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VA_Reallocation_with_management_final_with_CV.Rmd
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---
title: "Oyster Survey Effort Reallocation and Restratification"
author: "Robert Isdell"
date: "April 16, 2019"
output:
html_document: default
word_document: default
pdf_document: default
---
```{r set-options, echo=FALSE, cache=FALSE}
options(width = 175)
knitr::opts_chunk$set(warning = FALSE, message = FALSE, echo = FALSE, fig.align = 'center')
```
```{r, echo=FALSE, warning=FALSE}
# Install BIOSurvey2 package
# devtools::install("BIOSurvey2/BIOSurvey2")
# Load in required packages
pacman::p_load(tidyverse, readxl, BIOSurvey2, cluster, fpc, random, knitr)
# Read in data by year
site.info <- read_xlsx("Data/PTData_2006_2018.xlsx",1)
site.info <- site.info[1:199,] #trim off extra from the end
dat.2006 <- read_xlsx("Data/PTData_2006_2018.xlsx",2)
dat.2007 <- read_xlsx("Data/PTData_2006_2018.xlsx",3)
dat.2008 <- read_xlsx("Data/PTData_2006_2018.xlsx",4)
dat.2009 <- read_xlsx("Data/PTData_2006_2018.xlsx",5)
dat.2010 <- read_xlsx("Data/PTData_2006_2018.xlsx",6)
dat.2011 <- read_xlsx("Data/PTData_2006_2018.xlsx",7)
dat.2012 <- read_xlsx("Data/PTData_2006_2018.xlsx",8)
dat.2013 <- read_xlsx("Data/PTData_2006_2018.xlsx",9)
dat.2014 <- read_xlsx("Data/PTData_2006_2018.xlsx",10)
dat.2015 <- read_xlsx("Data/PTData_2006_2018.xlsx",11)
dat.2016 <- read_xlsx("Data/PTData_2006_2018.xlsx",12)
dat.2017 <- read_xlsx("Data/PTData_2006_2018.xlsx",13)
dat.2018 <- read_xlsx("Data/PTData_2006_2018.xlsx",14)
# Add year variable to each dataset
dat.2006$Year <- 2006
dat.2007$Year <- 2007
dat.2008$Year <- 2008
dat.2009$Year <- 2009
dat.2010$Year <- 2010
dat.2011$Year <- 2011
dat.2012$Year <- 2012
dat.2013$Year <- 2013
dat.2014$Year <- 2014
dat.2015$Year <- 2015
dat.2016$Year <- 2016
dat.2017$Year <- 2017
dat.2018$Year <- 2018
# merge all of the datasets into single dataframe
dat.years <- rbind(dat.2006, dat.2007, dat.2008, dat.2009, dat.2010, dat.2011, dat.2012,
dat.2013, dat.2014, dat.2015, dat.2016, dat.2017, dat.2018)
# Correct and merge various river names for continuity
dat.years$River_name[dat.years$River_name == "Mobjack"] <- "Mobjack Bay"
dat.years$River_name[dat.years$River_name == "Great Wicomico"] <- "Great Wicomico River"
dat.years$River_name[dat.years$River_name == "Piankatank"] <- "Piankatank River"
dat.years$River_name[dat.years$River_name == "Rappahannock"] <- "Rappahannock River"
dat.years$River_name[dat.years$River_name == "Elizabeth River Western Branch"] <- "Elizabeth River"
dat.years$River_name[dat.years$River_name == "Corrotoman River"] <- "Rappahannock River"
dat.years$River_name[dat.years$River_name == "Elizabeth River"] <- "James River"
dat.years$River_name[dat.years$River_name == "Lafayette River"] <- "James River"
dat.years$River_name[dat.years$River_name == "East River"] <- "York River"
dat.years$River_name[dat.years$River_name == "Mobjack Bay"] <- "York River"
dat.years$River_name[dat.years$River_name == "Pocomoke Sound"] <- "Tangier Sound"
dat.years$Station_ID[dat.years$Station_ID == 390] <- 395
dat.years$Station_ID[dat.years$Station_ID == 1131] <- 1128
# Select only relevant fields from site.info
site.acres <- site.info %>%
select(`Station ID`, `Management Subgroups`, Total, n_years)
# Convert Station ID field to numeric
site.acres$`Station ID` <- as.numeric(site.acres$`Station ID`)
# Join the site.acres dataset to the full dataset
dat.full <- left_join(dat.years, site.acres, by = c("Station_ID" = "Station ID"))
## Select stations with 5+ years of data
dat.full <- dat.full %>%
filter(n_years > 4)
```
The following document summarizes our efforts-to-date to restratify and reallocate effort within the annual oyster survey. This work has relied heavily on the R package **BioSurvey2** written by the late Stephen Smith and distributed by SCeMFIS ([http://www.scemfis.org/](http://www.scemfis.org/)).
# Introduction
After the initial meeting with Roger, Missy, Rob, Jim, Andrew, Vernon, John, and myself, we decided to make the following changes, which are reflected in this version of the document:
1) Only use reefs that have at least 5 years of data in the analysis
2) When available, use the predefined management subgroups within a given waterbody
3) If all reefs within a given waterbody do not fall into a predefined management subgroup, use the clustering and restratification method previously demonstrate.
Reefs that are <5 years old should continue to be sampled at their current levels until enough data has been collected to include in the analysis.
I am willing to make all data, code, and analyes publicly available via GitHub. Alternatively, the work could be hosted on WM Scholarworks where it would be assigned a DOI and useage statistics would be tracked.
# James River
The James River did not have predefined management subgroups, and so was stratified using the k-means method.
