# Application of Markov Structure of Genomes to
Outlier Identification and Read Classification

Alan F. Karr, Jason Hauzel, Adam A. Porter, Marcel Schaefer Fraunhofer USA
CMA<EMAIL_ADDRESS>Corresponding authorFraunhofer USA CMA;
<EMAIL_ADDRESS>of Maryland, Department of Computer Science;
<EMAIL_ADDRESS>USA CMA<EMAIL_ADDRESS>
###### Abstract

In this paper we apply the structure of genomes as second-order Markov
processes specified by the distributions of successive triplets of bases to
two bioinformatics problems: identification of outliers in genome databases
and read classification in metagenomics, using real coronavirus and adenovirus
data.

##  1 Introduction

That the sequential structure of genomes is important has been known since the
discovery of DNA. In this paper we employ a statistics and stochastic process
perspective on triplets of successive bases to address two important
applications: identifying outliers in genome databases, and classifying reads
in the metagenomic context of reference-guided assembly. From this stochastic
process perspective, triplets are a second-order Markov chain specified by the
distribution of each base conditional on its two immediate predecessors.

To be sure, studying genomes via base sequence distributions is not novel.
Previous papers have addressed genome signatures (Karlin et al.,, 1997;
Campbell et al.,, 1999; Takashi et al.,, 2003), as well as frequentist (Rosen
et al.,, 2008) and Bayesian (Wang et al.,, 2007) approaches to classification
problems. While we focus instead on triplet distributions, many extant works
employ quartets, also referred to as tetranucleotides, among them Pride et
al., (2003), Teeling et al., 2004a  and Teeling et al., 2004b .

In §3, we apply hierarchical cluster analysis of triplet distributions to
identification of outliers in genome databases. Genomes in small or highly
divergent clusters are statistical outliers. They may be data quality
problems, as some in our example clearly are, or simply aberrant. The
clustering is both statistically effective and scientifically interpretable.

The second application, in §4, is Bayesian classification of (short) reads in
metagenomics. The latter has consequences for uncertainty quantification, in
particular allowing assessment of the contributions to uncertainty of prior
knowledge, input data quality, and training data quality, which is also
interpretable as model quality. Throughout, we use real data, primarily a
coronavirus dataset downloaded from the National Center for Biotechnology
Information (NCBI), as well as a single adenovirus genome included in the
download package for the Art read simulator (Huang et al.,, 2012). Each genome
has length on the order of 30,000 base pairs (BP) , representing approximately
20 genes. Conclusions appear in §5.

##  2 Background

In this paper, a genome $G$ is a character string chosen from the base
(nucleotide) alphabet $\mathcal{B}=\\{A,C,G,T\\}$, and represents one strand
of the DNA in an organism. Given a genome $G$, the length of $G$ is $|G|$; the
$i^{\mathrm{th}}$ base in $G$ is $G(i)$; and the bases from location $i$ to
location $j>i$ are $G(i:j)$. For $n\geq 1$, let $\mathcal{B}(n)$ be the set of
all sequences of length $n$ from $\mathcal{B}$, of which there are $4^{n}$. We
focus on triplets, whose distributions are 64-dimensional summaries of
genomes;111Other cases of interest appear in the literature. Base
distributions ($n=1$) and pair distributions ($n=2$) are too coarse to be
useful for the problems we address. Quartets ($n=4$) have historical precedent
(Pride et al.,, 2003; Teeling et al., 2004a, ; Teeling et al., 2004b, ). Based
on informal experiments, for the applications in §3 and 4, quartets are more
cumbersome computationally than triplets, but do not seem to be significantly
more informative. for this focal case,
$\mathcal{B}(3)=\\{AAA,AAC,\dots,TTG,TTT\\}$, and the triplet distribution
$P_{3}(\cdot|G)$ is defined as

$P_{3}(b_{1}b_{2}b_{3}|G)=\mathrm{Prob}\big{\\{}G(k:k+2)=b_{1}b_{2}b_{3}\big{\\}}$
(1)

for each choice of $b_{1},b_{2},b_{3}$ from $\mathcal{B}$, where $k$ is chosen
at random from $1,\dots,|G|-2$. An equivalent perspective is that of a second-
order Markov chain. The information contained in $P_{3}(\cdot|G)$ is the same
as that contained in the pair distribution $P_{2}(\cdot|G)$ and the $16\times
4$ transition matrix

$T_{3}((b_{1},b_{2}),b_{3}|G)=\mathrm{Prob}\big{\\{}G(k+2)=b_{3}|G(k)=b_{1},G(k+1)=b_{2}\big{\\}},$
(2)

whose rows are indexed by $(b_{1},b_{2})$ and columns are indexed by $b_{3}$,
and which gives the distribution of each base conditional on its two
predecessors.

Appendix A contains examples of both distributions for the three virus genomes
underlying the read classification problem in §4.

##  3 Outlier Detection

In this section, we use triplet and amino acid distributions to detect
outliers in the NCBI database. Outliers may be true scientific anomalies, or
they may be data quality problems (Karr et al., 2021b, ). That large,
uncurated problem genome databases may have quality issues is of increasing
concern in a variety of contexts (Biggio and Roli,, 2018; Steinegger and
Salzberg,, 2020; Valdivia-Granda,, 2019).

