New version of the lecture

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:
+format: 
-  ioslides_presentation: 
+  revealjs:
-    widescreen: true
      smaller: true
-    css: slides.css
+     transition: slide 
-    mathjax: local
+     scrollable: true
-    self_contained: false
+     theme: simple 
-  slidy_presentation: default
+     html-math-method: mathjax
+editor: visual
 ---
 
 ```{r setup, include=FALSE}
@@ -18,15 +18,16 @@ library(knitr)
 library(tidyverse)
 library(kableExtra)
 library(latex2exp)
+library(MetabarSchool)
 
 opts_chunk$set(echo = FALSE,
                cache = TRUE,
                cache.lazy = FALSE)
 ```
 
-
 # Summary
 
+-   The MetabarSchool Package
 -   What do the reading numbers per PCR mean?
 -   Rarefaction vs. relative frequencies
 -   alpha diversity metrics
@@ -34,6 +35,28 @@ opts_chunk$set(echo = FALSE,
 -   multidimentionnal analysis
 -   comparison between datasets
 
+# The MetabarSchool Package
+
+## Installing the package
+
+You need the *devtools* package
+
+```{r eval=FALSE, echo=TRUE}
+install.packages("devtools",dependencies = TRUE)
+```
+
+Then you can install *MetabarSchool*
+
+```{r  eval=FALSE, echo=TRUE}
+devtools::install_git("https://git.metabarcoding.org/MetabarcodingSchool/biodiversity-metrics.git")
+```
+
+You will also need the *vegan* package
+
+```{r eval=FALSE, echo=TRUE}
+install.packages("vegan",dependencies = TRUE)
+```
+
 # The dataset
 
 ## The mock community {.flexbox .vcenter .smaller}
@@ -59,22 +82,21 @@ data("positive.samples")
 
 -   `r nrow(positive.samples)` PCR of the mock community using SPER02 trnL-P6-Loop primers
 
-  - `r length(table(positive.samples$dilution))` dilutions of the mock 
+    -   `r length(table(positive.samples$dilution))` dilutions of the mock community: `r paste0('1/',names(table(positive.samples$dilution)))`
-    community: `r paste0('1/',names(table(positive.samples$dilution)))` 
 
     -   `r as.numeric(table(positive.samples$dilution)[1])` repeats per dilution
 
 ## Loading data
 
 ```{r echo=TRUE}
+library(MetabarSchool)
+
 data("positive.count")
 data("positive.samples")
 data("positive.motus")
 ```
 
-- `positive.count` read count matrix 
+-   `positive.count` read count matrix $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$
-  $`r nrow(positive.count)` \; PCRs \; \times \;  `r ncol(positive.count)` \; MOTUs$
-  
 
 ```{r}
 knitr::kable(positive.count[1:5,1:5],
@@ -85,22 +107,22 @@ knitr::kable(positive.count[1:5,1:5],
 ```
 
 <br>
+
 ```{r echo=TRUE,eval=FALSE}
 positive.count[1:5,1:5]
 ```
 
-
 ## Loading data
 
 ```{r echo=TRUE}
+library(MetabarSchool)
+
 data("positive.count")
 data("positive.samples")
 data("positive.motus")
 ```
 
-- `positive.samples` a `r nrow(positive.samples)` rows `data.frame` of 
+-   `positive.samples` a `r nrow(positive.samples)` rows `data.frame` of `r ncol(positive.samples)` columns describing each PCR
-   `r ncol(positive.samples)` columns describing each PCR
-
 
 ```{r}
 knitr::kable(head(positive.samples,n=3),
@@ -110,21 +132,22 @@ knitr::kable(head(positive.samples,n=3),
 ```
 
 <br>
+
 ```{r echo=TRUE,eval=FALSE}
 head(positive.samples,n=3)
 ```
 
-
 ## Loading data
 
 ```{r echo=TRUE}
+library(MetabarSchool)
+
 data("positive.count")
 data("positive.samples")
 data("positive.motus")
 ```
 