```{r}
# Subset the James River data
all.james.dat <- dat.full %>%
filter(River_name == "James River" & Station_ID != 16) # Reef 16 now defunct, split into 410 and 411
# Create a vector of unique station IDs within the James
all.james.stations <- unique(all.james.dat$Station_ID)
# Make Station ID the same for both datasets
colnames(site.info)[2] <- "Station_ID"
colnames(site.acres)[1] <- "Station_ID"
# Subset the James River site info
all.james.site.info <- site.acres %>%
filter(Station_ID %in% all.james.stations) %>%
arrange(Station_ID)
all.james.site.info$Station_ID <- as.numeric(all.james.site.info$Station_ID)
# Multiply counts by subsample factor
all.james.dat$Lives <- rowSums(all.james.dat[,16:52]*all.james.dat$Subsample_Factor)
all.james.dat$Markets <- rowSums(all.james.dat[,31:52]*all.james.dat$Subsample_Factor)
all.james.dat$Spat1 <- rowSums(all.james.dat[,55:67]*all.james.dat$Subsample_Factor)
# Select only the variables needed for analysis
all.james.dat <- all.james.dat %>% select(Station_ID, Sample_ID, Lives, Markets, Spat1)
```
## Set up strata and summarize
The following table provides the number of grabs from each reef from 2006-2018 (Observed) and the number of grabs over that same time frame, given the same total effort within the James, that would have best addressed total catch and the variability of the catch relative to the area of the reef.
```{r}
# Specify Strata and total number of possible samples (NH = Reef acreage * years sampled * 4046.86 m^2/acre)
all.james.strata <- data.frame("Strata" = all.james.site.info$Station_ID, NH = all.james.site.info$Total*4046.86)
# Check current allocation using BIOSurvey2::Stratify()
all.james.numbers <- Stratify(all.james.dat, all.james.strata, "Station_ID", species = "Lives")
kable(summary(all.james.numbers, effic = TRUE, nopt = TRUE)$n.opt[,1:3])
```
```{r}
# Store the summary output
all.james.numbers.summary <- summary(all.james.numbers, effic = TRUE, nopt = TRUE)
# Extract the optimal number of samples per strata
all.james.numbers.nopt <- all.james.numbers.summary$n.opt
# Join the optimal sample numbers to the site info
all.james.reallocation <- left_join(all.james.numbers.nopt, site.info, by = c("Strata" = as.character("Station_ID")))
# Specify the column names
names(site.acres) <- c("Station_ID", "Subgroup", "Acreage", "Years")
# Set Station ID to numeric
site.acres$Station_ID <- as.numeric(site.acres$Station_ID)
# Join James counts and site info
all.james.full <- left_join(all.james.dat, site.acres, by = "Station_ID")
# Create a dataset of the mean and variance of Live counts, and reef acreage
all.james.mva <- all.james.full %>%
group_by(Station_ID) %>%
summarise(Mean = mean(Lives), Variance = var(Lives), Acreage = mean(Acreage))
row.names(all.james.mva) <- all.james.mva$Station_ID
# Scale and center the mva to prep for clustering
all.james.mva <- scale(all.james.mva)
```
## Clustering
One way to address the effort reallocation problem is to rexamine the current stratification, and see if we can reduce the number of strata while still getting accurate estimates of the mean and variation of the population within each strata. To do this, we can see if certain reefs cluster together based on similar features. I used the mean and variance of live oysters within each reef, and the acreage of each reef as factors for a K-means clustering analysis.
```{r, fig.align='center', fig.height=5, fig.width=7}
# Identify best number of k-means cluster groups by finding inflection point
# Iterate 100 times to avoid chances of spurious result
all.james.wss <- matrix(nrow = 100, ncol = 15)
for(j in 1:100) {
wss <- (nrow(all.james.mva[,2:4])-1)*sum(apply(all.james.mva[,2:4], 2, var))
for (i in 1:15) all.james.wss[j,i] <- sum(kmeans(all.james.mva[,2:4], centers = i, iter.max = 1000, nstart = 1)$withinss)
}
all.james.wss <- as.data.frame(all.james.wss)
all.james.wss.mean <- colMeans(all.james.wss)
# Plot the reduction in within-group sums-of-squares per cluster
plot(1:15, all.james.wss.mean, type = "b", xlab = "Number of Clusters",
ylab = "Within groups SS")
#################################################################
k <- 4 # number of clusters
#################################################################
```
In the above plot, you can see that the inflection point in the line indicates that based on these features, the 36 reefs within the James could be most confidently grouped into 4 clusters.
```{r, fig.align='center', fig.height=5}
# Check 1000 seeds to identify which provides the best clustering
best.seed <- matrix(nrow = 1000, ncol = 2)
for (i in 1:1000){
set.seed(i)
fit <- kmeans(all.james.mva[,2:4], k, iter.max = 1000, nstart = 1, algorithm = "Hartigan-Wong")
best.seed[i,1] <- i
best.seed[i,2] <- fit$betweenss/fit$totss*100
}
best.seed <- as.data.frame(best.seed)
seed <- best.seed$V1[best.seed$V2 == max(best.seed$V2)][1]
set.seed(seed)
# Cluster the reefs
fit.all.james <- kmeans(all.james.mva[,2:4], k, iter.max = 1000, nstart = 1, algorithm = "Hartigan-Wong")
# Create a dataset of the Station IDs and cluster groups
all.james.cluster.groups <- data.frame("Station_ID" = as.numeric(rownames(all.james.mva)))
all.james.cluster.groups$kmeans <- fit.all.james$cluster
# Generate a cluster plot of the results
clusplot(all.james.mva[,2:4], all.james.cluster.groups$kmeans, color = T, shade = T, labels = 2, lines = 0)
```
This plot shows how well those reefs cluster.
```{r}
# Show what cluster group each reef will be in
a2 <- fit.all.james$cluster
suggested.strata.all.james <- data_frame("Code" = as.numeric(names(a2)), "Cluster" = a2)
site.name <- dat.full %>%
select(Station_ID, Full_station_name) %>%
mutate("Code" = Station_ID) %>%
select(Code, Full_station_name)
site.name <- site.name[!duplicated(site.name),]
suggested.strata.all.james <- left_join(suggested.strata.all.james, site.name, by = "Code") %>%
arrange(Cluster)
kable(suggested.strata.all.james, caption = "Suggested Strata")
```
## Add new suggested groups and check allocation
```{r}
# Create new dataset of cluster groups as strata
names(all.james.site.info) <- c("Station_ID", "Subgroup", "NH", "Years")
all.james.site.info$NH <- all.james.site.info$NH*4046.86 #convert acres to square meters
all.james.site.info$Station_ID <- as.character(all.james.site.info$Station_ID)
all.james.site.info2 <- data.frame(all.james.site.info)
all.james.domain <- data_frame(NH = all.james.site.info$NH,
"kmeans" = all.james.cluster.groups$kmeans)
#Summarize NH by new strata (aka domains)
all.james.domain.info <- all.james.domain %>%
group_by(kmeans) %>%
summarise(NH = mean(NH))
all.james.domain.info2 <- data.frame(all.james.domain.info)
all.james.dat2 <- left_join(all.james.dat, all.james.cluster.groups)
all.james.dat2 <- arrange(all.james.dat2, Station_ID)
all.james.dat2$Station_ID <- as.character(all.james.dat2$Station_ID)
all.james.dat3 <- data.frame(all.james.dat2)
all.james.site.info2 <- data.frame(all.james.site.info)
# length(all.james.site.info2[, "Station_ID"])
# length(SFA29.Geophysical[, "GEOPHYS_ID"])
```
How well is effort allocated based on current strata?