Our strategy is to use hierarchical cluster analysis (Aggarwal and Reddy,,
2013; Everitt,, 2011) of the 64-dimensional triplet distributions, and then to
investigate clusters with small counts, which _are_ outliers in a statistical
sense, to see if they are outliers in other senses. As our experimental
platform, we employ a coronavirus database containing 26,953 genomes, which
was downloaded from the NCBI in November of 2020. To it, we added eleven
”known” outliers: one adenovirus genome and low-quality versions of ten
coronavirus genomes selected randomly from the 26,953. The latter were
produced using the degradation methodology described in Karr et al., 2021b ,
which consists of repeated iterations of the Mason_variator haplotype
generation software (Holtgrewe,, 2010). The clustering was performed on
standardized versions of the triplet distributions using the “complete” method
for hierarchical clustering (Müllner,, 2013) in the R software system (R Core
Team,, 2020).

We emphasize that some of what follows requires knowledge of the sources of
the genomes (original coronavirus dataset, adenovirus or degraded) that is not
available in some other contexts.

###  3.1 Clustering Based on Triplet Distributions

Figure 1 contains results for triplet-based hierarchical clustering in 64
dimensions.222Technically, in a 63-dimensional hyperplane, since the 64
probabilities sum to one. The number of clusters, which is 23, was determined
using standard heuristics that trade off complexity for explanatory power. The
top panel is a plot of the cluster centroids reduced to two dimensions using
multidimensional scaling (MDS) (Kruskal and Wish,, 1978; Cox and Cox,, 2001).
The greater the separation in the plot, the greater the dissimilarity between
clusters. The bottom panel of Figure 1 is the associated dendrogram, depicting
the dynamics of the clustering process, and is discussed in more detail below.

Statistically, the quality of the clustering is very high. The percentage of
the variation of each triplet frequency, over the entire 26,964-genome
dataset, that is explained by the cluster numbers alone is very large.
Specifically, for each triplet $x$, a linear model of the form
$P_{3}(x|\cdot)\sim\mathrm{ClusterNumber}(\cdot)$ was fitted, treating the
latter as a categorical variable. All of the coefficients of determination,
$R^{2}$, exceed 0.93, and the majority exceed 0.98.

Figure 1: _Top:_ Two-dimensional MDS plot of triplet distribution of cluster
centroids for the 26,964-element coronavirus dataset containing 26,953
coronavirus genomes and 11 deliberately added outliers. _Bottom:_
Corresponding dendrogram. Class in both plots is the same as in Table 1, and
is shown in the same colors in both graphics.

Table 1 complements the dendrogram in Figure 1. In the dendrogram, the
$x$-axis, _Height_ , is the stage is the clustering process at which splits
occur; the greater the height, the earlier the split. So, for example, the
first split in the dendrogram creates two branches, the upper one containing
the eleven added outliers and one coronavirus genome, and the lower containing
the remaining 26,952 coronavirus genomes. Although this is not a logical
necessity, every final cluster contains only one “kind” of genome, labeled
_Class_ , which appears in the MDS plot, the dendrogram and the
table.333“Kind” was termed source previously, and is one of coronavirus,
adenovirus or degraded coronavirus. _Separation Height_ Table 1 is that at
which the cluster is separated from all the others and not split further. The
higher this value, the earlier in the process a cluster is permanently split
off from the others. _Separation Rank_ orders the clusters by Separation
Height, with the greatest value of height having rank 1. Ties end in .5.

Cluster | Count | Class | Separation Height | Separation Rank
---|---|---|---|---
1 | 12 | Coronavirus | 68.60 | 3
2 | 15 | Coronavirus | 9.23 | 11.5
3 | 2 | Coronavirus | 9.23 | 11.5
4 | 141 | Coronavirus | 11.42 | 9
5 | 55 | Coronavirus | 6.98 | 14.5
6 | 27 | Coronavirus | 6.98 | 14.5
7 | 258 | Coronavirus | 15.03 | 8
8 | 3 | Coronavirus | 9.87 | 10
9 | 3 | Coronavirus | 7.99 | 13
10 | 3 | Coronavirus | 6.08 | 16.5
11 | 26433 | Coronavirus | 6.08 | 16.5
12 | 1 | Coronavirus | 71.43 | 2
13 | 1 | Adenovirus | 44.29 | 4
14 | 1 | Degraded Coronavirus | 0.00 | 22.5
15 | 1 | Degraded Coronavirus | 0.00 | 22.5
16 | 1 | Degraded Coronavirus | 88.92 | 1
17 | 1 | Degraded Coronavirus | 2.58 | 18.5
18 | 1 | Degraded Coronavirus | 2.58 | 18.5
19 | 1 | Degraded Coronavirus | 32.20 | 6.5
20 | 1 | Degraded Coronavirus | 32.20 | 6.5
21 | 1 | Degraded Coronavirus | 43.27 | 5
22 | 1 | Degraded Coronavirus | 0.92 | 20.5
23 | 1 | Degraded Coronavirus | 0.92 | 20.5
Table 1: Characteristics of the triplet distribution clusters in Figure 1. See
text for explanation of Separation Height and Separation Rank. The double
horizontal line divides original 26,953 coronavirus genomes from the 11
deliberate outliers.

We now interpret. The overwhelming majority of coronavirus genomes—26,433 of
the original 26,953, or 98.1%—are in cluster 11. Cluster 13 contains the
adenovirus genome alone; each of the 10 degraded coronavirus genomes appears
in a cluster by itself (clusters 14–23). Therefore, the deliberate outliers
are not only all detected, but also distinguished from one another.444We
caution against over-interpretation. Some of the behavior is a consequence of
the small number of deliberate outliers. The MDS plot in Figure 1 shows that
these 10 degraded coronavirus outliers differ more from the coronavirus
genomes (clusters 1–12) and the adenovirus (cluster 13) than from one another.
The adenovirus genome differs from both the coronavirus genomes and the
degraded coronavirus outliers.

