# Delete all objets of workspace
rm(list=ls(all=TRUE))
options(digits = 2, scipen=999)
# Working directory
setwd("C:/Users/christian.valero/Documents/SEGUIMIENTO_PELAGICOS/2018/CRUCERO/COMPOSICION/PRUEBA/PROCESOS_JUREL/PROCEDURES_DO")
# Libraries
library( DiagrammeR)
library(egg)
library(gdata)
library(dplyr)
library(tidyr)
library(ggplot2)
library(FSA) Growth is generally the first process analysed in studies of fish stock and results in the most immediate application of age data, as the majority of analytical stock assessments use age-based models (Panfili et al. 2002).
Age data derived from reading solid structures is generally limited due to the cost and complexity involved. In contrast, the collection of length information is relatively simpler and less costly, thus catch-at-length data are more frequently available. Considering age and length are correlated, it is possible to determine catch-at-age from catch-at-length data.
Figure 1: Diagram of catch at age estimation.In the second stage a sub-sample per size class is taken. In this sub-sample the length, weigth and the collectiother measures of each specimen was recordered. The otoliths of each specimen are also collected.
LenWeight <- read.csv2("lenWeight.csv", sep = ";")
LenWeight <- filter(LenWeight, !is.na(Month), !is.na(ForkLength), !is.na(Weight))
LenWeight$ForkLength <- as.numeric(as.character(LenWeight$ForkLength))
LenWeight$Weight <- as.numeric(as.character(LenWeight$Weight))
LenWeight$Zone <- 1 # Asignation a categorical variable indicating geographical area
LenWeight$Trimester <- as.numeric(cut(LenWeight$Month,c(0,3,6,9,12),c(1,2,3,4))) # Asignation a time categorical variable
headtail(LenWeight,10)## Month Year ForkLength Weight Otolith Zone Trimester
## 1 3 2018 11 14.7 2 1 1
## 2 3 2018 11 13.0 2 1 1
## 3 3 2018 15 33.7 2 1 1
## 4 3 2018 10 14.8 2 1 1
## 5 3 2018 12 9.7 2 1 1
## 6 3 2018 11 15.0 2 1 1
## 7 3 2018 10 11.0 2 1 1
## 8 3 2018 10 8.2 2 1 1
## 9 3 2018 12 14.8 2 1 1
## 10 3 2018 11 11.6 2 1 1
## 405 4 2018 13 25.0 2 1 2
## 406 4 2018 13 25.0 2 1 2
## 407 4 2018 12 20.0 2 1 2
## 408 4 2018 16 45.0 2 1 2
## 409 4 2018 21 120.0 2 1 2
## 410 4 2018 20 110.0 2 1 2
## 411 4 2018 22 140.0 2 1 2
## 412 4 2018 22 130.0 2 1 2
## 413 4 2018 23 130.0 2 1 2
## 414 4 2018 22 125.0 2 1 2Corresponds to the total catch frecuency differentiated by each size range.
LenCatch <- read.csv2("LenCatch.csv", sep =";")
LenCatch <- gather(LenCatch, Zone, CatchL,"a","b","c","d","e","f","g")# tyding data
LenCatch <- aggregate(CatchL~ForkLength, data = LenCatch, FUN = sum)
headtail(LenCatch,10)## ForkLength CatchL
## 1 5 0
## 2 6 0
## 3 7 0
## 4 8 0
## 5 9 29045221
## 6 10 270234305
## 7 11 785677931
## 8 12 1327220275
## 9 13 1927006478
## 10 14 2287199771
## 57 61 0
## 58 62 0
## 59 63 0
## 60 64 0
## 61 65 0
## 62 66 0
## 63 67 0
## 64 68 0
## 65 69 0
## 66 70 0This data contains each individual age estimation.
Age <- read.csv2("AgeAtLength.csv", sep =";")# individual age group estimation
Age <- filter(Age,!is.na(TotalRadio), !is.na(NumRings),!is.na(ForkLength))As a rule, if the marginal increment is translucent it is not counted because it is not fully formed. in the for cycle above, we calculate the proportion of the traslucent increment i in relation of the inncrement i-1.
