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index 0000000..20b4d91 --- /dev/null +++ b/Lecture.html @@ -0,0 +1,2125 @@ + + + + + + + + + + + + + Biodiversity metrics and metabarcoding + + + + + + + + + + + + + + + + + + +
+
+ +
+

Biodiversity metrics and metabarcoding

+ +
+
+
+Eric Coissac +
+
+
+ +

2/2/24

+
+
+

Summary

+
    +
  • The MetabarSchool Package
  • +
  • What do the reading numbers per PCR mean?
  • +
  • Rarefaction vs. relative frequencies
  • +
  • alpha diversity metrics
  • +
  • beta diversity metrics
  • +
  • multidimentionnal analysis
  • +
  • comparison between datasets
  • +
+
+ +
+
+

The MetabarSchool Package

+ +
+
+

Installing the package

+

You need the devtools package

+
+
install.packages("devtools",dependencies = TRUE)
+
+

Then you can install MetabarSchool

+
+
devtools::install_git("https://git.metabarcoding.org/MetabarcodingSchool/biodiversity-metrics.git")
+
+

You will also need the vegan package

+
+
install.packages("vegan",dependencies = TRUE)
+
+
+
+
+

The dataset

+ +
+
+

The mock community

+

A 16 plants mock community

+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
species taxid Relative aboundance
1 Taxus baccata 25629 1/2
2 Salvia pratensis 49216 1/4
3 Populus tremula 113636 1/8
4 Rumex acetosa 41241 1/16
5 Carpinus betulus 12990 1/32
6 Fraxinus excelsior 38873 1/64
7 Picea abies 3329 1/128
8 Lonicera xylosteum 439142 1/256
9 Abies alba 45372 1/512
10 Acer campestre 66205 1/1024
11 Briza media 281077 1/2048
12 Rosa canina 74635 1/4096
13 Capsella bursa-pastoris 3719 1/8192
14 Geranium robertianum 122183 1/16384
15 Rhododendron ferrugineum 49622 1/32768
16 Lotus corniculatus 47247 1/65536
+ +
+
+
+
+

The experiment

+
+ +
+
    +
  • 192 PCR of the mock community using SPER02 trnL-P6-Loop primers

    +
      +
    • 6 dilutions of the mock community: 1/1, 1/2, 1/4, 1/8, 1/16, 1/32

    • +
    • 32 repeats per dilution

    • +
  • +
+
+
+

Loading data

+
+
library(MetabarSchool)
+
+data("positive.count")
+data("positive.samples")
+data("positive.motus")
+
+
    +
  • positive.count read count matrix \(192 \; PCRs \; \times \; 24330 \; MOTUs\)
  • +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
P000001 P000002 P000003 P000004 P000005
sample.TM_POS_d16_1_a_A1 1167 4477 779 0 12
sample.TM_POS_d16_1_a_B1 1072 5077 985 2 8
sample.TM_POS_d16_1_b_A2 919 3599 601 0 10
sample.TM_POS_d16_1_b_B2 704 4129 835 2 15
sample.TM_POS_d16_2_a_A1 1155 5341 1023 2 6
+ +
+
+


+
+
positive.count[1:5,1:5]
+
+
+
+

Loading data

+
+
library(MetabarSchool)
+
+data("positive.count")
+data("positive.samples")
+data("positive.motus")
+
+
    +
  • positive.samples a 192 rows data.frame of 2 columns describing each PCR
  • +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + +
dilution repeats
sample.TM_POS_d16_1_a_A1 2 1.a.A1
sample.TM_POS_d16_1_a_B1 2 1.a.B1
sample.TM_POS_d16_1_b_A2 2 1.b.A2
+ +
+
+


+
+
head(positive.samples,n=3)
+
+
+
+

Loading data

+
+
library(MetabarSchool)
+
+data("positive.count")
+data("positive.samples")
+data("positive.motus")
+
+
    +
  • positive.motus : a 24330 rows data.frame of 4 columns describing each MOTU
  • +
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
dilution species taxid true
P000001 0.250 Salvia pratensis 49216 TRUE
P000002 0.125 Populus tremula 113636 TRUE
P000003 0.500 Taxus baccata 25629 TRUE
+ +
+
+


+
+
head(positive.motus,n=3)
+
+
+
+

Removing singleton sequences

+

Singleton sequences are observed only once over the complete dataset.

+
+
table(colSums(positive.count) == 1)
+
+
+
+ + + + + + + + + + + + + + +
FALSE TRUE
5579 18751
+ +
+
+


+

We discard them they are unanimously considered as rubbish.

+
+
are.not.singleton = colSums(positive.count) > 1
+positive.count = positive.count[,are.not.singleton]
+positive.motus = positive.motus[are.not.singleton,]
+
+
    +
  • positive.count is now a \(192 \; PCRs \; \times \; 5579 \; MOTUs\) matrix
  • +
+
+
+

Not all the PCR have the same number of reads

+

Despite all standardization efforts

+ +
+

Is it related to the amount of DNA in the extract ?

+
+
+
+

What do the reading numbers per PCR mean?

+
+
par(bg=NA) 
+boxplot(rowSums(positive.count) ~ positive.samples$dilution,log="y")
+abline(h = median(rowSums(positive.count)),lw=2,col="red",lty=2)
+ +
+
+ +
+
+
+Only 7.4% of the PCR read count variation is explain by dilution +
+
+
+
+

You must normalize your read counts

+

Two options:

+

Rarefaction

+

Randomly subsample the same number of reads for all the PCRs

+

Relative frequencies

+

Divide the read count of each MOTU in each sample by the total total read count of the same sample

+

\[ +\text{Relative fequency}(Motu_i,Sample_j) = \frac{\text{Read count}(Motu_i,Sample_j)}{\sum_{k=1}^n\text{Read count}(Motu_k,Sample_j)} +\]

+
+
library(vegan)
+
+
+
+

Rarefying read count (1)

+
    +
  • We look for the minimum read number per PCR
  • +
+
+
min(rowSums(positive.count))
+
+
[1] 2065
+
+
+
+
positive.count.rarefied = rrarefy(positive.count,2000)
+
+
+
+

Rarefying read count (2)

+ +
+
+

Rarefying read count (3)

+

Identifying the MOTUs with reads count greater than \(0\) after rarefaction.

