that specimen 6 has nearest neighbours 1, 2, and 8.
marker spatial values
marker value frequencies
specimen spatial distances
#
A
B
C
D
E
F
G
H
A
B
C
D
E
F
G
H
1
2
3
4
5
6
7
8
1
1
1
1
0
1
1
0
1
3/8
7/8
1/2
5/8
1/4
1/2
3/8
5/8
4
5
5
4
3
6
4
2
0
1
1
0
0
0
1
1
5/8
7/8
1/2
5/8
3/4
1/2
5/8
5/8
4
5
1
2
3
4
4
3
0
1
0
1
0
1
0
0
5/8
7/8
1/2
3/8
3/4
1/2
3/8
3/8
5
5
4
3
4
5
3
4
0
1
0
0
0
0
1
1
5/8
7/8
1/2
5/8
3/4
1/2
5/8
5/8
5
1
4
3
4
3
3
5
0
1
1
1
0
0
0
1
5/8
7/8
1/2
3/8
3/4
1/2
3/8
5/8
4
2
3
3
5
6
4
6
1
1
1
0
0
1
1
0
3/8
7/8
1/2
5/8
3/4
1/2
5/8
3/8
3
3
4
4
5
5
3
7
0
0
0
0
1
0
1
0
5/8
1/8
1/2
5/8
1/4
1/2
5/8
3/8
6
4
5
3
6
5
6
8
1
1
0
1
0
1
1
1
3/8
7/8
1/2
3/8
3/4
1/2
5/8
5/8
4
4
3
3
4
3
6
1.2. Use of synonyms in cluster analysis
When a metric produces a zero distance between two or more profiles, they are termed synonyms
under that metric. Investigators should make a note of such occurrences and then pick one as an
ambassador to represent the synonymous group going forward. Synonyms are a violation of the
metric positive definite axiom and should not be present when dissimilarity values are being
compared since they skew the analysis of connectivity among profiles (see Figure 1).
Figure 1. The effects of synonymy on cluster analysis of genetic distances. In the above graphs the marker
data for specimen Trojano is identical to the data for specimen Kadota and thus there is zero distance
between them. The presence and absence of Trojano produces dendrograms that are structurally different in
both clustering and depth of branch points. Structural differences are also apparent in the nearest neighbour
graphs on the right. Vertical hierarchy does not imply ancestry. See Supplemental Table S2 for marker data
and computed distances.
1.3. Flattening multi-dimensional data into vectors
Except for pattern-matching metrics, most distance measures available in software packages are
vector-based e.g. List in Mathematica® [6], pdist and clustergram in MATLAB® [7, 8],
and single spreadsheet rows in SPSS [9]. As such investigators and software packages often
“flatten” their tensor marker data (multiple primers per marker) into vectors by the following
procedure:
 

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where  is tensor data,  is the resulting vector,  is the number of markers and  the number of
primers per marker. This is sometimes done silently without the users knowledge e.g.
functionmat_gen_dist in R [10]. Doing so is equivalent to assuming all primers per marker
are independent. It also changes the problem under study when a metric with a non-trivial
normative expression (e.g. Euclidean) is used, since
󰇛󰇜
 󰇛󰇜

except in rare instances of , . Further, any distance 󰇛󰇜 computed by such metrics cannot
be viably projected back into the original problem space because of the nature of the transform.
In particular, the scalars of the normed   vector space would have to be inverted to scalars
of the normed  tensor space which is generally infeasible when   for non-trivial
norms due to the loss of dimensionality. Investigators desiring a normative metric for multi-allele
data should consider the spectral radius
󰇛󰇜
 󰈅󰇛󰇜󰇛󰇜󰈅
 󰇛 󰇜
Where 󰇛󰇜 is the ith eigenvalue of square matrix  and  denotes the modulus of the jth root
of .
1.4. Misuse of metrics on data with errors and omissions
Some investigators will use a pattern matching metric and incorporate all their markers into an
analysis of genetic distances – including those with missing values due to recording errors. This
malpractice has been previously discussed by Schlueter and Harris[11]. Investigators plus authors
of clustering and genetic distance software (e.g. [5, 8, 9, 12, 13]) should consider what happens
when profiles 󰇝󰇞,  󰇝󰇞, and  󰇝󰇞 where  is missing a final
value due to recording error. Jaccard's metric [14] will produce  , but without the
recording error the result would be  . Likewise if any metric is used to compare only the
primers with all values intact the result will be the same. Hence both approaches introduce at
least as many errors as are removed.