-- `positive.motus` : a `r nrow(positive.motus)` rows `data.frame` of 
+-   `positive.motus` : a `r nrow(positive.motus)` rows `data.frame` of `r ncol(positive.motus)` columns describing each MOTU
-   `r ncol(positive.motus)` columns describing each MOTU
 
 ```{r}
 knitr::kable(head(positive.motus,n=3),
@@ -134,6 +157,7 @@ knitr::kable(head(positive.motus,n=3),
 ```
 
 <br>
+
 ```{r echo=TRUE,eval=FALSE}
 head(positive.motus,n=3)
 ```
@@ -146,7 +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") %>% 
@@ -164,11 +187,9 @@ positive.count = positive.count[,are.not.singleton]
 positive.motus = positive.motus[are.not.singleton,]
 ```
 
-- `positive.count` is now a 
+-   `positive.count` is now a $`r nrow(positive.count)` \; PCRs \; \times \; `r ncol(positive.count)` \; MOTUs$ matrix
-  $`r nrow(positive.count)` \; PCRs \; \times \;  `r ncol(positive.count)` \; MOTUs$
-  matrix 
 
-## Not all the PCR have the number of reads  {.flexbox .vcenter}
+## Not all the PCR have the same number of reads {.flexbox .vcenter}
 
 Despite all standardization efforts
 
@@ -180,9 +201,9 @@ hist(rowSums(positive.count),
      main = "Number of read per PCR")
 ```
 
-<div class="green">
+::: green
 Is it related to the amount of DNA in the extract ?
-</div>
+:::
 
 ## What do the reading numbers per PCR mean? {.smaller}
 
@@ -192,17 +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`
 ```
 
-<div class="red2">
+::: red2
-<center>
+<center>Only `r round((SC/sum(SC)*100)[1],1)`% of the PCR read count variation is explain by dilution</center>
-  Only `r round((SC/sum(SC)*100)[1],1)`% of the PCR read count 
+:::
-  variation is explain by dilution
-</center>
-</div>
 
 ## You must normalize your read counts
 
@@ -212,7 +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
@@ -239,13 +255,19 @@ positive.count.rarefied = rrarefy(positive.count,2000)
 
 ## Rarefying read count (2) {.flexbox .vcenter}
 
-```{r fig.height=3}
+```{r fig.height=4}
 par(mfrow=c(1,2),bg=NA)
 hist(log10(colSums(positive.count)+1),
      main = "Not rarefied",
+     xlim = c(0,6),
+     ylim = c(0,2300),
+     breaks = 30,
      xlab = TeX("$\\log_{10}(reads per MOTUs)$"))
 hist(log10(colSums(positive.count.rarefied)+1),
      main = "Rarefied data",
+     xlim = c(0,6),
+     ylim = c(0,2300),
+     breaks = 30,
      xlab = TeX("$\\log_{10}(reads per MOTUs)$"))
 ```
 
@@ -282,9 +304,7 @@ positive.count.rarefied = positive.count.rarefied[,are.still.present]
 positive.motus.rare = positive.motus[are.still.present,]
 ```
 
-<center>
+<center>positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \times \; `r ncol(positive.count.rarefied)` \; MOTUs$</center>
-positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \times \;  `r ncol(positive.count.rarefied)` \; MOTUs$
-</center>  
 
 ## Why rarefying ? {.vcenter .columns-2}
 
@@ -292,8 +312,7 @@ positive.motus.rare is now a $`r nrow(positive.count.rarefied)` \; PCRs \; \time
 knitr::include_graphics("figures/subsampling.svg")
 ```
 
-<br><br><br><br>
+<br><br><br><br> 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
 
@@ -312,26 +331,21 @@ table(colSums(positive.count.relfreq) == 0)
 
 ## The different types of diversity {.vcenter}
 
-<div style="float: left; width: 40%;">
+::: {style="float: left; width: 40%;"}
 ```{r}
 knitr::include_graphics("figures/diversity.svg")
 ```
-</div>
+:::
 
-<div style="float: left; width: 60%;">
+::: {style="float: left; width: 60%;"}
+<br><br> @Whittaker:10:00 <br><br><br><br>
 