Age$Im <- NA
for(i in 1:dim(Age)[1]){
auxI1 <- numeric(); auxI1 <- Age$TotalRadio[i]
auxI2 <- numeric(); auxI2 <- Age[i,c(10:33)] # columns containg the ratio of each annulus
if(all(is.na(as.numeric(auxI2)))){
Age$Im[i] <- 0.249
}else{
auxI3 <- numeric(); auxI3 <- auxI2[which(is.na(as.numeric(auxI2)))[1]-2]
auxI4 <- numeric(); auxI4 <- auxI2[which(is.na(as.numeric(auxI2)))[1]-1]
if(length(auxI3) > 0 & length(auxI4) > 0){
if(as.numeric(auxI3) > 0 & as.numeric(auxI4) > 0){
Age$Im[i] <- (auxI1 - as.numeric(auxI4))/(as.numeric(auxI4) - as.numeric(auxI3))
}else{
Age$Im[i] <- 0.249
}
}else{
Age$Im[i] <- 0.249
}
Age$Im[which(Age$Im > 1)] <- 0.249
Age$Im[which(Age$Im < 0)] <- 0.249
}
}The age group is the number of calendar years after the date of capture, for this purpose each fish will be assigned depends on the year in wich it was spawned and on the date of capture, simplifying the identification of annual class per fish.
This estimate is calculated with the number of rings observed in the otolith, the type of edge of the otolith and an arbitrary date of birth. In Chile, the first date of birth is taken as a convention on January 1.
## Biological age (Cosidering the edge and the number of traslucient rings)
Age$Age <- NA
for(i in 1:length(Age$NumRings)){
Age$Age[i] <- as.factor( Age$NumRings[i])
if(Age$Border[i] > 2) {
Age$Age[i] <- as.numeric(sum(Age$NumRings[i],1 ))
} else {
Age$Age[i] <- Age$NumRings[i]
}
Age$Age[i] <- as.numeric(Age$Age[i])
}# Age group(considering the bioligical age, the nature of border, year of birth and the date of capture)
Age$AgeGroup <- NA
for(i in 1:length(Age$Month)){
if (Age$Month[i] <= 5 & Age$Border[i] <= 2) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 6 & Age$Border[i] == 1 & Age$Im[i] <= 0.249) {
Age$AgeGroup[i] <- Age$Age[i]
} else if (Age$Month[i] == 6 & Age$Border[i] == 1 & Age$Im[i] > 0.249) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 6 & Age$Border[i] == 2) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 7 & Age$Border[i] == 1 & Age$Im[i] <= 0.349) {
Age$AgeGroup[i] <- Age$Age[i]
} else if (Age$Month[i] == 7 & Age$Border[i] == 1 & Age$Im[i] > 0.349) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 7 & Age$Border[i] == 2) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 8 & Age$Border[i] %in% c(1,2) & Age$Im[i] <= 0.499) {
Age$AgeGroup[i] <- Age$Age[i]
} else if (Age$Month[i] == 8 & Age$Border[i] %in% c(1,2) & Age$Im[i] > 0.499) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] == 9 & Age$Border[i] == 1) {
Age$AgeGroup[i] <- Age$Age[i]
} else if (Age$Month[i] == 9 & Age$Border[i] == 2 & Age$Im[i] <= 0.699) {
Age$AgeGroup[i] <- Age$Age[i]
} else if (Age$Month[i] == 9 & Age$Border[i] == 2 & Age$Im[i] > 0.699) {
Age$AgeGroup[i] <- sum(Age$Age[i],1)
} else if (Age$Month[i] >= 10 & Age$Border[i] <= 2) {
Age$AgeGroup[i] <- Age$Age[i]
} else {
Age$AgeGroup[i] <- Age$Age[i]
}
}
# converting into a factor
Age$ForkLength <- factor(Age$ForkLength, levels = c(5:70), labels= c(5:70), ordered = F )
Age$AgeGroup <- factor(Age$AgeGroup, levels = c(0:24), labels= c(0:24),ordered = F )
headtail(Age,5)## Month Year Otolith Zone Trimester ForkLength NumRings Border
