Walkthrough — CCA, RDA, pRDA, db-RDA, LDA com vegan.

Walkthrough do código — Aula 5: Ordenações restritas: CCA, RDA, pRDA, db-RDA, LDA

Walkthrough do código usado em sala. Os comandos não são executados pelo renderizador — copie e cole no seu R local. Você pode baixar o script .R original e os dados em Aula 5.

# Wed May  8 22:12:42 2019 ------------------------------

#--Aula quinta : ordenações restritas
library(vegan)
library(ade4)

#### Aula 13 - CANONICAL CORRESPONDENCE ANALYSIS (CCA)

#13.1 - Preparation of data
#Import the data from CSV files
spe <- read.csv("DoubsSpe.csv", row.names=1)
env <- read.csv("DoubsEnv.csv", row.names=1)

data(doubs)
spa <- doubs$xy

head(spa)
#Remove empty site 8
spe <- spe[-8, ]
env <- env[-8, ]
spa <- spa[-8, ]

#Remove the 'das' variable from the env dataset
env <- env[, -1]
#Recode the slope variable (pen) into a factor (qualitative) 
#variable (to show how these are handled in the ordinations)
pen2 <- rep("very_steep", nrow(env))
pen2[env$pen <= quantile(env$pen)[4]] = "steep"
pen2[env$pen <= quantile(env$pen)[3]] = "moderate"
pen2[env$pen <= quantile(env$pen)[2]] = "low"
pen2 <- factor(pen2, levels=c("low", "moderate", "steep", "very_steep"))
table(pen2)
#Create an env2 data frame with slope as a qualitative variable
env$pen <- pen2
head(env)

#13.2 - CCA
head(spe)  #Utilizar dados n?o transformados
spe.cca <- cca(spe ~ ., env)
spe.cca
summary(spe.cca) #Scaling 2 (default)
summary(spe.cca)$species
summary(spe.cca)$sites
summary(spe.cca)$constraints
summary(spe.cca)$biplot

summary(spe.cca, scaling=1) #Scaling 1
summary(spe.cca, scaling=1)$species
summary(spe.cca, scaling=1)$sites
summary(spe.cca, scaling=1)$constraints
summary(spe.cca, scaling=1)$biplot

#13.3 - CCA triplots (using lc site scores)
#Scaling 1: A proje??o de um objeto em ?ngulo reto em uma vari?vel aproxima a posi??o do objeto ao longo da vari?vel.
plot(spe.cca, scaling=1, display=c("sp","lc","cn"), main="Triplot CCA spe ~ env2 - scaling 1")
plot(spe.cca, scaling=1, display=c("lc", "cn"), main="Biplot CCA spe ~ env2 - scaling 1") #without species

#Default scaling 2: O ?timo e uma esp?cie ao longo das vari?veis ambientais pode ser obtido projetando-a em ?ngulo reto em uma vari?vel.
plot(spe.cca, display=c("sp","lc","cn"), main="Triplot CCA spe ~ env2 - scaling 2")
plot(spe.cca, scaling=2, display=c("sp", "cn"), main="Biplot CCA spe ~ env2 - scaling 2") #without sites

#13.4 - Permutation tests of CCA results
anova(spe.cca, step=1000) #Permutation test of the overall analysis
set.seed(1001)
anova(spe.cca, by="axis", step=1000) #Permutation test of each axis

anova(spe.cca, by="margin", step=1000) #Permutation test of variables

#14.3 - Recuperando os R^2 ajustados
RsquareAdj(spe.cca) #Adjusted R^2

#13.5 - Three-dimensional interactive ordination plots
library(vegan3d)
spe.hel <- decostand(spe, "hell")
#Scaling=1
#Plot of the sites only (wa scores)
ordirgl(spe.cca, type="t", scaling=1)
#Plot the sites (wa scores) with a clustering result. Colour sites according to cluster membership
gr <- cutree(hclust(vegdist(spe.hel, "euc"), "ward.D"), 4)
ordirgl(spe.cca, type="t", scaling=1, ax.col="black", col=gr+1)
# Connect sites to cluster centroids
orglspider(spe.cca, gr, scaling=1)