-<br><br>
+-   $\alpha\text{-diversity}$ : Mean diversity per site ($species/site$)
-@Whittaker:10:00
-<br><br><br><br>
 
-- $\alpha-diversity$ : Mean diversity per site ($species/site$)
+-   $\gamma\text{-diversity}$ : Regional biodiversity ($species/region$)
-
-- $\gamma-diversity$ : Regional biodiversity   ($species/region$)
-
-- $\beta-diversity$  : $\beta = \frac{\gamma}{\alpha}$ ($site$)
-
-</div>
 
+-   $\beta\text{-diversity}$ : $\beta = \frac{\gamma}{\alpha}$ ($sites/region$)
+:::
 
 # $\alpha$-diversity
 
@@ -341,7 +355,6 @@ knitr::include_graphics("figures/diversity.svg")
 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)
@@ -352,7 +365,6 @@ kable(environments,
   kable_styling(position = "center")
 ```
 
-
 ## Richness {.flexbox .vcenter}
 
 The actual number of species present in your environement whatever their aboundances
@@ -374,17 +386,15 @@ kable(data.frame(S=S),
 
 ## Gini-Simpson's index {.smaller}
 
-<div style="float: left; width: 60%;">
+::: {style="float: left; width: 60%;"}
-The Simpson's index is the probability of having the same species twice when you randomly select two specimens.
+The Simpson's index is the probability of having the same species twice when you randomly select two specimens. <br> <br>
-<br>
+:::
-<br>
+
-</div>
+::: {style="float: right; width: 40%;"}
-<div style="float: right; width: 40%;">
 $$
 \lambda =\sum _{i=1}^{S}p_{i}^{2}
-$$
+$$ <br>
-<br>
+:::
-</div>
 
 <center>
 
@@ -409,30 +419,22 @@ kable(data.frame(`Gini-Simpson`=GS),
   kable_styling(position = "center")
 ```
 
-## Shanon entropy   {.smaller}
+## Shannon entropy {.smaller}
 
-<div style="float: left; width: 65%;">
+Shannon entropy is based on information theory:
-Shanon entropy is based on information theory. 
 
-Let $X$ be a uniformly distributed random variable with values in $A$ 
+<center>$H^{\prime }=-\sum _{i=1}^{S}p_{i}\log p_{i}$</center>
 
+if $A$ is a community where every species are equally represented then $$
+H(A) = \log|A|
 $$
-H(X) = \log|A|
-$$
-
-<br>
-</div>
-<div style="float: right; width: 35%;">
-$$
-H^{\prime }=-\sum _{i=1}^{S}p_{i}\log p_{i}
-$$
-<br>
-</div>
 
 <center>
+
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/alpha_diversity.svg")
 ```
+
 </center>
 
 ```{r echo=TRUE}
@@ -440,7 +442,7 @@ H = - rowSums(environments * log(environments),na.rm = TRUE)
 ```
 
 ```{r}
-kable(data.frame(`Shanon index`=H),
+kable(data.frame(`Shannon index`=H),
              format="html",
              align = 'rr') %>% 
   kable_styling(position = "center")
@@ -448,23 +450,24 @@ kable(data.frame(`Shanon index`=H),
 
 ## Hill's number {.smaller}
 
-<div style="float: left; width: 50%;">
+::: {style="float: left; width: 50%;"}
-As :
+As : $$
-$$
+H(A) = \log|A| \;\Rightarrow\; ^1D = e^{H(A)}
-H(X) = \log|A| \;\Rightarrow\; ^2D = e^{H(X)}
+$$ <br>
-$$
+:::
-<br>
+
-</div>
+::: {style="float: right; width: 50%;"}
-<div 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.
-where $^2D$ is the theoretical number of species in a evenly distributed community that would have the same Shanon's entropy than ours.
+:::
-</div>
 
 <center>
-<BR>
+
-<BR>
+<BR> <BR>
+
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/alpha_diversity.svg")
 ```
+
 </center>
 
