|
|
| (28 intermediate revisions by the same user not shown) |
| Line 1: |
Line 1: |
| =Prepare files=
| |
| angsd can work on remote bam files therefore first download a list with 23 unrelated europeans from the 1000genomes project.
| |
| <pre> | | <pre> |
| wget http://popgen.dk/netstuff/files.list
| | This page might be outdated, please see the 'Quick Start' shown in sidebar. |
| </pre> | | </pre> |
|
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed">
| |
| Contents of the file 'files.list'
| |
| <pre class="mw-collapsible-content">
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12003/alignment/NA12003.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12004/alignment/NA12004.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12005/alignment/NA12005.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12006/alignment/NA12006.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11830/alignment/NA11830.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11831/alignment/NA11831.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11832/alignment/NA11832.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11992/alignment/NA11992.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11993/alignment/NA11993.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11994/alignment/NA11994.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11995/alignment/NA11995.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12154/alignment/NA12154.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12155/alignment/NA12155.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12156/alignment/NA12156.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA06994/alignment/NA06994.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11840/alignment/NA11840.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12044/alignment/NA12044.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12813/alignment/NA12813.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12760/alignment/NA12760.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12762/alignment/NA12762.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA06985/alignment/NA06985.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA11881/alignment/NA11881.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| ftp://ftp.1000genomes.ebi.ac.uk/vol1/ftp/data/NA12249/alignment/NA12249.mapped.ILLUMINA.bwa.CEU.low_coverage.20120522.bam
| |
| </pre>
| |
| </div>
| |
|
| |
| ==Understaing angsd options==
| |
| As a simple reference for the program we have made most of the methods within angsd easy viewable by writing the associated command.
| |
| All options are given by
| |
| <pre>
| |
| -parameter value
| |
| </pre>
| |
| It's important that there are no space between the dash and the paramater, it is important that there are a space betwwen the parameter and the value. Futhermore the parameter is casesensitive.
| |
|
| |
| Simply writing angsd will give you the helpscreen.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd
| |
| <pre class="mw-collapsible-content">
| |
| Command:
| |
| angsd
| |
| -> angsd version: 0.515 build(Mar 23 2013 12:35:04)
| |
| -> Please use the website "http://www.popgen.dk/angsd" as reference
| |
| -> Use -nThreads for number of threads allocated to the program
| |
|
| |
| Overview of methods:
| |
| -GL estimate genotype likelihoods
| |
| -doCounts Calculate various counts statistics
| |
| -doDepth Do depth statistics
| |
| -doAsso perform association study
| |
| -doMaf estimate allele frequencies
| |
| -doError estimate the type specific error rates
| |
| -doAnsError estimate the errorrate based on perfect fastas
| |
| -doHWE Est inbreedning per site
| |
| -doGeno call genotypes
| |
| -realSFS Estimate the SFS and/or perform neutrality tests
| |
|
| |
| Below are options that can be usefull
| |
| -bam Options relating to bam reading
| |
| -doMajorMinor Infer the major/minor using different approaches
| |
| -ref/-anc Read reference or ancestral genome
| |
| many others
| |
|
| |
| For information of specific options type:
| |
| ./angsd METHODNAME eg
| |
| ./angsd -GL
| |
| ./angsd -doMaf
| |
| ./angsd -doAsso etc
| |
| Examples:
| |
| Estimate MAF for bam files in 'list'
| |
| './angsd -bam list -GL 2 -doMaf 2 -out RES -doMajorMinor 1'
| |
|
| |
| </pre>
| |
| </div>
| |
| An explanation for every parameter is shown beside the parameter, and for every of these options we can get additional information by typing that parameter solely without any options. An example below for the methods relating to genotype likelihood calculation.
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -GL
| |
| <pre class="mw-collapsible-content">
| |
| -> angsd version: 0.515 build(Mar 27 2013 11:15:58)
| |
| -> Analysis helpbox/synopsis information:
| |
| ---------------------
| |
| analysisEstLikes.cpp:
| |
| -GL=0:
| |
| 1: SAMtools
| |
| 2: GATK
| |
| 3: SOAPsnp
| |
| 4: SYK
| |
| -minQ 13 (remove bases with qscore<minQ)
| |
| -trim 0 (zero means no trimming)
| |
| -tmpdir angsd_tmpdir/ (used by SOAPsnp)
| |
| -errors (null) (used by SYK)
| |
| -minInd -1 (-1 indicates no filtering)
| |
|
| |
| Filedumping:
| |
| -doGlf=0
| |
| 1: binary glf (10 log likes) .glf
| |
| 2: beagle likelihood file .beagle.gz
| |
| 3: binary 3 times likelihood .glf
| |
| 4: text version (10 log likes) .glf
| |
| </pre>
| |
| </div>
| |
| This tells you that there are 4 different genotype likelihood models implemented and you can choose accordingly by writing -GL 1 for the SAMtools model. We also see that we can dump the genotype likelihoods in four different ways.