## 1 3 2018 2 1 1 11 0 2
## 2 3 2018 2 1 1 11 0 2
## 3 3 2018 2 1 1 15 0 3
## 4 3 2018 2 1 1 10 0 2
## 5 3 2018 2 1 1 12 0 2
## 194 3 2018 2 1 1 15 1 2
## 195 3 2018 2 1 1 16 1 1
## 196 3 2018 2 1 1 15 0 3
## 197 3 2018 2 1 1 15 0 3
## 198 3 2018 2 1 1 17 1 1
## TotalRadio R1 R2 R3 R4 R5 R6 R7 R8 R9 R10 R11 R12 R13 R14 R15 R16 R17
## 1 15 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 2 14 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 3 21 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 4 14 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 5 14 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 194 22 20 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 195 22 20 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 196 21 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 197 21 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## 198 23 21 NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA NA
## R18 R19 R20 R21 R22 R23 R24 Im Age AgeGroup
## 1 NA NA NA NA NA NA NA 0.25 0 1
## 2 NA NA NA NA NA NA NA 0.25 0 1
## 3 NA NA NA NA NA NA NA 0.25 1 1
## 4 NA NA NA NA NA NA NA 0.25 0 1
## 5 NA NA NA NA NA NA NA 0.25 0 1
## 194 NA NA NA NA NA NA NA 0.25 1 2
## 195 NA NA NA NA NA NA NA 0.25 1 2
## 196 NA NA NA NA NA NA NA 0.25 1 1
## 197 NA NA NA NA NA NA NA 0.25 1 1
## 198 NA NA NA NA NA NA NA 0.25 1 2The age-length key (ALK) is determined by ageing a sub-sample of the population, and is used to ascertain age data from larger length frecuency samples. ALKs describe distributions of size for each age, and relative number of individuals at each age. The ALK is presented as a table with ages in columns and lengths in rows, exhibiting probabilities of a fish belonging to a group of age “i” given it is of fork length “j”, with row totals summing to 1.
the probability of each size interval (j) to belong to a certain age group (i) is defined as:
\[q_{ij} = n_{ij}/n_j\] where:
# Finding length classes without age group estimation but with length frecuency data
Dif <- NA
auxD1 <- data.frame(table(Age$ForkLength));colnames(auxD1) <- c("ForkLength", "AgeGroup")# Vector with the size class without age group estimation
auxD1 <- cbind(LenCatch,auxD1["AgeGroup"])
auxD1 <- filter(auxD1, CatchL != 0 & AgeGroup == 0) # Vector with identifyng size classes whithout age group assigned
auxD2 <- read.csv2("Hist.csv", sep =";") # Vector containing historical reference data of age group estimation classes without ageing data
auxD2 <- filter(auxD2, ForkLength %in% unique(auxD1$ForkLength))
Dif <- cbind(auxD1["ForkLength"],auxD2["AgeGroup"])# Vector with data of age group estimation of the size classes without age group estimation but with length frecuency data
headtail(read.csv2("Hist.csv", sep =";"),5)## ForkLength AgeGroup
## 1 7 0
## 2 8 0
## 3 9 1
## 4 10 1
## 5 11 1
## 57 63 14
## 58 64 16
## 59 65 16
## 60 66 16
## 61 68 15headtail(Dif,10)## ForkLength AgeGroup
## 1 19 2
## 2 34 7
## 3 38 8
## 4 41 9
## 5 45 10#Age length key
Alk <- NA
auxAlk1 <- Age[,c("ForkLength","AgeGroup")]
auxAlk1 <- rbind(auxAlk1,Dif)
auxAlk1 <- table(auxAlk1)
Alk <- prop.table(auxAlk1, margin=1 ) #Row proportions table.
