backup

tex/Makefile

@@ -11,7 +11,10 @@
 
 all:minimap2.pdf
 
-minimap2.pdf:minimap2.tex minimap2.bib
+roc-color.eps:roc.gp
+		gnuplot roc.gp
+
+minimap2.pdf:minimap2.tex minimap2.bib roc-color.pdf
 		pdflatex minimap2; bibtex minimap2; pdflatex minimap2; pdflatex minimap2;
 
 clean:

tex/bioinfo.cls

@@ -455,7 +455,7 @@
   \vspace*{\aboveskipchk}%
   \vspace{\dropfromtop}%
   \hbox to \textwidth{%
-  {\helvetica\itshape\bfseries\fontsize{19}{12}\selectfont {\color{gray}TECHNICAL NOTES}
+  {\helvetica\itshape\bfseries\fontsize{19}{12}\selectfont {\color{gray}TECHNICAL REPORT}
     \hfil
     \if@appnotes APPLICATIONS NOTE\hfil\fi
     }%

tex/eval2roc.pl

@@ -4,17 +4,30 @@ use strict;
 use warnings;
 use Getopt::Std;
 
-my %opts = (n=>33088);
+my %opts = (n=>33088, s=>100);
 getopts('n:', \%opts);
 
 my $pseudo = .5;
 my $tot = $pseudo;
 my $err = $pseudo;
+my $tot_last_out = -$opts{s};
+my $state = 0;
+my $mapq = 0;
 while (<>) {
 	chomp;
 	if (/^Q\t(\d+)\t(\d+)\t(\d+)/) {
 		$tot += $2;
 		$err += $3;
-		print join("\t", $1, $err/$tot, $tot / $opts{n}), "\n";
+		if ($tot - $tot_last_out >= $opts{s}) {
+			print join("\t", $1, $err/$tot, $tot / $opts{n}), "\n";
+			$tot_last_out = $tot;
+			$state = 0;
+		} else {
+			$state = 1;
+			$mapq = $1;
+		}
 	}
 }
+if ($state) {
+	print join("\t", $mapq, $err/$tot, $tot / $opts{n}), "\n";
+}

tex/minimap2.bib

@@ -7,7 +7,7 @@
 	Year = {2012}}
 
 @article{Liu:2016ab,
-	Author = {Liu, Bo and Guan, Dengfeng and Teng, Mingxiang and Wang, Yadong},
+	Author = {Liu, Bo and others},
 	Journal = {Bioinformatics},
 	Pages = {1625-31},
 	Title = {{rHAT}: fast alignment of noisy long reads with regional hashing},
@@ -15,7 +15,7 @@
 	Year = {2016}}
 
 @article{Liu:2017aa,
-	Author = {Liu, Bo and Gao, Yan and Wang, Yadong},
+	Author = {Liu, Bo and others},
 	Journal = {Bioinformatics},
 	Pages = {192-201},
 	Title = {{LAMSA}: fast split read alignment with long approximate matches},
@@ -38,7 +38,7 @@
 	Year = {2013}}
 
 @article{Sovic:2016aa,
-	Author = {Sovi{\'c}, Ivan and {\v S}iki{\'c}, Mile and Wilm, Andreas and Fenlon, Shannon Nicole and Chen, Swaine and others},
+	Author = {Sovi{\'c}, Ivan and others},
 	Journal = {Nat Commun},
 	Pages = {11307},
 	Title = {Fast and sensitive mapping of nanopore sequencing reads with {GraphMap}},
@@ -68,6 +68,13 @@
 	howpublished = {\href{https://github.com/ocxtal/minialign}{https://github.com/ocxtal/minialign}},
 	year = {2016}}
 
+@misc{Ruan:2016,
+	title = {Ultra-fast de novo assembler using long noisy reads},
+	author = {Jue Ruan},
+	journal = {Unpulished},
+	howpublished = {\href{https://github.com/ruanjue/smartdenovo}{https://github.com/ruanjue/smartdenovo}},
+	year = {2016}}
+
 @article{Miller:1988aa,
 	Author = {Miller, W and Myers, E W},
 	Journal = {Bull Math Biol},
@@ -86,7 +93,7 @@
 	Year = {1990}}
 