One original coronavirus genome appears by itself, in cluster 12. Without
question it is an outlier; as noted above, it is separated from the other
coronavirus genomes at the first split. However, we see below that when
clustering is based on amino acids instead of triplets, it ceases to be an
outlier. The NCBI dataset ID of this genome is _MT451283.1 Severe acute
respiratory syndrome coronavirus 2 isolate SARS-CoV-2/human/AUS/VIC402/2020,
complete genome_. No other genome ID in the database is similar to it, which
we take as scientific confirmation that the triplet distribution-based
clustering truly detects outliers.

Cluster 16 has Separation Rank 1: that particular degraded coronavirus genome
is the first to be separated permanently from everything else. Cluster 12,
with the singleton outlier coronavirus genome, is second. Of the remaining
clusters (1–10), cluster 1 is the strongest candidate for containing outliers.
Not only does it have Separation Rank 3, but also its count is small. The next
two highest ranked coronavirus clusters (4 and 7) do not have similarly small
counts, but based on the results here and in §3.2 are nevertheless compelling
candidates to be outliers.

Returning to the ID-based interpretation of cluster 12, Table 2 shows that
every coronavirus cluster 1–10 (i.e., all except the the outlier constituting
cluster 12 and the cluster with 26,433 members) is essentially defined by a
substring of the ID, subject to two qualifications. First, some clusters (to
wit, 1, 4, and 7) contain multiple substrings that cover the cluster and
appear in no other cluster. And second, there are substrings that cover
multiple clusters but appear in no other clusters. For instance, “Human
coronavirus HKU1” covers all 15 elements of cluster 2 and all 3 elements of
cluster 10, and appears nowhere else. We again conclude that the clustering
captures scientific reality.

Cluster | Cluster | Substring of ID | Count in
---|---|---|---
| Count |  | This Cluster | Other Clusters
1 | 12 | TOTAL |  |
|  | SARS coronavirus Tor2 | 2 | 0
|  | SARS coronavirus GDH-BJH01 | 1 | 0
|  | SARS coronavirus P2 | 1 | 0
|  | SARS coronavirus HKU-39849 isolate TCVSP-HARROD | 2 | 0
|  | SARS coronavirus Urbani isolate icSARS | 6 | 0
2 | 15 | Human coronavirus HKU1 | 15 | 3
10 | 3 |  | 3 | 15
3 | 2 | Human enteric coronavirus […] 4408 | 2 | 0
4 | 141 | TOTAL |  |
|  | Human coronavirus OC43 strain | 138 | 0
|  | Coronavirus cy[abc]-BetaCoV/2019 | 3 | 0
5 | 55 | Human coronavirus NL63 strain | 55 | 6
8 | 3 |  | 3 | 58
9 | 3 |  | 3 | 58
6 | 27 | Human coronavirus 229E | 27 | 0
7 | 258 | TOTAL |  |
|  | Middle East respiratory syndrome | 253 | 0
|  | [Bb]etacoronavirus | 5 | 0
12 | 1 | SARS-CoV-2/human/AUS/VIC402/2020 | 1 | 0
Table 2: Substrings of NCBI IDs that characterize the clusters containing
coronavirus genomes. Single-member clusters comprised of the adenovirus genome
or one of the ten degraded coronavirus genomes are excluded.

###  3.2 Clustering Based on Amino Acids

Clustering was also performed on amino acid distributions,555Generated
directly from triplet distributions using the known many-to-one mapping of
triplets to amino acids. in 21 dimensions rather than 64, resulting in the MDS
plot and dendrogram shown in Figure 2 and the cluster characteristics in Table
3. There are fewer clusters than for triplets—13 as compared to 23. The
statistical explanatory power matches that for triplets: all coefficients of
determination exceed 0.95.

Figure 2: _Top:_ Two-dimensional MDS plot of amino acid distribution of cluster centroids for the 26,964-element coronavirus dataset containing 26,953 coronavirus genomes and 11 deliberately added outliers. _Bottom:_ Corresponding dendrogram. Class in the dendrogram is the same as in Table 3, and shown the the same colors in both graphics. Cluster | Count | Class | Separation Height | Separation Rank
---|---|---|---|---
1 | 12 | Coronavirus | 33.0 | 3.0
2 | 18 | Coronavirus | 4.6 | 11.0
3 | 143 | Coronavirus | 0.0 | 12.5
4 | 55 | Coronavirus | 0.0 | 12.5
5 | 27 | Coronavirus | 34.9 | 2.0
6 | 258 | Coronavirus | 11.5 | 7.5
7 | 3 | Coronavirus | 11.5 | 7.5
8 | 3 | Coronavirus | 12.8 | 6.0
9 | 26434 | Coronavirus | 7.2 | 9.5
10 | 1 | Adenovirus | 7.2 | 9.5
11 | 4 | Degraded Coronavirus | 37.5 | 1.0
12 | 5 | Degraded Coronavirus | 18.5 | 4.5
13 | 1 | Degraded Coronavirus | 18.5 | 4.5
Table 3: Characteristics of the amino acid distribution clusters in Figure 2.
See text for explanation of Separation Height and Separation Rank. The double
horizontal line divides coronavirus genomes from the deliberate outliers.