+
+
are.still.present = colSums(positive.count.rarefied)>0
+are.still.present[1:5]
+
+
P000001 P000002 P000003 P000004 P000005 
+   TRUE    TRUE    TRUE    TRUE    TRUE 
+
+
+
+
table(are.still.present)
+
+
are.still.present
+FALSE  TRUE 
+ 1876  3703 
+
+
+
+
+

Rarefying read count (4)

+
+
par(bg=NA) 
+boxplot(colSums(positive.count) ~ are.still.present, log="y")
+ +
+

The MOTUs removed by rarefaction were at most occurring 20 times

+

The MOTUs kept by rarefaction were at least occurring 2 times

+
+
+

Rarefying read count (5)

+

Keep only sequences with reads after rarefaction

+
+
positive.count.rarefied = positive.count.rarefied[,are.still.present]
+positive.motus.rare = positive.motus[are.still.present,]
+
+
+positive.motus.rare is now a \(192 \; PCRs \; \times \; 3703 \; MOTUs\) +
+
+
+

Why rarefying ?

+ +





Increasing the number of reads just increase the description of the subpart of the PCR you have sequenced.

+
+
+

Transforming read counts to relative frequencies

+
+
positive.count.relfreq = decostand(positive.count,
+                                   method = "total")
+
+

No sequences will be set to zero

+
+
table(colSums(positive.count.relfreq) == 0)
+
+

+FALSE 
+ 5579 
+
+
+
+
+
+

Measuring diversity

+ +
+
+

The different types of diversity

+
+
+
+

+
+
+
+
+



Whittaker (2010)



+
    +
  • \(\alpha\text{-diversity}\) : Mean diversity per site (\(species/site\))

  • +
  • \(\gamma\text{-diversity}\) : Regional biodiversity (\(species/region\))

  • +
  • \(\beta\text{-diversity}\) : \(\beta = \frac{\gamma}{\alpha}\) (\(sites/region\))

  • +
+
+
+
+
+

\(\alpha\)-diversity

+ +
+
+

Which is th most diverse environment ?

+ +
+
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
A B C D E F G
Environment.1 0.25 0.25 0.25 0.25 0.00 0.00 0.00
Environment.2 0.55 0.07 0.02 0.17 0.07 0.07 0.03
+ +
+
+
+
+

Richness

+

The actual number of species present in your environement whatever their aboundances

+ +
+
S = rowSums(environments > 0)
+
+
+
+ + + + + + + + + + + + + + + + + + +
S
Environment.1 4
Environment.2 7
+ +
+
+
+
+

Gini-Simpson’s index

+
+

The Simpson’s index is the probability of having the same species twice when you randomly select two specimens.

+
+
+

\[ +\lambda =\sum _{i=1}^{S}p_{i}^{2} +\]

+
+
+

\(\lambda\) decrease when complexity of your ecosystem increase.

+

Gini-Simpson’s index defined as \(1-\lambda\) increase with diversity

+
+
+

+
+
+
+
+
GS = 1 - rowSums(environments^2)
+
+
+
+ + + + + + + + + + + + + + + + + + +
Gini.Simpson
Environment.1 0.7500
Environment.2 0.6526
+ +
+
+
+
+

Shannon entropy

+

Shannon entropy is based on information theory:

+
+\(H^{\prime }=-\sum _{i=1}^{S}p_{i}\log p_{i}\) +
+

if \(A\) is a community where every species are equally represented then \[ +H(A) = \log|A| +\]

+
+
+
+

+
+
+
+
+
H = - rowSums(environments * log(environments),na.rm = TRUE)
+
+
+
+ + + + + + + + + + + + + + + + + + +
Shannon.index
Environment.1 1.386294
Environment.2 1.371925
+ +
+
+
+
+

Hill’s number

+
+

As : \[ +H(A) = \log|A| \;\Rightarrow\; ^1D = e^{H(A)} +\]

+
+
+

where \(^1D\) is the theoretical number of species in a evenly distributed community that would have the same Shannon’s entropy than ours.

+
+
+



+
+
+

+
+
+
+
+
D2 = exp(- rowSums(environments * log(environments),na.rm = TRUE))
+
+
+
+ + + + + + + + + + + + + + + + + + +
Hill.Numbers
Environment.1 4.000000
Environment.2 3.942933
+ +
+
+
+
+

Generalized logaritmic function

+

Based on the generalized entropy Tsallis (1994) we can propose a generalized form of logarithm.

+

\[ +^q\log(x) = \frac{x^{(1-q)}-1}{1-q} +\]

+

The function is not defined for \(q=1\) but when \(q \longrightarrow 1\;,\; ^q\log(x) \longrightarrow \log(x)\)

+

\[ +^q\log(x) = \left\{ + \begin{align} + \log(x),& \text{if } q = 1\\ + \frac{x^{(1-q)}-1}{1-q},& \text{otherwise} + \end{align} + \right. +\]

+
+
log_q = function(x,q=1) {
+  if (q==1)
+    log(x)
+  else 
+    (x^(1-q)-1)/(1-q)
+}
+
+
+
+

Impact of \(q\) on the log_q function

+ +
+
+

And its inverse function

+

\[ +^qe^x = \left\{ + \begin{align} + e^x,& \text{if } x = 1 \\ + (1 + x(1-q))^{(\frac{1}{1-q})},& \text{otherwise} + \end{align} + \right. +\]

+
+
exp_q = function(x,q=1) {
+  if (q==1)
+    exp(x)
+  else
+    (1 + (1-q)*x)^(1/(1-q))
+}
+
+
+
+

Generalised Shannon entropy

+

\[ +^qH = - \sum_{i=1}^S p_i \; ^q\log p_i +\]

+
+
H_q = function(x,q=1) {
+  sum(x * log_q(1/x,q),na.rm = TRUE)
+}
+
+

and generalized the previously presented Hill’s number

+

\[ +^qD=^qe^{^qH} +\]

+
+
 D_q = function(x,q=1) {
+  exp_q(H_q(x,q),q)
+}
+
+
+
+

Biodiversity spectrum (1)

+
+
H_spectrum = function(x,q=1) {
+  sapply(q,function(Q) H_q(x,Q))
+}
+
+
+
D_spectrum = function(x,q=1) {
+  sapply(q,function(Q) D_q(x,Q))
+}
+
+
+
+

Biodiversity spectrum (2)

+
+
library(MetabarSchool)
+qs = seq(from=0,to=3,by=0.1)
+environments.hq = apply(environments,MARGIN = 1,H_spectrum,q=qs)
+environments.dq = apply(environments,MARGIN = 1,D_spectrum,q=qs)
+
+ +
+
+

Generalized entropy \(vs\) \(\alpha\)-diversity indices

+
    +
  • \(^0H(X) = S - 1\) : the richness minus one.