 @article{Wu:1996aa,
-	Author = {Wu, Sun and Manber, U and Myers, Gene},
+	Author = {Wu, Sun and others},
 	Journal = {Algorithmica},
 	Pages = {50-67},
 	Title = {A subquadratic algorithm for approximate limited expression matching},
@@ -98,22 +105,22 @@
 	Journal = {BMC Bioinformatics},
 	Month = {Feb},
 	Pages = {81},
-	Title = {Parasail: SIMD C library for global, semi-global, and local pairwise sequence alignments},
+	Title = {Parasail: {SIMD C} library for global, semi-global, and local pairwise sequence alignments},
 	Volume = {17},
 	Year = {2016}}
 
 @article{Sedlazeck169557,
-	author = {Sedlazeck, Fritz J and Rescheneder, Philipp and Smolka, Moritz and Fang, Han and Nattestad, Maria and others},
+	author = {Sedlazeck, Fritz J and others},
 	title = {Accurate detection of complex structural variations using single molecule sequencing},
 	note = {doi:10.1101/169557},
 	journal = {bioRxiv},
 	year = {2017}}
 
 @article{Altschul:1997vn,
-	Author = {Altschul, S F and Madden, T L and Sch{\"a}ffer, A A and Zhang, J and Zhang, Z and others},
+	Author = {Altschul, S F and others},
 	Journal = {Nucleic Acids Res},
 	Pages = {3389-402},
-	Title = {Gapped BLAST and PSI-BLAST: a new generation of protein database search programs},
+	Title = {Gapped {BLAST} and {PSI-BLAST}: a new generation of protein database search programs},
 	Volume = {25},
 	Year = {1997}}
 
@@ -132,3 +139,27 @@
 	Title = {Chaining algorithms for multiple genome comparison},
 	Volume = {3},
 	Year = {2005}}
+
+@article{Ono:2013aa,
+	Author = {Ono, Yukiteru and others},
+	Journal = {Bioinformatics},
+	Pages = {119-21},
+	Title = {{PBSIM}: {PacBio} reads simulator--toward accurate genome assembly},
+	Volume = {29},
+	Year = {2013}}
+
+@article {Jain128835,
+	author = {Jain, Miten and others},
+	title = {Nanopore sequencing and assembly of a human genome with ultra-long reads},
+	year = {2017},
+	note = {doi:10.1101/128835},
+	publisher = {Cold Spring Harbor Labs Journals},
+	journal = {bioRxiv}}
+
+@article{Lau:2016aa,
+	Author = {Lau, Bayo and others},
+	Journal = {Bioinformatics},
+	Pages = {3829-3832},
+	Title = {LongISLND: in silico sequencing of lengthy and noisy datatypes},
+	Volume = {32},
+	Year = {2016}}

tex/minimap2.tex

@@ -2,6 +2,7 @@
 \copyrightyear{2017}
 \pubyear{2017}
 
+\usepackage{graphicx}
 \usepackage{hyperref}
 \usepackage{url}
 \usepackage{amsmath}
@@ -19,7 +20,7 @@
 \begin{document}
 \firstpage{1}
 
-\title[Long-read and assembly alignment with minimap2]{Minimap2: fast sequence alignment for long noisy reads and assembly contigs}
+\title[Long sequence alignment with minimap2]{Minimap2: fast pairwise alignment for long noisy sequences}
 \author[Li]{Heng Li}
 \address{Broad Institute, 415 Main Street, Cambridge, MA 02142, USA}
 
@@ -29,7 +30,7 @@
 \section{Summary:} Minimap2 is a program to align long noisy sequences against
 a large reference database. It targets query sequences of 1kb--100Mb in length
 with sequence divergence typically below 25\%. Minimap2 is $\sim$30 times
-faster than most existing long-read aligners and achieves higher accuracy on
+faster than many mainstream long-read aligners and achieves higher accuracy on
 simulated data. It also employs concave gap cost and rescues inversions for
 improved alignment around potential structural variations.
 
@@ -60,10 +61,10 @@ towards higher accuracy.
 