Table 4, a cross-tabulation of triplet distribution clusters (rows) and amino
acid distribution clusters (columns), shows that each amino acid cluster is
either identical to a triplet distribution cluster or an amalgam of triplet
distribution clusters. Specifically, for the deliberate outliers, the
adenovirus genome remains in a cluster of its own (13 for triplets; 10 for
amino acids), while the 10 degraded coronavirus genomes (triplet clusters
14–23) are condensed to three amino acid clusters (11–13). Interestingly, the
coronavirus genome that alone comprised triplet cluster 12 is absorbed into
the massive cluster (triplet cluster 11), which now has size 26,434 (amino
acid cluster 9). Triplet cluster 1 (count = 12) and amino acid cluster 1 are
identical, as are triplet cluster 5 (count = 55) and amino acid cluster 12, as
well as triplet cluster 6 (count = 27) and amino acid cluster 5, triplet
cluster 7 (count = 258) and amino acid cluster 6, triplet cluster 8 (count =
3) and amino acid cluster 7 and, finally, triple cluster 9 (count = 3) and
amino acid cluster 10. Triplet clusters 2 and 10 are merged to create amino
acid cluster 2, while triplet clusters 3 and 4 merge into amino acid cluster
3.

| Amino Acid Cluster
---|---
Triplet Cluster | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | Sum
1 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 12
2 | 0 | 15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 15
3 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2
4 | 0 | 0 | 141 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 141
5 | 0 | 0 | 0 | 55 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 55
6 | 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27
7 | 0 | 0 | 0 | 0 | 0 | 258 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 258
8 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 3
9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 3
10 | 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3
11 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 26433 | 0 | 0 | 0 | 0 | 26433
12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 1
13 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1
14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1
15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1
17 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
19 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
20 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1
21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1
22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1
23 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 1
Sum | 12 | 18 | 143 | 55 | 27 | 258 | 3 | 3 | 26434 | 1 | 4 | 5 | 1 | 26964
Table 4: Cross-tabulation of triplet-based clusters and amino acid-based
clusters. Entries corresponding to identical clusters are highlighted in
enlarged boldface.

The dendrogram in Figure 2 and the separation heights and ranks in Table 3 add
insightful nuances. In the dendrogram, the first split isolates amino acid
clusters 1–4 from everything else, including the massive coronavirus cluster
and the deliberate outliers. The next split isolates the ten degraded
coronavirus genomes from the latter, but the adenovirus genome remains coupled
with the dominant coronavirus cluster until relatively late in the process
(height = 7.2), which is the separation height for both. Ignoring the
deliberate outliers, there are are two extreme separation heights—34.9 for
cluster 5 and 33.0 for cluster 1. The latter reinforces our hypothesis in §3.1
that cluster 1 contains outliers. The analogous message regarding cluster 5 is
attenuated, so using amino acids provides information not available from
triplets alone. The very last split in the amino acid dendrogram, which
separates cluster 3 (count = 143) and cluster 4 (count = 55) is less
discernible in the dendrogram in Figure 1.

##  4 Bayesian Classification of Simulated Illumina Reads

In this section, we apply triplet distributions to the metagenomics problem of
classifying short genome reads in mixtures comprising multiple genomes, a
common step in reference-guided assembly.

###  4.1 The Experiment

The components of the experiment are introduced here.

Reference Genomes. There are three reference genomes: the adenovirus genome of
length 34,125, downloaded with the read simulator Art, which we also call
Adeno; a SARS-CoV-2 genome of length 29,926 contained in the NCBI coronavirus
dataset employed in §3, which we call COVID; and a severe acute respiratory
syndrome (SARS) genome of length 29,751 from the same database, which we call
SARS. Tables 10 and 11, in Appendix A, show the triplet and amino acid
distributions for these three genomes A natural question is “How different are
these triplet distributions?” Measured by Hellinger distance (Nikulin,, 2001)
they are very different. The distances are 0.234 for adenovirus/COVID, 0.125
for adenovirus/SARS and 0.161 for COVID/SARS. To gauge these differences, the
empirical 0.001 $p$-values for distributions of genomes of the same size
simulated from each distribution are 0.01941755 for adenovirus, 0.02094808 for
COVID, and 0.02065539 for SARS. The Hellinger distances for amino acid
distributions are 0.189 for adenovirus/COVID, 0.090 for adenovirus/SARS and
0.130 for COVID/SARS. All these distances exceed empirical 0.1% $p$-values.

Read Generation. The Mason_simulator read simulator (Holtgrewe,, 2010) was
used to simulate Illumina666Illumina is a major manufacturer of high-end
instruments for genome sequencing; their technology is optical in nature; see
https://www.illumina.com/. reads of length 101 from each genome, with
approximate 6X coverage. The numbers of reads are 1966, 1996 and 1907,
respectively; the total number of reads is 5869. The Mason_simulator
introduces errors in the form of transpositions (single nucleotide
polymorphisms (SNPs)), insertions, deletions and undetermined bases, which,
following convention, appear in the simulated reads as “N” and must be
accommodated in computation of likelihood functions. Parameters of the
Mason_simulator were set at default values.