1.5. Use of marker frequencies to distinguish individuals
Investigators who choose non-pattern-matching metrics on data composed of amino letter
sequences are forced to use marker frequencies due to the lack of numeric values. To do so, the
distribution of letters within a specimen marker profile is considered a population sample whose
positional frequencies (Table 1, Part 2) are compared by distance metric or pseudo-metric to the
letter distribution of another specimen profile. The first problem that occurs here is with multi-
dimensional data: investigators and software packages are flattening tensors as discussed above
instead of comparing multi-dimensional distribution samples. The second, more general problem
is that distances computed between specimen marker distributions cannot be considered valid
when the correlation matrix (or tensor) of all profile frequencies is singular – demonstrating that a
valid encompassing distribution has not been established for the positional frequencies. Singular
frequency correlation objects are typically the case with data from genetic profiles e.g., the
correlation matrices and tensors of Supplemental Tables S2, S3, S4 are singular as discussed in
the Table legends. Investigators desiring a dissimilarity measure for single-value marker
frequencies should consider Mahalanobis’ metric [15]

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Mahalanobis 
where   󰇝󰇞 contains frequency vectors of the  specimen spatial vectors and  is the
correlation matrix of . Be sure to check the numerical condition [16] of  before computing
distances.
1.6. Structures derived from profile enrichment
To classify genetic profiles into clades or distance clusters, some software packages use the
sample distribution of primer values per marker to generate additional profiles for an “enriched”
population, sometimes referred to as “burn-in”, “bootstrap”, or “Bayesian clustering”. Partitions
of the enriched set are then used to decide cluster memberships for the original sample e.g., [17-
22]. Some implementations provide a questionable confidence interval calculated from within the
enriched profile set.
It is unclear whether the enriched sets are relevant to the original data [23] or whether the
resulting partition should be considered as anything more than one of several possibilities [24].
The enrichments are commonly of magnitude 1k to 100k. However, a non-trivial set of biased
profiles with  markers and  primers per marker will imply a full population of at least
magnitude  to , requiring a statistical sample size   to  [25]. This is an
intractable situation in terms of computing order  distances or order  metric tests for
nontrivial size . Implementers of enrichment methods claim that smaller magnitudes are
sufficient. To verify this claim they need to provide an analytic function of the distance metric
and the marker distributions of all specimen profiles that for a given profile and radial
displacement yields an unbiased measurable set of enriched profiles that fill (topologically cover)
the enclosed hypersphere. This will enable computation of an accurate confidence interval for the
profile enrichment results.
1.7. False nearest neighbours
Nearest neighbour analysis is preferred for cluster determination among genetic distances due to
the complex topology of multi-dimensional genetic distance spaces (the alternative is to establish
an eigenbasis with a Lie algebra). It is important for investigators to realize that a genetic profile
can have multiple nearest neighbours (n.n.) of the same distance – a condition termed multiplicity
(Table 1, Part 3). Unfortunately, some of the n.n. algorithms used internally in biostatistical
software only pick the first element returned by sorting instead of the entire multiplicity group – a
behaviour inherited from graph traversals (see Figure 2). When multiplicity exists and is ignored
by the n.n. algorithm the distance to the selected neighbour is effectively shortened and the
problem under study is changed. Consequently, many published sets of genetic distance clusters
are erroneous.
1.8. Misuse of pair group analysis
The concordance correlation of pair group analysis forces elements into pairs by design [3]. As a
result, pair group analysis cannot accurately partition distance sets containing odd-ranked
multiplicity – a condition due to the nature of genetic profiles (see Figure 2). Another concern is
the interpretation of branching points in pair group dendrograms as mutation or procreation
points. This might be true in a system with only binary branching and no re-entrant breeding, but
statements of these qualifications are not found in the genomic literature even though ample
evidence is available for both.

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1.9. Misapplication of graph partitioning software
Among the many graph partitioning algorithms available from computer science and graph
theory, only a subset is applicable to distances, and only those that do not ignore multiplicity are
viable for genetic profiles with significant number of markers. Further, just because an
investigator picks a good algorithm does not mean the results will have any relevance as
population subgroups or clades. To check this, examine a distance-limited nearest neighbour
graph using either the component maximal or an empirically known upper bound of distance
separation for  generations. The result will be one of 3 outcomes: viable clusters, lack of
cohesion, or too much cohesion (Figure 3). These conditions can be due to the choice of metric,
choice of markers, or the reality of the specimens.