 Minimap2 is the successor of minimap~\citep{Li:2016aa}. It uses similar
 indexing and seeding algorithms except that minimap2 optionally uses
-homopolymer-compressed (HPC; cite) $k$-mers in addition to normal $k$-mers.
+homopolymer-compressed (HPC; \citealp{Ruan:2016,Lau:2016aa}) $k$-mers in
-Indexing with HPC $k$-mers leads to higher mapping sensitivity for SMRT reads.
+addition to normal $k$-mers.  Indexing with HPC $k$-mers leads to higher
-Minimap2 further implements a more accurate chaining algorithm and adds
+mapping sensitivity for SMRT reads.  Minimap2 further implements a more
-the ability to produce detailed alignment.
+accurate chaining algorithm and adds the ability to produce detailed alignment.
 
 \subsection{Chaining}
 
@@ -84,20 +85,19 @@ distance between two anchors is too large); otherwise
 \gamma(j,i)=\gamma'(\max\{y_i-y_j,x_i-x_j\}-\min\{y_i-y_j,x_i-x_j\})
 \]
 In implementation, a gap of length $l$ costs $\gamma'(l)=\alpha\cdot
-l+\beta\log_2(l)$. For $m$ anchors, computing all $f(\cdot)$
+l+\beta\log_2(l)$. For $m$ anchors, directly computing all $f(\cdot)$ with
-with Eq.~(\ref{eq:chain}) takes $O(m^2)$ time. We note that if anchor $i$ is
+Eq.~(\ref{eq:chain}) takes $O(m^2)$ time. Although theoretically faster
-appended to $j$, appending $i$ to a predecessor of $j$ is likely to yield a
+chaining algorithms exist~\citep{Abouelhoda:2005aa}, they
-lower score. When evaluating Eq.~(\ref{eq:chain}), we start from anchor $i-1$
+are inapplicable to generic gap cost, complex to implement and usually
-and stop the evaluation if we cannot find a better score after up to $h$
+associated with a large constant. We introduced a simple heurstic to accelerate
-iterations. This heuristic reduces the average time to $O(h\cdot m)$. In
+chaining.
-practical, we can almost always find the optimal chain with $h=50$; even if the
-heuristic fails, the optimal chain often looks dubious.
 
-%Although theoretically faster chaining algorithms exist for simple gap
+We note that if anchor $i$ is appended to $j$, appending $i$ to a predecessor
-%cost~\citep{Abouelhoda:2005aa}, they are not as flexible as DP and may not lead
+of $j$ is likely to yield a lower score. When evaluating Eq.~(\ref{eq:chain}),
-%to better performance than our approach in practice.  Furthermore, chaining
+we start from anchor $i-1$ and stop the evaluation if we cannot find a better
-%takes much less computing time than alignment. It is not critical to the
+score after up to $h$ iterations. This heuristic reduces the average time to
-%performance of minimap2.
+$O(h\cdot m)$. In practice, we can almost always find the optimal chain with
+$h=50$; even if the heuristic fails, the optimal chain often looks dubious.
 
 \subsubsection{Backtracking}
 Let $P(i)$ be the index of the best predecessor of anchor $i$. It equals 0 if
@@ -120,9 +120,9 @@ banded alignment, which is critical to performance. In practice, our
 implementation is three times as fast as Parasail's 4-way
 vectorization~\citep{Daily:2016aa} for global alignment.
 Without banding, our implementation is slower than Edlib~\citep{Sosic:2017aa},
-but with a 1000bp band, it is much faster. When performing global alignment
+but with a 1000bp band, it is considerably faster. When performing global
-between anchors, we expect the alignment to stay close to the diagonal of the
+alignment between anchors, we expect the alignment to stay close to the
-DP matrix. Banding is applicable most of time.
+diagonal of the DP matrix. Banding is often applicable.
 
 Minimap2 uses a 2-piece affine gap cost
 $\gamma(l)=\min\{q+l\cdot e,\tilde{q}+l\cdot\tilde{e}\}$.
@@ -151,57 +151,154 @@ alignment, but this time with the one subsequence reverse complemented. This
 additional alignment step may identify short inversions that are missed during
 chaining.
 