Likelihood Functions. Three likelihood functions, one for each genome and
denoted by $L_{A}(\cdot)$, $L_{C}(\cdot)$ and $L_{S}(\cdot)$, were calculated,
representing the triplet distributions. To illustrate for adenovirus,

$\displaystyle L_{A}(R)=P_{2}(R(1)R(2)|A)\times T_{3}((R(1),R(2)),R(3)|A)$ (3)
$\displaystyle\times T_{3}((R(2),R(3)),R(4)|A)\times\dots\times
T_{3}\bigg{(}\big{(}R(|R|-2),R(|R|-1)\big{)},R(|R|)|A\bigg{)},$

where $R(k)$ is the $k^{\mbox{th}}$ element of the read $R$, $P_{2}(\cdot|A)$
is the pair distribution, and $T_{3}$ is given by (2). In (3), we have ignored
Ns for simplicity; when they are present, they lead to sums over all possible
bases. Below, we also explore alterative likelihood functions representing
deliberate degradation of quality of the training data by means of the
Mason_variator.

Prior and Posterior Probabilities. For each read $R$, we specify a prior
probability distribution $\pi_{R}$ over the set $\mathcal{A}=\\{\mbox{Adeno,
COVID, SARS}\\}$ of the three genomes, and then use Bayes’ theorem and the
three likelihoods to calculate posterior probabilities over $\mathcal{A}$,
which we then analyze. Specifically, for each read $R$, the posterior
probability of $x\in\mathcal{A}$ is given by

$p(x|R)=\frac{\pi_{R}(x)L_{x}(R)}{\pi_{R}(A)L_{A}(R)+\pi_{R}(C)L_{C}(R)+\pi_{R}(S)L_{S}(R)}.$
(4)

Experimental Protocol. All three components in the calculation can be varied,
and their contributions to the final output resolved:

The input data,

that is, the reads themselves, which contain Mason_simulator-generated errors.
Because the Mason_simulator records the starting location of each read, we
have as well the error-free reads, which are of higher quality. Conversely, we
can use the Mason_variator to degrade read quality, as in Karr et al., 2021b .

The prior distributions,

which in most cases we take to be (mildly) informative distributions generated
randomly from a Dirichlet distribution. An alternative representing no
external prior knowledge is the non-informative priors
$\pi_{R}=(1/3,1/3,1/3)$.

The models,

embedded in the likelihood functions for the three genomes. The three genomes
themselves are, in effect, the training data. As discussed further below,
model quality can be decreased by degrading them using the Mason_variator.

###  4.2 The Base Case

As base case, we use the informative priors, the actual reads and the correct
models based on the triplet distribution likelihood functions from the three
genomes. Figure 3 contains the results. Because similar figures follow, we
discuss it in some detail. First, three-dimensional probabilities (barycentric
coordinates) are represented in the figure using Cartesian coordinates, as
points in a two-dimensional simplex—an equilateral triangle. Pure adenovirus,
in the sense that $\mathrm{Prob}(\mbox{Adeno})=1$, is the top vertex, pure
COVID is the lower left vertex, and pure SARS is the lower right vertex.
Because we know the sources of the reads, we create separate displays for each
source—adenovirus at the left, COVID in the center and SARS at the right. The
upper three panels show prior probabilities, while the lower three panels show
posterior probabilities. The green/red/blue coloring depicting read source is
redundant, but useful. The white diamond in each scatterplot is the centroid
of the probabilities it contains.

The interior black lines, which are often not completely visible, are the
decision boundaries for the MAP (maximum _a posteriori_ probability)
classifier, which assigns to each read the genome with the highest posterior
probability. The value of the largest posterior probability cannot be less
than 1/3, but can be arbitrarily close to it. Points that lie near any of
these boundaries are close calls, but are not distinguished by the MAP
classifier from clear-cut cases.777To note an extreme example, the MAP
classifier does not distinguish read $R$ with $p(\cdot|R)=(0.34,0.33,0.33)$—an
extremely close call—from read $R^{\prime}$ with
$p(\cdot|R^{\prime})=(0.99,0.005,0.005)$—a near-certainty to be from the
adenovirus genome. In some contexts, this may be problematic. Alternative
decision strategies in Karr et al., 2021a  implement a concept known as
Specified Certainty Classification (SSC) that enforces a user-specified lower
bound on the certainty of each decision.

Figure 3: Prior and posterior probabilities for the base case of actual reads,
informative prior distributions and correct triplet distribution-based models.
_Top left:_ Prior probabilities for reads from adenovirus. _Top middle:_ Prior
probabilities for reads from COVID. _Top right:_ Prior probabilities for reads
from SARS. _Bottom left:_ Posterior probabilities for reads from adenovirus.
_Bottom middle:_ Posterior probabilities for reads from COVID. _Bottom right:_
Posterior probabilities for reads from SARS. White symbols in each plot are
centroids.

Now to interpretations. That the prior distributions are informative is
manifested in that each, while scattered, gravitates toward the vertex
representing the truth. From the prior to the posterior, all probabilities
shift in the direction of the truth–which is what data are supposed to do.
Centroids move in response. We caution that there is massive overplotting. For
instance, the posterior probabilities for adenovirus (lower left) appear
scattered, yet the centroid is very close to the truth—the upper vertex. Table
5 shows the base case confusion matrix for the MAP classifier. The performance
is imperfect but respectable: the correct classification rate is
$(1757+1762+1549)/5869=86.35$%.

| Decision |
---|---|---
Source | Adeno | COVID | SARS | Sum
Adeno | 1757 | 74 | 135 | 1966
COVID | 64 | 1762 | 170 | 1996
SARS | 214 | 144 | 1549 | 1907
Sum | 2035 | 1980 | 1854 | 5869
Table 5: Confusion matrix for the MAP classifier for the base case. Rows are
read sources; columns are MAP classifier decisions. The correct classification
rate is 86.35%.