  • +
  • \(^1H(X) = H^{\prime}\) : the Shannon’s entropy.

  • +
  • \(^2H(X) = 1 - \lambda\) : Gini-Simpson’s index.

  • +
+

When computing the exponential of entropy : Hill’s number

+
    +
  • \(^0D(X) = S\) : The richness.

  • +
  • \(^1D(X) = e^{H^{\prime}}\) : The number of species in an even community having the same \(H^{\prime}\).

  • +
  • \(^2D(X) = 1 / \lambda\) : The number of species in an even community having the same Gini-Simpson’s index.

  • +
+


+
+

\(q\) can be considered as a penality you give to rare species

+

when \(q=0\) all the species have the same weight

+
+
+
+

Biodiversity spectrum of the mock community

+
+
H.mock = H_spectrum(plants.16$dilution,qs)
+D.mock = D_spectrum(plants.16$dilution,qs)
+
+ +
+
+

Biodiversity spectrum and metabarcoding (1)

+
+
positive.H = apply(positive.count.relfreq,
+                   MARGIN = 1,
+                   FUN = H_spectrum,
+                   q=qs)
+
+ +
+
+

Biodiversity spectrum and metabarcoding (2)

+ +
+
+

Biodiversity spectrum and metabarcoding (3)

+
+
positive.D = apply(positive.count.relfreq,
+                   MARGIN = 1,
+                   FUN = D_spectrum,
+                   q=qs)
+
+ +
+
+

Impact of data cleaning on \(\alpha\)-diversity (1)

+

We realize a basic cleaning:

+
    +
  • removing signletons
  • +
  • too short or long sequences
  • +
  • clustering data using obiclean
  • +
+
+
obigrep -p 'count > 1' \
+        positifs.uniq.annotated.fasta \
+      > positifs.uniq.annotated.no.singleton.fasta
+
+obigrep -l 10 -L 150 \
+        positifs.uniq.annotated.no.singleton.fasta \
+      > positifs.uniq.annotated.good.length.fasta
+
+obiclean -s merged_sample -H -C -r 0.1 \
+        positifs.uniq.annotated.good.length.fasta \
+      > positifs.uniq.annotated.clean.fasta
+
+
+
+

Impact of data cleaning on \(\alpha\)-diversity (2)

+
+
data(positive.clean.count)
+
+positive.clean.count.relfreq = decostand(positive.clean.count,
+                                         method = "total")
+
+positive.clean.H = apply(positive.clean.count.relfreq,
+                         MARGIN = 1,
+                         FUN = H_spectrum,
+                         q=qs)
+
+ +
+
+

Impact of data cleaning on \(\alpha\)-diversity (3)

+
+
positive.clean.D = apply(positive.clean.count.relfreq,
+                         MARGIN = 1,
+                         FUN = D_spectrum,
+                         q=qs)
+
+ +
+
+
+

\(\beta\)-diversity

+ +
+
+

Dissimilarity indices or non-metric distances

+
+A dissimilarity index \(d(A,B)\) is a numerical measurement
of how far apart objects \(A\) and \(B\) are. +
+

Properties

+

\[ +\begin{align} +d(A,B) \geqslant& 0 \\ +d(A,B) =& d(B,A) \\ +d(A,B) =& 0 \iff A = B \\ +\end{align} +\]

+
+
+

Some dissimilarity indices

+

Bray-Curtis

+

Relying on contengency table (quantitative data)

+

\[ +{\displaystyle BC(A,B)=1-{\frac {2\sum _{i=1}^{p}min(N_{Ai},N_{Bi})}{\sum _{i=1}^{p}(N_{Ai}+N_{Bi})}}}, \; \text{with }p\text{ the total number of species} +\]

+

Jaccard indices

+

Relying on presence absence data

+

\[ +J(A,B) = {{|A \cap B|}\over{|A \cup B|}} = {{|A \cap B|}\over{|A| + |B| - |A \cap B|}}. +\]

+
+
+

Metrics or distances

+
+
+
+

+
+
+
+
+

A metric is a dissimilarity index verifying the subadditivity also named triangle inequality

+

\[ +\begin{align} +d(A,B) \geqslant& 0 \\ +d(A,B) =& \;d(B,A) \\ +d(A,B) =& \;0 \iff A = B \\ +d(A,B) \leqslant& \;d(A,C) + d(C,B) +\end{align} +\]

+
+
+
+

Some metrics

+
+
+
+
+

+
+
+
+

Computing

+

\[ +\begin{align} +d_e =& \sqrt{(x_A - x_B)^2 + (y_A - y_B)^2} \\ +d_m =& |x_A - x_B| + |y_A - y_B| \\ +d_c =& \max(|x_A - x_B| , |y_A - y_B|) \\ +\end{align} +\]

+
+
+
+
+

Generalizable on a n-dimension space

+

Considering 2 points \(A\) and \(B\) defined by \(n\) variables

+

\[ +\begin{align} +A :& (a_1,a_2,a_3,...,a_n) \\ +B :& (b_1,b_2,b_3,...,b_n) +\end{align} +\]

+

with \(a_i\) and \(b_i\) being respectively the value of the \(i^{th}\) variable for \(A\) and \(B\).

+

\[ +\begin{align} +d_e =& \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2 } \\ +d_m =& \sum_{i=1}^{n}\left| a_i - b_i \right| \\ +d_c =& \max\limits_{1\leqslant i \leqslant n}\left|a_i - b_i\right| \\ +\end{align} +\]

+
+
+

For the fun… ;-)

+

You can generalize those distances as a norm of order \(k\)

+

\[ +d^k = \sqrt[k]{\sum_{i=1}^n|a_i - b_i|^k} +\]

+
    +
  • \(k=1 \Rightarrow D_m\) Manhatan distance
  • +
  • \(k=2 \Rightarrow D_e\) Euclidean distance
  • +
  • \(k=\infty \Rightarrow D_c\) Chebychev distance
  • +
+
+
+

Metrics and ultrametrics

+
+
+
+
+

+
+
+
+

Metric

+

\[ +d(x,z)\leqslant d(x,y)+d(y,z) +\]

+

Ultrametric

+

\[ +d(x,z)\leq \max(d(x,y),d(y,z)) +\]

+
+
+
+
+

Why it is nice to use metrics ?