-%\begin{equation}\label{eq:ae86}
-%\left\{\begin{array}{l}
-%H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\
-%E_{i+1,j}= \max\{H_{ij}-q,E_{ij}\}-e\\
-%F_{i,j+1}= \max\{H_{ij}-q,F_{ij}\}-e\\
-%\tilde{E}_{i+1,j}= \max\{H_{ij}-\tilde{q},\tilde{E}_{ij}\}-\tilde{e}\\
-%\tilde{F}_{i,j+1}= \max\{H_{ij}-\tilde{q},\tilde{F}_{ij}\}-\tilde{e}
-%\end{array}\right.
-%\end{equation}
-%where $s(i,j)$ is the score between the $i$-th reference base and $j$-th query
-%base. If we define~\citep{Wu:1996aa,Suzuki:2016}
-%\[
-%\left\{\begin{array}{ll}
-%u_{ij}\triangleq H_{ij}-H_{i-1,j} & v_{ij}\triangleq H_{ij}-H_{i,j-1} \\
-%x_{ij}\triangleq E_{i+1,j}-H_{ij} & \tilde{x}_{ij}\triangleq \tilde{E}_{i+1,j}-\tilde{H}_{ij} \\
-%y_{ij}\triangleq F_{i,j+1}-H_{ij} & \tilde{y}_{ij}\triangleq \tilde{F}_{i,j+1}-\tilde{H}_{ij}
-%\end{array}\right.
-%\]
-%we can transform Eq.~(\ref{eq:ae86}) to
-%\[
-%\left\{\begin{array}{lll}
-%z_{ij}&=&\max\{s(i,j),x_{i-1,j}+v_{i-1,j},y_{i,j-1}+u_{i,j-1},\\
-%&&\tilde{x}_{i-1,j}+v_{i-1,j},\tilde{y}_{i,j-1}+u_{i,j-1}\}\\
-%u_{ij}&=&z_{ij}-v_{i-1,j}\\
-%v_{ij}&=&z_{ij}-u_{i,j-1}\\
-%x_{ij}&=&\max\{0,x_{i-1,j}+v_{i-1,j}-z_{ij}+q\}-q-e\\
-%y_{ij}&=&\max\{0,y_{i,j-1}+u_{i,j-1}-z_{ij}+q\}-q-e\\
-%\tilde{x}_{ij}&=&\max\{0,\tilde{x}_{i-1,j}+v_{i-1,j}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
-%\tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}
-%\end{array}\right.
-%\]
-%with boundary conditions
-%\[
-%\left\{\begin{array}{l}
-%x_{-1,\cdot}=y_{\cdot,-1}=-q-e\\
-%\tilde{x}_{-1,\cdot}=\tilde{y}_{\cdot,-1}=-\tilde{q}-\tilde{e}\\
-%u_{i,-1}=\eta(i)\\
-%v_{-1,j}=\eta(j)
-%\end{array}\right.
-%\]
-%where
-%\[
-%\eta(k)=\left\{\begin{array}{ll}
-%-q-e & (k=0) \\
-%-e & (k<\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
-%i\cdot(e-\tilde{e})-(\tilde{q}-q)-\tilde{e} & (k=\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
-%-\tilde{e} & (k>\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil)
-%\end{array}\right.
-%\]
-
 \end{methods}
 