###  4.3 Variations on the Base Case

In terms of understanding all the decisions the classifier _can make_ , as
opposed to the decisions it _does make_ in a specific application (Karr et
al.,, 2019), the Bayesian framework allows investigation and quantification of
the relative contributions to uncertainty of the priors, the data and the
models. Here we explore the effect of each.

Changing the Priors. The priors are relatively the least important of the
three, with one important _caveat_ discussed momentarily.. Figure 4 is the
analog of Figure 3 for the “no prior knowledge” case that
$\pi_{R}=(1/3,1/3,1/3)$ for all reads $R$. The visual message that there is no
dramatic difference is corroborated by the confusion matrix in Table 6; the
correct classification rate has declined only modestly, to
$(1601+1717+1470)/5869=81.58$%.

Figure 4: Prior and posterior probabilities for the case of actual reads, “no knowledge priors” and correct triplet distribution-based models. _Top left:_ Prior probabilities for reads from adenovirus. _Top middle:_ Prior probabilities for reads from COVID. _Top right:_ Prior probabilities for reads from SARS. _Bottom left:_ Posterior probabilities for reads from adenovirus. _Bottom middle:_ Posterior probabilities for reads from COVID. _Bottom right:_ Posterior probabilities for reads from SARS. White symbols in each plot are centroids. | Decision |
---|---|---
Source | Adeno | COVID | SARS | Sum
Adeno | 1601 | 115 | 250 | 1966
COVID | 64 | 1717 | 215 | 1996
SARS | 268 | 169 | 1470 | 1907
Sum | 1933 | 2001 | 1935 | 5869
Table 6: Confusion matrix for the MAP classifier for the case of actual reads,
“no knowledge” priors, and correct triplet distribution-based models. The
correct classification rate is 81.58%.

Zeros in priors, however, can be highly problematic. The “no adenovirus”
priors $\pi_{R}=(0,1/2,1/2)$ instantiate a mistaken belief that no adenovirus
reads are present. Figure 5 contains results for these priors, actual reads,
and the correct triplet distribution-based models. All posterior probabilities
lie on the COVID–SARS axis—the bottom edge of the triangle. Table 7 contains
the associated confusion matrix. Because, as (4) makes clear, a zero in a
prior distribution is inherited by the posterior, all adenovirus reads are
misclassified, most of them as SARS. The operational lesson is that “if it
might be there, allow for it in the priors.” See §4.4 for further discussion
in the context of contamination.

Figure 5: Prior and posterior probabilities for the case of actual reads, “no adenovirus priors” and correct triplet distribution-based models. _Top left:_ Prior probabilities for reads from adenovirus. _Top middle:_ Prior probabilities for reads from COVID. _Top right:_ Prior probabilities for reads from SARS. _Bottom left:_ Posterior probabilities for reads from adenovirus. _Bottom middle:_ Posterior probabilities for reads from COVID. _Bottom right:_ Posterior probabilities for reads from SARS. | Decision |
---|---|---
| COVID | SARS | Sum
Adeno | 163 | 1803 | 1966
COVID | 1728 | 268 | 1996
SARS | 177 | 1730 | 1907
Sum | 2068 | 3801 | 5869
Table 7: Confusion matrix for the MAP classifier for the case of actual reads,
“no adenovirus” priors, and correct triplet distribution-based models. The
correct classification rate is 58.92%

Hereafter, we consider only the informative priors, and omit prior
probabilities from the graphics.

Changing Read (= Input Data) Quality. Figure 6 shows the effects of increasing
and decreasing the quality of the simulated reads. To enable comparisons, the
top panel there is the base case posterior probabilities appearing at the
bottom of Figure 3. The middle panel shows that using the error-free reads in
place of the Mason_simulator-generated reads makes no meaningful difference.
This is because the default error rates in the Mason_simulator are not large.

More dramatic is the effect of degrading the reads using 1000 iterations of
the Mason_variator, in the manner of Karr et al., 2021b , which is shown in
the bottom panel in Figure 6. The associated MAP confusion matrix is in Table
8. The capability to classify COVID and SARS reads correctly decreases
significantly, but the same is not true for adenovirus reads: the number of
correctly classified adenovirus reads is higher than in the base case.

Figure 6: Posterior probabilities when input data quality is changed. _Top:_ Base case (actual reads, informative priors, correct models). _Middle:_ Error-free reads, informative priors, correct models. _Bottom:_ Degraded reads, informative priors, correct models. | Decision |
---|---|---
| Adeno | COVID | SARS | Sum
Adeno | 1827 | 22 | 117 | 1966
COVID | 821 | 644 | 531 | 1996
SARS | 1042 | 65 | 800 | 1907
Sum | 3690 | 731 | 1448 | 5869
Table 8: Confusion matrix for the MAP classifier for the case of reads
degraded by 1000 Mason_variator iterations, informative priors, and correct
triplet distribution-based model. The correct classification rate is 54.80%

Changing Model (= Training Data) Quality. Finally, we consider the effects of
decreased model quality resulting from low quality training data.
Specifically, before creating the triplet distribution likelihood functions,
we subject each of the three genomes to degradation by 2000 iterations of the
Mason_variator. Figure 7 shows the results. As in the preceding variation,
adenovirus suffers more than COVID or SARS. Comparison of the lower panel
there with the bottom panel in Figure 6 shows that the effect of poor input
data in the former exceeds the effect of poor training data in the latter.