 ```{r echo=TRUE}
@@ -483,7 +486,7 @@ 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 $q \longrightarrow 1\;,\; ^q\log(x) \longrightarrow \log(x)$
@@ -491,14 +494,14 @@ The function is not defined for $q=1$ but when $q \longrightarrow 1\;,\; ^q\log(
 $$
 ^q\log(x) = \left\{ 
              \begin{align}
-               \log(x),& \text{if } x = 1\\
+               \log(x),& \text{if } q = 1\\
-               \frac{x^{(1-q)}}{1-q},& \text{otherwise}
+               \frac{x^{(1-q)}-1}{1-q},& \text{otherwise}
              \end{align}
            \right.
 $$
 
 ```{r echo=TRUE, eval=FALSE}
-log.q = function(x,q=1) {
+log_q = function(x,q=1) {
   if (q==1)
     log(x)
   else 
@@ -506,6 +509,28 @@ log.q = function(x,q=1) {
 }
 ```
 
+## Impact of $q$ on the `log_q` function
+
+```{r}
+layout(matrix(c(1,1,2),nrow=1))
+qs = seq(0,5,by=1)
+x = seq(0.001,1,length.out = 100)
+plot(x,log_q(x,0),lty=2,lwd=3,col=0,type="l",
+     ylab = TeX("$^q\\log$"),
+     ylim=c(-10,0))
+
+for (i in seq_along(qs)) {
+  points(x,log_q(x,qs[i]),lty=1,lwd=1,col=i,type="l")
+}
+
+points(x,log(x),lty=2,lwd=3,col="red",type="l")
+
+plot(0,type='n',axes=FALSE,ann=FALSE)
+legend("topleft",legend = qs,fill =  seq_along(qs),cex=1.5)
+
+
+```
+
 ## And its inverse function {.flexbox .vcenter}
 
 $$
@@ -516,8 +541,9 @@ $$
              \end{align}
            \right.
 $$
+
 ```{r echo=TRUE, eval=FALSE}
-exp.q = function(x,q=1) {
+exp_q = function(x,q=1) {
   if (q==1)
     exp(x)
   else
@@ -525,41 +551,41 @@ exp.q = function(x,q=1) {
 }
 ```
 
-## Generalised Shanon entropy
+## Generalised Shannon entropy
 
 $$
-^qH = - \sum_{i=1}^S pi \times ^q\log pi
+^qH = - \sum_{i=1}^S p_i \; ^q\log p_i
 $$
 
 ```{r echo=TRUE, eval=FALSE}
-H.q = function(x,q=1) {
+H_q = function(x,q=1) {
-  sum(x * log.q(1/x,q),na.rm = TRUE)
+  sum(x * log_q(1/x,q),na.rm = TRUE)
 }
 ```
 
-
 and generalized the previously presented Hill's number
 
 $$
 ^qD=^qe^{^qH}
 $$
+
 ```{r echo=TRUE, eval=FALSE}
- D.q = function(x,q=1) {
+ D_q = function(x,q=1) {
-  exp.q(H.q(x,q),q)
+  exp_q(H_q(x,q),q)
 }
 ```
 
 ## Biodiversity spectrum (1) {.flexbox .vcenter}
 
 ```{r echo=TRUE, eval=FALSE}
-H.spectrum = function(x,q=1) {
+H_spectrum = function(x,q=1) {
-  sapply(q,function(Q) H.q(x,Q))
+  sapply(q,function(Q) H_q(x,Q))
 }
 ```
 
 ```{r echo=TRUE, eval=FALSE}
-D.spectrum = function(x,q=1) {
+D_spectrum = function(x,q=1) {
-  sapply(q,function(Q) D.q(x,Q))
+  sapply(q,function(Q) D_q(x,Q))
 }
 ```
 