+
    +
  • A metric induce a metric space
  • +
  • In a metric space rotations are isometries
  • +
  • This means that rotations are not changing distances between objects
  • +
  • Multidimensional scaling (PCA, PCoA, CoA…) are rotations
  • +
+
+
+

The data set

+

We analyzed two forest sites in French Guiana

+
    +
  • Mana : Soil is composed of white sands.

  • +
  • Petit Plateau : Terra firme (firm land). In the Amazon, it corresponds to the area of the forest that is not flooded during high water periods. The terra firme is characterized by old and poor soils.

  • +
+

At each site, twice sixteen samples where collected over an hectar

+
    +
  • Sixteen samples of soil. Each of them is constituted by a mix of five cores of 50g from the 10 first centimeters of soil covering half square meter.

  • +
  • Sixteen samples of litter. Each of them is constituted by the total litter collecter over the same half square meter where soil was sampled

  • +
+
+
data("guiana.count")
+data("guiana.motus")
+data("guiana.samples")
+
+
+
+

Clean out bad PCR cycle 1

+
+
s = tag_bad_pcr(guiana.samples$sample,guiana.count)
+ +
guiana.count.clean = guiana.count[s$keep,]
+guiana.samples.clean = guiana.samples[s$keep,]
+
+
+
table(s$keep)
+
+

+FALSE  TRUE 
+   42   299 
+
+
+
+
+

Clean out bad PCR cycle 2

+
+
s = tag_bad_pcr(guiana.samples.clean$sample,guiana.count.clean)
+ +
guiana.count.clean = guiana.count.clean[s$keep,]
+guiana.samples.clean = guiana.samples.clean[s$keep,]
+
+
+
table(s$keep)
+
+

+FALSE  TRUE 
+    8   291 
+
+
+
+
+

Clean out bad PCR cycle 3

+
+
s = tag_bad_pcr(guiana.samples.clean$sample,guiana.count.clean)
+ +
guiana.count.clean = guiana.count.clean[s$keep,]
+guiana.samples.clean = guiana.samples.clean[s$keep,]
+
+
+
table(s$keep)
+
+

+FALSE  TRUE 
+    4   287 
+
+
+
+
+

Averaging good PCR replicates (1)

+
+
guiana.samples.clean = cbind(guiana.samples.clean,s[rownames(guiana.samples.clean),])
+
+guiana.count.mean = aggregate(decostand(guiana.count.clean,method = "total"),
+                              by = list(guiana.samples.clean$sample),
+                              FUN=mean)
+
+n = guiana.count.mean[,1]
+guiana.count.mean = guiana.count.mean[,-1]
+rownames(guiana.count.mean)=as.character(n)
+guiana.count.mean = as.matrix(guiana.count.mean)
+
+dim(guiana.count.mean)
+
+
[1]   84 7884
+
+
+
+
+

Averaging good PCR replicates (2)

+
+
guiana.samples.mean = aggregate(guiana.samples.clean,
+                              by = list(guiana.samples.clean$sample),
+                              FUN=function(i) i[1])
+n = guiana.samples.mean[,1]
+guiana.samples.mean = guiana.samples.mean[,-1]
+rownames(guiana.samples.mean)=as.character(n)
+
+dim(guiana.samples.mean)
+
+
[1] 84 17
+
+
+

Keep only samples

+
+
guiana.samples.final = guiana.samples.mean[! is.na(guiana.samples.mean$site_id),]
+guiana.count.final   = guiana.count.mean[! is.na(guiana.samples.mean$site_id),]
+
+
+
+

Estimating similarity between samples

+
+
guiana.hellinger.final = decostand(guiana.count.final,method = "hellinger")
+guiana.relfreq.final   = decostand(guiana.count.final,method = "total")
+guiana.presence.1.final  = guiana.relfreq.final > 0.001
+guiana.presence.10.final   = guiana.relfreq.final > 0.01
+guiana.presence.50.final   = guiana.relfreq.final > 0.05
+
+guiana.bc.dist = vegdist(guiana.relfreq.final,method = "bray")
+guiana.euc.dist = vegdist(guiana.hellinger.final,method = "euclidean")
+guiana.jac.1.dist = vegdist(guiana.presence.1.final,method = "jaccard")
+guiana.jac.10.dist = vegdist(guiana.presence.10.final,method = "jaccard")
+guiana.jac.50.dist = vegdist(guiana.presence.50.final,method = "jaccard")
+
+
+
+

Euclidean distance on Hellinger transformation

+
+
xy = guiana.count.final[,order(-colSums(guiana.count.final))]
+xy = xy[,1:2]
+xy.hellinger = decostand(xy,method = "hellinger")
+
+
+
+
+
+

+
+
+
+
+
+

+
+
+
+
+
+
+

Bray-Curtis distance on relative frequencies

+

\[ +BC_{jk}=1-{\frac {2\sum _{i=1}^{p}min(N_{ij},N_{ik})}{\sum _{i=1}^{p}(N_{ij}+N_{ik})}} +\]

+

\[ +BC_{jk}=\frac{\sum _{i=1}^{p}(N_{ij}+N_{ik})-\sum _{i=1}^{p}2\;min(N_{ij},N_{ik})}{\sum _{i=1}^{p}(N_{ij}+N_{ik})} +\]

+

\[ +BC_{jk}=\frac{\sum _{i=1}^{p}(N_{ij} - min(N_{ij},N_{ik}) + (N_{ik} - min(N_{ij},N_{ik}))}{\sum _{i=1}^{p}(N_{ij}+N_{ik})} +\]

+

\[ +BC_{jk}=\frac{\sum _{i=1}^{p}|N_{ij} - N_{ik}|}{\sum _{i=1}^{p}N_{ij}+\sum _{i=1}^{p}N_{ik}} +\]

