Performance of individual de novo assemblers on simulated RNAseq library using default parameters or pooled across multiple kmer lengths.
\\n\\n
More than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\\n\\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\\n\\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\\n\\nAdditionally, each book published by IntechOpen contains original content and research findings.
\\n\\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\\n\\n\\n\\n
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'
Simba Information has released its Open Access Book Publishing 2020 - 2024 report and has again identified IntechOpen as the world’s largest Open Access book publisher by title count.
\n\nSimba Information is a leading provider for market intelligence and forecasts in the media and publishing industry. The report, published every year, provides an overview and financial outlook for the global professional e-book publishing market.
\n\nIntechOpen, De Gruyter, and Frontiers are the largest OA book publishers by title count, with IntechOpen coming in at first place with 5,101 OA books published, a good 1,782 titles ahead of the nearest competitor.
\n\nSince the first Open Access Book Publishing report published in 2016, IntechOpen has held the top stop each year.
\n\n\n\nMore than half of the publishers listed alongside IntechOpen (18 out of 30) are Social Science and Humanities publishers. IntechOpen is an exception to this as a leader in not only Open Access content but Open Access content across all scientific disciplines, including Physical Sciences, Engineering and Technology, Health Sciences, Life Science, and Social Sciences and Humanities.
\n\nOur breakdown of titles published demonstrates this with 47% PET, 31% HS, 18% LS, and 4% SSH books published.
\n\n“Even though ItechOpen has shown the potential of sci-tech books using an OA approach,” other publishers “have shown little interest in OA books.”
\n\nAdditionally, each book published by IntechOpen contains original content and research findings.
\n\nWe are honored to be among such prestigious publishers and we hope to continue to spearhead that growth in our quest to promote Open Access as a true pioneer in OA book publishing.
\n\n\n\n
\n'}],latestNews:[{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"},{slug:"all-intechopen-books-available-on-perlego-20201215",title:"All IntechOpen Books Available on Perlego"},{slug:"oiv-awards-recognizes-intechopen-s-editors-20201127",title:"OIV Awards Recognizes IntechOpen's Editors"},{slug:"intechopen-joins-crossref-s-initiative-for-open-abstracts-i4oa-to-boost-the-discovery-of-research-20201005",title:"IntechOpen joins Crossref's Initiative for Open Abstracts (I4OA) to Boost the Discovery of Research"},{slug:"intechopen-hits-milestone-5-000-open-access-books-published-20200908",title:"IntechOpen hits milestone: 5,000 Open Access books published!"},{slug:"intechopen-books-hosted-on-the-mathworks-book-program-20200819",title:"IntechOpen Books Hosted on the MathWorks Book Program"}]},book:{item:{type:"book",id:"3729",leadTitle:null,fullTitle:"Orbital Integrals on Reductive Lie Groups and Their Algebras",title:"Orbital Integrals on Reductive Lie Groups and Their Algebras",subtitle:null,reviewType:"peer-reviewed",abstract:"The purpose is to present a complete course on global analysis topics and establish some orbital applications of the integration on topological groups and their algebras to harmonic analysis and induced representations in representation theory.",isbn:null,printIsbn:"978-953-51-1007-1",pdfIsbn:"978-953-51-5029-9",doi:"10.5772/55875",price:119,priceEur:129,priceUsd:155,slug:"orbital-integrals-on-reductive-lie-groups-and-their-algebras",numberOfPages:196,isOpenForSubmission:!1,isInWos:null,hash:"0e1538fba70d1a9ed222ff4e8d5b8d90",bookSignature:"Francisco Bulnes",publishedDate:"March 6th 2013",coverURL:"https://cdn.intechopen.com/books/images_new/3729.jpg",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,hasAltmetrics:null,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"January 16th 2013",dateEndSecondStepPublish:"February 6th 2013",dateEndThirdStepPublish:"May 13th 2013",dateEndFourthStepPublish:"August 11th 2013",dateEndFifthStepPublish:"September 10th 2013",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,7",editedByType:"Authored by",kuFlag:!1,editors:[{id:"92918",title:"Dr.",name:"Francisco",middleName:null,surname:"Bulnes",slug:"francisco-bulnes",fullName:"Francisco Bulnes",profilePictureURL:"https://mts.intechopen.com/storage/users/92918/images/system/92918.jpeg",biography:"Dr. Francisco Bulnes, PhD, PostDocs, Doctor H. 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Transcriptome assembly from high-throughput sequencing of mRNA (RNAseq) is a powerful tool for detecting variations in gene expression and sequences between conditions, tissues, or strains/species for both model and non-model organisms [1, 2]. However, the ability to accurately perform such analyses is crucially dependent on the quality of the underlying assembly [3]. Especially for the detection of sequence variations, but also for isoform detection and transcript quantification, mis-assembly of genes of interest can increase both the false positive and false negative rates, depending on the nature of the mis-assembly [4]. These problems are exacerbated in non-model organisms where genomic sequences that can be used as the references, if available at all, are sufficiently different than those from the individuals sequenced [5].
Transcripts can be mis-assembled in several ways [6]. Two of the most drastic assembly errors are fragmentation, where a single transcript is assembled as one or more smaller contigs, and chimeras, where a contig is assembled using part or all of more than one transcript. Fragmentation errors tend to result from fluctuations in the read coverage along a transcript, with the breaks in the transcript sequence occurring in regions that have lower coverage. By contrast, chimera errors often occur because of ambiguous overlaps within the reads, coupled with algorithms that choose the longest possible contig represented by the data, or by adjacent genes on the genome being merged. Both of these types of errors can have major impacts especially on gene identification. Small (single or few) nucleotide alterations to the contig sequence also happen as mis-assemblies. Sequence mistakes are often the result of mis-sequenced reads, but can also result from ambiguity for highly similar reads e.g. from heterozygous genes and from duplicated genes. In some cases, these errors can shift the reading frame for the contig, which can have significant impacts on the translated protein sequence. Finally, transcripts can be mis-assembled when alternative transcripts are collapsed into a single contig [6].
In the following sections, we will first review strategies used for transcriptome assembly as well as how their performance can be assessed. We then compare the performance of representative transcriptome assembly methods using a simulated human transcriptome and RNAseq. Finally we discuss a possible strategy to improve transcriptome assembly accuracy.
De novo assemblers generate contigs based solely on the RNAseq data [7, 8, 9, 10, 11, 12, 13]. Most of the de novo assemblers rely on de Bruijn graphs generated from kmer decompositions of the reads in the RNAseq data [14]. The reads are subdivided into shorter sequences of a given length k (the kmers) and the original sequence is reconstructed by the overlap of these kmer sequences. One major limitation of the de Bruijn graphs is the need for a kmer to start at every position along the original sequence in order for the graph to cover the full sequence [13]. This limitation creates a tradeoff in regard to the length of the kmers. Shorter kmers are more likely to fully cover the original sequence, but are more likely to be ambiguous, with a single kmer corresponding to multiple reads from multiple transcripts. While by using longer kmers such ambiguity can be avoided, those kmers may not cover the entire sequence of some transcripts causing e.g. fragmented assembly. Consequently, each transcript, with its unique combination of expression level (corresponding to the number of reads in the RNAseq data generated from that transcript) and sequence will have a different best kmer length for its assembly [15]. As a result, even using the same de novo assembly algorithm, performing two assemblies with different kmer lengths will generate a different set of contigs, inevitably with a varying set of correctly assembled contigs [16].