+\section{Results and discussions}
+\begin{figure}[!tb]
+\centering
+\includegraphics[width=.5\textwidth]{roc-color.pdf}
+\caption{Evaluation on simulated SMRT reads aligned against human genome
+GRCh38. (a) ROC-like curve. (b) Accumulative mapping error rate as a function
+of mapping quality. 33,088 $\ge$1000bp reads were simulated using
+pbsim~\citep{Ono:2013aa} with error profile sampled from file
+`m131017\_060208\_42213\_*.1.*' downloaded at
+\href{http://bit.ly/chm1p5c3}{http://bit.ly/chm1p5c3}. The N50 read length is
+11,628. A read is considered correctly mapped if the true position overlaps
+with the best mapping position by 10\% of the read length. All aligners were
+run under the default setting for SMRT reads.}\label{fig:eval}
+\end{figure}
+
+As a sanity check, we evaluated minimap2 on simulated human reads along with
+BLASR~\citep{Chaisson:2012aa}, 
+BWA-MEM~\citep{Li:2013aa},
+GraphMap~\citep{Sovic:2016aa},
+minialign~\citep{Suzuki:2016} and
+NGMLR~\citep{Sedlazeck169557}. We excluded rHAT~\citep{Liu:2016ab},
+LAMSA~\citep{Liu:2017aa} and Kart~\citep{Lin:2017aa} because they either
+crashed or produced malformatted SAM.  In this evaluation, Minimap2 has a
+higher power to distinguish unique and repetitive hits, and achieves overall
+higher mapping accuracy (Fig.~\ref{fig:eval}a). Minimap2 and NGMLR provide
+better mapping quality estimate: they rarely give repetitive hits high mapping
+quality (Fig.~\ref{fig:eval}b).  Apparently, other aligners may occasionally
+miss close suboptimal hits and be overconfident in wrong mappings. On run time,
+minialign is slightly faster than minimap2. They are over 30 times faster than
+the rest.
+
+On real SMRT reads from human, the relative performance and sensitivity of
+these aligners are broadly similar to those on simulated data. We are unable to
+provide a good estimate of mapping error rate due to the lack of the truth.  On
+ONT ultra-long human reads~\citep{Jain128835}, BWA-MEM failed. Minialign and
+minimap2 are over 70 times faster than others. In addition to reference-based
+read mapping, minimap2 can also find overlaps between long reads and align
+long-read assemblies.
+
 \bibliography{minimap2}
+
+\pagebreak
+
+\begin{methods}
+\section*{Appendix}
+A 2-piece gap cost function is
+\[
+\gamma(l)=\min\{q+l\cdot e,\tilde{q}+l\cdot\tilde{e}\}
+\]
+Without losing generality, we assume $q+e\le\tilde{q}+\tilde{e}$. The equation
+to compute the optimal alignment under such a gap cost is
+\begin{equation}\label{eq:ae86}
+\left\{\begin{array}{l}
+H_{ij} = \max\{H_{i-1,j-1}+s(i,j),E_{ij},F_{ij},\tilde{E}_{ij},\tilde{F}_{ij}\}\\
+E_{i+1,j}= \max\{H_{ij}-q,E_{ij}\}-e\\
+F_{i,j+1}= \max\{H_{ij}-q,F_{ij}\}-e\\
+\tilde{E}_{i+1,j}= \max\{H_{ij}-\tilde{q},\tilde{E}_{ij}\}-\tilde{e}\\
+\tilde{F}_{i,j+1}= \max\{H_{ij}-\tilde{q},\tilde{F}_{ij}\}-\tilde{e}
+\end{array}\right.
+\end{equation}
+where $s(i,j)$ is the score between the $i$-th reference base and $j$-th query
+base. If we define
+\[
+\left\{\begin{array}{ll}
+u_{ij}\triangleq H_{ij}-H_{i-1,j} & v_{ij}\triangleq H_{ij}-H_{i,j-1} \\
+x_{ij}\triangleq E_{i+1,j}-H_{ij} & \tilde{x}_{ij}\triangleq \tilde{E}_{i+1,j}-\tilde{H}_{ij} \\
+y_{ij}\triangleq F_{i,j+1}-H_{ij} & \tilde{y}_{ij}\triangleq \tilde{F}_{i,j+1}-\tilde{H}_{ij}
+\end{array}\right.
+\]
+we can transform Eq.~(\ref{eq:ae86}) to