| |
|
| |
| ==Understanding angsd output==
| |
| Program catches system signals, if you press ctrl+c, it will therefore stop the filereading, but will let the threads already running finish their jobs. You can therefore press ctrl+c at anytime at expect to get proper output files.
| |
| After a run has been completed the program will printout a list of the generated files.
| |
|
| |
| An example is below.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1 -GL 1 -nInd 20 -minMaf 0.005 -nThreads 10 -doCounts 1 -dumpCounts 3
| |
| <pre class="mw-collapsible-content">
| |
| Command:
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1 -GL 1 -nInd 20 -minMaf 0.005 -nThreads 10 -doCounts 1 -dumpCounts 3
| |
| -> angsd version: 0.529 build(May 2 2013 14:09:03)
| |
| ->Starting analysis
| |
| [uppile] parsing 20 number of samples
| |
| Change of chromo detected Waiting for nThreads:0
| |
| ->printing at chr: chr1 pos:3756580 chunknumber 500^CCaught SIGNAL: 2 will try to exit nicely (no more threads are created, we will wait for the current threads to finish)
| |
| SEnding NULL this is a killswitch
| |
| -> Done reading data waiting for calculations to finish
| |
| -> Calling destroy
| |
| -> Waiting for nThreads:9
| |
| -> Waiting for nThreads:3
| |
|
| |
| -> Done waiting for threads
| |
| ->Output filenames:
| |
| ->"tstMaf.arg"
| |
| ->"tstMaf.pos"
| |
| ->"tstMaf.counts"
| |
| ->"tstMaf.mafs"
| |
|
| |
| Thu May 2 15:11:31 2013
| |
| [ALL done] cpu-time used = 77.95 sec
| |
| [ALL done] walltime used = 21.00 sec
| |
| </pre>
| |
| </div>
| |
|
| |
| =Getting simple Counts/depth=
| |
| For some analysis simply getting the sequencing depth for all sites could be of interest, this can of analysis is grouped in the '-doCounts' methods.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -doCounts
| |
| <pre class="mw-collapsible-content">
| |
| </pre>
| |
| </div>
| |
|
| |
| So if we wanted the sum of ACGTS across all samples we could write
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -bam files.list -doCounts 1 -dumpCounts 3 -out tstCounts -nInd 10
| |
| <pre class="mw-collapsible-content">
| |
| ---------------
| |
| head -n tstCounts.counts
| |
| totA totC totG toft
| |
| 0 0 1 0
| |
| 0 1 0 0
| |
| 1 0 0 0
| |
| 1 0 0 0
| |
| 2 0 0 0
| |
| 0 0 0 2
| |
| 0 0 0 2
| |
| 0 0 0 2
| |
| 0 0 2 0
| |
| </pre>
| |
| </div>
| |
|
| |
|
| |
|
| |
| Or if we wanted the sequencing depth per sample but only for the good quality data
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -bam files.list -doCounts 1 -dumpCounts 2 -out tstCounts -nInd 10 -minQ 20 -minMapQ 30
| |
| <pre class="mw-collapsible-content">
| |
| ---------------
| |
| head tstCounts.counts
| |
| ind0 ind1 ind2 ind3 ind4 ind5 ind6 ind7 ind8 ind9
| |
| 0 0 0 0 0 0 0 0 0 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| 0 0 0 0 0 0 0 0 1 0
| |
| </pre>
| |
| </div>
| |
|
| |
| =Frequencies=
| |
| We can also estimate the allele frequencies. This we do by using the -doMaf option.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -doMaf
| |
| <pre class="mw-collapsible-content">
| |
| analysisMaf.cpp:
| |
| -doMaf 0
| |
| 1: BFGS frequency (known major minor)
| |
| 2: EM frequency (known major minor)
| |
| 4: BFGS frequency (unknown major minor)
| |
| 8: EM frequency (unknown major minor)
| |
| 16: frequency from genotype probabilities
| |
| 32: alleleCounts based method (known major minor)
| |
| -doSNP 0
| |
| -minMaf 0.010000 0
| |
| -minLRT 24.000000 0
| |
| -ref (null)
| |
| -anc (null)
| |
| -doZ 0
| |
| -eps 0.001000 [Only used for -doMaf &32]
| |
| </pre>
| |
| </div>
| |
| So if we try to use -doMaf 2 angsd will complain!
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf
| |
| <pre class="mw-collapsible-content">
| |
| Error: you need to specify doMajorMinor in order to doMaf
| |
| </pre>
| |
| </div>
| |
| So lets decide also estimate the major and minor, what are the options.