Examples of popularly used de novo assemblers include idba-Tran [9], SOAPdenovo-Trans [8], rnaSPAdes [12], and Trinity [7]. Idba-Tran is unique among these de novo assemblers, as it runs individual assemblies across a range of kmer lengths and merges the results to form the final prediction. The remaining assemblers use only the results of a single kmer length. For SOAPdenovo-Trans and Trinity, a kmer length needs to be chosen (default kmer: 23 and 25, respectively), while rnaSPAdes dynamically determines the kmer length to be used based on the read data. While all of these tools use the same fundamental strategies to construct, revise, and parse the de Bruijn graph for the assemblies, each method uses different thresholds and different assumptions to make decisions. These differences lead to different subsets of transcripts being correctly assembled by each method. An example of how these tools produce different sets of contigs is shown in Section 4.2.
Genome-guided assemblers avoid the ambiguity of kmer decompositions used in de Bruijn graphs by mapping the RNAseq data to the reference genome. In order to account of introns, mapping of the reads for genome-guided assembly needs to allow them to be split, where the first part of the read maps to one location (an exon), and the other half maps to a downstream location (another exon). This mapping is done by split-read mappers such as TopHat [17], STAR [18], HISAT [19], or HPG-aligner [20]. Each of these methods maps the reads slightly differently, which may impact the quality of subsequent assembly.
This read mapping greatly reduces the complexity of transcript assembly by clustering the reads based on genomic location rather than relying solely on overlapping sequences within the reads themselves [3]. However, this approach still has some major drawbacks. The most obvious drawback is that genome-guided assemblers require a reference genome, which is not available for all organisms. The quality of the reference genome, if it is available, also impacts the quality of the read mapping and, by extension, the assembly. This impact is particularly noteworthy when genes of interest contain gaps in the genome assembly, preventing the reads necessary to assemble those genes from mapping to part or all of the transcript sequence. Ambiguity occurs also when reads map to multiple places within a genome. How the specific algorithm handles choosing which potential location a read should map to can have a large impact on the final transcripts predicted [6]. This problem is expounded when working with organisms different from the reference, where not all of reads map to the reference without gaps or mismatches.
Examples of popularly used genome-guided assemblers include Bayesembler [21], Cufflinks [22], and StringTie [23]. While each of these methods uses the mapped reads to create a graph representing the splice junctions of the transcripts, how they select which splice junctions are real differs fundamentally. Cufflinks constructs transcripts based on using the fewest number of transcripts to cover the highest percentage of mapped reads. StringTie uses the number of reads that span each splice junction to construct a flow graph, constructing the transcripts based in order of the highest flow. Bayesembler constructs all viable transcripts for each splice junction and uses a Bayesian likelihood estimation based on the read coverage of each potential transcript to determine which combination of transcripts is most likely. Due to these fundamentally different approaches, each of these tools produces different sets of transcripts from the same set of reads. An example of assemblies produced by these methods and how the assembled contigs differ is described in Section 4.3.
While a core set of transcripts are expected to be assembled correctly by many different assemblers, many transcripts will be missed by any individual tool [24] (also see Section 4). Through combining the assemblies produced by multiple methods, ensemble assemblers such as EvidentialGene [25] and Concatenation [26] attempt to address the limitations of individual assemblers, ideally keeping contigs that are more likely to be correctly assembled and discarding the rest. Both of EvidentialGene and Concatenation filter the contigs obtained from multiple assemblers (usually de novo) by clustering the contigs based on their sequences, predicting the coding region of the contig, and using features of the overall contig and the coding region to determine the representative sequence for each cluster. EvidentialGene recommends using several different tools across a wide range of kmer lengths. It uses the redundancy from multiple tools generating nearly identical sequences, clusters them, scores the sequences in each cluster based of the features of the sequence (e.g. lengths of the 5′ and 3′ untranslated regions), and returns one representative sequence from each cluster (keeping also some alternative sequences). In contrast, Concatenation recommends using only three assemblers, with one kmer length each. Concatenation merges nucleotide sequences that are identical or perfect subsets, only filters contigs with no predicted coding region.
These approaches greatly reduce the number of contigs by removing redundant and highly similar sequences. However, there is no guarantee that the correct representative sequence is kept for a given cluster or that each cluster represents one unique gene. Because they require multiple assemblies to merge, they also come at a far greater computational cost. An example of how these ensemble assembly strategies perform compared to individual de novo and genome-guided methods is shown in Section 4.4.
All of the methods described so far primarily use short but highly accurate reads from Illumina sequencing for assembly, with or without a reference. With the rise of third-generation sequencing technologies from Pacific Biosciences (PacBio SMRT) and Oxford Nanopore Technologies (ONT MinION), it is becoming possible to sequence entire mRNA molecules as one very long read, though with a high error rate [27]. The ability to sequence the entire mRNA molecule is especially beneficial for detecting alternative splice forms, which remain a challenge for short-read only assembly, and potentially for more accurate transcript quantification if there is no bias in the mRNA molecules sequenced.
While many tools exist to perform genome assemblies using either these long reads alone or by combining long reads and Illumina reads, at present no short read transcriptome assemblers take advantage of long-reads in transcriptome assembly. If these long reads can be sufficiently error-corrected (e.g. [28, 29]), they can be used for a snapshot of the expressed transcriptome, without requiring assembly or external references [30, 31]. Alternatively, after an independent de novo assembly of short reads, the long reads can be used to confirm alternative splice forms present in the assembly [32]. The long reads can be also mapped to a reference genome similar to the split-read mapping methods used for genome-guided short-read assemblers discussed above [27, 33, 34, 35]. With their accuracy increasing, in the future, long reads can be used more to improve transcriptome assembly quality.
In this section, we will discuss commonly used metrics to assess the quality of transcriptome assemblies.
The most straightforward assembly metrics are those based on the number and lengths of the sequences produced [36]. The number of sequences can be presented either or both of:
the number of contigs
the number of scaffolds
where for contigs no further joining of the sequences is performed after assembly, and for scaffold contigs that have some support for being from the same original sequence are combined together with a certain number of gaps between them.