```{r}
# How well is effort allocated based on current strata?
names(all.james.domain.info2) <- c("Strata","NH")
all.james.numbers2 <- Stratify(all.james.dat3, all.james.domain.info2, "kmeans", species = "Lives")
kable(summary(all.james.numbers2, effic = T, nopt = T)$n.opt)
```
Overall, effort allocation is inefficient within the new strata.
## Estimate the population mean and variance given the new strata and allocation
### Simulate the data
We can resample the data at the proportions suggested by BIOSurvey2, sequentially reduce the effort by 10% across all strata, and then plot the results. We are effectively creating 100 datasets per proportional effort reduction for a total of 1000 datasets. Each one is the result of resampling (with replacement). Then, we can plot the mean of each strata within the datasets for each proportional effort reduction, and get a sense of what the spread in the data would look like. By plotting the 95% CI of the data, a wedge forms as the estimates become less accurate with increasing proportional reduction. Identifying the point where our overall interpretation of the data remains more-or-less the same while maximixing effort reduction should provide support for how much we could reduce effort within a given waterbody.
```{r}
# Create a blank dataframe for the mean, SD, and variance of each cluster group
# sampled (with replacement) 100 times at the optimal level suggested by BIOSurvey2
varest.boot.dat.all.james <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat.all.james) <- c("kmeans", "MEAN", "SD", "VAR")
varest.boot.dat.all.james <- as.data.frame(varest.boot.dat.all.james)
# Cummulative number of samples for each cluster
n.opt.test.cumm.all.james <- cumsum(summary(all.james.numbers2, effic = T, nopt = T)$n.opt$Optimal[1:k])
# Optimal number of samples per cluster
n.opt.test.all.james <- summary(all.james.numbers2, effic = T, nopt = T)$n.opt$Optimal[1:k]
# Identify the starting position for each cluster in the dataframe
start.pos.all.james <- c(1, n.opt.test.cumm.all.james[1:k-1]+1)
# Blank dataframe of the BIOSurvey2::Stratify() output for each resampled dataset
nopt.boot.dat.all.james <- matrix(ncol = 7, nrow = k*100)
colnames(nopt.boot.dat.all.james) <- c("Strata", "Observed", "Optimal", "Perc.Increase.Var.Opt",
"Compromise", "Perc.Increase.Var.comp", "run")
nopt.boot.dat.all.james <- as.data.frame(nopt.boot.dat.all.james)
# Start and end position for each cluster summary in varest.boot.dat.all.james
j.start.all.james <- seq(1,k*100, by = k)
j.end.all.james <- seq(k, k*100, by = k)
# Estimating the mean and variance of 100 resampled datasets at the original effort level
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6,nrow = n.opt.test.cumm.all.james[k])
colnames(sub.dat1) <- names(all.james.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- all.james.dat3 %>%
filter(kmeans == i)
rand.n <- sample(1:nrow(temp),n.opt.test.all.james[i], replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos.all.james[i]:n.opt.test.cumm.all.james[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(kmeans) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat.all.james[j.start.all.james[j]:j.end.all.james[j],] <- sub.summary
}
#kable(head(varest.boot.dat.all.james, 10), caption = "Population mean for each strata with a random resampling with new effort")
```
```{r, warning=FALSE}
# Same as section above, except with sequential 10% reduction in effort from 0% to 90% reduction
all.james.sample <- n.opt.test.all.james
all.james.sample.cumm <- n.opt.test.cumm.all.james
all.james.plot.dat <- matrix(ncol = 3, nrow = k*1000)
colnames(all.james.plot.dat) <- c("ER","KMEAN", "MEAN")
all.james.plot.dat <- as.data.frame(all.james.plot.dat)
all.james.plot.dat$ER <- rep(seq(0, 0.9, 0.1), each = k*100)
all.james.plot.dat$KMEAN <- rep(1:k, 1000)
all.james.plot.dat$KMEAN <- factor(all.james.plot.dat$KMEAN)
h.start <- seq(1, 1+(k*1000 - k*100), by = k*100)
h.end <- seq(k*100, k*1000, by = k*100)
all.james.sub.sample <- all.james.sample/sum(all.james.sample)
all.james.sub.cumm <- round(seq(1, 0.1, by = -0.1)*all.james.sample.cumm[k], 0)
j.start <- seq(1,k*100, by = k)
j.end <- seq(k, k*100, by = k)
for(h in 1:10){
varest.boot.dat <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat) <- c("kmeans", "MEAN", "SD", "VAR")
varest.boot.dat <- as.data.frame(varest.boot.dat)
sample.cumm <- cumsum(round(all.james.sub.sample*all.james.sub.cumm[h],0))
start.pos <- c(1, sample.cumm[1:k-1]+1)
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6, nrow = all.james.sub.cumm[h])
colnames(sub.dat1) <- names(all.james.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- all.james.dat3 %>%
filter(kmeans == i)
rand.n <- sample(1:nrow(temp),round(all.james.sub.sample[i]*all.james.sub.cumm[h],0), replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos[i]:sample.cumm[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(kmeans) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat[j.start[j]:j.end[j],] <- sub.summary
}
all.james.plot.dat$MEAN[h.start[h]:h.end[h]] <- varest.boot.dat$MEAN
}
#kable(head(all.james.plot.dat, 10), caption = "Population mean for each strata with a random resampling with new effort")
```
```{r}
# Create a function to calculate the mean and CI of the mean
mean_ci <- function(x, mult = 1) {
x <- na.omit(x)
SD <- mult * sd(x)
MEAN <- mean(x)
data.frame(y = MEAN, ymin = MEAN - qt(1 - (0.05 / 2), 99) * SD, ymax = MEAN + qt(1 - (0.05 / 2), 99) * SD)
}
# Plot the change in quality of the data with each sequential reduction in effort
jreffortplot <- ggplot(all.james.plot.dat, aes(y = MEAN, x = ER, colour = KMEAN, fill = KMEAN)) +
geom_smooth(se = FALSE) +
geom_point(size = 2) +
geom_abline(intercept = c(148,163,366,415,30,36, 27, 32),
size = 1, linetype = "dashed", alpha = 0.7) +
stat_summary(fun.data = mean_ci, geom = "ribbon", alpha = 0.5) +
theme_light() +
scale_color_brewer(palette = "Paired") +
scale_fill_brewer(palette = "Paired") +
theme(axis.text = element_text(size = 15, colour = "black")) +
theme(axis.title = element_text(size = 15)) +
theme(legend.text = element_text(size = 15)) +
theme(legend.title = element_text(size = 15)) +
labs(y = expression(paste("Mean Oysters m"^-2 %+-% "CI"))) +
labs(x = "Proportional Effort Reduction")
```
In this plot, you can see that the mean and CI remain relatively constant through a 40% reduction in effort.