@@ -568,8 +594,8 @@ D.spectrum = function(x,q=1) {
 ```{r echo=TRUE,warning=FALSE,error=FALSE}
 library(MetabarSchool)
 qs = seq(from=0,to=3,by=0.1)
-environments.hq = apply(environments,MARGIN = 1,H.spectrum,q=qs)
+environments.hq = apply(environments,MARGIN = 1,H_spectrum,q=qs)
-environments.dq = apply(environments,MARGIN = 1,D.spectrum,q=qs)
+environments.dq = apply(environments,MARGIN = 1,D_spectrum,q=qs)
 ```
 
 ```{r}
@@ -593,7 +619,7 @@ abline(v=c(0,1,2),lty=2,col=4:6)
 
 -   $^0H(X) = S - 1$ : the richness minus one.
 
-- $^1H(X) = H^{\prime}$ : the Shanon's entropy.
+-   $^1H(X) = H^{\prime}$ : the Shannon's entropy.
 
 -   $^2H(X) = 1 - \lambda$ : Gini-Simpson's index.
 
@@ -606,18 +632,20 @@ abline(v=c(0,1,2),lty=2,col=4:6)
 -   $^2D(X) = 1 / \lambda$ : The number of species in an even community having the same Gini-Simpson's index.
 
 <br>
+
 <center>
+
 $q$ 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
 
 </center>
 
 ## Biodiversity spectrum of the mock community
 
 ```{r echo=TRUE}
-H.mock = H.spectrum(plants.16$dilution,qs)
+H.mock = H_spectrum(plants.16$dilution,qs)
-D.mock = D.spectrum(plants.16$dilution,qs)
+D.mock = D_spectrum(plants.16$dilution,qs)
 ```
 
 ```{r}
@@ -640,9 +668,10 @@ abline(v=c(0,1,2),lty=2,col=4:6)
 ```{r echo=TRUE}
 positive.H = apply(positive.count.relfreq,
                    MARGIN = 1,
-                   FUN = H.spectrum,
+                   FUN = H_spectrum,
                    q=qs)
 ```
+
 ```{r}
 par(bg=NA) 
 boxplot(t(positive.H),
@@ -654,7 +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],
@@ -672,7 +700,7 @@ positive.H.means = rowMeans(positive.H)
 ```{r echo=TRUE}
 positive.D = apply(positive.count.relfreq,
                    MARGIN = 1,
-                   FUN = D.spectrum,
+                   FUN = D_spectrum,
                    q=qs)
 ```
 
@@ -709,7 +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}
@@ -720,7 +747,7 @@ positive.clean.count.relfreq = decostand(positive.clean.count,
 
 positive.clean.H = apply(positive.clean.count.relfreq,
                          MARGIN = 1,
-                         FUN = H.spectrum,
+                         FUN = H_spectrum,
                          q=qs)
 ```
 
@@ -738,7 +765,7 @@ points(H.mock,col="red",type="l")
 ```{r echo=TRUE}
 positive.clean.D = apply(positive.clean.count.relfreq,
                          MARGIN = 1,
-                         FUN = D.spectrum,
+                         FUN = D_spectrum,
                          q=qs)
 ```
 
@@ -753,16 +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}
-<center>
+
-A dissimilarity index $d(A,B)$ is a numerical measurement 
+<center>A dissimilarity index $d(A,B)$ is a numerical measurement <br> of how far apart objects $A$ and $B$ are.</center>
-<br>
-of how far apart  objects $A$ and $B$ are.
-</center>
 
 ### Properties
 
@@ -794,17 +816,15 @@ $$
 
 ## Metrics or distances
 
-<div style="float: left; width: 50%;">
+::: {style="float: left; width: 50%;"}
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/metric.svg")
 ```
-</div>
+:::
-
-<div style="float: right; width: 50%;">
 
+::: {style="float: right; width: 50%;"}
 A metric is a dissimilarity index verifying the *subadditivity* also named *triangle inequality*
 
-
 $$
 \begin{align}
 d(A,B) \geqslant& 0 \\
@@ -813,20 +833,18 @@ d(A,B) =& \;0 \iff A = B \\
 d(A,B) \leqslant& \;d(A,C) + d(C,B)
 \end{align}
 $$
-
+:::
-</div>
 
 ## Some metrics
 
-<div style="float: left; width: 50%;">
+::: columns
-
+::: {.column width="40%"}
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/Distance.svg")
 ```
+:::
 