Several different statistics are available for presenting the lengths of the sequences (either contigs or scaffolds). The most commonly reported metrics are:
minimum length (bp): the length of the shortest sequence produced
maximum length (bp): the length of the longest sequence produced
mean length (bp): the average length of the sequences produced
median length (bp): the length where half of the sequences are shorter, and half of the sequences are longer
N50 (bp): a weighted median where the sum of the lengths of all sequences longer than the N50 is at least half of the total length of the assembly
L50: the smallest number of sequences whose combined length is longer than the N50
Additional metrics similar to N50 (e.g. N90) based on different thresholds are also used.
For genome assemblies where the target number of sequences is known (one circular genome plus any smaller plasmids for prokaryotic organisms and the number of chromosomes for eukaryotic organisms), these metrics provide an estimate for the thoroughness of the assembly [36]. For instance, in prokaryotic assemblies, the vast majority of the sequence is expected to be in one long sequence, and having many shorter sequences indicates fragmentation of the assembly [15]. In this context, longer sequences (e.g. larger N50) tend to indicate higher quality assemblies. For transcriptome assemblies, however, the length of the assembled contigs varies depending on the lengths of the transcripts being assembled. For the human transcriptome, for example, while the longest transcript (for the gene coding the Titin protein) is over 100 kb, the shortest is only 186 bp, with a median length of 2787 bp [37]. Emphasizing longer contigs also rewards assemblers that over-assemble sequences, either by including additional sequence incorrectly within a gene, or by joining multiple genes together to form chimeric contigs. Therefore, for transcriptome assembly, metrics based on contig lengths do not necessarily reflect its quality.
Rather than focusing on the number or length of the sequences produced by the assembly, performing similarity searches with the assembled sequences can provide an estimate of the quality of the contigs or scaffolds [24, 38]. Typically, the process consists of either similarity searches against well annotated databases (such as the protein datasets of related genomes or targeted orthologs, the BLAST non-redundant protein database [39] or the UniProt/Swiss-Prot database [40]), conserved domain search within the contig sequence that determines the potential function of the gene (such as PFAM or Panther [41, 42]), or a search against a lineage specific conserved single-copy protein database (such as BUSCO [43]). These similarity searches are usually performed on the predicted protein sequences for the contigs (e.g. using GeneMarkS [44]), but can also be performed directly from the assembled nucleotide sequences using BLASTX where translated nucleotide sequences are used to search against a protein database [38]. If the organism being sequenced is closely related to a model organism with a well-defined transcriptome, nearly all of the contigs that are not erroneously assembled and code proteins should have identifiable potential homologs in the database. If a large percentage of the contigs do not have similar proteins identified in the database, there is a high probability that the sequences are incorrectly assembled, regardless of the length of the sequences. By performing similarity searches, over-assemblies or chimera contigs (those covering more than one gene) can be also detected as large gaps in the alignment between the query and the hits. As protein sequence annotations are necessary for most downstream analyses, they also provide a convenient metric without the need for additional, otherwise unnecessary analyses.
Despite these advantages, there are some limitations to using protein-similarity based metrics for assembler performance. First, the more divergent the organism being sequenced is from the sequences in the database searched and the more species-specific genes in the transcriptome, the lower the percentage of contigs with hits will be. This can result in some organisms appearing to have a lower quality assembly solely due to their divergence from those well represented in the databases. By extension, assemblies that recover more transcripts whose coded proteins have few similar sequences in the database will appear worse than assemblies that only recover conserved genes. This limitation can be somewhat mitigated by comparing only genes that are universally single-copy across different species, which are more likely to be conserved and similar enough to be identified. This is the strategy used in BUSCO [43]. However, this comparison at best uses only a subset of the assembled contigs. Second, and more problematic, this metric rewards assemblies that artificially duplicate conserved genes with only small differences in the nucleotide sequence. In the extreme, this can result in several times as many contigs in the assembly than were present in the actual transcriptome, but with nearly all of the contigs coding conserved protein sequences. This is particularly an issue when the analysis depends on identifying the gene copy numbers in the assembly. It also has a large impact on the accuracy of contig quantification and differential expression analyses [45].
The only way to overcome the limitations of the metrics described in the previous sections is to compare the assembly output against a benchmark transcriptome where correct sequences of all transcripts are known. When an RNAseq data generated from a well-established model organism is used for assembly, many of correctly assembled contigs can be identified. However, variability in the transcriptome among e.g. cell types limits the amount of information that can be gained for incorrectly assembled contigs. It is also not possible to determine whether sequences from the reference that are missing from the assembled transcriptome are due to assembly errors, or whether they were not expressed in the library sequenced. Transcriptome sequences may also vary between the individual under study and the reference. Such variations can mask assembly errors that affect the contig sequences. Although this limitation can be mitigated by sequencing an individual that is genetically identical to the reference, it severely limits the types of organisms that can be used for the benchmark.
To comprehensively assess all of the assembly errors, we need to obtain RNAseq data from a transcriptome where all transcript sequences and expression patterns are known. Ideally, such a benchmark transcriptome would be synthetically produced and sequenced using standard protocols. However, currently no such synthetic mRNA library exists. An alternative approach is to simulate the sequencing of a given benchmark transcriptome. There are several tools that can generate simulated reads modeling short Illumina reads [46, 47] and/or long third-generation sequencing reads such as PacBio SMRT and ONT MinION [48, 49]. These tools typically either focus on identifying the statistical distribution of reads across the sequences and errors within the reads, as is the case for RSEM [46], PBSIM [48], and Nanosim [49], or attempt to reconstruct each step of the library preparation and sequencing pipeline, mimicking the errors and biases introduced at each step, as is the case for Flux Simulator [47].
Using simulated RNAseq data with a known transcriptome as a benchmark gives the most detailed and close to true performance metric for assemblies. Specifically, this strategy allows the quantification of each of the following categories:
correctly assembled sequences (true positives or TPs)
sequences that are assembled with errors (false positives or FPs)
sequences in the reference that are missing from the assembly (false negatives or FNs)
“Correctness” and “incorrectness” (or error) can be defined using varying degrees of sequence similarities. Using the strictest threshold, a contig sequence is assembled “correctly” only if the entire nucleotide or coded protein sequence is identical to a reference transcript. All other contigs found in the assembly, including those whose sequences have no similarity in the reference transcriptome (missing contigs), are considered to be assembled “incorrectly” (FPs) regardless of the similarity against the reference sequences.
Note that true negatives (TNs) can be counted only if the assembly experiments are done including reads that are derived from transcripts that are not part of the reference transcriptome (negative transcripts). Using these categories, following assembly metrics can be calculated:
Accuracy =
Sensitivity (or recall) =
Specificity =
Precision =
F-measure (or F1 score) =
False discovery rate (FDR) =
Often in an RNAseq simulation, negative transcripts are not included; hence TN cannot be counted. In such cases, we can calculate an alternative metric as the accuracy:
Accuracy* =
Despite the added benefits of simulation for measuring the performance of assemblers, these metrics assume that the simulation accurately reflects the nature of real RNAseq data. Differences in the distribution of reads or errors between the simulations and real data can impact the relative performance of the assemblers. Assemblers that perform well on simulated data may perform poorly on real data if those assumptions are not met. Consequently, great care must be taken to ensure that the simulated data captures the features of real data as accurately as possible to best characterize the performance of different assembly strategies.