+

\[ +BC_{jk}=\frac{\sum _{i=1}^{p}|N_{ij} - N_{ik}|}{1+1} +\]

+

\[ +BC_{jk}=\frac{1}{2}\sum _{i=1}^{p}|N_{ij} - N_{ik}| +\]

+
+
+

Principale coordinate analysis (1)

+
+
guiana.bc.pcoa  = cmdscale(guiana.bc.dist,k=3,eig = TRUE)
+guiana.euc.pcoa = cmdscale(guiana.euc.dist,k=3,eig = TRUE)
+guiana.jac.1.pcoa = cmdscale(guiana.jac.1.dist,k=3,eig = TRUE)
+guiana.jac.10.pcoa = cmdscale(guiana.jac.10.dist,k=3,eig = TRUE)
+guiana.jac.50.pcoa = cmdscale(guiana.jac.50.dist,k=3,eig = TRUE)
+
+
+
+

Principale coordinate analysis (2)

+ +
+
+

Principale composante analysis

+
+
guiana.hellinger.pca = prcomp(guiana.hellinger.final,center = TRUE, scale. = FALSE)
+
+ + +
+
+

Comparing diversity of the environments

+
+ +
+ +
+
+

Bibliography

+ +
+
+Tsallis, Constantino. 1994. “What Are the Numbers That Experiments Provide.” Quim. Nova 17 (6): 468–71. +
+
+Whittaker, Robert J. 2010. “Meta-Analyses and Mega-Mistakes: Calling Time on Meta-Analysis of the Species Richness-Productivity Relationship.” Ecology 91 (9): 2522–33. +
+
+
+
+
+ + + + + + + + + + + + + + + + + + + + + + + + + \ No newline at end of file diff --git a/index.Rmd b/Lecture.qmd similarity index 82% rename from index.Rmd rename to Lecture.qmd index 9f911cd..6801115 100644 --- a/index.Rmd +++ b/Lecture.qmd @@ -1,16 +1,16 @@ --- title: "Biodiversity metrics \ and metabarcoding" author: "Eric Coissac" -date: "28/01/2019" +date: "02/02/2024" bibliography: inst/REFERENCES.bib -output: - ioslides_presentation: - widescreen: true - smaller: true - css: slides.css - mathjax: local - self_contained: false - slidy_presentation: default +format: + revealjs: + smaller: true + transition: slide + scrollable: true + theme: simple + html-math-method: mathjax +editor: visual --- ```{r setup, include=FALSE} @@ -25,20 +25,19 @@ opts_chunk$set(echo = FALSE, cache.lazy = FALSE) ``` - # Summary -- The MetabarSchool Package -- What do the reading numbers per PCR mean? -- Rarefaction vs. relative frequencies -- alpha diversity metrics -- beta diversity metrics -- multidimentionnal analysis -- comparison between datasets +- The MetabarSchool Package +- What do the reading numbers per PCR mean? +- Rarefaction vs. relative frequencies +- alpha diversity metrics +- beta diversity metrics +- multidimentionnal analysis +- comparison between datasets # The MetabarSchool Package -## Instaling the package +## Installing the package You need the *devtools* package @@ -62,7 +61,7 @@ install.packages("vegan",dependencies = TRUE) ## The mock community {.flexbox .vcenter .smaller} -A 16 plants mock community +A 16 plants mock community ```{r} data("plants.16") @@ -81,13 +80,12 @@ knitr::kable(x[,-(4:5)], data("positive.samples") ``` -- `r nrow(positive.samples)` PCR of the mock community using SPER02 trnL-P6-Loop primers +- `r nrow(positive.samples)` PCR of the mock community using SPER02 trnL-P6-Loop primers - - `r length(table(positive.samples$dilution))` dilutions of the mock - community: `r paste0('1/',names(table(positive.samples$dilution)))` + - `r length(table(positive.samples$dilution))` dilutions of the mock community: `r paste0('1/',names(table(positive.samples$dilution)))` + + - `r as.numeric(table(positive.samples$dilution)[1])` repeats per dilution - - `r as.numeric(table(positive.samples$dilution)[1])` repeats per dilution - ## Loading data ```{r echo=TRUE} @@ -98,9 +96,7 @@ data("positive.samples") data("positive.motus") ``` -- `positive.count` read count matrix - $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$ - +- `positive.count` read count matrix $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$ ```{r} knitr::kable(positive.count[1:5,1:5], @@ -111,11 +107,11 @@ knitr::kable(positive.count[1:5,1:5], ```
+ ```{r echo=TRUE,eval=FALSE} positive.count[1:5,1:5] ``` - ## Loading data ```{r echo=TRUE} @@ -126,9 +122,7 @@ data("positive.samples") data("positive.motus") ``` -- `positive.samples` a `r nrow(positive.samples)` rows `data.frame` of - `r ncol(positive.samples)` columns describing each PCR - +- `positive.samples` a `r nrow(positive.samples)` rows `data.frame` of `r ncol(positive.samples)` columns describing each PCR ```{r} knitr::kable(head(positive.samples,n=3), @@ -138,11 +132,11 @@ knitr::kable(head(positive.samples,n=3), ```
+ ```{r echo=TRUE,eval=FALSE} head(positive.samples,n=3) ``` - ## Loading data ```{r echo=TRUE} @@ -153,8 +147,7 @@ data("positive.samples") data("positive.motus") ``` -- `positive.motus` : a `r nrow(positive.motus)` rows `data.frame` of - `r ncol(positive.motus)` columns describing each MOTU +- `positive.motus` : a `r nrow(positive.motus)` rows `data.frame` of `r ncol(positive.motus)` columns describing each MOTU ```{r} knitr::kable(head(positive.motus,n=3), @@ -164,11 +157,12 @@ knitr::kable(head(positive.motus,n=3), ```
+ ```{r echo=TRUE,eval=FALSE} head(positive.motus,n=3) ``` -## Removing singleton sequences {.flexbox .vcenter} +## Removing singleton sequences {.flexbox .vcenter} Singleton sequences are observed only once over the complete dataset. @@ -176,7 +170,6 @@ Singleton sequences are observed only once over the complete dataset. table(colSums(positive.count) == 1) ``` - ```{r} kable(t(table(colSums(positive.count) == 1)), format = "html") %>% @@ -194,11 +187,9 @@ positive.count = positive.count[,are.not.singleton] positive.motus = positive.motus[are.not.singleton,] ``` -- `positive.count` is now a - $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$ - matrix - -## Not all the PCR have the same number of reads {.flexbox .vcenter} +- `positive.count` is now a $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$ matrix + +## Not all the PCR have the same number of reads {.flexbox .vcenter} Despite all standardization efforts @@ -210,9 +201,9 @@ hist(rowSums(positive.count), main = "Number of read per PCR") ``` -
+::: green Is it related to the amount of DNA in the extract ? -