Figure 2. Three renderings of the example specimen data from Table 1 using Jaccard’s metric on spatial
values. In the first graph, notice the nearest-neighbor multiplicities of length 3 for specimens 6, 8, and 3,
while specimen 4’s n.n. is at length 1. The second graph is from an algorithm that ignores multiplicity and
leads to false structural conclusions. The dendrogram on the right illustrates the inability of Sokal’s pair-
joining algorithm to parse odd-ranked multiplicity. Vertical hierarchy does not imply ancestry.

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Figure 3. Cohesion extremes in distance limited nearest neighbour graphs. Distance limits were empirically
determined by comparing specimen ancestry records to 2nd generation distances computed with Jaccard’s
metric. Vertical hierarchy does not imply ancestry. The left graph illustrates lack of cohesion in SSR data
(see Supplemental Table S3. The right graph shows excess cohesion in SSR data (see Supplemental Table
S5). Thick black edges are length 1/9, thin grey edges are length 2/9. Note that distance partitioning cannot
improve the right-hand graph because to cut any grey line one must cut them all.
2. A TOPOLOGICAL APPROACH
If  is a collection of genetic profiles plus a set of some or all the distances between them and
these qualify as a metric, then  is termed a distance graph with profiles for vertices and
distances for edges. The traditional nearest neighbor graph  is the subset of  containing only
nearest neighbor edges and their vertices. (Note: some computer programs ignore multiplicity so
it is recommended to check automated results.) A least bridges graph  of  will have overlap
with  but offers more insight to components. The construction method is hierarchical. Vertices
are first added as disconnected components. The shortest available edge connections are then
added incrementally. Edges are only added between disconnected components and thus termed
"bridges" [26]. A new component is created each time an edge is added, replacing the prior two.
If there are multiple edges of the same distance that qualify then the entire set is added, possibly
engulfing multiple components. The distances among components must be re-evaluated after an
edge or edge set is added. Inter-component distances are determined by selecting the shortest
vertex-to-vertex distance between them. This is often but not always a nearest neighbor edge. The
process is continued incrementally until a prescribed limit is reached (e.g. a maximum distance)
or a connected graph is achieved.
When  or  is distance-limited by a non-trivial amount , the number of edges will be
reduced and hence the number of components can increase. In any such graph, consider the
function  of the product of the # of components  with the # of vertices : 󰇛󰇜 . The
value  which produces a graph that maximizes  is termed the component maximal (Figure
4). Since this value maximizes the number of graph components with respect to vertices, it
produces elemental clusters of original graph  for its given distance metric. An example is
shown in Figure 5.

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Figure 4. Variation of c of components (isolated clusters), v of vertices (specimens), and fcv
in least bridges graphs of 28 Moraceae specimens limited to genetic distance . The component maximal
occurs at f󰇛󰇜 where  is used for illustration purposes (see Figure 5).
Figure 5. Distance-limited least bridges graph showing elemental clusters of 28 Moraceae specimens [27].
Solid lines denote nearest neighbors and dashed lines are least bridges. 26 specimens are present in the
graph with 2 remaining cladeless. In the bottom left component Seizuro and Sabbawala-2 are mutual
nearest neighbors, while Seizuro is the single nearest neighbor of Mysore Local and SRDC-1. A complete
set of distance values are available in Supplemental Table S6 and the referenced publication. Distance limit
is the component maximal  . Vertical hierarchy does not imply ancestry.

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[32] K. W. Pomper et al., "Characterization and identification of pawpaw cultivars and advanced
selections by simple sequence repeat markers," Journal of the American Society for Horticultural
Science, vol. 135, no. 2, pp. 143-149, 2010, doi: https://doi.org/10.21273/JASHS.135.2.143.
[33] K. Vinod, "Structured association mapping using STRUCTURE and TASSEL," Bioinformatics Tools
for Genomics Research, p. 103, 2011.
https://www.academia.edu/706699/Structured_Association_Mapping_using_STRUCTURE_and_TA
SSEL.
[34] A. Wünsch and J. Hormaza, "Molecular characterisation of sweet cherry (Prunus avium L.)
genotypes using peach [Prunus persica (L.) Batsch] SSR sequences," Heredity, vol. 89, no. 1, pp. 56-
63, 2002, doi: https://doi.org/10.1038/sj.hdy.6800101.