#Scaling=2
ordirgl(spe.cca, type="t", scaling=2)
orgltext(spe.cca.pars, display="species", type="t", scaling=2, col="cyan")
#Plot species groups (Jaccard similarity, useable in R mode)
gs <- cutree(hclust(vegdist(t(spe), method="jaccard"), "ward"), 4)
ordirgl(spe.cca.pars, display="species", type="t", col=gs+1)

#-------------------------------------------------------------------------------

#### Aula 14 - RDA

#Hellinger-transform the species dataset
spe.hel <- decostand(spe, "hellinger")

#14.2 - RDA using Vegan
#RDA of the Hellinger-transformed fish species data, constrained by all the environmental variables contained in env2
spe.rda <- rda(spe.hel~., env) # Observe the shortcut formula
summary(spe.rda) # Scaling 2 (default) #Here the default choices are used, i.e. scale=FALSE (RDA on a covariance matrix) and scaling=2.

#How to obtain canonical coefficients from an rda() object. Canonical coefficients are the equivalent of regression coefficients 
#for each explanatory variable on each canonical axis.
coef(spe.rda)

#14.3 - Recuperando os R^2 ajustados
RsquareAdj(spe.rda) #Adjusted R^2

#The numerical output shows that the first two canonical axes explain together 56.1% of the total variance of
#the data, the first axis alone explaining 45.4%. These are unadjusted values, however. Since adj R2 = 0.5224,
#the percentages of accumulated constrained eigenvalues show that the first axis alone explains
#0.5224 ? 0.6242 = 0.326 or 32.6% variance, and the two first 0.5224 ? 0.7712 = 0.4029 or 40.3% variance

#14.4 - Triplots of the rda results. #Site scores as linear combinations of the environmental variables (lc)
#Scaling 1
plot(spe.rda, scaling=1, display=c("sp", "lc", "cn"),   main="Triplot RDA spe.hel ~ env2 - scaling 1 - lc scores")
arrows(0, 0, spe.sc[, 1], spe.sc[, 2], length=0, lty=1, col="red")

#Scaling 2
plot(spe.rda, display=c("sp", "lc", "cn"), main="Triplot RDA spe.hel ~ env2 - scaling 2 - lc scores")
arrows(0, 0, spe2.sc[,1], spe2.sc[,2], length=0, lty=1, col="red")

#OBS: Em ambas, a proje??o dos objetos (ua) em ?ngulo reto sobre uma vari?vel resposta (sp) ou explanat?ria (env)
#aproxima a posi??o do objeto ao longo daquela vari?vel.

#14.5 - Global test of the RDA result
anova.cca(spe.rda, step=1000)
#Tests of all canonical axes
anova.cca(spe.rda, by="axis", step=1000)
anova.cca(spe.rda, by="margin", step=1000)
#Apply Kaiser-Guttman criterion to residual axes
spe.rda$CA$eig[spe.rda$CA$eig > mean(spe.rda$CA$eig)]  
#Of course, given that these tests are available, it is useless to apply other criteria
#like the broken-stick or Kaiser?Guttman?s criterion to canonical axes. These could
#be applied to the residual, unconstrained axes, however.

#-------------------------------------------------------------------------------

#### Aula 15 - Partial RDA
#15.1 - Preparation of data

#Create two subsets of explanatory variables
#Physiography (upstream-downstream gradient)
envtopo <- env[, c(1:3)]
names(envtopo)
#Water quality
envchem <- env[, c(4:9)]
names(envchem)

#Hellinger-transform the species dataset
spe.hel <- decostand(doubs$fish[-8,], "hellinger")

#15.2 - Explanation of fraction labels
windows(title="Symbols of variation partitioning fractions",12,4)
par(mfrow=c(1,3))
showvarparts(2) # Two explanatory matrices
showvarparts(3) # Three explanatory matrices
showvarparts(4) # Four explanatory matrices

#15.3 - Variation partitioning with all explanatory variables
spe.part.all <- varpart(spe.hel, envchem, envtopo)
spe.part.all
windows(title="Variation partitioning - all variables")
plot(spe.part.all, digits=2)  # Plot of the partitioning results

#Tests of all testable fractions
#Test of fractions [a+b]
anova.cca(rda(spe.hel, envchem), step=1000)
#Test of fractions [b+c]
anova.cca(rda(spe.hel, envtopo), step=1000)
#Test of fractions [a+b+c]
env<- cbind(envchem, envtopo)
anova.cca(rda(spe.hel, env), step=1000)
#Test of fraction [a]
anova.cca(rda(spe.hel, envchem, envtopo), step=1000)
#Test of fraction [c]
anova.cca(rda(spe.hel, envtopo, envchem), step=1000)