In this section, as an example, we compare the performance of transcriptome assemblers using a simulated benchmark transcriptome dataset.
RNAseq datasets were generated by Flux Simulator [47] using the hg38 human genome (available at
The read pairs generated by Flux Simulator were quality filtered using Erne-filter version 2.0 [50]. The reads were filtered using ultra-sensitive settings with a minimum average quality of q20 (representing a 99% probability that the nucleotide is correctly reported). The filtering was performed in paired-end mode to ensure that both reads of the pair were either kept or discarded concurrently to keep the pairs together. The remaining reads were normalized using Khmer [51] with a kmer size of 32 and an expected coverage of 50×. The normalization was also performed in paired-end mode to maintain pairs.
We compared the performance among four de novo transcriptome assemblers: idba-Tran (version 1.1.1) [9], SOAPdenovo-Trans (version 1.03) [8], rnaSPAdes (version 3.11.0) [12], and Trinity (version 2.5.1) [7], using the simulated human RNAseq dataset as described in the previous section. The resulted assemblies were compared against the benchmark transcriptome. As shown in Table 1, all of the tools underestimated the number of transcripts present, generating fewer contigs than the number of transcripts expected (14,040). The best performing tool among the four compared was Trinity with the most correct contigs (5782) and the highest correct/incorrect ratio (C/I = 0.84). However, even with Trinity, still only 41% (5782/14,040) of transcripts in the benchmark were correctly assembled; the remaining almost 60% of contigs either contained errors in the sequence or were missed entirely. rnaSPAdes assembled the largest number of transcripts (874 more unique transcripts compared to Trinity). The number of unique transcripts generated, 13,513, is also the closest to the expected total number of transcripts (96% of 14,040). However, fewer of those sequences (36%) were correctly assembled, lowering the overall performance across all statistics than Trinity.
Methods | Totala | Uniquea | Correcta (%)b | Incorrecta | C/Ic |
---|---|---|---|---|---|
[Default] | |||||
idba-Tran | 11,943 | 11,941 | 3504 (24.96) | 8437 | 0.4153 |
SOAPdenovo-Trans | 12,902 | 11,830 | 3754 (26.74) | 8076 | 0.4648 |
rnaSPAdes | 15,670 | 13,513 | 5014 (35.71) | 8499 | 0.5900 |
Trinity | 14,044 | 12,639 | 5782 (41.18) | 6857 | 0.8432 |
[Pooled]d | |||||
idba-Tran | 170,358 | 41,849 | 6391 (45.52) | 35,458 | 0.1802 |
SOAPdenovo-Trans | 297,192 | 50,504 | 6059 (43.16) | 44,445 | 0.1363 |
rnaSPAdes | 765,525 | 113,975 | 6665 (47.47) | 107,310 | 0.0621 |
Trinity | 89,126 | 25,045 | 6452 (45.95) | 18,593 | 0.3470 |
Performance of individual de novo assemblers on simulated RNAseq library using default parameters or pooled across multiple kmer lengths.
Number of contigs assembled.
Proportion (%) of transcripts in the benchmark that were correctly assembled.
(Number of correctly assembled contigs)/(number of incorrectly assembled contigs).
Pooled results from using multiple kmers as follows: 15, 19, 23, 27, and 31 for Trinity; 15 kmer values ranging from 15 to 75 in increments of 4 for SOAPdenovo-Trans and rnaSPAdes; 20, 30, 40, 50, and 60 for idba-Tran.
Performance statistics for each assembler is given in Table 2. Precision is a measure of how likely an assembled contig is to be correct, and recall is a measure of how likely the assembler is to correctly assemble a contig. In these terms, for assemblers with high precision, the contigs produced are more likely to be correct, but the assembly may miss a large number of sequences present in the sample. Conversely, assemblers with high recall values correctly assemble more of the sequences present in the sample, but may do so at the cost of accumulating a large number of incorrectly assembled contigs. In these statistics, both the modified accuracy score (accuracy*; see Section 3.3) and the F1 score are a measure of the number of correctly assembled contigs relative to the number of missing and incorrectly assembled contigs. FDR is the proportion of assembled reads that are incorrect. Based on these statistics, Trinity is the best performing de novo assembler with the highest precision, recall, accuracy* and F1 score, and the lowest FDR, followed by rnaSPAdes then SOAPdenovo-Trans. Despite idba-Tran running multiple kmers and merging the results, it performed worst across every metric.
Methods | Precision | Recall | Accuracy* | F1 | FDR |
---|---|---|---|---|---|
idba-Tran | 0.2934 | 0.2496 | 0.1559 | 0.2697 | 0.7066 |
SOAPdenovo-Trans | 0.3173 | 0.2674 | 0.1697 | 0.2902 | 0.6827 |
rnaSPAdes | 0.3711 | 0.3571 | 0.2225 | 0.3640 | 0.6289 |
Trinity | 0.4575 | 0.4118 | 0.2767 | 0.4334 | 0.5425 |
Performance statistics of individual de novo assemblers using default parameters on simulated RNAseq library.
In Table 1, the results from pooling (taking the union of) the outputs of multiple runs of each assembler across a range of kmer lengths are also shown. With these pooled assemblies, the proportion of correctly assembled transcripts in the benchmark for Trinity increased from 41 to 46%, and for rnaSPAdes from 36 to 47%. However, the pooling process also accumulated several times more unique incorrect sequences than additional correct sequences recovered. For Trinity, the C/I decreased from 0.8432 to 0.3470, and for rnaSPAdes this ratio decreased from 0.5900 to 0.0621.
Although the four de novo assembly methods use the same core approach, each method assembled a different set of sequences correctly (Figure 1A). Only a set of 5331 contigs were correctly assembled by all of the four de novo assemblers with at least one kmer length. Additional 813, 567, and 670 contigs were correctly assembled by at least three, at least two, and only one of the assemblers, respectively. In contrast, the vast majority of the incorrectly assembled contigs were produced by only one assembler (Figure 1B). For these contigs, 3764 were produced by all four assemblers, while an additional 2692, 7977 and 166,720 were produced by at least three, at least two or only one of the assemblers, respectively.
Comparisons of the contigs correctly (A) and incorrectly (B) assembled among four de novo assemblers. For each assembler, results from multiple kmers were pooled. Correctly assembled sequences were identified when the protein sequence of the contig matched the protein sequence in the benchmark transcriptome. Incorrectly assembled sequences were identified when the protein sequence of the contig did not exactly match any protein sequence in the benchmark transcriptome.
We next compared the transcriptome assembly performance among three genome-guided assemblers: Bayesembler (version 1.2.0) [21], Cufflinks (version 2.2.1) [22], and StringTie (version 1.0.4) [23]. To demonstrate the impact of using different reference genomes on genome-guided transcriptome assemblies, we used both of the hg38 as well as hg19 genomes as the references. Assembly assessment was done against the hg38 benchmark transcriptome.