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd -doMajorMinor
| |
| <pre class="mw-collapsible-content">
| |
| -------------------
| |
| analysisMajorMinor.cpp:
| |
|
| |
| -doMajorMinor 0
| |
| 1: Infer major and minor from GL
| |
| 2: Infer major and minor from allele counts
| |
| 3: use major and minor from bim file (requires -filter afile.bim)
| |
| 4: Use reference allele as major (requires -ref)
| |
| 5: Use ancestral allele as major (requires -anc)
| |
| </pre>
| |
| </div>
| |
|
| |
| Let us infer the major and minor using the genotype likelihoods.
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1
| |
| <pre class="mw-collapsible-content">
| |
| -doMajorMinor 1 is based on genotype likelihoods, you must specify a genotype likelihood model -GL
| |
| </pre>
| |
| </div>
| |
|
| |
| So now we need to specify which genotype likelihood model we want to use, let us see what our options are
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -GL
| |
| <pre class="mw-collapsible-content">
| |
| ---------------------
| |
| analysisEstLikes.cpp:
| |
| -GL=0:
| |
| 1: SAMtools
| |
| 2: GATK
| |
| 3: SOAPsnp
| |
| 4: SYK
| |
| -minQ 13 (remove bases with qscore<minQ)
| |
| -trim 0 (zero means no trimming)
| |
| -tmpdir angsd_tmpdir/ (used by SOAPsnp)
| |
| -errors (null) (used by SYK)
| |
| -minInd -1 (-1 indicates no filtering)
| |
|
| |
| Filedumping:
| |
| -doGlf 0
| |
| 1: binary glf (10 log likes) .glf
| |
| 2: beagle likelihood file .beagle.gz
| |
| 3: binary 3 times likelihood .glf
| |
| 4: text version (10 log likes) .glf
| |
|
| |
|
| |
| </pre>
| |
| </div>
| |
| We pick the same model they use in samtools '-GL 1'.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1 -GL 1 -nInd 20
| |
| <pre class="mw-collapsible-content">
| |
| head tstMaf.mafs
| |
| chromo position major minor knownEM nInd
| |
| chr1 11782 G A 0.000008 1
| |
| chr1 11783 C A 0.000008 1
| |
| chr1 11784 A C 0.000008 1
| |
| chr1 11785 A C 0.000008 2
| |
| chr1 11786 A C 0.000008 3
| |
| chr1 11787 T A 0.000008 3
| |
| chr1 11788 T A 0.000008 3
| |
| chr1 11789 T A 0.000008 4
| |
| chr1 11790 G A 0.000008 4
| |
| </pre>
| |
| </div>
| |
| These sites are all invariable so lets filter out the sites with a maf below 0.5%
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1 -GL 1 -nInd 20 -minMaf 0.005 -nThreads 10
| |
| <pre class="mw-collapsible-content">
| |
| head tstMaf.mafs
| |
| chromo position major minor knownEM nInd
| |
| chr1 13032 A T 0.016541 9
| |
| chr1 13038 T C 0.073424 9
| |
| chr1 13309 G T 0.207854 4
| |
| chr1 13396 T A 0.012727 16
| |
| chr1 13482 G C 0.019104 15
| |
| chr1 13502 G A 0.025732 14
| |
| chr1 13519 T C 0.018256 17
| |
| chr1 14933 A G 0.487768 1
| |
| chr1 16259 T G 0.060360 7
| |
| </pre>
| |
| </div>
| |
|
| |
| The filters works across the different analysis classes, so if we supply the dumpCounts we will only get the sites with a maf >0.5%
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam files.list -doMaf 2 -out tstMaf -doMajorMinor 1 -GL 1 -nInd 20 -minMaf 0.005 -nThreads 10 -doCounts 1 -dumpCounts 3
| |
| <pre class="mw-collapsible-content">
| |
| paste tstMaf.pos tstMaf.counts |head
| |
| chr pos totDepth totA totC totG totT
| |
| chr1 13032 21 20 0 0 1
| |
| chr1 13038 21 0 1 0 20
| |
| chr1 13309 8 0 0 7 1
| |
| chr1 13396 34 1 0 0 33
| |
| chr1 13482 30 0 1 29 0
| |
| chr1 13502 25 1 0 24 0
| |
| chr1 13519 26 0 1 0 25
| |
| chr1 14933 2 1 0 1 0
| |
| chr1 16259 9 0 0 1 8
| |
| </pre>
| |
| </div>
| |
| =Estimating the SFS=
| |
| We can also use angsd for estimating the site frequency spectrum.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -realSFS
| |
| <pre class="mw-collapsible-content">
| |
| --------------
| |
| angsd_realSFS.cpp:
| |
| -realSFS 0
| |
| 1: perform multisample GL estimation
| |
| 2: use an inbreeding version
| |
| -doThetas 0 (calculate thetas)
| |
| -underFlowProtect 0
| |
| -fold 0 (deprecated)
| |
| -anc (null) (ancestral fasta)
| |
| -noTrans 0 (remove transitions)
| |
| -pest (null) (prior SFS)
| |
|
| |
| </pre>
| |
| </div>
| |
| We have butterfly dataset for which we want to find the SFS.