+::: ## What do the reading numbers per PCR mean? {.smaller} @@ -222,17 +213,13 @@ boxplot(rowSums(positive.count) ~ positive.samples$dilution,log="y") abline(h = median(rowSums(positive.count)),lw=2,col="red",lty=2) ``` - ```{r} SC = summary(aov((rowSums(positive.count)) ~ positive.samples$dilution))[[1]]$`Sum Sq` ``` -
-
- Only `r round((SC/sum(SC)*100)[1],1)`% of the PCR read count - variation is explain by dilution -
-
+::: red2 +
Only `r round((SC/sum(SC)*100)[1],1)`% of the PCR read count variation is explain by dilution
+::: ## You must normalize your read counts @@ -242,7 +229,6 @@ Two options: Randomly subsample the same number of reads for all the PCRs - ### Relative frequencies Divide the read count of each MOTU in each sample by the total total read count of the same sample @@ -255,9 +241,9 @@ $$ library(vegan) ``` -## Rarefying read count (1) {.flexbox .vcenter} +## Rarefying read count (1) {.flexbox .vcenter} -- We look for the minimum read number per PCR +- We look for the minimum read number per PCR ```{r echo=TRUE} min(rowSums(positive.count)) @@ -267,7 +253,7 @@ min(rowSums(positive.count)) positive.count.rarefied = rrarefy(positive.count,2000) ``` -## Rarefying read count (2) {.flexbox .vcenter} +## Rarefying read count (2) {.flexbox .vcenter} ```{r fig.height=4} par(mfrow=c(1,2),bg=NA) @@ -285,7 +271,7 @@ hist(log10(colSums(positive.count.rarefied)+1), xlab = TeX("$\\log_{10}(reads per MOTUs)$")) ``` -## Rarefying read count (3) {.flexbox .vcenter} +## Rarefying read count (3) {.flexbox .vcenter} Identifying the MOTUs with reads count greater than $0$ after rarefaction. @@ -298,7 +284,7 @@ are.still.present[1:5] table(are.still.present) ``` -## Rarefying read count (4) {.flexbox .vcenter} +## Rarefying read count (4) {.flexbox .vcenter} ```{r echo=TRUE, fig.height=3.5} par(bg=NA) @@ -309,7 +295,7 @@ The MOTUs removed by rarefaction were at most occurring `r max(colSums(positive. The MOTUs kept by rarefaction were at least occurring `r min(colSums(positive.count[,are.still.present]))` times -## Rarefying read count (5) {.vcenter} +## Rarefying read count (5) {.vcenter} ### Keep only sequences with reads after rarefaction @@ -318,9 +304,7 @@ positive.count.rarefied = positive.count.rarefied[,are.still.present] positive.motus.rare = positive.motus[are.still.present,] ``` -
-positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \times \; `r ncol(positive.count.rarefied)` \; MOTUs$ -
+
positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \times \; `r ncol(positive.count.rarefied)` \; MOTUs$
## Why rarefying ? {.vcenter .columns-2} @@ -328,8 +312,7 @@ positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \time knitr::include_graphics("figures/subsampling.svg") ``` -



-Increasing the number of reads just increase the description of the subpart of the PCR you have sequenced. +



Increasing the number of reads just increase the description of the subpart of the PCR you have sequenced. ## Transforming read counts to relative frequencies @@ -348,36 +331,30 @@ table(colSums(positive.count.relfreq) == 0) ## The different types of diversity {.vcenter} -
+::: {style="float: left; width: 40%;"} ```{r} knitr::include_graphics("figures/diversity.svg") ``` -
+::: -
+::: {style="float: left; width: 60%;"} +

@Whittaker:10:00



-

-@Whittaker:10:00 -



+- $\alpha\text{-diversity}$ : Mean diversity per site ($species/site$) -- $\alpha\text{-diversity}$ : Mean diversity per site ($species/site$) - -- $\gamma\text{-diversity}$ : Regional biodiversity ($species/region$) - -- $\beta\text{-diversity}$ : $\beta = \frac{\gamma}{\alpha}$ ($sites/region$) - -
+- $\gamma\text{-diversity}$ : Regional biodiversity ($species/region$) +- $\beta\text{-diversity}$ : $\beta = \frac{\gamma}{\alpha}$ ($sites/region$) +::: # $\alpha$-diversity -## Which is th most diverse environment ? {.flexbox .vcenter} +## Which is th most diverse environment ? {.flexbox .vcenter} ```{r out.width = "400px"} knitr::include_graphics("figures/alpha_diversity.svg") ``` - ```{r out.width = "400px"} E1 = c(A=0.25,B=0.25,C=0.25,D=0.25,E=0,F=0,G=0) E2 = c(A=0.55,B=0.07,C=0.02,D=0.17,E=0.07,F=0.07,G=0.03) @@ -388,8 +365,7 @@ kable(environments, kable_styling(position = "center") ``` - -## Richness {.flexbox .vcenter} +## Richness {.flexbox .vcenter} The actual number of species present in your environement whatever their aboundances @@ -410,21 +386,19 @@ kable(data.frame(S=S), ## Gini-Simpson's index {.smaller} -
-The Simpson's index is the probability of having the same species twice when you randomly select two specimens. -
-
-
-
+::: {style="float: left; width: 60%;"} +The Simpson's index is the probability of having the same species twice when you randomly select two specimens.

+::: + +::: {style="float: right; width: 40%;"} $$ \lambda =\sum _{i=1}^{S}p_{i}^{2} -$$ -
-
+$$
+:::
-$\lambda$ decrease when complexity of your ecosystem increase. +$\lambda$ decrease when complexity of your ecosystem increase. Gini-Simpson's index defined as $1-\lambda$ increase with diversity @@ -445,24 +419,22 @@ kable(data.frame(`Gini-Simpson`=GS), kable_styling(position = "center") ``` -## Shannon entropy {.smaller} +## Shannon entropy {.smaller} Shannon entropy is based on information theory: -
-$H^{\prime }=-\sum _{i=1}^{S}p_{i}\log p_{i}$ -
+
$H^{\prime }=-\sum _{i=1}^{S}p_{i}\log p_{i}$
- -if $A$ is a community where every species are equally represented then -$$ +if $A$ is a community where every species are equally represented then $$ H(A) = \log|A| $$
+ ```{r out.width = "400px"} knitr::include_graphics("figures/alpha_diversity.svg") ``` +
```{r echo=TRUE} @@ -476,25 +448,26 @@ kable(data.frame(`Shannon index`=H), kable_styling(position = "center") ``` -## Hill's number {.smaller} +## Hill's number {.smaller} -
-As : -$$ +::: {style="float: left; width: 50%;"} +As : $$ H(A) = \log|A| \;\Rightarrow\; ^1D = e^{H(A)} -$$ -
-
-
+$$
+::: + +::: {style="float: right; width: 50%;"} where $^1D$ is the theoretical number of species in a evenly distributed community that would have the same Shannon's entropy than ours. -
+:::
-
-
+ +