```{r}
# Create a function to calculate the coefficient of variation (CV)
cv <- function(x) {
m <- mean(x)
v <- sd(x)
cfvar <- v/m
return(cfvar)
}
# Plot the change in quality of the data using the CV with each sequential reduction in effort
all.james.plot.dat.cv <- all.james.plot.dat %>%
group_by(KMEAN, ER) %>%
summarise(CV = cv(MEAN))
## Create a sequence of the no effort reduction CV for comparison
all.james.plot.dat.cv$PctCV <- all.james.plot.dat.cv$CV / rep(pull(all.james.plot.dat.cv[c(seq(1,((k-1)*10)+1,10)),3]), each = 10) * 100 - 100
ggplot(all.james.plot.dat.cv, aes(y = PctCV, x = ER, color = KMEAN, linetype = KMEAN)) +
geom_smooth(size = 1.2, se = F) +
geom_abline(intercept = 40, slope = 0, size = 1.2, linetype = "dashed") +
theme_light() +
scale_color_brewer(palette = "Paired") +
theme(axis.text = element_text(size = 15, colour = "black")) +
theme(axis.title = element_text(size = 15)) +
theme(legend.text = element_text(size = 15)) +
theme(legend.title = element_text(size = 15)) +
labs(y = "Relative Coefficient of Variation") +
labs(x = "Proportional Effort Reduction")
```
This plot shows how the coefficient of variation (CV) changes with each proportion reduction in effort relative to the starting CV.
```{r}
# Population summary for each cluster
strat.kmeans.summary.all.james <- all.james.dat3 %>%
group_by(kmeans) %>%
summarise(pMEAN = mean(Lives), pSD = sd(Lives), pVAR = var(Lives), nOBS = n()) %>%
mutate(lower.ci.mean = pMEAN - qt(1 - (0.05 / 2), nOBS) * (pSD/sqrt(nOBS)),
upper.ci.mean = pMEAN + qt(1 - (0.05 / 2), nOBS) * (pSD/sqrt(nOBS)))
kable(strat.kmeans.summary.all.james, caption = "Population Stats based on all data")
all.james.ci.summary <- all.james.plot.dat %>%
group_by(ER, KMEAN) %>%
summarise(m.MEAN = mean(MEAN), sd.MEAN = sd(MEAN)) %>%
mutate(lower.ci.mean = m.MEAN - qt(1 - (0.05 / 2), 99) * sd.MEAN,
upper.ci.mean = m.MEAN + qt(1 - (0.05 / 2), 99) * sd.MEAN) %>%
arrange(KMEAN, ER)
kable(all.james.ci.summary, caption = "Mean of the means")
```
Based on the information above, it appears that the total effort in Burwell Bay could be reduced by 40% and still obtain similar population parameters. Each year, total effort (i.e., number of grabs) in each cluster would be as follows:
```{r}
n.reefs <- suggested.strata.all.james %>%
group_by(Cluster) %>%
summarise(N = n())
n.suggested <- round(all.james.sub.sample*all.james.sub.cumm[5]/13,0)
effort <- 1:k
for (i in 1:k) {
effort[i] <- ifelse(n.suggested[i] < n.reefs$N[i], n.reefs$N[i], n.suggested[i])
}
kable(data_frame("Cluster" = 1:k, "Future Effort" = effort,
"Current Effort" = round(summary(all.james.numbers2, effic = TRUE, nopt = TRUE)$n.opt[,2][1:4]/13,0)))
```
This would mean a total of 310 grabs in the James Bay compared to 518 in previous years.
# Lynnhaven River
Lynnhaven remains unchanged from the previous iteration. If you all decide to continue, then there's no need to restratify given how few reefs are down there and that it's a full day either way. However, if you do continue sampling, BIOSurvey2 does recommend reallocating effort.