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam
| |
| <pre class="mw-collapsible-content">
| |
| --------------
| |
| angsd_realSFS.cpp:
| |
| -realSFS 0
| |
| 1: perform multisample GL estimation
| |
| 2: use an inbreeding version
| |
| -doThetas 0 (calculate thetas)
| |
| -underFlowProtect 0
| |
| -fold 0 (deprecated)
| |
| -anc (null) (ancestral fasta)
| |
| -noTrans 0 (remove transitions)
| |
| -pest (null) (prior SFS)
| |
| </pre>
| |
| </div>
| |
|
| |
| So let us try to estimate the sfs
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| .angsd0.530/angsd -bam bom.bam -realSFS 1 -out testSFS
| |
| <pre class="mw-collapsible-content">
| |
| --------------
| |
| -> angsd version: 0.529 build(May 2 2013 14:09:03)
| |
| Must supply -anc for polarizing the spectrum
| |
| </pre>
| |
| </div>
| |
| So we also need to supply a fasta which should contain our ancestral states. So lets do that.
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| ./angsd0.530/angsd -bam bom.bam -realSFS 1 -out testSFS -anc bombyx/input/referenceseq.fasta -r ref_contig:-100000
| |
| <pre class="mw-collapsible-content">
| |
| --------------
| |
| Command:
| |
| angsd0.530/angsd -bam bom.bam -realSFS 1 -out testSFS -anc bombyx/input/referenceseq.fasta -r ref_contig:-100000
| |
| -> angsd version: 0.529 build(May 2 2013 14:09:03)
| |
| Must supply genotype likelihoods (-GL [INT])
| |
| </pre>
| |
| </div>
| |
|
| |
| So we also need to pick a genotype likelihood model
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| angsd0.530/angsd -bam bom.bam -realSFS 1 -out testSFS -anc bombyx/input/referenceseq.fasta -r ref_contig:-100000 -GL 1
| |
| <pre class="mw-collapsible-content">
| |
| --------------
| |
| -> angsd version: 0.529 build(May 2 2013 14:09:03)
| |
| [getFasta.cpp.init:24] bombyx/input/referenceseq.fasta
| |
| ->Starting analysis
| |
| [uppile] parsing 20 number of samples
| |
| region lookup 1/1
| |
| Change of chromo detected Waiting for nThreads:0
| |
| [magic] chaning to chr: 0
| |
| ->printing at chr: ref_contig pos:91269 chunknumber 100SEnding NULL this is a killswitch
| |
|
| |
| -> Done reading data waiting for calculations to finish
| |
| -> Calling destroy
| |
| -> Done waiting for threads
| |
| ->Output filenames:
| |
| ->"testSFS.arg"
| |
| ->"testSFS.sfs"
| |
| ->"testSFS.sfs.pos"
| |
|
| |
| Thu May 2 16:57:57 2013
| |
| [ALL done] cpu-time used = 4.45 sec
| |
| [ALL done] walltime used = 5.00 sec
| |
|
| |
| </pre>
| |
| </div>
| |
| We know use the external angsd program, for finding the MLE of the sfs
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| angsd0.530/misc/emOptim
| |
| <pre class="mw-collapsible-content">
| |
| -> -binput -nChr -maxIter -nThread -outnames
| |
| </pre>
| |
| </div>
| |
| we need to supply the binary .sfs file and the number of chromosomes.
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| angsd0.530/misc/emOptim -binput testSFS.sfs -nChr 40 -nThread 2
| |
| <pre class="mw-collapsible-content">
| |
| -> Thu May 2 17:03:03 2013
| |
| dumped:testSFS.sfs.em testSFS.sfs.em.ml testSFS.sfs.em.log
| |
| </pre>
| |
|
| |
| </div>
| |
| Let us visualise the results
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
| > a<-scan("testSFS.sfs.em.ml")
| |
| Read 41 items
| |
| > barplot(a[-1])
| |
| > barplot(a[-1])
| |
| > norm<-function(x) x/sum(x)
| |
| > barplot(norm(a[-1]))
| |
| <pre class="mw-collapsible-content">
| |
|
| |
| </pre>
| |
| </div>
| |
|
| |
|
| |
| </div>
| |
|
| |
| <div class="toccolours mw-collapsible mw-collapsed" >
| |
|
| |
| <pre class="mw-collapsible-content">
| |
|
| |
| </pre>
| |
| </div>
| |