Figure 7: Posterior probabilities showing the effect of low quality models
resulting from low quality training data. _Top:_ Base case—actual reads,
informative priors, correct models. _Bottom:_ Actual reads, informative
priors, degraded models.

###  4.4 Detecting Contamination

We sketch here how our strategy can be applied to detect contamination, a
major issue in some genomics and metagenomics settings.888For example, Reich,
(2018) cites microbial and experimenter contamination as major problems in the
context of ancient DNA. To maintain continuity, we use the same three
reference genomes as in §4, but with adenovirus viewed as the contaminant.

The key point is addressed in Figure 5: contamination not allowed for in the
priors cannot be detected. If we replace the “no adenovirus” priors
$\pi_{R}=(0,1/2,1/2)$ underlying Figure 5 by “almost no adenovirus” priors
$\pi_{R}=(0.0001,0.9999/2,0.9999/2)$, the result is as shown in Figure 8. For
the original task of classifying (roughly comparable numbers of) reads, the
performance is noticeably less that in than the base case of Figure 3, which
is confirmed by comparison of the confusion matrices—that for the base case in
Table 5, and that for the case at hand in Table 9. The correct classification
rate drops from 86.35% to 62.57%, which is sobering, but not frightening. The
performance for COVID and SARS reads is undiminished. Crucial for what
follows, the presence of adenovirus reads is confirmed.

Figure 8: Posterior probabilities for the case of actual reads, “almost no
adenovirus priors” and correct triplet distribution-based models. _Left:_
Posterior probabilities for reads from adenovirus. _Middle:_ Posterior
probabilities for reads from COVID. _Right:_ Posterior probabilities for reads
from SARS.

| Decision |
---|---|---
Source | Adeno | COVID | SARS | Sum
Adeno | 215 | 163 | 1589 | 1966
COVID | 0 | 1728 | 268 | 1996
SARS | 1 | 177 | 1729 | 1907
Sum | 469 | 2063 | 3337 | 5869
Table 9: Confusion matrix for the MAP classifier for the case of actual reads,
“almost no adenovirus” prior and correct triplet distribution-based models.
Rows are read sources; columns are MAP decisions. The correct classification
rate is 62.57%.

Can much smaller numbers of adenovirus reads be detected by our methodology? A
plausible strategy is to establish a threshold on the posterior probability of
adenovirus and to declare, or at least suspect, contamination if sufficiently
many—which might be as few as one—adenovirus posterior probabilities exceed
that threshold. One candidate for the threshold is the maximum of the
adenovirus posterior probabilities over the 3903 = 1996 + 1907 COVID and SARS
reads in the dataset, which is

$T^{*}=\max\big{\\{}P(\textrm{Adeno|R}):\mbox{Source}(R)\neq\textrm{Adeno}\big{\\}}=0.5219,$
(5)

where $p(\cdot|R)$ is the posterior probability given by (4). Note that false
positives do not occur if the threshold is greater than $T^{*}$, so there is
no benefit from using a larger threshold.

Arguably, a traditional receiver operating characteristic (ROC) curve approach
(Fawcett,, 2006; Karr et al.,, 2019) is too aggressive; while values of area
under the curve (AUC) are high, unless the utility function is skewed, there
are too many false positives. Figure 9 points toward an alternative decision
strategy. It shows the mean probability of detection, averaged over 2000
replicates, for numbers of contaminating adenovirus reads ranging from 1 to
20, as a function of threshold. The vertical black line is the value $T^{*}$
given by (5). As noted previously, there are no false positives for thresholds
exceeding $T^{*}$. For 20 reads, the probability of detecting contamination at
this threshold is nearly 0.9, and even for only one read, it still exceeds
0.1.

Figure 9: Detection probabilities for adenovirus contamination as a function
of threshold and number of contamination reads.

##  5 Conclusion

Triplet distributions are the “sweet spot” means of dimension reduction for
virus genomes. In the applications to outlier identification and read
classification—the major contributions of the paper, the results are not only
statistically strong but also scientifically interpretable.

To understand the extent to which these results apply to longer genomes, or
larger genome databases, requires further research. Computationally, there are
not major impediments: almost all of the tools employed here have memory and
running time demands that are linear in genome length and database size. The
cluster analyses in §3 are an exception, because they require distance
matrices whose size is the square of the database size (divided by 2, of
course). The read classification problem is linear in the number of classes
and the number of reads. The big question is whether, for instance, a 64- or
21-dimensional summary of a 3 million BP genome (bacteria) or 3 billion BP
genome (humans) captures enough information to be both statistically useful
and scientifically credible.

## Acknowledgements

This research was supported in part by NIH grant 5R01AI100947–06, “Algorithms
and Software for the Assembly of Metagenomic Data,” to the University of
Maryland College Park (Mihai Pop, PI).

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##  Appendix A Example Triplet and Amino Acid Distributions

Table 10 contains the triplet distributions for the three virus genomes
underlying the read classification problem in §4. The corresponding amino acid
distributions are in Table 11. We discuss in §4 the extent to which these
differ across the three genomes.