Alk[is.nan(Alk)] <- 0
headtail(Alk,10)## 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
## 5 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 6 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 7 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 8 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 9 0 1.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 10 0 1.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 11 0 1.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 12 0 0.93 0.067 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 13 0 0.91 0.087 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 14 0 0.89 0.105 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 61 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 62 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 63 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 64 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 65 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 66 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 67 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 68 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 69 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
## 70 0 0.00 0.000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0The length-weight relationship is widely used in fisheries to estimate weight from the length of an individual and also to estimate condition indices. The most frequently used expression for this relationship corresponds to the allometric equation where the weight is expressed as a function of length, and its parameters are estimated by linear regression of log-transformed data. Since variability in weight usually increases with length, this transformation has the advantage that it tends to stabilize the weight variance, but introduces a bias factor in back-transformed predictions which requires correction.
The weight/length model is:
\[W_i = a*L_i^b\] We can derive a simple linear regression model applying a logarithmic transformation:
\[ln(W_i) = ln(a)+b*ln(L_i)\]
where Wi represents the weight and Li the length of the i-th fish; a and b correspond to the intercept and slope, respectively.
# Length weigth relationship paramethers
LenWeight$logL <- log(LenWeight$ForkLength)
LenWeight$logW <- log(LenWeight$Weight)
lm <- lm(logW ~ logL,data = LenWeight)
Vb <- data.frame(coef(lm))
a <- exp(Vb[1,1])# y-intercept
b <- Vb[2,1]# Slope
R <-cor(LenWeight$logW, LenWeight$logL)# R-squared
n <- dim(LenWeight)[1]# Number of individualsSystematic bias for each given average length occurs when the mean weight at a given age is estimated from the mean length. This bias increases with the variability in length within the sample (Ricker 1958). Pienaar & Ricker (1968) addressed this issue, presenting a method to correct this bias.
# Pending error adjustment table
SlopeAdj <- read.csv2("PendingError.csv", sep = ";")
SlopeAdj## Pend a1 a2
## 1 1.0 0.00 0.0
## 2 1.1 0.08 0.0
## 3 1.2 0.15 0.0
## 4 1.3 0.23 0.0
## 5 1.4 0.32 0.0
## 6 1.5 0.42 0.0
## 7 1.6 0.52 0.0
## 8 1.7 0.63 0.0
## 9 1.8 0.74 0.0
## 10 1.9 0.87 0.0
## 11 2.0 1.00 0.0
## 12 2.1 1.14 0.0
## 13 2.2 1.31 0.0
## 14 2.3 1.48 0.0
## 15 2.4 1.66 0.0
## 16 2.5 1.86 0.0
## 17 2.6 2.07 0.0
## 18 2.7 2.29 0.0
## 19 2.8 2.52 0.0
## 20 2.9 2.75 0.0
## 21 3.0 3.00 0.0
## 22 3.1 3.25 0.2
## 23 3.2 3.51 0.4
## 24 3.3 3.77 0.6
## 25 3.4 4.04 0.9
## 26 3.5 4.32 1.2
## 27 3.6 4.62 1.5
## 28 3.7 4.95 1.8
## 29 3.8 5.29 2.2
## 30 3.9 5.64 2.6
## 31 4.0 6.00 3.0
## 32 4.1 6.36 3.5
## 33 4.2 6.73 4.1
## 34 4.3 7.11 4.9
## 35 4.4 7.50 5.9
## 36 4.5 7.90 7.1
## 37 4.6 8.31 8.4
## 38 4.7 8.72 9.7
## 39 4.8 9.14 11.3
## 40 4.9 9.57 13.0# Ricker and piennar imput formula
auxbmax <- round(b,1)
auxbmin <- auxbmax - 0.1
auxP1 <- filter(SlopeAdj, Pend == auxbmax)["a1"]
auxP2 <- filter(SlopeAdj, Pend == auxbmin)["a1"]
auxP3 <- filter(SlopeAdj, Pend == auxbmax)["a2"]
auxP4 <- filter(SlopeAdj, Pend == auxbmin)["a2"]
# Piennar and ricker slope adjustments values (a1 and a2)
a1 <- as.numeric(auxP1 - ((auxbmax - b) * (auxP1 - auxP2)) / (auxbmax - auxbmin))
a2 <- as.numeric(auxP3 - ((auxbmax - b) * (auxP3 - auxP4)) / (auxbmax - auxbmin))Table 1: Length weight relationship parameters
ab <- cbind("a" = a,"b" = b, "R2" = R,
"n" = n, "a1" = a1, "a2" = a2)
ab## a b R2 n a1 a2
## [1,] 0.0072 3.1 0.99 414 3.4 0.3Once a key is available, samples of fish that were only measured for length can be distributed to age groups accordingly.