#MEMs
library(adespatial)
mems_doubs <- dbmem(spa, MEM.autocor = "positive")
mems.doubs <- data.frame(mems_doubs$MEM1, mems_doubs$MEM2, mems_doubs$MEM3, mems_doubs$MEM4, mems_doubs$MEM5, mems_doubs$MEM6, mems_doubs$MEM7)

vp <- varpart(spe.hel, env, mems.doubs)
plot(vp)

# Linear discriminant analysis (LDA) ==============================

# Ward clustering result of Hellinger-transformed species data, 
# cut into 4 groups
gr <- cutree(hclust(vegdist(spe.hel, "euc"), "ward.D2"), k = 4)

# Environmental matrix with only 3 variables (ele, oxy and bod)
env.pars2 <- as.matrix(env[, c(3, 10, 11)])

# Verify multivariate homogeneity of within-group covariance
# matrices using the betadisper() function {vegan}
env.pars2.d1 <- dist(env.pars2)
(env.MHV <- betadisper(env.pars2.d1, gr))
anova(env.MHV)
permutest(env.MHV)  # Permutational test

# Log transform ele and bod
env.pars3 <- cbind(log(env$slo), env$oxy, log(env$bdo))
colnames(env.pars3) <- c("ele.ln", "oxy", "bod.ln") 
rownames(env.pars3) <- rownames(env)
env.pars3.d1 <- dist(env.pars3)
(env.MHV2 <- betadisper(env.pars3.d1, gr))
permutest(env.MHV2)

# Preliminary test :  do the means of the explanatory variable 
# differ among groups?
# Compute Wilk'S lambda test
# First way: with function Wilks.test() of package rrcov, χ2 test
library(rrcov)
library(MASS)
Wilks.test(env.pars3, gr)
# Second way: with function manova() of stats, which uses
#             an F-test approximation
lw <-  manova(env.pars3 ~ as.factor(gr))
summary(lw, test = "Wilks")

## Computation of LDA - identification functions (on unstandardized 
## variables)

env.pars3.df <- as.data.frame(env.pars3)
(spe.lda <- lda(gr ~ ele.ln + oxy + bod.ln, data = env.pars3.df))
# Alternate coding without formula interface :  
#    spe.lda <- lda(env.pars3.df, gr)
# The result object contains the information necessary to interpret 
# the LDA

# Display the group means for the 3 variables
spe.lda$means

# Extract the unstandardized identification functions (matrix C, 
# eq. 11.33 in Legendre and Legendre 2012)
(C <- spe.lda$scaling)

# Classification of two new objects (identification)
# A new object is created with two sites: 
#     (1) ln(ele) = 6.8, oxygen = 9 and ln(bod) = 0.8 
# and (2) ln(ele) = 5.5, oxygen = 10 and ln(bod) = 1.0
newo <- data.frame(c(6.8, 5.5), c(9, 10), c(0.8, 1))
colnames(newo) <- colnames(env.pars3)
newo
(predict.new <- predict(spe.lda, newdata = newo))

## Computation of LDA - discrimination functions (on standardized 
## variables)

env.pars3.sc <- as.data.frame(scale(env.pars3.df))
spe.lda2 <- lda(gr ~ ., data = env.pars3.sc)

# Display the group means for the 3 variables
spe.lda2$means

# Extract the classification functions
(C2 <- spe.lda2$scaling)

# Compute the canonical eigenvalues
spe.lda2$svd^2

# Position the objects in the space of the canonical variates
(Fp2 <- predict(spe.lda2)$x)
# alternative way :  Fp2 <- as.matrix(env.pars3.sc) %*% C2

# Classification of the objects
(spe.class2 <- predict(spe.lda2)$class)

# Posterior probabilities of the objects to belong to the groups
# (rounded for easier interpretation)
(spe.post2 <- round(predict(spe.lda2)$posterior, 2))

# Contingency table of prior versus predicted classifications
(spe.table2 <- table(gr, spe.class2))

# Proportion of correct classification (classification success)
diag(prop.table(spe.table2, 1))

# Plot the LDA results using the homemade function plot.lda()

plot(spe.lda2, 
         groups = gr, 
         plot.sites = 2, 
         plot.centroids = 1, 
         mul.coef = 2.35)