+\begin{equation}\label{eq:suzuki}
+\left\{\begin{array}{lll}
+z_{ij}&=&\max\{s(i,j),x_{i-1,j}+v_{i-1,j},y_{i,j-1}+u_{i,j-1},\\
+&&\tilde{x}_{i-1,j}+v_{i-1,j},\tilde{y}_{i,j-1}+u_{i,j-1}\}\\
+u_{ij}&=&z_{ij}-v_{i-1,j}\\
+v_{ij}&=&z_{ij}-u_{i,j-1}\\
+x_{ij}&=&\max\{0,x_{i-1,j}+v_{i-1,j}-z_{ij}+q\}-q-e\\
+y_{ij}&=&\max\{0,y_{i,j-1}+u_{i,j-1}-z_{ij}+q\}-q-e\\
+\tilde{x}_{ij}&=&\max\{0,\tilde{x}_{i-1,j}+v_{i-1,j}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
+\tilde{y}_{ij}&=&\max\{0,\tilde{y}_{i,j-1}+u_{i,j-1}-z_{ij}+\tilde{q}\}-\tilde{q}-\tilde{e}
+\end{array}\right.
+\end{equation}
+where $z_{ij}$ is a temporary variable that does not need to be stored. We can
+see that
+\[
+x_{ij}=E_{i+1,j}-H_{ij}=\max\{-q,E_{ij}-H_{ij}\}-e
+\]
+With $E_{ij}\le H_{ij}$, we have
+\[
+-q-e\le x_{ij}\le\max\{-q,0\}-e=-e
+\]
+and similar inequations for $y_{ij}$, $\tilde{x}_{ij}$ and $\tilde{y}_{ij}$.
+In addition,
+\[
+u_{ij}=z_{ij}-v_{i-1,j}\ge\max\{x_{i-1,j},\tilde{x}_{i-1,j}\}\ge-q-e
+\]
+We also note that the maximum possible $z_{ij}=H_{ij}-H_{i-1,j-1}$ is $M$, the
+maximal matching score. As a result,
+\[
+u_{ij}\le M-v_{i-1,j}\le M+q+e
+\]
+In conclusion, all values in Eq.~(\ref{eq:suzuki}) are bounded: $x$ and $y$ by
+$[-q-e,-e]$ and $\tilde{x}$, $\tilde{y}$ by
+$[-\tilde{q}-\tilde{e},-\tilde{e}]$, and $u$ and $v$ by $[-q-e,M+q+e]$. When
+matching score and gap cost are small, each of them can be stored as a 8-bit
+integer. This enables efficient SSE vectorization regardless of the peak score
+of the alignment.
+
+For more efficient SSE implementation, we transform the row-column coordinate
+to diagonal-anti-diagonal coordinate by letting $r\gets i+j$ and $t\gets i$.
+Eq.~(\ref{eq:suzuki}) becomes:
+\begin{equation*}
+\left\{\begin{array}{lll}
+z_{rt}&=&\max\{s(t,r-t),x_{r-1,t-1}+v_{r-1,t-1},y_{r-1,t}+u_{r-1,t},\\
+&&\tilde{x}_{r-1,t-1}+v_{r-1,t-1},\tilde{y}_{r-1,t}+u_{r-1,t}\}\\
+u_{rt}&=&z_{rt}-v_{r-1,t-1}\\
+v_{rt}&=&z_{rt}-u_{r-1,t}\\
+x_{rt}&=&\max\{0,x_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+q\}-q-e\\
+y_{rt}&=&\max\{0,y_{r-1,t}+u_{r-1,t}-z_{rt}+q\}-q-e\\
+\tilde{x}_{rt}&=&\max\{0,\tilde{x}_{r-1,t-1}+v_{r-1,t-1}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}\\
+\tilde{y}_{rt}&=&\max\{0,\tilde{y}_{r-1,t}+u_{r-1,t}-z_{rt}+\tilde{q}\}-\tilde{q}-\tilde{e}
+\end{array}\right.
+\end{equation*}
+In this formulation, cells with the same row index $r$ are independent of each
+other. This allows us to vectorize the computation of all cells on the same
+anti-diagonal in one inner loop.
+
+On the condition that $q+e<\tilde{q}+\tilde{e}$ and $e>\tilde{e}$, the boundary
+condition of this equation in the diagonal-anti-diagonal coordinate is
+\[
+\left\{\begin{array}{l}
+x_{r-1,-1}=y_{r-1,r}=-q-e\\
+\tilde{x}_{r-1,-1}=\tilde{y}_{r-1,r}=-\tilde{q}-\tilde{e}\\
+u_{r-1,r}=v_{r-1,-1}=\eta(r)\\
+\end{array}\right.
+\]
+where
+\[
+\eta(r)=\left\{\begin{array}{ll}
+-q-e & (r=0) \\
+-e & (r<\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
+r\cdot(e-\tilde{e})-(\tilde{q}-q)-\tilde{e} & (r=\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil) \\
+-\tilde{e} & (r>\lceil\frac{\tilde{q}-q}{e-\tilde{e}}-1\rceil)
+\end{array}\right.
+\]
+\citet{Suzuki:2016} first derived a similar set of equations under affine gap
+cost but with different notations.
+\end{methods}
 \end{document}