-</div>
+::: {.column width="60%"}
-<div style="float: right; width: 50%;">
-
 ### Computing
 
 $$
@@ -836,8 +854,8 @@ d_m =& |x_A - x_B| + |y_A - y_B| \\
 d_c =& \max(|x_A - x_B| , |y_A - y_B|) \\
 \end{align}
 $$
-
+:::
-</div>
+:::
 
 ## Generalizable on a n-dimension space {.smaller}
 
@@ -852,7 +870,6 @@ $$
 
 with $a_i$ and $b_i$ being respectively the value of the $i^{th}$ variable for $A$ and $B$.
 
-
 $$
 \begin{align}
 d_e =& \sqrt{\sum_{i=1}^{n}(a_i - b_i)^2 } \\
@@ -875,14 +892,14 @@ $$
 
 ## Metrics and ultrametrics
 
-<div style="float: left; width: 50%;">
+::: columns
+::: {.column width="40%"}
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/ultrametric.svg")
 ```
-</div>
+:::
-
-<div style="float: right; width: 50%;">
 
+::: {.column width="60%"}
 ### Metric
 
 $$
@@ -894,9 +911,8 @@ $$
 $$
 d(x,z)\leq \max(d(x,y),d(y,z))
 $$
-
+:::
-
+:::
-</div>
 
 ## Why it is nice to use metrics ? {.flexbox .vcenter}
 
@@ -905,7 +921,6 @@ $$
 -   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**
@@ -926,7 +941,6 @@ data("guiana.motus")
 data("guiana.samples")
 ```
 
-
 ## Clean out bad PCR cycle 1 {.flexbox .vcenter .smaller}
 
 ```{r echo=TRUE,fig.height=2.5}
@@ -934,6 +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)
 ```
@@ -965,7 +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),
@@ -1023,18 +1038,20 @@ xy = xy[,1:2]
 xy.hellinger = decostand(xy,method = "hellinger")
 ```
 
-<div style="float: left; width: 50%;">
+::: columns
-
+::: {.column width="40%"}
 ```{r, fig.width=4,fig.height=4}
 par(bg=NA)
 plot(xy.hellinger,asp=1)
 ```
-</div>
+:::
-<div style="float: right; width: 50%;">
+
+::: {.column width="60%"}
 ```{r out.width = "400px"}
 knitr::include_graphics("figures/euclidean_hellinger.svg")
 ```
-</div>
+:::
+:::
 
 ## Bray-Curtis distance on relative frequencies
 
@@ -1051,15 +1068,15 @@ BC_{jk}=\frac{\sum _{i=1}^{p}(N_{ij} - min(N_{ij},N_{ik}) + (N_{ik} - min(N_{ij}
 $$
 
 $$
-BC_{jk}=\frac{\sum _{i=1}^{p}]N_{ij} - N_{ik}|}{\sum _{i=1}^{p}N_{ij}+\sum _{i=1}^{p}N_{ik}} 
+BC_{jk}=\frac{\sum _{i=1}^{p}|N_{ij} - N_{ik}|}{\sum _{i=1}^{p}N_{ij}+\sum _{i=1}^{p}N_{ik}} 
 $$
 
 $$
-BC_{jk}=\frac{\sum _{i=1}^{p}]N_{ij} - N_{ik}|}{1+1}
+BC_{jk}=\frac{\sum _{i=1}^{p}|N_{ij} - N_{ik}|}{1+1}
 $$
 
 $$
-BC_{jk}=\frac{1}{2}\sum _{i=1}^{p}]N_{ij} - N_{ik}|
+BC_{jk}=\frac{1}{2}\sum _{i=1}^{p}|N_{ij} - N_{ik}|
 $$
 
 ## Principale coordinate analysis (1) {.flexbox .vcenter}
@@ -1086,7 +1103,7 @@ plot(guiana.bc.pcoa$points[,1:2],
      xlab="Axis 1",
      ylab="Axis 2",
      main = "Bray Curtis on Rel. Freqs")
-plot(guiana.euc.pcoa$points[,1:2],
+plot(guiana.euc.pcoa$points[,1],-guiana.euc.pcoa$points[,2],
      col = samples.type,
      asp = 1,
      xlab="Axis 1",
@@ -1100,7 +1117,7 @@ plot(guiana.jac.1.pcoa$points[,1:2],
      xlab="Axis 1",
      ylab="Axis 2",
      main = "Jaccard on presence (0.1%)")
-plot(guiana.jac.10.pcoa$points[,1:2],
+plot(-guiana.jac.10.pcoa$points[,1],guiana.jac.10.pcoa$points[,2],
      col = samples.type,
      asp = 1,
      xlab="Axis 1",
@@ -1139,12 +1156,66 @@ plot(0,type='n',axes=FALSE,ann=FALSE)
 legend("topleft",legend = levels(samples.type),fill = 1:4,cex=1.2)
 ```
 