--- title: "Walkthrough — CCA, RDA, pRDA, db-RDA, LDA com vegan." subtitle: "Walkthrough do código — Aula 5: Ordenações restritas: CCA, RDA, pRDA, db-RDA, LDA" format: html: toc: true toc-depth: 3 code-tools: true code-overflow: scroll execute: eval: false --- ::: {.callout-note appearance="minimal"} Walkthrough do código usado em sala. Os comandos **não são executados** pelo renderizador — copie e cole no seu R local. Você pode baixar o script `.R` original e os dados em [Aula 5](./). ::: ```{r} # Wed May 8 22:12:42 2019 ------------------------------ #--Aula quinta : ordenações restritas library(vegan) library(ade4) #### Aula 13 - CANONICAL CORRESPONDENCE ANALYSIS (CCA) #13.1 - Preparation of data #Import the data from CSV files spe <- read.csv("DoubsSpe.csv", row.names=1) env <- read.csv("DoubsEnv.csv", row.names=1) data(doubs) spa <- doubs$xy head(spa) #Remove empty site 8 spe <- spe[-8, ] env <- env[-8, ] spa <- spa[-8, ] #Remove the 'das' variable from the env dataset env <- env[, -1] #Recode the slope variable (pen) into a factor (qualitative) #variable (to show how these are handled in the ordinations) pen2 <- rep("very_steep", nrow(env)) pen2[env$pen <= quantile(env$pen)[4]] = "steep" pen2[env$pen <= quantile(env$pen)[3]] = "moderate" pen2[env$pen <= quantile(env$pen)[2]] = "low" pen2 <- factor(pen2, levels=c("low", "moderate", "steep", "very_steep")) table(pen2) #Create an env2 data frame with slope as a qualitative variable env$pen <- pen2 head(env) #13.2 - CCA head(spe) #Utilizar dados n?o transformados spe.cca <- cca(spe ~ ., env) spe.cca summary(spe.cca) #Scaling 2 (default) summary(spe.cca)$species summary(spe.cca)$sites summary(spe.cca)$constraints summary(spe.cca)$biplot summary(spe.cca, scaling=1) #Scaling 1 summary(spe.cca, scaling=1)$species summary(spe.cca, scaling=1)$sites summary(spe.cca, scaling=1)$constraints summary(spe.cca, scaling=1)$biplot #13.3 - CCA triplots (using lc site scores) #Scaling 1: A proje??o de um objeto em ?ngulo reto em uma vari?vel aproxima a posi??o do objeto ao longo da vari?vel. plot(spe.cca, scaling=1, display=c("sp","lc","cn"), main="Triplot CCA spe ~ env2 - scaling 1") plot(spe.cca, scaling=1, display=c("lc", "cn"), main="Biplot CCA spe ~ env2 - scaling 1") #without species #Default scaling 2: O ?timo e uma esp?cie ao longo das vari?veis ambientais pode ser obtido projetando-a em ?ngulo reto em uma vari?vel. plot(spe.cca, display=c("sp","lc","cn"), main="Triplot CCA spe ~ env2 - scaling 2") plot(spe.cca, scaling=2, display=c("sp", "cn"), main="Biplot CCA spe ~ env2 - scaling 2") #without sites #13.4 - Permutation tests of CCA results anova(spe.cca, step=1000) #Permutation test of the overall analysis set.seed(1001) anova(spe.cca, by="axis", step=1000) #Permutation test of each axis anova(spe.cca, by="margin", step=1000) #Permutation test of variables #14.3 - Recuperando os R^2 ajustados RsquareAdj(spe.cca) #Adjusted R^2 #13.5 - Three-dimensional interactive ordination plots library(vegan3d) spe.hel <- decostand(spe, "hell") #Scaling=1 #Plot of the sites only (wa scores) ordirgl(spe.cca, type="t", scaling=1) #Plot the sites (wa scores) with a clustering result. Colour sites according to cluster membership gr <- cutree(hclust(vegdist(spe.hel, "euc"), "ward.D"), 4) ordirgl(spe.cca, type="t", scaling=1, ax.col="black", col=gr+1) # Connect sites to cluster centroids orglspider(spe.cca, gr, scaling=1) #Scaling=2 ordirgl(spe.cca, type="t", scaling=2) orgltext(spe.cca.pars, display="species", type="t", scaling=2, col="cyan") #Plot species groups (Jaccard similarity, useable in R mode) gs <- cutree(hclust(vegdist(t(spe), method="jaccard"), "ward"), 4) ordirgl(spe.cca.pars, display="species", type="t", col=gs+1) #------------------------------------------------------------------------------- #### Aula 14 - RDA #Hellinger-transform the species dataset spe.hel <- decostand(spe, "hellinger") #14.2 - RDA using Vegan #RDA of the Hellinger-transformed fish species data, constrained by all the environmental variables contained in env2 spe.rda <- rda(spe.hel~., env) # Observe the shortcut formula summary(spe.rda) # Scaling 2 (default) #Here the default choices are used, i.e. scale=FALSE (RDA on a covariance matrix) and scaling=2. #How to obtain canonical coefficients from an rda() object. Canonical coefficients are the equivalent of regression coefficients #for each explanatory variable on each canonical axis. coef(spe.rda) #14.3 - Recuperando os R^2 ajustados RsquareAdj(spe.rda) #Adjusted R^2 #The numerical output shows that the first two canonical axes explain together 56.1% of the total variance of #the data, the first axis alone explaining 45.4%. These are unadjusted values, however. Since adj R2 = 0.5224, #the percentages of accumulated constrained eigenvalues show that the first axis alone explains #0.5224 ? 0.6242 = 0.326 or 32.6% variance, and the two first 0.5224 ? 0.7712 = 0.4029 or 40.3% variance #14.4 - Triplots of the rda results. #Site scores as linear combinations of the environmental variables (lc) #Scaling 1 plot(spe.rda, scaling=1, display=c("sp", "lc", "cn"), main="Triplot RDA spe.hel ~ env2 - scaling 1 - lc scores") arrows(0, 0, spe.sc[, 1], spe.sc[, 2], length=0, lty=1, col="red") #Scaling 2 plot(spe.rda, display=c("sp", "lc", "cn"), main="Triplot RDA spe.hel ~ env2 - scaling 2 - lc scores") arrows(0, 0, spe2.sc[,1], spe2.sc[,2], length=0, lty=1, col="red") #OBS: Em ambas, a proje??o dos objetos (ua) em ?ngulo reto sobre uma vari?vel resposta (sp) ou explanat?ria (env) #aproxima a posi??o do objeto ao longo daquela vari?vel. #14.5 - Global test of the RDA result anova.cca(spe.rda, step=1000) #Tests of all canonical axes anova.cca(spe.rda, by="axis", step=1000) anova.cca(spe.rda, by="margin", step=1000) #Apply Kaiser-Guttman criterion to residual axes spe.rda$CA$eig[spe.rda$CA$eig > mean(spe.rda$CA$eig)] #Of course, given that these tests are available, it is useless to apply other criteria #like the broken-stick or Kaiser?Guttman?s criterion to canonical axes. These could #be applied to the residual, unconstrained axes, however. #------------------------------------------------------------------------------- #### Aula 15 - Partial RDA #15.1 - Preparation of data #Create two subsets of explanatory variables #Physiography (upstream-downstream gradient) envtopo <- env[, c(1:3)] names(envtopo) #Water quality envchem <- env[, c(4:9)] names(envchem) #Hellinger-transform the species dataset spe.hel <- decostand(doubs$fish[-8,], "hellinger") #15.2 - Explanation of fraction labels windows(title="Symbols of variation partitioning fractions",12,4) par(mfrow=c(1,3)) showvarparts(2) # Two explanatory matrices showvarparts(3) # Three explanatory matrices showvarparts(4) # Four explanatory matrices #15.3 - Variation partitioning with all explanatory variables spe.part.all <- varpart(spe.hel, envchem, envtopo) spe.part.all windows(title="Variation partitioning - all variables") plot(spe.part.all, digits=2) # Plot of the partitioning results #Tests of all testable fractions #Test of fractions [a+b] anova.cca(rda(spe.hel, envchem), step=1000) #Test of fractions [b+c] anova.cca(rda(spe.hel, envtopo), step=1000) #Test of fractions [a+b+c] env<- cbind(envchem, envtopo) anova.cca(rda(spe.hel, env), step=1000) #Test