+ ```{r out.width = "400px"} knitr::include_graphics("figures/alpha_diversity.svg") ``` +
```{r echo=TRUE} @@ -513,16 +486,16 @@ kable(data.frame(`Hill Numbers`=D2), Based on the generalized entropy @Tsallis:94:00 we can propose a generalized form of logarithm. $$ -^q\log(x) = \frac{x^{(1-q)}}{1-q} +^q\log(x) = \frac{x^{(1-q)}-1}{1-q} $$ -The function is not defined for $q=1$ but when $q \longrightarrow 1\;,\; ^q\log(x) \longrightarrow \log(x)$ +The function is not defined for $q=1$ but when $q \longrightarrow 1\;,\; ^q\log(x) \longrightarrow \log(x)$ $$ ^q\log(x) = \left\{ \begin{align} - \log(x),& \text{if } x = 1\\ - \frac{x^{(1-q)}}{1-q},& \text{otherwise} + \log(x),& \text{if } q = 1\\ + \frac{x^{(1-q)}-1}{1-q},& \text{otherwise} \end{align} \right. $$ @@ -568,6 +541,7 @@ $$ \end{align} \right. $$ + ```{r echo=TRUE, eval=FALSE} exp_q = function(x,q=1) { if (q==1) @@ -589,12 +563,12 @@ H_q = function(x,q=1) { } ``` - and generalized the previously presented Hill's number $$ ^qD=^qe^{^qH} $$ + ```{r echo=TRUE, eval=FALSE} D_q = function(x,q=1) { exp_q(H_q(x,q),q) @@ -641,27 +615,29 @@ points(qs,environments.dq[,1],type="l",col="blue") abline(v=c(0,1,2),lty=2,col=4:6) ``` -## Generalized entropy $vs$ $\alpha$-diversity indices +## Generalized entropy $vs$ $\alpha$-diversity indices -- $^0H(X) = S - 1$ : the richness minus one. +- $^0H(X) = S - 1$ : the richness minus one. -- $^1H(X) = H^{\prime}$ : the Shannon's entropy. +- $^1H(X) = H^{\prime}$ : the Shannon's entropy. -- $^2H(X) = 1 - \lambda$ : Gini-Simpson's index. +- $^2H(X) = 1 - \lambda$ : Gini-Simpson's index. ### When computing the exponential of entropy : Hill's number {.smaller} -- $^0D(X) = S$ : The richness. +- $^0D(X) = S$ : The richness. -- $^1D(X) = e^{H^{\prime}}$ : The number of species in an even community having the same $H^{\prime}$. +- $^1D(X) = e^{H^{\prime}}$ : The number of species in an even community having the same $H^{\prime}$. -- $^2D(X) = 1 / \lambda$ : The number of species in an even community having the same Gini-Simpson's index. +- $^2D(X) = 1 / \lambda$ : The number of species in an even community having the same Gini-Simpson's index.
+
+ $q$ can be considered as a penality you give to rare species -**when $q=0$ all the species have the same weight** +**when** $q=0$ all the species have the same weight
@@ -695,6 +671,7 @@ positive.H = apply(positive.count.relfreq, FUN = H_spectrum, q=qs) ``` + ```{r} par(bg=NA) boxplot(t(positive.H), @@ -706,7 +683,6 @@ points(H.mock,col="red",type="l") ## Biodiversity spectrum and metabarcoding (2) {.flexbox .vcenter .smaller} - ```{r} par(bg=NA) boxplot(t(positive.H)[,11:31], @@ -743,9 +719,9 @@ positive.D.means = rowMeans(positive.D) We realize a basic cleaning: -- removing signletons -- too short or long sequences -- clustering data using `obiclean` +- removing signletons +- too short or long sequences +- clustering data using `obiclean` ```{bash eval=FALSE,echo=TRUE} obigrep -p 'count > 1' \ @@ -761,7 +737,6 @@ obiclean -s merged_sample -H -C -r 0.1 \ > positifs.uniq.annotated.clean.fasta ``` - ## Impact of data cleaning on $\alpha$-diversity (2) ```{r echo=TRUE} @@ -805,16 +780,11 @@ points(D.mock,col="red",type="l") positive.clean.D.means = rowMeans(positive.D) ``` - # $\beta$-diversity - ## Dissimilarity indices or non-metric distances {.flexbox .vcenter} -
-A dissimilarity index $d(A,B)$ is a numerical measurement -
-of how far apart objects $A$ and $B$ are. -
+ +
A dissimilarity index $d(A,B)$ is a numerical measurement
of how far apart objects $A$ and $B$ are.
### Properties @@ -844,19 +814,17 @@ $$ J(A,B) = {{|A \cap B|}\over{|A \cup B|}} = {{|A \cap B|}\over{|A| + |B| - |A \cap B|}}. $$ -## Metrics or distances +## Metrics or distances -
+::: {style="float: left; width: 50%;"} ```{r out.width = "400px"} knitr::include_graphics("figures/metric.svg") ``` -
- -
+::: +::: {style="float: right; width: 50%;"} A metric is a dissimilarity index verifying the *subadditivity* also named *triangle inequality* - $$ \begin{align} d(A,B) \geqslant& 0 \\ @@ -865,20 +833,18 @@ d(A,B) =& \;0 \iff A = B \\ d(A,B) \leqslant& \;d(A,C) + d(C,B) \end{align} $$ +::: -
- -## Some metrics - -
+## Some metrics +::: columns +::: {.column width="40%"} ```{r out.width = "400px"} knitr::include_graphics("figures/Distance.svg") ``` +::: -
-
- +::: {.column width="60%"} ### Computing $$ @@ -888,8 +854,8 @@ d_m =& |x_A - x_B| + |y_A - y_B| \\ d_c =& \max(|x_A - x_B| , |y_A - y_B|) \\ \end{align} $$ - -