```{r, echo=FALSE}
lynnhaven.dat <- dat.full %>%
filter(River_name == "Lynnhaven")
lynnhaven.stations <- unique(lynnhaven.dat$Station_ID)
colnames(site.info)[2] <- "Station_ID"
colnames(site.acres)[1] <- "Station_ID"
lynnhaven.site.info <- site.acres %>%
filter(Station_ID %in% lynnhaven.stations) %>%
arrange(Station_ID)
lynnhaven.site.info$Station_ID <- as.numeric(lynnhaven.site.info$Station_ID)
lynnhaven.dat$Lives <- rowSums(lynnhaven.dat[,16:52]*lynnhaven.dat$Subsample_Factor)
lynnhaven.dat$Markets <- rowSums(lynnhaven.dat[,31:52]*lynnhaven.dat$Subsample_Factor)
lynnhaven.dat$Spat1 <- rowSums(lynnhaven.dat[,55:67]*lynnhaven.dat$Subsample_Factor)
lynnhaven.dat <- lynnhaven.dat %>% select(Station_ID, Sample_ID, Lives, Markets, Spat1)
```
## Set up strata and summarize
```{r, echo=FALSE}
lynnhaven.strata <- data_frame("Strata" = lynnhaven.site.info$Station_ID, NH = lynnhaven.site.info$Acreage*4046.86)
lynnhaven.numbers <- Stratify(lynnhaven.dat, lynnhaven.strata, "Station_ID", species = "Lives")
kable(summary(lynnhaven.numbers, effic = TRUE, nopt = TRUE)$n.opt[,1:3])
```
Each year, total effort (i.e., number of grabs) in each reef would be as follows:
```{r, echo=FALSE}
lynnhaven.summary <- summary(lynnhaven.numbers, effic = TRUE, nopt = TRUE)$n.opt[,1:3]
lynnhaven.summary <- mutate(lynnhaven.summary, `Annual Effort` = ifelse(Optimal/13 < 1, 1, ceiling(Optimal/13)))
kable(lynnhaven.summary[1:12,c(1,4)])
```
This would be a total of 78 grabs per year in the Lynnhaven.
# Rappahannock River
The Rappahannock has predefined management subgroups, thereby making any cluster analyses unneccesary. The results shown here demonstrate how the effort should be allocated among management subgroups.
```{r}
Rappahannock.dat <- dat.full %>%
filter(River_name == "Rappahannock River")
Rappahannock.stations <- unique(Rappahannock.dat$Station_ID)
colnames(site.info)[2] <- "Station_ID"
colnames(site.acres)[1] <- "Station_ID"
Rappahannock.site.info <- site.acres %>%
filter(Station_ID %in% Rappahannock.stations) %>%
arrange(Station_ID)
Rappahannock.site.info$Station_ID <- as.numeric(Rappahannock.site.info$Station_ID)
Rappahannock.site.info2 <- Rappahannock.site.info %>%
group_by(Subgroup) %>%
summarise(Acreage2 = sum(Acreage))
Rappahannock.dat$Lives <- rowSums(Rappahannock.dat[,16:52]*Rappahannock.dat$Subsample_Factor)
Rappahannock.dat$Markets <- rowSums(Rappahannock.dat[,31:52]*Rappahannock.dat$Subsample_Factor)
Rappahannock.dat$Spat1 <- rowSums(Rappahannock.dat[,55:67]*Rappahannock.dat$Subsample_Factor)
Rappahannock.dat <- Rappahannock.dat %>% select(Station_ID, Sample_ID, Lives, Markets, Spat1)
Rappahannock.dat <- left_join(Rappahannock.dat, Rappahannock.site.info)
```
## Set up strata and summarize
```{r}
Rappahannock.sites <- site.info %>%
filter(Station_ID %in% Rappahannock.stations) %>%
select(Station_ID, Site, `Management Subgroups`) %>%
arrange(`Management Subgroups`)
kable(Rappahannock.sites, caption = "Rappahannock Reef and Management Subgroup")
```
```{r}
Rappahannock.strata <- data_frame("Strata" = Rappahannock.site.info2$Subgroup, NH = Rappahannock.site.info2$Acreage2*4046.86)
Rappahannock.numbers <- Stratify(Rappahannock.dat, Rappahannock.strata, "Subgroup", species = "Lives")
kable(summary(Rappahannock.numbers, effic = TRUE, nopt = TRUE)$n.opt[,1:3], caption = "Observed vs. Optimal Effort")
```
```{r}
Rappahannock.numbers.summary <- summary(Rappahannock.numbers, effic = TRUE, nopt = TRUE)
Rappahannock.numbers.nopt <- Rappahannock.numbers.summary$n.opt
Rappahannock.reallocation <- left_join(Rappahannock.numbers.nopt, site.info, by = c("Strata" = as.character("Management Subgroups")))
site.acres$Station_ID <- as.numeric(site.acres$Station_ID)
Rappahannock.full <- left_join(Rappahannock.dat, site.acres, by = "Station_ID")
```
## Estimate the population mean and variance given the new strata and allocation
```{r}
k <- length(unique(Rappahannock.dat$Subgroup))
Rappahannock.dat3 <- Rappahannock.dat %>%
select(1:6)
varest.boot.dat.Rappahannock <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat.Rappahannock) <- c("Subgroup", "MEAN", "SD", "VAR")
varest.boot.dat.Rappahannock <- as.data.frame(varest.boot.dat.Rappahannock)
n.opt.test.cumm.Rappahannock <- cumsum(summary(Rappahannock.numbers, effic = T, nopt = T)$n.opt$Optimal[1:k])
n.opt.test.Rappahannock <- summary(Rappahannock.numbers, effic = T, nopt = T)$n.opt$Optimal[1:k]
start.pos.Rappahannock <- c(1, n.opt.test.cumm.Rappahannock[1:k-1]+1)
nopt.boot.dat.Rappahannock <- matrix(ncol = 7, nrow = k*100)
colnames(nopt.boot.dat.Rappahannock) <- c("Strata", "Observed", "Optimal", "Perc.Increase.Var.Opt",
"Compromise", "Perc.Increase.Var.comp", "run")
nopt.boot.dat.Rappahannock <- as.data.frame(nopt.boot.dat.Rappahannock)
j.start.Rappahannock <- seq(1,k*100, by = k)
j.end.Rappahannock <- seq(k, k*100, by = k)
subgroup.Rappahannock <- sort(unique(Rappahannock.dat3$Subgroup))
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6,nrow = n.opt.test.cumm.Rappahannock[k])
colnames(sub.dat1) <- names(Rappahannock.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- Rappahannock.dat3 %>%
filter(Subgroup == subgroup.Rappahannock[i])