This is possible because it assumes that the sample of aged fish and the sample of fish measured for length are simple random samples from the same population, thus the probability that a fish is of a particular age, given its length is the same in both samples follows.
The individuals present in each length interval (Nj) are assigned to different ages according to a size-age key.
The number of individuals belonging to each age group according to size interval is estimated as:
\[N_{ij} = q_{ij}/N_j\] where:
The abundance by age group is obtained by:
\[N_i = \sum_{i=1}^{n} N_{ij} \]
where:
Comp <- NA
auxC1 <- LenCatch$CatchL * Alk
Comp <- as.data.frame(cbind("Zone" = 1,
"Year" = 2018,
LenCatch$ForkLength,
LenCatch$CatchL,
auxC1))#Catch at age by length
names(Comp) <- c("Zone","Year","ForkLength","Abun_n","GE0","GE1","GE2","GE3",
"GE4","GE5","GE6","GE7",
"GE8","GE9","GE10","GE11","GE12","GE13","GE14",
"GE15","GE16","GE17","GE18","GE19","G20","G21",
"G22","G23","G24")
headtail(Comp[,1:17],10)## Zone Year ForkLength Abun_n GE0 GE1 GE2 GE3 GE4 GE5
## 5 1 2018 5 0 0 0 0 0 0 0
## 6 1 2018 6 0 0 0 0 0 0 0
## 7 1 2018 7 0 0 0 0 0 0 0
## 8 1 2018 8 0 0 0 0 0 0 0
## 9 1 2018 9 29045221 0 29045221 0 0 0 0
## 10 1 2018 10 270234305 0 270234305 0 0 0 0
## 11 1 2018 11 785677931 0 785677931 0 0 0 0
## 12 1 2018 12 1327220275 0 1238738923 88481352 0 0 0
## 13 1 2018 13 1927006478 0 1759440697 167565781 0 0 0
## 14 1 2018 14 2287199771 0 2046441901 240757871 0 0 0
## 61 1 2018 61 0 0 0 0 0 0 0
## 62 1 2018 62 0 0 0 0 0 0 0
## 63 1 2018 63 0 0 0 0 0 0 0
## 64 1 2018 64 0 0 0 0 0 0 0
## 65 1 2018 65 0 0 0 0 0 0 0
## 66 1 2018 66 0 0 0 0 0 0 0
## 67 1 2018 67 0 0 0 0 0 0 0
## 68 1 2018 68 0 0 0 0 0 0 0
## 69 1 2018 69 0 0 0 0 0 0 0
## 70 1 2018 70 0 0 0 0 0 0 0
## GE6 GE7 GE8 GE9 GE10 GE11 GE12
## 5 0 0 0 0 0 0 0
## 6 0 0 0 0 0 0 0
## 7 0 0 0 0 0 0 0
## 8 0 0 0 0 0 0 0
## 9 0 0 0 0 0 0 0
## 10 0 0 0 0 0 0 0
## 11 0 0 0 0 0 0 0
## 12 0 0 0 0 0 0 0
## 13 0 0 0 0 0 0 0
## 14 0 0 0 0 0 0 0
## 61 0 0 0 0 0 0 0
## 62 0 0 0 0 0 0 0
## 63 0 0 0 0 0 0 0
## 64 0 0 0 0 0 0 0
## 65 0 0 0 0 0 0 0
## 66 0 0 0 0 0 0 0
## 67 0 0 0 0 0 0 0
## 68 0 0 0 0 0 0 0
## 69 0 0 0 0 0 0 0
## 70 0 0 0 0 0 0 0Summary <- NA
auxS1 <- data.frame("CatchAgeNum" = apply(Comp[,c(5:29)], 2, sum))#Catch at age by length
auxS2 <-data.frame("AgeGroup" = c(0:24))
Summary <- cbind(auxS2,auxS1)Summary$PercNum <- Summary$CatchAgeNum / sum(Summary$CatchAgeNum) * 100the average length for each age group is expressed as