AUTHOR
R.B. Frost is an old-school numerical analyst with academic and vocational experience
in applied mathematics, computer science, and horticulture. He is currently pursuing
research in the genomics of lesser-studied fruits.
SUPPLEMENTARY INFORMATION
Supplemental Table S1. Non-metric dissimilarity measures erroneously presented as distances in software
packages commonly used for biostatistics research [12, 28-30]. Symmetric pairs of zero distances were
removed prior to analysis.
Metric Tests
Software
Measure
Name
Data Type
Domain
Dataset
Reflexive
Com-
mutative
Triangle
Inequality
MATLAB R2022a
Correlation
Numerical
Spatial
Table x
Passed
Passed
Failed
MATLAB R2022a
Correlation
Numerical
Frequency
Table x
Passed
Passed
Failed
MATLAB R2022a
Cosine
Numerical
Spatial
Table x
Passed
Passed
Failed
MATLAB R2022a
Cosine
Numerical
Frequency
Table x
Passed
Passed
Failed
Mathematica v13.0
Bray-Curtis
Numerical
Spatial
Table x
Passed
Passed
Failed
Mathematica v13.0
Bray-Curtis
Numerical
Frequency
Table x
Passed
Passed
Failed
Mathematica v13.0
Correlation
Numerical
Spatial
Table x
Passed
Passed
Failed
Mathematica v13.0
Correlation
Numerical
Frequency
Table x
Passed
Passed
Failed
Mathematica v13.0
Cosine
Numerical
Spatial
Table x
Passed
Passed
Failed
Mathematica v13.0
Cosine
Numerical
Frequency
Table x
Passed
Passed
Failed

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
10
NTSYSpc v2.21w
Hillis
Multi-allele
Counts
Spatial
Table S2
Passed
Passed
Failed
NTSYSpc v2.21w
Hillis
Multi-allele
Counts
Frequency
Table S2
Passed
Passed
Failed
NTSYSpc v2.21w
Nei 1972
Multi-allele
Counts
Spatial
Table S2
Passed
Failed
Failed
NTSYSpc v2.21w
Nei 1972
Multi-allele
Counts
Frequency
Table S2
Passed
Failed
Failed
NTSYSpc v2.21w
Nei 1978
Multi-allele
Counts
Spatial
Table S2
Passed
Passed
Failed
NTSYSpc v2.21w
Nei 1978
Multi-allele
Counts
Frequency
Table S2
Passed
Passed
Failed
SPSS v28.0.1.1
Size
Difference
0,1 Binary
Spatial
Table 1
Passed
Passed
Failed
SPSS v28.0.1.1
Pattern
Difference
0,1 Binary
Spatial
Table 1
Passed
Passed
Failed
SPSS v28.0.1.1
Binary
Shape
0,1 Binary
Spatial
Table 1
Passed
Passed
Failed
SPSS v28.0.1.1
Lance and
Williams
0,1 Binary
Spatial
Table 1
Passed
Passed
Failed
Supplemental Table S2. Multi-dimensional SSR and genetic distance data from 8 Ficus carica specimens
at NCGR Davis [31]. Distances computed with spectral radius of , where  are the  tensors
below each specimen name. The correlation tensor of frequencies for this data is singular due to marker
frequencies of M8N1.2 all having value 1/2, and several other markers having no variation e.g., C22F1.1.
specimens
Adriatic
Archipel
Calimyrna
Kadota
Mission
Panachee
Trojano
Vernino
SSR markers
C22F1
283
283
283
283
283
283
283
283
283
283
283
283
285
283
283
283
C24H1
272
270
272
272
270
272
272
270
272
272
272
272
272
272
272
272
C26N1
234
234
234
234
234
234
234
234
234
236
234
234
236
234
234
234
C31F1
224
224
239
224
224
224
224
224
239
224
239
239
239
239
239
239
C35H1
252
254
254
254
254
252
254
254
254
254
256
254
254
254
254
254
C37N1
204
204
204
204
204
204
204
204
204
208
204
208
204
204
208
204
LM12H1
214
214
214
214
214
233
214
233
243
243
243
243
243
233
243
243
LM14H1
200
200
200
200
200
200
200
198
200
200
200
200
200
200
200
200
LM30N1
245
245
243
243
245
237
243
231
251
251
245
245
247
251
245
251
LM36N1
248
248
248
248
248
248
248
248
248
248
250
248
250
248
248
250
M1F1
172
189
172
172
189
155
172
172
184
189
184
189
189
184
189
188
M2H1
155
161
153
153
161
153
153
155
161
167
161
167
167
153
167
155
M3N1
124
120
132
120
122
124
120
132
132
132
132
132
122
132
132
132
M4F1
194
194
194
194
194
194
194
214

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
11
214
214
214
218
218
214
218
214
M8N1
171
171
171
171
171
171
171
171
171
175
171
175
175
171
175
171
distances
Adriatic
21.383
17.378
12.042
20.772
27.715
12.042
32.14
Archipel
21.383
30.989
19.046
19.546
40.948
19.046
37.917
Calimyrna
17.378
30.989
19.209
27.995
32.223
19.209
33.892
Kadota
12.042
19.046
19.209
19.134
27.599
0.