rand.n <- sample(1:nrow(temp),n.opt.test.Rappahannock[i], replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos.Rappahannock[i]:n.opt.test.cumm.Rappahannock[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(Subgroup) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat.Rappahannock[j.start.Rappahannock[j]:j.end.Rappahannock[j],] <- sub.summary
}
```
```{r}
Rappahannock.sample <- n.opt.test.Rappahannock
Rappahannock.sample.cumm <- n.opt.test.cumm.Rappahannock
Rappahannock.plot.dat <- matrix(ncol = 3, nrow = k*1000)
colnames(Rappahannock.plot.dat) <- c("ER","Subgroup", "MEAN")
Rappahannock.plot.dat <- as.data.frame(Rappahannock.plot.dat)
Rappahannock.plot.dat$ER <- rep(seq(0, 0.9, 0.1), each = k*100)
Rappahannock.plot.dat$Subgroup <- rep(subgroup.Rappahannock, 1000)
Rappahannock.plot.dat$Subgroup <- factor(Rappahannock.plot.dat$Subgroup)
h.start <- seq(1, 1+(k*1000 - k*100), by = k*100)
h.end <- seq(k*100, k*1000, by = k*100)
Rappahannock.sub.sample <- Rappahannock.sample/sum(Rappahannock.sample)
Rappahannock.sub.cumm <- round(seq(1, 0.1, by = -0.1)*Rappahannock.sample.cumm[k], 0)
j.start <- seq(1,k*100, by = k)
j.end <- seq(k, k*100, by = k)
for(h in 1:10){
varest.boot.dat <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat) <- c("Subgroup", "MEAN", "SD", "VAR")
varest.boot.dat <- as.data.frame(varest.boot.dat)
sample.cumm <- cumsum(round(Rappahannock.sub.sample*Rappahannock.sub.cumm[h],0))
start.pos <- c(1, sample.cumm[1:k-1]+1)
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6, nrow = Rappahannock.sub.cumm[h])
colnames(sub.dat1) <- names(Rappahannock.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- Rappahannock.dat3 %>%
filter(Subgroup == subgroup.Rappahannock[i])
rand.n <- sample(1:nrow(temp),round(Rappahannock.sub.sample[i]*Rappahannock.sub.cumm[h],0), replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos[i]:sample.cumm[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(Subgroup) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat[j.start[j]:j.end[j],] <- sub.summary
}
Rappahannock.plot.dat$MEAN[h.start[h]:h.end[h]] <- varest.boot.dat$MEAN
}
```
```{r}
ggplot(Rappahannock.plot.dat, aes(y = MEAN, x = ER, colour = Subgroup, fill = Subgroup)) +
geom_smooth(se = FALSE) +
geom_point(size = 2) +
stat_summary(fun.data = mean_ci, geom = "ribbon", alpha = 0.5) +
theme_light() +
scale_color_brewer(palette = "Paired") +
scale_fill_brewer(palette = "Paired") +
theme(axis.text = element_text(size = 15, colour = "black")) +
theme(axis.title = element_text(size = 15)) +
theme(legend.text = element_text(size = 15)) +
theme(legend.title = element_text(size = 15)) +
labs(y = expression(paste("Mean Oysters m"^-2 %+-% "CI"))) +
labs(x = "Proportional Effort Reduction")
# Plot the change in quality of the data using the CV with each sequential reduction in effort
Rappahannock.plot.dat.cv <- Rappahannock.plot.dat %>%
group_by(Subgroup, ER) %>%
summarise(CV = cv(MEAN))
## Create a sequence of the no effort reduction CV for comparison
Rappahannock.plot.dat.cv$PctCV <- Rappahannock.plot.dat.cv$CV / rep(pull(Rappahannock.plot.dat.cv[c(seq(1,((k-1)*10)+1,10)),3]), each = 10) * 100 - 100
ggplot(Rappahannock.plot.dat.cv, aes(y = PctCV, x = ER, color = Subgroup, linetype = Subgroup)) +
geom_line(size = 1.2) +
geom_abline(intercept = 40, slope = 0, size = 1.2, linetype = "dashed") +
theme_light() +
scale_color_brewer(palette = "Paired") +
theme(axis.text = element_text(size = 15, colour = "black")) +
theme(axis.title = element_text(size = 15)) +
theme(legend.text = element_text(size = 15)) +
theme(legend.title = element_text(size = 15)) +
labs(y = "Relative Coefficient of Variation") +
labs(x = "Proportional Effort Reduction")
```
The current subgroups are *very* messy in terms of overlapping population estimates. However, it does still appear that a 40% reduction is possible in the Rappahannock.
```{r}
strat.kmeans.summary.Rappahannock <- Rappahannock.dat3 %>%
group_by(Subgroup) %>%
summarise(pMEAN = mean(Lives), pSD = sd(Lives), pVAR = var(Lives), nOBS = n()) %>%
mutate(lower.ci.mean = pMEAN - qt(1 - (0.05 / 2), nOBS) * (pSD/sqrt(nOBS)),
upper.ci.mean = pMEAN + qt(1 - (0.05 / 2), nOBS) * (pSD/sqrt(nOBS)))
kable(strat.kmeans.summary.Rappahannock, caption = "Population Stats based on all data")
Rappahannock.ci.summary <- Rappahannock.plot.dat %>%
group_by(ER, Subgroup) %>%
summarise(m.MEAN = mean(MEAN), sd.MEAN = sd(MEAN)) %>%
mutate(lower.ci.mean = m.MEAN - qt(1 - (0.05 / 2), 99) * sd.MEAN,
upper.ci.mean = m.MEAN + qt(1 - (0.05 / 2), 99) * sd.MEAN)
# kable(Rappahannock.ci.summary, caption = "Mean of the means")
```
Each year, total effort (i.e., number of grabs) in each cluster would be as follows:
```{r}
kable(data_frame("Subgroup" = subgroup.Rappahannock, "Future Effort" = round(Rappahannock.sub.sample*Rappahannock.sub.cumm[5]/13,0),
"Current Effort" = round(summary(Rappahannock.numbers, effic = TRUE, nopt = TRUE)$n.opt[,2][1:13]/13,0)))
```
This would mean a total of 234 grabs in the Rappahannock compared to 393 in previous years.
# Piankatank River
The Piankatank also lacked management subgroups, so clusters were identified.