Table 3 shows the performance of each of these tools in the two scenarios (RNAseq data and the reference were derived from the same or different genomes). As observed with de novo methods, all of these genome-guided methods underestimated the number of transcripts present, even more severely than de novo methods. In terms of the number of contigs correctly assembled, StringTie performed slightly better than other two methods. All three methods had comparable percent correct (36–41% with the same reference) and C/I (0.87–0.88 with the same reference). While none of the genome-guided assemblers produced as many correctly assembled contigs as the best performing de novo assembler (Trinity), proportions of correctly assembled contigs were higher with genome-guided methods (C/I = 0.87–0.88 with the same reference) than with the four de novo methods (C/I = 0.41–0.84). When the performance metrics are compared between the best performing de novo assembler (Trinity) and genome-guided assembler (StringTie) (Table 4), while both methods showed similar accuracy, StringTie (when using the same reference) showed slightly higher precision, accuracy* and F1 and lower FDR compared to Trinity, but a slightly lower recall. It reflects fewer FPs and FNs produced by StringTie.
Methods | Total | Unique | Correct (%) | Incorrect | C/I |
---|---|---|---|---|---|
[Same reference] | |||||
Bayesembler | 12,989 | 11,482 | 5327 (37.94) | 6155 | 0.8655 |
Cufflinks | 11,257 | 10,733 | 4992 (35.56) | 5741 | 0.8695 |
StringTie | 13,218 | 12,147 | 5696 (40.57) | 6451 | 0.8830 |
[Different reference] | |||||
Bayesembler | 8536 | 7479 | 3345 (23.82) | 4134 | 0.8091 |
Cufflinks | 7234 | 6906 | 3078 (21.92) | 3828 | 0.8041 |
StringTie | 8608 | 7867 | 3466 (24.69) | 4401 | 0.7875 |
Performance of individual genome-guided assemblers using default parameters on simulated RNAseq library with both the same and different references genome as the benchmark. See Table 1 for the description of numbers shown.
Methods | Precision | Recall | Accuracy* | F1 | FDR |
---|---|---|---|---|---|
[Same reference] | |||||
Bayesembler | 0.4639 | 0.3794 | 0.2638 | 0.4174 | 0.5361 |
Cufflinks | 0.4651 | 0.3556 | 0.2524 | 0.4030 | 0.5349 |
StringTie | 0.4689 | 0.4057 | 0.2780 | 0.4350 | 0.5311 |
[Different reference] | |||||
Bayesembler | 0.4473 | 0.2382 | 0.1841 | 0.3109 | 0.5527 |
Cufflinks | 0.4457 | 0.2192 | 0.1723 | 0.2939 | 0.5543 |
StringTie | 0.4406 | 0.2469 | 0.1880 | 0.3164 | 0.5594 |
Performance statistics of individual genome-guided assemblers using default parameters on simulated RNAseq library with both the same and different references genome as the benchmark.
As with the de novo assemblers, each of these tools correctly assembled a different set of transcripts (Figure 2A and C). When the assemblies were performed using the same reference as the simulation, all of the genome-guided tools correctly assembled a core set of 4013 transcripts (Figure 2A). There were nearly a quarter as many (936) that were unique to only one genome-guided tool. When a different reference was used, the number of sequences correctly assembled by all of the tools dropped to 2546 (Figure 2C). Similar to the de novo assemblers, most of the incorrectly assembled contigs produced by each of the genome-guided assemblers were produced by only one assembler regardless of the reference genome used (Figure 2B and D). For assemblies using the same reference genome, 2013 incorrectly assembled contigs were produced by all of the tools, while an additional 2382 and 7546 were produced by any two or only one tool, respectively (Figure 2B). For assemblies using a different reference genome, 1420 incorrectly assembled contigs were produced by all of the tools, while an additional 1667 and 4772 were produced by any two or only one tool, respectively (Figure 2D).
Comparisons of the contigs correctly (A and C) and incorrectly (B and D) assembled among three genome-guided assemblers. Correctly assembled sequences were identified when the protein sequence of the contig matches the protein sequence in the same (A) or different (C) reference genome. Incorrectly assembled sequences were identified when the protein sequence of the contig does not exactly match any protein sequence in the same (B) or different (D) reference genome.
While the overall statistics are comparable between the best de novo assemblies and the genome-guided assemblies using the same reference genome, these tools produced different sets of contigs. The overlap of correctly assembled contigs between the assemblers from de novo with pooled kmers lengths and the three genome-guided assemblers are shown in Figure 3A. All of the de novo assemblers and at least one genome-guided assembler correctly assembled 4605 contigs. An additional 629 were assembled by at least three de novo and at least one genome-guided assembler and 427 assembled by at least two de novo and at least one genome-guided assembler. Conversely, 3861 contigs were correctly assembled by all of the three genome-guided assemblers and at least one de novo assembler, with 1338 assembled by at least two genome-guided assemblers and at least one de novo assembler (Figure 3B). Additionally, these tools produced only 602 correctly assembled contigs that were not predicted by any de novo assembly, while 1514 sequences were correctly assembled by at least one de novo assembly, but no genome-guided assemblies.
Comparisons of the results among de novo and genome-guided transcriptome assemblers. For each de novo assembler, results from multiple kmers were pooled. Correctly (A) and incorrectly (C) assembled sequences for each de novo assembler are compared with the combined results from genome-guided assemblers. Correctly (B) and incorrectly (D) assembled sequences for each genome-guided assembler are compared with the combined results from de novo assemblers.
As with the individual assemblies, fewer incorrectly assembled contigs were produced by all of the tools, and most are assembler specific (Figure 3C and D). In particular, only 1387 incorrectly assembled contigs were produced by all of the de novo assemblers and at least one genome-guided assembler (Figure 3C), and only 1593 contigs were produced all of the genome-guided assemblers and at least one de novo assembler (Figure 3D). In contrast, 4823 incorrectly assemblers were produced by at least one genome-guided assembler but no de novo assemblers, and 176,397 incorrectly assembled contigs were produced by at least one de novo assembler but no genome-guided assemblers.
Overall, these results suggest that genome-guided assemblies provide relatively few correctly assembled contigs relative to performing multiple de novo assemblies, even when using the same reference genome. However, they produce far fewer incorrectly assembled contigs than the pooled de novo assemblies. If the correctly assembled contigs produced by each of the de novo assemblies can be retained while filtering out the incorrectly assembled contigs, de novo assemblies can outperform all of the genome-guided assemblies. This result forms the motivation of ensemble assembly strategies, discussed in the next section.