\[ \bar{p_{i}} = \frac{\sum_{i=1}^{n} l_{j}n_{ij}}{ni} \]
Donde:
auxML1 <- LenCatch$ForkLength * Comp[,c(5:29)]
auxML1 <- data.frame("MeanLength" = apply(auxML1,2, sum))
Summary <- cbind(Summary,auxML1)
Summary$MeanLength <- Summary$MeanLength/Summary$CatchAgeNum
Summary$MeanLength[is.nan(Summary$MeanLength)] <- 0 #Replacing nanThe variance of the mean length is expressed as
\[ P_i = \sum_{j=1} p_jq_{ij} \]
\[\sigma_{p_j} = \sum_{j=1}^{L} (\frac{p_j^{2}q_{ij}(1-q_{ij})}{n_{j}-1}+ \frac{p_{j}(q_{ij}-P_{i})^{2}}{N}) \]
Donde:
Summary$Variance <- NA
auxV1 <- as.data.frame(LenCatch$CatchL * Alk)
auxV1$ForkLength <- as.numeric(as.character(auxV1$ForkLength))
auxV1$AgeGroup <- as.numeric(as.character(auxV1$AgeGroup))
auxV1$Tmed <- auxV1$ForkLength * auxV1$Freq
auxV1$Varl <- (auxV1$ForkLength^2) * auxV1$Freq
auxV2 <- aggregate(Varl ~ AgeGroup, data = auxV1, FUN = sum)
auxV3 <- aggregate(Tmed ~ AgeGroup, data = auxV1, FUN = sum)
Summary <- cbind(Summary, auxV2[2], auxV3[2])
Summary$Df <- abs(Summary$CatchAgeNum - 1)# degree of freedom
for(i in 1:length(Summary$Tmed)){
if(Summary$Tmed[i] > 0){
Summary$Variance[i] <- abs((Summary$Varl[i] - (Summary$MeanLength[i] *
Summary$Tmed[i]))/Summary$Df[i])
}else{
Summary$Variance[i] <- 0
}
}Assuming that the length is a normal random variable the expected function value of \(W_L\), will be defined as:
\[E(W) = a(\bar{p_{i}}^b+a_{1}\bar{p_{i}}^{b-2}\sigma^2+a_{2}\bar{p_{i}}^{b-4}\sigma^4)\]
where:
Summary$MeanWeight <- NA
for(i in 1:length(Summary$MeanLength)){
if(Summary$MeanLength[i] > 0){
Summary$MeanWeight[i] <- (a * ((Summary$MeanLength[i]^b) +
a1 * (Summary$MeanLength[i]^(b-2)) * Summary$Variance[i] +
a2 * (Summary$MeanLength[i]^(b-4)) * Summary$Variance[i]^2))/1000
}else{
Summary$MeanWeight[i] <- 0
}
}Summary$CatchAgeWeight <- (Summary$CatchAgeNum * Summary$MeanWeight) / 1000Summary$PercWeight <- (Summary$CatchAgeWeight / sum(Summary$CatchAgeWeight)) * 100auxC1 <- Summary[,c("CatchAgeNum","PercNum","MeanLength",
"Variance", "MeanWeight",
"CatchAgeWeight", "PercWeight" )]
auxC2 <- t(auxC1)
colnames(auxC2) <- c("GE0","GE1","GE2","GE3","GE4",
"GE5","GE6","GE7","GE8","GE9",
"GE10","GE11","GE12","GE13","GE14",
"GE15","GE16","GE17","GE18","GE19",
"GE20","GE21","GE22","GE23","GE24")
auxC3 <- data.frame(rowSums(auxC2))
# Total catch in number of individuals
TotalCatchN <- auxC3[1,1]
# Percent of catch in number
PercentCatchN <- auxC3[2,1]