Triplet | Adeno | COVID | SARS | Triplet | Adeno | COVID | SARS
---|---|---|---|---|---|---|---
AAA | 0.031826 | 0.026367 | 0.025581 | GAA | 0.017496 | 0.012899 | 0.015261
AAC | 0.020016 | 0.010660 | 0.018051 | GAC | 0.012220 | 0.005982 | 0.012337
AAG | 0.018463 | 0.016776 | 0.018891 | GAG | 0.013832 | 0.007252 | 0.013278
AAT | 0.018551 | 0.030176 | 0.021917 | GAT | 0.011927 | 0.021621 | 0.015597
ACA | 0.020016 | 0.012699 | 0.026219 | GCA | 0.017847 | 0.008221 | 0.014421
ACC | 0.015327 | 0.006851 | 0.013076 | GCC | 0.015063 | 0.004278 | 0.007866
ACG | 0.010316 | 0.003709 | 0.005210 | GCG | 0.015327 | 0.002373 | 0.004975
ACT | 0.016265 | 0.016609 | 0.022018 | GCT | 0.016499 | 0.013868 | 0.020908
AGA | 0.014213 | 0.016208 | 0.018085 | GGA | 0.016704 | 0.005514 | 0.011698
AGC | 0.017701 | 0.007519 | 0.011765 | GGC | 0.014243 | 0.005247 | 0.009849
AGG | 0.015591 | 0.008421 | 0.013984 | GGG | 0.012279 | 0.003409 | 0.004975
AGT | 0.014711 | 0.019015 | 0.015059 | GGT | 0.013041 | 0.018346 | 0.013916
ATA | 0.012836 | 0.024429 | 0.013345 | GTA | 0.013334 | 0.020151 | 0.014723
ATC | 0.011224 | 0.010694 | 0.011294 | GTC | 0.010228 | 0.007419 | 0.009984
ATG | 0.017378 | 0.029475 | 0.026085 | GTG | 0.014067 | 0.016308 | 0.018488
ATT | 0.018990 | 0.038765 | 0.024438 | GTT | 0.016294 | 0.037562 | 0.019698
CAA | 0.020573 | 0.012565 | 0.024471 | TAA | 0.018961 | 0.032148 | 0.019160
CAC | 0.014008 | 0.006015 | 0.016202 | TAC | 0.015679 | 0.017244 | 0.019933
CAG | 0.019342 | 0.008956 | 0.014891 | TAG | 0.010579 | 0.018179 | 0.011832
CAT | 0.017115 | 0.012030 | 0.018589 | TAT | 0.012836 | 0.039533 | 0.019026
CCA | 0.019400 | 0.006149 | 0.013311 | TCA | 0.013774 | 0.012498 | 0.020202
CCC | 0.014067 | 0.002573 | 0.004773 | TCC | 0.013744 | 0.006115 | 0.007059
CCG | 0.010257 | 0.001604 | 0.003059 | TCG | 0.008059 | 0.003676 | 0.005849
CCT | 0.014506 | 0.009491 | 0.011631 | TCT | 0.014301 | 0.019416 | 0.019093
CGA | 0.009055 | 0.002406 | 0.004672 | TGA | 0.015503 | 0.023593 | 0.022018
CGC | 0.015503 | 0.001771 | 0.004471 | TGC | 0.017290 | 0.014203 | 0.022085
CGG | 0.010257 | 0.001571 | 0.002756 | TGG | 0.018170 | 0.019115 | 0.018723
CGT | 0.009143 | 0.005614 | 0.007194 | TGT | 0.017027 | 0.038464 | 0.026724
CTA | 0.013012 | 0.017544 | 0.018723 | TTA | 0.018873 | 0.044981 | 0.023160
CTC | 0.011459 | 0.007085 | 0.012000 | TTC | 0.016968 | 0.016508 | 0.018925
CTG | 0.017525 | 0.014203 | 0.018891 | TTG | 0.019019 | 0.035390 | 0.026085
CTT | 0.019576 | 0.020552 | 0.024034 | TTT | 0.030595 | 0.059985 | 0.027463
Table 10: Triplet distributions for the adenovirus, COVID and SARS genomes underlying §4. Amino Acid | Adeno | COVID | SARS
---|---|---|---
Alanine | 0.043959 | 0.011362 | 0.019093
Arginine | 0.092782 | 0.054271 | 0.074322
Asparagine | 0.037892 | 0.080370 | 0.049245
Aspartate | 0.030537 | 0.031747 | 0.037615
Cysteine | 0.030331 | 0.016408 | 0.031430
Glutamate | 0.031035 | 0.027637 | 0.036035
Glutomine | 0.024294 | 0.023727 | 0.028472
Glycine | 0.058231 | 0.019817 | 0.032774
Histidine | 0.029628 | 0.057713 | 0.034421
Isoleucine | 0.042376 | 0.089861 | 0.050018
Leucine | 0.094042 | 0.088591 | 0.096440
Lysine | 0.047563 | 0.076494 | 0.046388
Methionine; START | 0.015679 | 0.017244 | 0.019933
Phenylalanine | 0.050289 | 0.043143 | 0.044472
Proline | 0.056267 | 0.032516 | 0.040438
Serine | 0.084049 | 0.067337 | 0.084944
STOP | 0.046479 | 0.066067 | 0.057750
Threonine | 0.067989 | 0.095375 | 0.089549
Tryptophan | 0.015327 | 0.006851 | 0.013076
Tyrosine | 0.030214 | 0.053903 | 0.039430
Valine | 0.071037 | 0.039567 | 0.074154
Table 11: Amino acid distributions for the adenovirus, COVID and SARS genomes
underlying §4.