+````{=html}
+<!---
+## Computation of norms 
+
+```{r guiana_norm, echo=TRUE}
+guiana.n1.dist = norm(guiana.relfreq.final,l=1)
+guiana.n2.dist = norm(guiana.relfreq.final^(1/2),l=2)
+guiana.n3.dist = norm(guiana.relfreq.final^(1/3),l=3)
+guiana.n4.dist = norm(guiana.relfreq.final^(1/100),l=100)
+```
+
+## pCoA on norms 
+
+```{r dependson="guiana_norm"}
+guiana.n1.pcoa  = cmdscale(guiana.n1.dist,k=3,eig = TRUE)
+guiana.n2.pcoa  = cmdscale(guiana.n2.dist,k=3,eig = TRUE)
+guiana.n3.pcoa  = cmdscale(guiana.n3.dist,k=3,eig = TRUE)
+guiana.n4.pcoa  = cmdscale(guiana.n4.dist,k=3,eig = TRUE)
+```
+
+```{r}
+par(mfrow=c(2,3),bg=NA)
+plot(guiana.n1.pcoa$points[,1],guiana.n1.pcoa$points[,2],
+     col = samples.type,
+     asp = 1,
+     xlab="Axis 1",
+     ylab="Axis 2",
+     main = "Norm 1 on Hellinger")
+plot(guiana.n2.pcoa$points[,1],-guiana.n2.pcoa$points[,2],
+     col = samples.type,
+     asp = 1,
+     xlab="Axis 1",
+     ylab="Axis 2",
+     main = "Norm 2 on Hellinger")
+plot(0,type='n',axes=FALSE,ann=FALSE)
+legend("topleft",legend = levels(samples.type),fill = 1:4,cex=1.2)
+plot(-guiana.n3.pcoa$points[,1],-guiana.n3.pcoa$points[,2],
+     col = samples.type,
+     asp = 1,
+     xlab="Axis 1",
+     ylab="Axis 2",
+     main = "Norm 3 on Hellinger")
+plot(-guiana.n4.pcoa$points[,1],-guiana.n4.pcoa$points[,2],
+     col = samples.type,
+     asp = 1,
+     xlab="Axis 1",
+     ylab="Axis 2",
+     main = "Norm 4 on Hellinger")
+
+```
+
+--->
+````
+
 ## Comparing diversity of the environments
 
 ```{r}
 guiana.relfreq.final = apply(guiana.relfreq.final,
                              MARGIN = 1,
-                             FUN = H.spectrum,
+                             FUN = H_spectrum,
                              q=qs)
 ```
 
@@ -1174,7 +1245,4 @@ boxplot(t(guiana.relfreq.final[,samples.type=="soil.Petit Plateau"]),log="y",
         names=qs,las=2,col=4,add=TRUE)
 ```
 
-
-
-
 ## Bibliography

Lecture_cache/revealjs/__packages

@@ -0,0 +1,17 @@
+knitr
+tidyverse
+ggplot2
+tibble
+tidyr
+readr
+purrr
+dplyr
+stringr
+forcats
+lubridate
+kableExtra
+latex2exp
+MetabarSchool
+permute
+lattice
+vegan