```{r}
Piankatank.dat <- dat.full %>%
filter(River_name == "Piankatank River")
Piankatank.stations <- unique(Piankatank.dat$Station_ID)
colnames(site.info)[2] <- "Station_ID"
colnames(site.acres)[1] <- "Station_ID"
Piankatank.site.info <- site.acres %>%
filter(Station_ID %in% Piankatank.stations) %>%
arrange(Station_ID)
Piankatank.site.info$Station_ID <- as.numeric(Piankatank.site.info$Station_ID)
Piankatank.dat$Lives <- rowSums(Piankatank.dat[,16:52]*Piankatank.dat$Subsample_Factor)
Piankatank.dat$Markets <- rowSums(Piankatank.dat[,31:52]*Piankatank.dat$Subsample_Factor)
Piankatank.dat$Spat1 <- rowSums(Piankatank.dat[,55:67]*Piankatank.dat$Subsample_Factor)
Piankatank.dat <- Piankatank.dat %>% select(Station_ID, Sample_ID, Lives, Markets, Spat1)
```
## Set up strata and summarize
```{r}
Piankatank.strata <- data_frame("Strata" = Piankatank.site.info$Station_ID, NH = Piankatank.site.info$Acreage*4046.86)
Piankatank.numbers <- Stratify(Piankatank.dat, Piankatank.strata, "Station_ID", species = "Lives")
kable(summary(Piankatank.numbers, effic = TRUE, nopt = TRUE)$n.opt[,1:3], caption = "Observed vs. Optimal Effort")
```
```{r}
Piankatank.numbers.summary <- summary(Piankatank.numbers, effic = TRUE, nopt = TRUE)
Piankatank.numbers.nopt <- Piankatank.numbers.summary$n.opt
Piankatank.reallocation <- left_join(Piankatank.numbers.nopt, site.info, by = c("Strata" = as.character("Station_ID")))
site.acres$Station_ID <- as.numeric(site.acres$Station_ID)
Piankatank.full <- left_join(Piankatank.dat, site.acres, by = "Station_ID")
Piankatank.mv <- Piankatank.full %>%
group_by(Station_ID) %>%
summarise(Mean = mean(Lives), Variance = var(Lives))
Piankatank.mv2 <- Piankatank.dat %>%
group_by(Station_ID) %>%
summarise(Mean = mean(Lives), Variance = var(Lives))
Piankatank.site.info$Station_ID <- as.numeric(Piankatank.site.info$Station_ID)
Piankatank.for.mclust <- left_join(Piankatank.mv2, Piankatank.site.info, by=c("Station_ID" = "Station_ID"))
row.names(Piankatank.for.mclust) <- Piankatank.for.mclust$Station_ID
Piankatank.mva <- Piankatank.full %>%
group_by(Station_ID) %>%
summarise(Mean = mean(Lives), Variance = var(Lives), Acreage = mean(Acreage))
row.names(Piankatank.mva) <- Piankatank.mva$Station_ID
Piankatank.mva <- scale(Piankatank.mva)
```
## Clustering
```{r, fig.align='center', fig.height=5, fig.width=7}
Piankatank.wss <- matrix(nrow = 100, ncol = 10)
for(j in 1:100) {
wss <- (nrow(Piankatank.mva[,2:4])-1)*sum(apply(Piankatank.mva[,2:4], 2, var))
for (i in 1:10) Piankatank.wss[j,i] <- sum(kmeans(Piankatank.mva[,2:4], centers = i, iter.max = 1000, nstart = 1)$withinss)
}
Piankatank.wss <- as.data.frame(Piankatank.wss)
Piankatank.wss.mean <- colMeans(Piankatank.wss)
plot(1:10, Piankatank.wss.mean, type = "b", xlab = "Number of Clusters",
ylab = "Within groups SS")
#################################################################
k <- 3
#################################################################
```
```{r, fig.align='center', fig.height=5}
best.seed <- matrix(nrow = 1000, ncol = 2)
for (i in 1:1000){
set.seed(i)
fit <- kmeans(Piankatank.mva[,2:4], k, iter.max = 1000, nstart = 1, algorithm = "Hartigan-Wong")
best.seed[i,1] <- i
best.seed[i,2] <- fit$betweenss/fit$totss*100
}
best.seed <- as.data.frame(best.seed)
seed <- best.seed$V1[best.seed$V2 == max(best.seed$V2)][1]
set.seed(seed)
fit.Piankatank <- kmeans(Piankatank.mva[,2:4], k, iter.max = 1000, nstart = 1, algorithm = "Hartigan-Wong")
Piankatank.cluster.groups <- data.frame("Station_ID" = as.numeric(rownames(Piankatank.mva)))
Piankatank.cluster.groups$kmeans <- fit.Piankatank$cluster
clusplot(Piankatank.mva[,2:4], Piankatank.cluster.groups$kmeans, color = T, shade = T, labels = 2, lines = 0)
```
This plot shows how well those reefs cluster.
```{r}
a2 <- fit.Piankatank$cluster
suggested.strata.Piankatank <- data_frame("Code" = as.numeric(names(a2)), "Cluster" = a2)
site.name <- dat.full %>%
select(Station_ID, Full_station_name) %>%
mutate("Code" = Station_ID) %>%
select(Code, Full_station_name)
site.name <- site.name[!duplicated(site.name),]
suggested.strata.Piankatank <- left_join(suggested.strata.Piankatank, site.name, by = "Code") %>%
arrange(Cluster)
kable(suggested.strata.Piankatank, caption = "Suggested Strata")
```
## Add new suggested groups
```{r}
names(Piankatank.site.info) <- c("Station_ID", "Subgroup", "NH")
Piankatank.site.info$NH <- Piankatank.site.info$NH*4046.86 #convert acres to square meters
Piankatank.site.info$Station_ID <- as.character(Piankatank.site.info$Station_ID)
Piankatank.site.info2 <- data.frame(Piankatank.site.info)
Piankatank.domain <- data_frame(NH = Piankatank.site.info$NH,
"kmeans" = Piankatank.cluster.groups$kmeans)
#Summarize NH by new strata (aka domains)
Piankatank.domain.info <- Piankatank.domain %>%
group_by(kmeans) %>%
summarise(NH = mean(NH))
Piankatank.domain.info2 <- data.frame(Piankatank.domain.info)
Piankatank.dat2 <- left_join(Piankatank.dat, Piankatank.cluster.groups)
Piankatank.dat2 <- arrange(Piankatank.dat2, Station_ID)
Piankatank.dat2$Station_ID <- as.character(Piankatank.dat2$Station_ID)
Piankatank.dat3 <- data.frame(Piankatank.dat2)
Piankatank.site.info2 <- data.frame(Piankatank.site.info)
```
## Testing Stratification
How well is effort allocated based on current strata?