of fraction [a] anova.cca(rda(spe.hel, envchem, envtopo), step=1000) #Test of fraction [c] anova.cca(rda(spe.hel, envtopo, envchem), step=1000) #MEMs library(adespatial) mems_doubs <- dbmem(spa, MEM.autocor = "positive") mems.doubs <- data.frame(mems_doubs$MEM1, mems_doubs$MEM2, mems_doubs$MEM3, mems_doubs$MEM4, mems_doubs$MEM5, mems_doubs$MEM6, mems_doubs$MEM7) vp <- varpart(spe.hel, env, mems.doubs) plot(vp) # Linear discriminant analysis (LDA) ============================== # Ward clustering result of Hellinger-transformed species data, # cut into 4 groups gr <- cutree(hclust(vegdist(spe.hel, "euc"), "ward.D2"), k = 4) # Environmental matrix with only 3 variables (ele, oxy and bod) env.pars2 <- as.matrix(env[, c(3, 10, 11)]) # Verify multivariate homogeneity of within-group covariance # matrices using the betadisper() function {vegan} env.pars2.d1 <- dist(env.pars2) (env.MHV <- betadisper(env.pars2.d1, gr)) anova(env.MHV) permutest(env.MHV) # Permutational test # Log transform ele and bod env.pars3 <- cbind(log(env$slo), env$oxy, log(env$bdo)) colnames(env.pars3) <- c("ele.ln", "oxy", "bod.ln") rownames(env.pars3) <- rownames(env) env.pars3.d1 <- dist(env.pars3) (env.MHV2 <- betadisper(env.pars3.d1, gr)) permutest(env.MHV2) # Preliminary test : do the means of the explanatory variable # differ among groups? # Compute Wilk'S lambda test # First way: with function Wilks.test() of package rrcov, χ2 test library(rrcov) library(MASS) Wilks.test(env.pars3, gr) # Second way: with function manova() of stats, which uses # an F-test approximation lw <- manova(env.pars3 ~ as.factor(gr)) summary(lw, test = "Wilks") ## Computation of LDA - identification functions (on unstandardized ## variables) env.pars3.df <- as.data.frame(env.pars3) (spe.lda <- lda(gr ~ ele.ln + oxy + bod.ln, data = env.pars3.df)) # Alternate coding without formula interface : # spe.lda <- lda(env.pars3.df, gr) # The result object contains the information necessary to interpret # the LDA # Display the group means for the 3 variables spe.lda$means # Extract the unstandardized identification functions (matrix C, # eq. 11.33 in Legendre and Legendre 2012) (C <- spe.lda$scaling) # Classification of two new objects (identification) # A new object is created with two sites: # (1) ln(ele) = 6.8, oxygen = 9 and ln(bod) = 0.8 # and (2) ln(ele) = 5.5, oxygen = 10 and ln(bod) = 1.0 newo <- data.frame(c(6.8, 5.5), c(9, 10), c(0.8, 1)) colnames(newo) <- colnames(env.pars3) newo (predict.new <- predict(spe.lda, newdata = newo)) ## Computation of LDA - discrimination functions (on standardized ## variables) env.pars3.sc <- as.data.frame(scale(env.pars3.df)) spe.lda2 <- lda(gr ~ ., data = env.pars3.sc) # Display the group means for the 3 variables spe.lda2$means # Extract the classification functions (C2 <- spe.lda2$scaling) # Compute the canonical eigenvalues spe.lda2$svd^2 # Position the objects in the space of the canonical variates (Fp2 <- predict(spe.lda2)$x) # alternative way : Fp2 <- as.matrix(env.pars3.sc) %*% C2 # Classification of the objects (spe.class2 <- predict(spe.lda2)$class) # Posterior probabilities of the objects to belong to the groups # (rounded for easier interpretation) (spe.post2 <- round(predict(spe.lda2)$posterior, 2)) # Contingency table of prior versus predicted classifications (spe.table2 <- table(gr, spe.class2)) # Proportion of correct classification (classification success) diag(prop.table(spe.table2, 1)) # Plot the LDA results using the homemade function plot.lda() plot(spe.lda2, groups = gr, plot.sites = 2, plot.centroids = 1, mul.coef = 2.35) ```