33.119
Mission
20.772
19.546
27.995
19.134
40.773
19.134
37.264
Panachee
27.715
40.948
32.223
27.599
40.773
27.599
28.505
Trojano
12.042
19.046
19.209
0.
19.134
27.599
33.119
Vernino
32.14
37.917
33.892
33.119
37.264
28.505
33.119
Adriatic
Archipel
Calimyrna
Kadota
Mission
Panachee
Trojano
Vernino
Supplemental Table S3. SSR data values for 5 loci from 2010 Asimina triloba (Pawpaw) study of
Pomper et al [32]. Suffix _.F refers to forward SSR primer, _.R to reverse. Note that the zeros are missing
values which skew the analysis as discussed in the main text. The correlation tensor of frequencies for this
data is singular due to the interdependence of primers C104.F and C104.R.
Genotype
B3.F
B3.R
B103.F
B103.R
B129.F
B129.R
C104.F
C104.R
G119.F
G119.R
10-35
183
191
266
339
166
172
184
0
158
164
11-13
191
0
264
305
166
172
184
0
158
0
1-23
185
189
290
310
158
0
184
0
158
176
1-68
185
187
268
341
158
179
175
184
158
164
2-10
191
0
264
270
170
172
184
0
161
164
2-54
191
0
264
270
162
166
184
0
161
0
3-11
191
0
272
288
158
172
184
0
158
161
3-21
189
191
266
305
166
170
184
0
161
164
5-5
183
189
270
305
166
168
184
0
161
0
7-90
185
191
305
342
170
176
184
0
161
164
8-20
189
191
264
270
162
0
184
0
158
167
9-47
183
0
272
274
158
166
184
0
158
161
9-58
183
191
264
339
170
176
184
0
158
164
BH10
189
0
319
321
162
170
184
0
144
161
Cales Creek
175
183
266
274
156
158
184
0
158
164
Davis
185
189
264
268
158
164
175
184
158
164
Greenriver Belle
183
189
264
266
162
172
184
0
158
161
IXL
187
189
274
309
158
162
175
184
158
164
M. Gordon
185
195
270
312
164
170
184
0
161
164
Middletown
183
193
270
321
170
0
184
0
158
161
Mitchell
0
0
266
321
158
172
184
0
158
167
NC-1
185
193
266
0
158
162
184
0
158
161
Overleese
185
189
264
0
158
164
175
184
158
0
PA-Golden#1
191
193
336
343
172
176
184
0
158
164
PA-Golden#3
189
191
336
343
158
172
184
0
161
170
PA-Golden#4
175
183
319
326
164
0
184
0
158
161
Potomac
183
191
264
324
170
0
184
0
158
164
Prolific
189
191
309
323
158
162
184
0
158
0
Rappahannock
183
191
266
0
166
0
184
0
164
170
Rebeccas Gold
185
193
266
0
158
162
184
0
158
161
Shenandoah
185
187
264
274
162
164
184
0
158
164
Sue
175
189
266
329
166
180
184
0
161
164
Sunflower
187
0
274
341
162
180
175
184
164
0
Susquehanna
189
191
264
270
162
0
184
0
158
167
Sweet Alice
175
183
260
324
166
182
184
0
144
164
Taylor
183
185
268
322
173
193
184
0
167
170
Taytwo
175
185
252
290
158
0
184
0
164
176
Wabash
183
0
266
324
170
172
184
0
158
170
Wells
175
191
276
290
177
0
184
0
161
164
Wilson
183
185
268
321
173
193
184
0
167
170
Zimmerman
191
195
303
324
164
177
184
0
158
0

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
12
Supplemental Table S4. Example SSR data table from Vinod 2011 [33]. Row “GENO6” has been deleted
due to a missing data value in marker SSR2. The correlation matrix of frequencies for this data is singular
due to the lack of variation in columns 4, 6, and 8.