We compared the two ensemble transcriptome assembly methods, EvidentialGene (version 2017.03.09) [25] and Concatenation (version 1) [26] using the simulated RNAseq data. The strategies for these assemblies followed the recommendations by each method. For EvidentialGene, the pooled results from all of the four de novo assemblies performed across the full range of kmer lengths (described in Section 4.2) were used. For Concatenation, the results of a single assembly each from idba-Tran (using kmer length of 50), rnaSPAdes (with default kmer selection), and Trinity (with default kmer length) were used. These assemblers were chosen to match the assemblies used in [26], substituting the commercial CLC Assembly Cell (
In addition to the two ensemble methods, we also included three “consensus” approaches taking the consensus of the pooled de novo methods. These consensus assemblies involve keeping all of the unique protein sequences produced by any two, three and four tools (named Consensus 2, Consensus 3 and Consensus 4, respectively). Note that Consensus 4 is a subset of Consensus 3, and Consensus 3 is a subset of Consensus 2.
The performance of these ensemble strategies is shown in Table 5. Both of EvidentialGene and Concatenation resulted in an over-estimation in the number of transcripts present. Interestingly, while Concatenation produced a larger total number of transcripts (19,767) than EvidentialGene (19,177), ~2300 of those sequences were redundant, leading to fewer unique sequences (17,497 by Concatenation). Additionally, Concatenation both kept more of the correctly assembled contigs from the individual de novo assemblies, and removed more of the incorrectly assembled contigs than EvidentialGene. These differences lead Concatenation to outperform EvidentialGene across every statistic (Table 6). The performance of the consensus approach varied based on the number of assemblers required.
Methods | Total | Unique | Correct (%) | Incorrect | C/I |
---|---|---|---|---|---|
EvidentialGene | 19,177 | 19,175 | 2267 (16.15) | 16,908 | 0.1341 |
Concatenation | 19,767 | 17,497 | 4697 (33.45) | 12,800 | 0.3670 |
Consensus 2 | 21,444 | 21,444 | 6711 (47.80) | 14,433 | 0.4650 |
Consensus 3 | 12,600 | 12,600 | 6144 (43.76) | 6456 | 0.9517 |
Consensus 4 | 9095 | 9095 | 5331 (37.97) | 3764 | 1.416 |
Performance of individual genome-guided assemblers using default parameters on simulated RNAseq library with both the same and different references genome as the benchmark transcriptome. See Table 1 for the description of numbers shown.
Methods | Precision | Recall | Accuracy* | F1 | FDR |
---|---|---|---|---|---|
EvidentialGene | 0.1182 | 0.1615 | 0.0733 | 0.1365 | 0.8818 |
Concatenation | 0.2684 | 0.3345 | 0.1750 | 0.2979 | 0.7316 |
Consensus 2 | 0.3174 | 0.4780 | 0.2357 | 0.3815 | 0.6826 |
Consensus 3 | 0.4876 | 0.4376 | 0.2998 | 0.4613 | 0.5124 |
Consensus 4 | 0.5861 | 0.3797 | 0.2994 | 0.4609 | 0.4139 |
Performance statistics of ensemble assembly strategies using de novo assemblies on simulated RNAseq library.
Consensus 2 produced the most correctly assembled contigs of any method (6711), but at the cost of more incorrectly assembled contigs than Concatenation (14,433). However, both Consensus 3 and Consensus 4 kept the majority of the correctly assembled contigs while reducing the number of incorrectly assembled contigs by roughly half or three quarters, respectively. Consensus 4 had the highest precision (0.5861) and lowest FDR (0.4139) of any method. However, the additional reduction in the number of correctly assembled contigs lead to Consensus 3 having slightly higher accuracy* (0.2998) and F1 score (0.4613).
In Figure 4 all individual methods (both de novo and genome-guided) as well as ensemble methods are compared. Concatenation performed more poorly than Trinity despite the Trinity assembly forming part of the ensemble. In contrast, Consensus 3 kept more correctly assembled contigs than any individual assembly, with fewer incorrectly assembled than any approach except Consensus 4. This test highlights the weakness of ensemble assembly strategies to retain the incorrect version of a transcript, even if the correct version of the transcript exists in the individual assemblies. More robust methods, such as the consensus approaches we presented here, are needed to reliably improve over individual assemblies.
Performance comparison among all assemblers including de novo, genome-guided, and ensemble strategies. Simulated RNAseq data were used for testing, and the default parameters were used for each assembler. See Tables 1, 3, and 5 for the actual numbers. The expected number of contigs is 14,040.
Transcriptome assembly can be approached from multiple different strategies. Historically, these approaches have revolved around assembling short but highly accurate Illumina reads with or without an existing genome assembly as a reference, referred to as genome-guided or de novo assemblers, respectively. All of the widely used de novo assemblers decompose the short reads into smaller kmers and use de Bruijn graphs built on these kmers to attempt to reconstruct the original transcripts. Due to the limitations of the de Bruijn graphs, this approach presents a trade-off between the uniqueness of the longer kmers and increased coverage of the shorter kmers. As a result, different kmer lengths can produce drastically different graphs, leading to large differences in the final assemblies.
Genome-guided assemblers avoid the limitations of the de Bruijn graphs by mapping the reads to the reference genome. This mapping, however, introduces its own limitations and trade-offs. Reads that are ambiguous between splice forms in the same genomic locations or across multiple genomic locations create similar challenges to the de Bruijn graphs. These ambiguities are compounded when the mapping must take into account mismatches due to sequencing errors as well as biological variations.
The limitations of the individual tools can potentially be overcome by combining multiple different assemblies in ensemble. As each tool and set of parameters results in a different set of correctly assembled contigs, accurately selecting these correctly assembled contigs without selecting any redundant incorrectly assembled contigs would leverage the strengths of each methods without the weaknesses of any. However, currently available ensemble strategies cannot guarantee that the correct sequence is chosen, leading to ensemble assemblies that are less accurate than individual assemblies. As the selection criteria for ensemble methods improve, such as with the “Consensus” approach shown here, these methods can also leverage new assembly approaches that can better handle certain subsets of transcripts (e.g. alternative splice forms) that may have other weaknesses that prevent them from being competitive as a general transcript assembly tool.
Overall, as our results demonstrated, transcriptome assemblers can still be improved, regardless of the approach used. While the genome-guided assemblers generally perform best when the assembly is performed against the same reference sequence that the RNAseq data was generated from, this is not always possible. When these sequences differ, the genome-guided assemblers may have lower accuracy than the de novo assemblers. While ensemble assembly strategies can potentially improve on accuracy over individual assemblies, it is also possible that they instead reduce the accuracy. Improving the performance of these tools, whether individual assemblers, ensemble strategies, or combined with long-read sequencing, will improve not only the accuracy of the reconstructed transcriptome but also the accuracy of downstream analyses, such as sequence annotation, quantification, and differential expression.
This work was supported by a grant from the National Science Foundation to ENM (Award #: 1339385).