#Mean Length
auxC4 <- colSums(Summary)["Tmed"]
Mean_Length <- auxC4/TotalCatchN
#Variance
Variance <- abs((sum(Summary$Varl) - (Mean_Length*auxC4))/(TotalCatchN - 1))
MeanWeight <- (a * ((Mean_Length^b) +
a1 * (Mean_Length^(b-2)) * Variance+
a2 * (Mean_Length^(b-4)) * Variance^2))/1000
# Total catch in weigth(tonnes)
TotalCatchW <- rowSums(auxC2)[6]
# Percent of catch in weigth
PercentCatchW <- rowSums(auxC2)[7]CatchAtAge <- data.frame()
auxC5 <- data.frame("Abun_n" = c(TotalCatchN,
PercentCatchN,
Mean_Length,
Variance,
MeanWeight,
TotalCatchW,
PercentCatchW))
auxC6 <- data.frame("ForkLength" = c("CatchAgeNum","PercNum",
"MeanLength","Variance",
"MeanWeight","CatchAgeWeight",
"PercWeight"))
auxC7 <- data.frame(cbind(auxC6, auxC5, auxC2))
auxC7$ForkLength <- as.character(auxC7$ForkLength)
auxC8 <- data.frame(Comp[,c(3:29)])
auxC8$ForkLength <- as.character(auxC8$ForkLength)
colnames(auxC8) <- colnames(auxC7)
auxC9 <- matrix(c(rep.int(NA,length(auxC8))),nrow = 1,ncol = length(auxC8))
colnames(auxC9) <- colnames(auxC8)
CatchAtAge <- rbind(auxC8, auxC9, auxC7)
rownames(CatchAtAge) <- NULL
# deleting all auxiliar objects
rm(list=ls(pattern="aux"))Table 2: Table required for JM technical report
headtail(CatchAtAge[,1:5],10)## ForkLength Abun_n GE0 GE1 GE2
## 1 5 0.000 0 0.000 0.000
## 2 6 0.000 0 0.000 0.000
## 3 7 0.000 0 0.000 0.000
## 4 8 0.000 0 0.000 0.000
## 5 9 29045220.616 0 29045220.616 0.000
## 6 10 270234304.594 0 270234304.594 0.000
## 7 11 785677930.922 0 785677930.922 0.000
## 8 12 1327220274.591 0 1238738922.952 88481351.639
## 9 13 1927006478.169 0 1759440697.459 167565780.710
## 10 14 2287199771.304 0 2046441900.640 240757870.664
## 65 69 0.000 0 0.000 0.000
## 66 70 0.000 0 0.000 0.000
## 67 <NA> NA NA NA NA
## 68 CatchAgeNum 10053739366.706 0 7852187551.781 1564023045.326
## 69 PercNum 100.000 0 78.102 15.557
## 70 MeanLength 14.318 0 13.338 15.132
## 71 Variance 10.137 0 2.581 2.446
## 72 MeanWeight 0.037 0 0.026 0.039
## 73 CatchAgeWeight 380762.704 0 207559.915 60763.955
## 74 PercWeight 100.000 0 54.512 15.958Panfili, J., Morales-Nin, B. 2002. Manual of fish sclerochronology. IFREMER-IRD coedition, Brest. pp. 129-134.
Pienaar L. and W. Ricker, 1968. Estimating mean weight from length statistic. J. Fish. Res. Board of Can. 25: 2743 - 2747.
Ricker W., 1958. Handbook of computations for biological statistics of fish populations. Bull. Fish. Res. Bd. Can., Nº 119.