SSR1
SSR2
SSR3
SSR4
SSR5
SSR6
SSR7
SSR8
SSR9
GENO1
110
330
190
140
220
140
240
160
200
GENO2
110
330
190
140
230
140
240
160
190
GENO3
110
320
190
140
220
140
240
160
200
GENO4
110
320
190
140
220
140
240
160
200
GENO5
110
330
180
140
220
140
240
160
200
GENO7
110
330
190
140
220
140
240
160
200
GENO8
110
320
180
140
220
140
240
160
200
GENO9
110
330
190
140
220
140
250
160
200
GENO10
120
320
180
140
220
140
240
160
200
Supplemental Table S5. SSR data values for 9 loci from 2002 Prunus avium (Sweet Cherry) study of
Wünsch & Hormaza [34]. Dual values are treated as rational numbers. Note that single values are
instances of missing scores which skew the analysis as discussed in the main text. The correlation matrix
of frequencies for this data is not singular, with determinant 0.280843. This would not be the case if the
missing scores were recorded as zeros (see Table S3).
Cultivar
Index
Pchcms1
Pchcms3
Pchcms5
UDP96-
005
UDP98-
409
UPD98-
021
UPD98-
022
UPD97-
402
UPD98-
412
Ambrunes
1
140
180/160
290
150/125
160/130
110/100
110/90
125
130
Arcina
2
140
180
260
150/130
130
110/100
100/90
140/125
130
Beige
3
190/140
180
260
150/120
130
110/100
110/90
130
130
Bing
4
190/140
180
260
150/120
130
110/100
110/90
130
130
Blanca de
Provenza
5
140
180/160
290/260
120
130
100
105/90
145/125
130/100
Brooks
6
140
180
290
150/120
130
110/100
90
135/125
130
Burlat
7
140
180
290
150/130
130
110/100
100/90
130/120
130
Burlat C-1
8
140
180
290
150/130
130
110/100
100/90
130/120
130
Celeste
9
140
180
290
150/130
130
110/100
100
135/125
130
Chinook
10
190/140
180
260
150/120
130
110/100
110
145/130
130
Compact
Stella
11
190/140
180
290/260
150/120
130
110/100
110/100
145/125
130
Coralise
12
140
180
260
150/130
130
110/100
100/90
135/125
130
Corum
13
190/140
180
290/260
150/120
130
110/100
110/100
135/125
140/130
Cristalina
14
190/140
180/160
260
150/120
130
110
110/100
135/125
130
Cristo-
balina
15
190/140
180
290/260
150/115
130
110/100
110/90
135/125
130
Duroni 3
16
140
180
290/260
150/115
130
110/100
100/90
125
130
Earlise
17
140
180
290/260
150/130
130
100
90
130/120
130
Earlystar
18
190/140
180
290
150/120
130
100
100/90
145/125
130
Early Van
Compact
19
190/140
180
290/260
150/120
130
110
100/90
135/125
130/120
Ferrovia
20
140
180
290/260
150/120
130
110
100
145/130
140/130
Garnet
21
140
180
260
120
130
110/100
110/90
130
130
Gil Peck
22
190/140
180/160
260
150/120
130
110
110
145/130
130
Giorgia
23
140
180
290/260
150/120
130
110
100/90
125
130/120
Hartland
24
140
180
260
150/120
130
110
100/90
140/130
130
Hedel-
finger
25
140
180
290/260
150/120
130
110/100
110/100
135/125
130/100
Lambert
26
190/140
180/160
290/260
150/120
130
110/100
110
125
130
Lamida
27
140
180
260
150/120
130
110
110
145/125
130

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
13
Lapins
28
190/140
180
290
150/120
130
110/100
100/90
135/125
130
Larian
29
190/140
180
260
150/120
130
110
110/100
145/130
130
Marmotte
30
190/140
180
290/260
150/120
130
110/100
110/100
135/125
130/100
Marvin
31
140
180
290/260
120
130
110/100
100/90
130/125
130
Moreau
32
140
180
290/260
150/115
130
100
100/90
145/125
130
Napoleon
33
190/140
180
260
150/120
130
110/100