tex/roc.gp

@@ -1,28 +1,52 @@
-set t po eps enh co so "Helvetica,18"
+set t po eps enh co so "Helvetica,26"
 
-set style line 1 lt 1 pt 1 lc rgb "#FF0000" lw 2;
+set style line 1 lt 1 pt 1 lc rgb "#e41a1c" lw 2;
-set style line 2 lt 1 pt 2 lc rgb "#00C000" lw 2;
+set style line 2 lt 1 pt 2 lc rgb "#377eb8" lw 2;
-set style line 3 lt 1 pt 3 lc rgb "#0080FF" lw 2;
+set style line 3 lt 1 pt 3 lc rgb "#4daf4a" lw 2;
-set style line 4 lt 1 pt 4 lc rgb "#C000FF" lw 2;
+set style line 4 lt 1 pt 4 lc rgb "#984ea3" lw 2;
-set style line 5 lt 1 pt 5 lc rgb "#00EEEE" lw 2;
+set style line 5 lt 1 pt 6 lc rgb "#ff7f00" lw 2;
-set style line 6 lt 1 pt 6 lc rgb "#C04000" lw 2;
+set style line 6 lt 1 pt 8 lc rgb "#f781bf" lw 2;
-set style line 7 lt 1 lc rgb "#C8C800" lw 2;
-set style line 8 lt 1 lc rgb "#FF80FF" lw 2;
-set style line 9 lt 1 lc rgb "#4E642E" lw 2;
-set style line 10 lt 1 lc rgb "#800000" lw 2;
-set style line 11 lt 1 lc rgb "#67B7F7" lw 2;
-set style line 12 lt 1 lc rgb "#FFC127" lw 2;
-
-set xlab "False positive rate"
-set ylab "Sensitivity"
-set yran [0.9:1]
 
 set out "roc-color.eps"
+
+set pointsize 2.0
+set size 1.59,1.04
+set multiplot layout 1,2
+
+set label "(a)" at graph -0.245,1.06 font "Helvetica-bold,40"
+set xlab "Error rate of mapped reads"
+set ylab "Fraction of mapped reads" off +1.8
+set ytics 0.02
+set yran [0.9:1]
+
+set size 0.8,1
 set log x
 set format x "10^{%L}"
 set key bot right
-plot "<./eval2roc.pl blasr-mc.eval" u 2:3 t "blasr-mc" w lp ls 1, \
+plot "<./eval2roc.pl blasr-mc.eval" u 2:3 t "blasr-mc" w lp ls 4, \
      "<./eval2roc.pl bwa.eval" u 2:3 t "bwa-mem" w lp ls 2, \
-     "<./eval2roc.pl minialign.eval" u 2:3 t "minialign" w lp ls 3, \
+     "<./eval2roc.pl graphmap.eval" u 2:3 t "graphmap" w lp ls 3, \
-     "<./eval2roc.pl mm2.eval" u 2:3 t "minimap2" w lp ls 4, \
+     "<./eval2roc.pl minialign.eval" u 2:3 t "minialign" w lp ls 1, \
+     "<./eval2roc.pl mm2.eval" u 2:3 t "minimap2" w lp ls 6, \
      "<./eval2roc.pl ngmlr.eval" u 2:3 t "ngm-lr" w lp ls 5
+unset label
+
+set origin 0.8,0
+set size 0.79,1
+set label "(b)" at graph -0.245,1.06 font "Helvetica-bold,40"
+unset log
+unset format
+unset key
+set log y
+set ylab "Accumulative mapping error rate" off +0
+set xlab "Mapping quality"
+set yran [1e-5:0.1]
+set ytics 1e-5,0.1
+set format y "10^{%L}"
+set xran [60:0] reverse
+plot "<./eval2roc.pl blasr-mc.eval" u 1:2 w lp ls 4, \
+     "<./eval2roc.pl bwa.eval" u 1:2 t "bwa-mem" w lp ls 2, \
+     "<./eval2roc.pl graphmap.eval" u 1:2 t "graphmap" w lp ls 3, \
+     "<./eval2roc.pl minialign.eval" u 1:2 t "minialign" w lp ls 1, \
+     "<./eval2roc.pl mm2.eval" u 1:2 t "minimap2" w lp ls 6, \
+     "<./eval2roc.pl ngmlr.eval" u 1:2 t "ngm-lr" w lp ls 5