It is well known that a rotational transformation in the complex plane is equivalent to multiplying the complex number by a factor
When Hamilton introduced his quaternion algebra, German high school teacher Hermann Gunther Grassmann (1809–1877) was constructing his exterior algebra [2]. He defined the exterior product or outer product
British mathematician William Kingdon Clifford (1845–1879) was one of the few mathematicians who read and understood Grassmann’s work. In 1878, he combined the algebraic rules of Hamilton and Grassmann to define a new algebraic system, which he himself called geometric algebra [3]. In this algebra, both the inner and exterior products of vectors can be uniquely represented by a linear combination of geometric product. In addition, geometric algebra is always isomorphic to some special matrix algebra.
Clifford algebra combines all the advantages of quaternion with the advantages of vector algebra and uniformly and succinctly describes the contents of geometry and physics. However, the vector calculus introduced by Gibbs had also successfully described the mathematical physics problem in three-dimensional space [4]. Clifford died prematurely at the age of 34, so that the theory of geometric algebra was not deeply researched and fully developed, and people still could not see the superiority of this algebra at that time. Thus, the important insights of Grassmann and Clifford were lost in the late nineteenth century papers. Mathematicians abstracted Clifford algebra from its geometric origins, and, for the most part of a century, it languished as a minor subdiscipline of mathematics and became one more algebra among so many others.
With the establishment of relativity, especially the introduction of Pauli and Dirac’s matrix algebra for spin and the successful application in quantum theory [5], it was felt that there is an urgent need for a mathematical system to deal with problems in high-dimensional space-time. In the 1920s, Clifford algebra re-entered the field of vision and was paid attention and researched by some of the famous mathematicians and physicists such as R. Lipschitz, T. Vahlen, E. Cartan, E. Witt, C. Chevalley, and M. Riesz [6, 7, 8]. When only formal algebra is involved, we usually use the term “Clifford algebra,” but more often use the “geometric algebra” named by Clifford himself if applied to geometric problems.
The first person who realized that Clifford algebra is a unified language in geometry and physics should be David Hestenes. By the 1960s, Hestenes began to restore the geometric meaning behind Pauli and Dirac algebra. Although his initial motivation was to gain insight into the nature of quantum mechanics, he quickly realized that Clifford algebra was a unified language and tool for mathematics, physics, and engineering. He published “space-time algebra” in 1966 and has been working on the promotion of Clifford algebra in teaching and research [9, 10, 11, 12]. Because representation and algorithm in geometric algebra are seemingly as ordinary as arithmetic, his work has been neglected by the scientific community for more than 20 years. Only with the joint impetus of computer-aided design, computer vision and robotics, protein folding, neural networks, modern differential geometry, mathematical physics [13, 14, 15, 16, 17], and especially the Journal “Advances in Applied Clifford algebras” founded by Professor Jaime Keller, geometric algebra began to move towards popularity and prosperity.
As a unified and universal language of natural science, Clifford algebra is developed by many mathematicians, physicists, and engineers according to their different requirements and knowledge background. Such situation leads to “There are a thousand Hamlets in a thousand people’s eyes.” In this chapter, by introducing typical application of Clifford algebra in geometry, we show some special feature and elegance of the algebra.
In Euclidean space, we have several important concepts such as vector, length, angle, area, volume, and tensor. The study of relationship between these concepts constitutes the whole content of Euclidean geometry. The mathematical tools previously used to discuss these contents are vector algebra and geometrical method, which are complex and require much fundamental knowledge. Clifford algebra exactly and faithfully describes the intrinsic properties of vector space by introducing concepts such as inner, exterior, and geometric products of vectors and thus becomes a unified language and standard tool for dealing with geometric and physical problems. Clifford algebra has the characteristics of simple concept, standard operation, completeness in conclusion, and easy understanding.
Definition 1 For Minkowski space
the algebra is called Grassmann algebra and
The Grassmann is also called exterior algebra. The geometrical meaning of
Geometric meaning of exterior products of vectors.
By the definition, we can easily check:
Theorem 1 For exterior algebra defined in
The dimension of the algebra is
Under the orthonormal basis
in which
The exterior product of vectors contains alternating combinations of basis, for example:
Definition 2 For any vectors
Clifford product is also calledgeometric product.
Similarly, we can define Clifford algebra for many vectors as
Definition 3 For Minkowski space
then the algebra
is called as Clifford algebra or geometric algebra, which is denoted as
There are several definitions for Clifford algebra [18, 19]. The above definition is the original definition of Clifford. Clifford algebra has also
In physics, we often use curvilinear coordinate system or consider problems in curved space-time. In this case, we must discuss problems in
Definition 4 In
where
in which
is Grassmann basis. The following Clifford-Grassmann number with basis
defines real universal Clifford algebra
The definitions and treatments in this chapter make the corresponding subtle and fallible concepts in differential geometry much simpler. For example, in spherical coordinate system of
We have the total area of the sphere
The above definition involves a number of concepts, some more explanations are given in the following:
The geometrical meanings of elements
which is the most important relation in Clifford algebra. Since Clifford algebra is isomorphic to some matrix algebra, by (17)
Assume
In which
By (12) and (13) we find that, using Clifford algebra to deal with the problems on a manifold or in the tangent space, the method is the same. Unless especially mentioned, we always use the Greek alphabet to stand for the index in curved space-time, and the Latin alphabet for the index in tangent space. We use Einstein summation convention.
In Eq. (15), each grade-
Geometric meaning of vectors dx,dy and dx∧dy.
The real difficulty in learning modern mathematics is that in order to get a little result, we need a long list of subtle concepts. Mathematicians are used to defining concepts over concepts, but if the chain of concepts breaks down, the subsequent contents will not be understandable. Except for the professionals, the common readers impossibly have so much time to check and understand all concepts carefully. Fortunately, the Clifford algebra can avoid this problem, because Clifford algebra depends only on a few simple concepts, such as numbers, vectors, derivatives, and so on. The only somewhat new concept is the Clifford product of the vector bases, which is isomorphic to some special matrix algebra; and the rules of Clifford algebra are also standardized and suitable for brainless operations, which can be well mastered by high school students.
Definition 5 For vector
in which
Theorem 2 For basis of Clifford algebra, we have the following relations
Proof. Clearly
Permuting the indices
Again by
Likewise, we can define multi-inner product
We use
In
We use
Based on the above preliminaries, we can display some enlightening examples of application, which show how geometric algebra works efficiently. For a skew-symmetrical torsion
and then
where
So, the skew-symmetrical torsion is equivalent to a pseudo vector in
The following example discusses the absolute differential of tensors. The definition of vector, tensor, and spinor in differential geometry involving a number of refined concepts such as vector bundle and dual bundle, which are too complicated for readers in other specialty. Here, we inherit the traditional definitions based on the bases
For simplicity, we denote tensor basis by:
In general, a tensor of rank
The geometrical information of the tensor such as transformation law and differential connection are all recorded by basis
For the absolute differential of vector field
We call
It is a real linear transformation of basis
It satisfies metric consistent condition
Thus, the differential connection can be generally expressed as:
For metric
By (43), we have:
Since
(45) is a linear nonhomogeneous algebraic equation of
Solving (45), we get the symmetrical particular solution “Christoffel symbols” as follows;
in which
where
By (48),
Theorem 3 For post-metric connections we have the following relations
and consistent condition
Proof If we represent
If we represent
in which
So this part of torsion is a skew-symmetrical tensor
Calculating the summation of (55) for circulation of
Substituting (42) into
we get
To understand the meaning of
The term
Theorem 4 In the case
in which
where
Similarly, we can calculate the absolute differential for any tensor. The example also shows the advantages to combine variable with basis.