```{r}
# How well is effort allocated based on current strata?
names(Piankatank.domain.info2) <- c("Strata","NH")
Piankatank.numbers2 <- Stratify(Piankatank.dat3, Piankatank.domain.info2, "kmeans", species = "Lives")
kable(summary(Piankatank.numbers2, effic = T, nopt = T)$n.opt[,1:3])
```
## Estimate the population mean and variance given the new strata and allocation
```{r}
varest.boot.dat.Piankatank <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat.Piankatank) <- c("kmeans", "MEAN", "SD", "VAR")
varest.boot.dat.Piankatank <- as.data.frame(varest.boot.dat.Piankatank)
n.opt.test.cumm.Piankatank <- cumsum(summary(Piankatank.numbers2, effic = T, nopt = T)$n.opt$Optimal[1:k])
n.opt.test.Piankatank <- summary(Piankatank.numbers2, effic = T, nopt = T)$n.opt$Optimal[1:k]
start.pos.Piankatank <- c(1, n.opt.test.cumm.Piankatank[1:k-1]+1)
nopt.boot.dat.Piankatank <- matrix(ncol = 7, nrow = k*100)
colnames(nopt.boot.dat.Piankatank) <- c("Strata", "Observed", "Optimal", "Perc.Increase.Var.Opt",
"Compromise", "Perc.Increase.Var.comp", "run")
nopt.boot.dat.Piankatank <- as.data.frame(nopt.boot.dat.Piankatank)
j.start.Piankatank <- seq(1,k*100, by = k)
j.end.Piankatank <- seq(k, k*100, by = k)
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6,nrow = n.opt.test.cumm.Piankatank[k])
colnames(sub.dat1) <- names(Piankatank.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- Piankatank.dat3 %>%
filter(kmeans == i)
rand.n <- sample(1:nrow(temp),n.opt.test.Piankatank[i], replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos.Piankatank[i]:n.opt.test.cumm.Piankatank[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(kmeans) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat.Piankatank[j.start.Piankatank[j]:j.end.Piankatank[j],] <- sub.summary
}
#kable(head(varest.boot.dat.Piankatank, 10), caption = "Population mean for each strata with a random resampling with new effort")
```
```{r}
Piankatank.sample <- n.opt.test.Piankatank
Piankatank.sample.cumm <- n.opt.test.cumm.Piankatank
Piankatank.plot.dat <- matrix(ncol = 3, nrow = k*1000)
colnames(Piankatank.plot.dat) <- c("ER","KMEAN", "MEAN")
Piankatank.plot.dat <- as.data.frame(Piankatank.plot.dat)
Piankatank.plot.dat$ER <- rep(seq(0, 0.9, 0.1), each = k*100)
Piankatank.plot.dat$KMEAN <- rep(1:k, 1000)
Piankatank.plot.dat$KMEAN <- factor(Piankatank.plot.dat$KMEAN)
h.start <- seq(1, 1+(k*1000 - k*100), by = k*100)
h.end <- seq(k*100, k*1000, by = k*100)
Piankatank.sub.sample <- Piankatank.sample/sum(Piankatank.sample)
Piankatank.sub.cumm <- round(seq(1, 0.1, by = -0.1)*Piankatank.sample.cumm[k], 0)
j.start <- seq(1,k*100, by = k)
j.end <- seq(k, k*100, by = k)
for(h in 1:10){
varest.boot.dat <- matrix(ncol = 4, nrow = k*100)
colnames(varest.boot.dat) <- c("kmeans", "MEAN", "SD", "VAR")
varest.boot.dat <- as.data.frame(varest.boot.dat)
sample.cumm <- cumsum(round(Piankatank.sub.sample*Piankatank.sub.cumm[h],0))
start.pos <- c(1, sample.cumm[1:k-1]+1)
for(j in 1:100) {
sub.dat1 <- matrix(ncol=6, nrow = Piankatank.sub.cumm[h])
colnames(sub.dat1) <- names(Piankatank.dat3)
sub.dat1 <- as.data.frame(sub.dat1)
for(i in 1:k) {
temp <- Piankatank.dat3 %>%
filter(kmeans == i)
rand.n <- sample(1:nrow(temp),round(Piankatank.sub.sample[i]*Piankatank.sub.cumm[h],0), replace = T)
temp2 <- temp[rand.n,]
sub.dat1[start.pos[i]:sample.cumm[i],] <- temp2
}
sub.summary <- sub.dat1 %>%
group_by(kmeans) %>%
summarise("MEAN" = mean(Lives), "SD" = sd(Lives), "VAR" = var(Lives))
varest.boot.dat[j.start[j]:j.end[j],] <- sub.summary
}
Piankatank.plot.dat$MEAN[h.start[h]:h.end[h]] <- varest.boot.dat$MEAN
}
#kable(head(Piankatank.plot.dat, 10), caption = "Population mean for each strata with a random resampling with new effort")
```
```{r}
ggplot(Piankatank.plot.dat, aes(y = MEAN, x = ER, colour = KMEAN, fill = KMEAN)) +
geom_smooth(se = FALSE) +
geom_point(size = 2) +
stat_summary(fun.data = mean_ci, geom = "ribbon", alpha = 0.5) +
theme_light() +
scale_color_brewer(palette = "Paired") +
scale_fill_brewer(palette = "Paired") +
theme(axis.text = element_text(size = 15, colour = "black")) +
theme(axis.title = element_text(size = 15)) +
theme(legend.text = element_text(size = 15)) +
theme(legend.title = element_text(size = 15)) +
labs(y = expression(paste("Mean Oysters m"^-2 %+-% "CI"))) +