+::: +::: ## Generalizable on a n-dimension space {.smaller} @@ -904,7 +870,6 @@ $$ with $a_i$ and $b_i$ being respectively the value of the $i^{th}$ variable for $A$ and $B$. - $$ \begin{align} d_e =& \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2 } \\ @@ -921,20 +886,20 @@ $$ d^k = \sqrt[k]{\sum_{i=1}^n|a_i - b_i|^k} $$ -- $k=1 \Rightarrow D_m$ Manhatan distance -- $k=2 \Rightarrow D_e$ Euclidean distance -- $k=\infty \Rightarrow D_c$ Chebychev distance +- $k=1 \Rightarrow D_m$ Manhatan distance +- $k=2 \Rightarrow D_e$ Euclidean distance +- $k=\infty \Rightarrow D_c$ Chebychev distance -## Metrics and ultrametrics +## Metrics and ultrametrics -
+::: columns +::: {.column width="40%"} ```{r out.width = "400px"} knitr::include_graphics("figures/ultrametric.svg") ``` -
- -
+::: +::: {.column width="60%"} ### Metric $$ @@ -946,31 +911,29 @@ $$ $$ d(x,z)\leq \max(d(x,y),d(y,z)) $$ - - -
+::: +::: ## Why it is nice to use metrics ? {.flexbox .vcenter} -- A metric induce a metric space -- In a metric space rotations are isometries -- This means that rotations are not changing distances between objects -- Multidimensional scaling (PCA, PCoA, CoA...) are rotations - +- A metric induce a metric space +- In a metric space rotations are isometries +- This means that rotations are not changing distances between objects +- Multidimensional scaling (PCA, PCoA, CoA...) are rotations ## The data set {.flexbox .vcenter} **We analyzed two forest sites in French Guiana** -- Mana : Soil is composed of white sands. +- Mana : Soil is composed of white sands. -- Petit Plateau : Terra firme (firm land). In the Amazon, it corresponds to the area of the forest that is not flooded during high water periods. The terra firme is characterized by old and poor soils. +- Petit Plateau : Terra firme (firm land). In the Amazon, it corresponds to the area of the forest that is not flooded during high water periods. The terra firme is characterized by old and poor soils. **At each site, twice sixteen samples where collected over an hectar** -- Sixteen samples of soil. Each of them is constituted by a mix of five cores of 50g from the 10 first centimeters of soil covering half square meter. +- Sixteen samples of soil. Each of them is constituted by a mix of five cores of 50g from the 10 first centimeters of soil covering half square meter. -- Sixteen samples of litter. Each of them is constituted by the total litter collecter over the same half square meter where soil was sampled +- Sixteen samples of litter. Each of them is constituted by the total litter collecter over the same half square meter where soil was sampled ```{r echo=TRUE} data("guiana.count") @@ -978,7 +941,6 @@ data("guiana.motus") data("guiana.samples") ``` - ## Clean out bad PCR cycle 1 {.flexbox .vcenter .smaller} ```{r echo=TRUE,fig.height=2.5} @@ -986,6 +948,7 @@ s = tag_bad_pcr(guiana.samples$sample,guiana.count) guiana.count.clean = guiana.count[s$keep,] guiana.samples.clean = guiana.samples[s$keep,] ``` + ```{r echo=TRUE} table(s$keep) ``` @@ -1017,7 +980,7 @@ table(s$keep) ## Averaging good PCR replicates (1) {.flexbox .vcenter} ```{r echo=TRUE} -guiana.samples.clean = cbind(guiana.samples.clean,s) +guiana.samples.clean = cbind(guiana.samples.clean,s[rownames(guiana.samples.clean),]) guiana.count.mean = aggregate(decostand(guiana.count.clean,method = "total"), by = list(guiana.samples.clean$sample), @@ -1075,18 +1038,20 @@ xy = xy[,1:2] xy.hellinger = decostand(xy,method = "hellinger") ``` -
- +::: columns +::: {.column width="40%"} ```{r, fig.width=4,fig.height=4} par(bg=NA) plot(xy.hellinger,asp=1) ``` -
-
+::: + +::: {.column width="60%"} ```{r out.width = "400px"} knitr::include_graphics("figures/euclidean_hellinger.svg") ``` -
+::: +::: ## Bray-Curtis distance on relative frequencies @@ -1167,13 +1132,13 @@ plot(guiana.jac.50.pcoa$points[,1:2], ``` -## Principale composante analysis {.flexbox .vcenter} +## Principale composante analysis {.flexbox .vcenter} ```{r echo=TRUE} guiana.hellinger.pca = prcomp(guiana.hellinger.final,center = TRUE, scale. = FALSE) ``` -```{r fig.height=4,fig.width=12} +```{r fig.height=4,fig.width=12} par(mfrow=c(1,3),bg=NA) plot(guiana.euc.pcoa$points[,1:2], col = samples.type, @@ -1191,6 +1156,7 @@ plot(0,type='n',axes=FALSE,ann=FALSE) legend("topleft",legend = levels(samples.type),fill = 1:4,cex=1.2) ``` +````{=html} +```` ## Comparing diversity of the environments @@ -1278,7 +1245,4 @@ boxplot(t(guiana.relfreq.final[,samples.type=="soil.Petit Plateau"]),log="y", names=qs,las=2,col=4,add=TRUE) ``` - - - ## Bibliography diff --git a/Lecture_cache/revealjs/__packages b/Lecture_cache/revealjs/__packages new file mode 100644 index 0000000..a7c0014 --- /dev/null +++ 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