Now we take spinor connection as example to show the power of Clifford algebra. For Dirac equation in curved space-time without torsion, we have [23, 25, 26]:
where
where
where
If the gravitational field is generated by a rotating ball, the corresponding metric, like the Kerr one, cannot be diagonalized. In this case, the spin-gravity coupling term has nonzero coupling effect. In asymptotically flat space-time, we have the line element in quasi-spherical coordinate system [28]:
in which
where
Substituting it into (65) we get
By (70), we find that the intensity of
(71) shows that the force lines of
The matrix representation of Clifford algebra is an old problem with a long history. As early as in 1908, Cartan got the following periodicity of 8 [18, 19].
Theorem 5 For real universal Clifford algebra
For
By (73), we find
Similarly, for
By (74), the basis is equivalent to
Thus, (75) means
In geometry and physics, the matrix representation of generators of Clifford algebra is more important and fundamental than the representation of whole algebra. Define
which forms the generator or grade-1 basis of Clifford algebra
Theorem 6 Assuming the matrices
then there is a natural number
This means in equivalent sense, we have unique representation (76) for generator of
Theorem 7 Let
Other
In order to express the general representation of generators, we introduce some simple notations.
in which the direct product of matrix is Kronecker product. Obviously, we have
Theorem 8
In equivalent sense, for
in which
For
If and only if
For any
For
Then, we get all complex matrix representations for generators of real
The real representation of
By the construction of generators, we have only two kinds of
Denote
Theorem 9
For
For
For
Obviously we have
Proof. By calculating rules of block matrix, it is easy to check the following relations:
By these relations, Theorem 9 becomes a direct result of Theorem 8.
For example, we have
It is easy to check
Assume
Definition 6 The conjugation of element in
The main involution of element is defined by
The norm and inverse of element are defined by
By the definition, it is easy to check
Definition 7 The Pin group and Spin group of
The transformation
For
In all transformations of vector, the reflection and rotation transformations are important in geometry. Here, we discuss the transformation in detail. Let
Let
Eq. (107) clearly shows the geometrical meaning of reflection. By (106), we learn reflection transformation belongs to
Reflection transformation X′=X⊥−X∥.
The rotation transformation
The group elements of elementary transformation in
The total transformation can be expressed as multiplication of elementary transformations as follows:
(111) has
in which
If
Thus,
Suppose the basic space of projective geometry is
Diagram of parameter setting for projective geometry.
The equation of projective ray is given by:
where
Let
In the above equation
Theorem 10 For 4 different points
Proof Substituting (119) into (120) we get
where
(123) is independent of
Now we examine affine transformation. In this case, the polar
Since
Theorem 11 Assume
is an affine invariant.
Proof By equation of transformation (124) we get
In (126), only the parameters
Denote the unit directional vector of line
Substituting them into (127) we get:
This proves the simple ratio
The treatment of image information by computer requires concise and general algebraic representation for geometric modeling as well as fast and robust algebraic algorithm for geometric calculation. Conformal geometry algebra was introduced in this context. By establishing unified covariant algebra representation of classical geometry, the efficient calculation of invariant algebra is realized [13, 14, 15]. It provides a unified and concise homogeneous algebraic framework for classical geometry and algorithms, which can thus be used for complicated symbolic geometric calculations. This technology is currently widely applied in high-tech fields such as computer graphics, vision calculation, geometric design, and robots.
The algebraic representation of a geometric object is homogeneous, which means that any two algebraic expressions representing this object differ by only one nonzero factor and any such algebraic expressions with different nonzero multiple represent the same geometric object. The embedding space provided by conformal geometric algebra for
In conformal geometry algebra, an additional Minkowski plane
In practical application,
They satisfy
A unit pseudo-scalar
In conformal geometric algebra, we work with
Define the horosphere of
which is a bijective mapping
Now we examine how conformal geometric algebra represents geometric objects. For a line passing through points
Since
Again by using (135) and (136), we get
We recognize
For a sphere with radius
Using
where
From these properties, the form (139) and center
The examples given above are only applications of Clifford algebra in geometry, but we have seen the power of Clifford algebra in solving geometrical problems. In fact, Clifford algebra is more widely used in physics. Why does Clifford algebra work so well? As have been seen from the above examples, the power of Clifford algebra comes from the following features:
In the geometry of flat space, the basic concepts are only length, angle, area, and volume, which are already implicitly included in the definition of Clifford algebra. So, Clifford algebra summarizes these contents of classical geometry and algebraize them all. By introducing the concepts of inner, exterior, and direct products of vector, Clifford algebra summarizes the operations of scalars, vectors, and tensors and then can represent all the physical variables in classical physics, because only these variables are included in classical physics.
By localizing the basis or frame of space-time, Clifford algebra is naturally suitable for the tangent space in a manifold. If the differential
If the above contents seem to be very natural, Clifford algebra still has another unusual advantage, that is, it includes the theory of spinor. So, Clifford algebra also contains quantum theory and spinor connection. These things are far beyond the human intuition and have some surprising properties.
There are many reasons to make Clifford algebra become a unified and efficient language and tool for mathematics, physics, and engineering, such as Clifford algebra generalizes real number, complex number, quaternion, and vector algebra; Clifford algebra is isomorphic to matrix algebra; the derivative operator
“But, if geometric algebra is so good, why is it not more widely used?” As Hestenes replied in [11]: “Its time will come!” The published geometric algebra literature is more than sufficient to support instruction with geometric algebra at intermediate and advanced levels in physics, mathematics, engineering, and computer science. Though few faculty are conversant with geometric algebra now, most could easily learn what they need while teaching. At the introductory level, geometric algebra textbooks and teacher training will be necessary before geometric algebra can be widely taught in the schools. There is steady progress in this direction, but funding is needed to accelerate it. Malcolm Gladwell has discussed social conditions for a “tipping point” when the spread of an idea suddenly goes viral. Place your bets now on a Tipping Point for Geometric Algebra!
I would like to thank Dr. Min Lei for her kind invitation and help. The discussion on torsion is completed under the inspiration and guidance of Prof. James M. Nester. The content of conformal geometric algebra is added according to the suggestion of Dr. Isiah Zaplana. The chapter has been improved according to the comments of a referee.
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