110/100
135/125
130
Newstar
34
140
180
290/260
150/120
130
110/100
100
135/125
130/120
Pico
Colorado
35
140
180/160
290/260
150/130
160/130
110/100
110
140/125
130
Pico Negro
36
140
180/160
290/260
150/115
160/130
100
110/90
140/120
130
Precoce
Bernard
37
140
180
290/260
150/130
130
100
100/90
145/130
130
Rainier
38
190/140
180
290/260
120
130
110/100
90
130
130
Ramon
Oliva
39
140
180/160
290
150/130
130
100
90
120
130
Reverchon
40
140
180
290/260
150/115
130
110/100
100
125
130
Royalton
41
190/140
180
260
150/130
130
100
110/100
135/125
130
Ruby
42
190/140
180
260
120
130
110/100
110/90
130
130
Sam
43
140
180/160
290/260
150/120
130
100
110/100
145/125
130
Samba
44
190/140
180
290/260
150/120
130
110
110/90
135/130
130
Santina
45
190/140
180/160
290/260
150/130
130
100
110/100
135/125
130
Skeena
46
190
180
290/260
150/120
130
100
90
145/125
130
Somerset
47
190/140
180
260
150/120
130
110/100
100
135/125
130/120
Sonata
48
190
180
290
150/120
130
100
100/90
145/125
130/120
Spalding
49
190/140
180
260
150/130
130
110/100
110/100
135/125
140/130
Star
50
190/140
180/160
260
150/120
130
110/100
110/100
135/125
130
Starky
Hardy
Giant
51
140
180
260
150/120
130
110/100
100/90
135/125
130
Sue
52
140
180/160
260
150/120
130
110
110/100
135/125
140/130
Sumesi
53
140
180
290/260
150/120
130
110/100
110/100
145/130
130
Summit
54
140
180/160
290/260
150/120
130
110/100
100
125
130
Sunburst
55
140
180
290
150/120
130
110/100
100/90
145/130
130/120
Sweetheart
56
190/140
180
290/260
150/120
130
110/100
100/90
135/125
130/120
Sylvia
57
140
180/160
290
150/120
130
110/100
100
145/125
130
Taleguera
Brilliante
58
140
180/160
290/260
150/115
130
100
100/90
135
130
Tigre
59
140
180
290
150/115
130
110
110/90
120
130
Van
60
190/140
180
290/260
150/120
130
110
100/90
135/125
130/120
Van Spur
61
190/140
180
290/260
150/120
130
110
100/90
135/125
130/120
Vega
62
190/140
180
260
150/130
130
110
100/90
135/125
140/130
Vic
63
140
180
260
150/120
130
110/100
100/90
135/125
140/130
Vignola
64
140
180
260
150/130
130
110
100
140/125
130
Vittoria
65
140
180
290/260
150/130
130
110
100/90
145/130
130/120
13N.7.19
66
140
180
290/260
150/120
130
110/100
100/90
135/125
130/120
13S.17.20
67
190/140
180
290/260
150/120
130
110
110/90
130
130
13S.18.10
68
190/140
180
290/260
150/120
130
110/100
110/100
135/125
130
13S.18.15
69
190/140
180
290/260
150/120
130
110
110/100
135/125
130/120
13S.21.7
70
140
180
290
150/130
130
110/100
100
135/125
130
13S.27.17
71
140
180
260
150/120
130
110
100/90
135/125
140/130
13S.3.13
72
190/140
180
290
150/120
130
100
110/100
145/130
130
44W.11.8
73
190/140
180
260
150/120
130
110
110/100
145/130
130
83703007
74
190/140
180
290
150/120
130
100
100
145/120
130
84703002
75
190/140
180
290
150/120
130
110/100
100/90
130/120
130
84704006
76
140
180
290
150/130
130
110/100
100/90
130/120
130

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
14
Supplemental Table S6. Distance matrix from 2016 Moraceae SSR study by Mathi Thumilan et al [27].
The investigators computed distances with one of the metrics in the Darwin v.5.0 program [5]. It is
unknown whether the original SSR data was vector-valued or flattened from a tensor.
Genot
ype
M.
macro
ura
M.
nigra
ME-
107
T-12
C-
1725
T-21
Moul
ai
T-08
C-776
ACC-
118
ACC-
115
Gajap
athipu
ra
SRDC
-1
M.
nigra
0.286
ME-
107
0.302
0.298
T-12
0.322
0.319
0.308
C-
1725
0.353
0.349
0.339
0.277
T-21
0.36
0.357
0.346
0.285
0.313
Moula
i
0.355
0.351
0.341
0.279
0.307
0.259
T-08
0.354
0.35
0.339
0.278
0.306
0.258
0.235
C-776
0.341
0.338
0.327
0.266
0.294
0.246
0.234
0.232
ACC-
118
0.364
0.36
0.35
0.288
0.316
0.288
0.283
0.282
0.27
ACC-
115
0.357
0.354
0.343
0.282
0.31
0.282
0.276
0.275
0.263
0.243
Gajap
athipu
ra
0.36
0.356
0.346
0.285
0.313
0.303
0.297
0.296
0.284
0.306
0.3
SRDC
-1
0.394
0.39
0.38
0.318
0.346
0.336
0.331
0.33
0.318
0.34
0.333
0.319
Sabba
wala-2
0.397
0.393
0.383
0.321
0.349
0.339
0.334
0.333
0.32
0.343
0.336
0.322
0.288
Seizur
o
0.388
0.384
0.374
0.313
0.34
0.331
0.325
0.324
0.312
0.334
0.328
0.313
0.279
Mysor
e
Local
0.404
0.4
0.39
0.328
0.356
0.346
0.341
0.34
0.327
0.35
0.343
0.329
0.295
Kanva
-2
0.398
0.394
0.384
0.323
0.35
0.341
0.335
0.334
0.322
0.344
0.338
0.323
0.309
S-1
0.397
0.393
0.383
0.322
0.349
0.34
0.334
0.333
0.321
0.343
0.337
0.322
0.308
RF-
175
0.407
0.403
0.393
0.331
0.359
0.349
0.344
0.343
0.33
0.353
0.346
0.332
0.317
S-54
0.417
0.414
0.403
0.342
0.37
0.36
0.354
0.353
0.341
0.363
0.357
0.342
0.328
S-13
0.399
0.395
0.385
0.324
0.352
0.342
0.336
0.335
0.323
0.345
0.339
0.324
0.31
DD-1
0.398
0.394
0.384
0.322
0.35
0.34
0.335
0.334
0.321
0.344
0.337
0.323
0.308
V-1
0.406
0.403
0.392
0.331
0.359
0.349
0.343
0.342
0.33
0.353
0.346
0.332
0.317
S-36
0.413
0.41
0.399
0.338
0.366
0.356
0.35
0.349
0.337
0.36
0.353
0.339
0.324
F.
bengh
alensis
0.613
0.609
0.599
0.538
0.565
0.556
0.55
0.549
0.537
0.559
0.553
0.538
0.547
F.
carica
0.612
0.608
0.598
0.536
0.564
0.555
0.549
0.548
0.536
0.558
0.551
0.537
0.546
Jackfr
uit
0.612
0.608
0.597
0.536
0.564
0.554
0.549
0.547
0.535
0.558
0.551
0.537
0.546
Dudia
white
0.866
0.863
0.852
0.791
0.819
0.809
0.803
0.802
0.79
0.813
0.806
0.792
0.801
Genot
ype
Sabba
wala-2
Seizur
o
Mysor
e
Local
Kanv
a-2
S-1
RF-
175
S-54
S-13
DD-1
V-1
S-36
F.
bengh
alensis
F.
carica
Jackfr
uit
Seizur
o
0.214

International Journal on Computational Science & Applications (IJCSA) Vol.12, No.4, August 2022
15
Mysor
e
Local
0.264
0.255
Kanva
-2
0.312
0.303
0.318
S-1
0.311
0.302
0.318
0.223
RF-
175
0.32
0.312
0.327
0.257
0.256
S-54
0.331
0.322
0.338
0.268
0.267
0.228
S-13
0.313
0.304
0.32
0.29
0.289
0.298
0.309
DD-1
0.311
0.303
0.318
0.288
0.287
0.297
0.307
0.229
V-1
0.32
0.311
0.327
0.297
0.296
0.305
0.316
0.238
0.217
S-36
0.327
0.318
0.334
0.304
0.303
0.312
0.323
0.245
0.224
0.192
F.
bengh
alensis
0.55
0.542
0.557
0.552
0.551
0.56
0.571
0.553
0.551
0.56
0.567
F.
carica
0.549
0.541
0.556
0.55
0.549
0.559
0.57
0.552
0.55
0.559
0.566
0.231
Jackfr
uit
0.549
0.54
0.556
0.55
0.549
0.559
0.569
0.551
0.55
0.559
0.566
0.246
0.245
Dudia
white
0.804
0.795
0.811
0.805
0.804
0.814
0.824
0.806
0.805
0.813
0.821
0.861
0.859
0.859