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```
#' Identify Point-Based Distance Threshold for Contact
#'
#' Sample from a multivariate normal distribution to create "in-contact" n
#' point pairs based on real-time-location systems accuracy, and
#' generate a distribution describing observed distances between point
#' pairs.
#'
#' This function is for adjusting contact-distance thresholds (spTh) to account
#' for positional accuracy of real-time-location systems, assuming random
#' (non-biased) error in location-fix positions relative to true locations.
#' Essentially this function can be used to determine an adjusted spTh value
#' that likely includes the majority of true contacts defined using the
#' initial spTh.
#'
#'
#' @param n Integer. Number of "in-contact" point-pairs used in the
#' expected-distance distribution(s). Defaults to 1000.
#' @param acc.Dist1 Numerical. Accuracy distance for point 1.
#' @param acc.Dist2 Numerical. Accuracy distance for point 2. If == NULL,
#' defaults to acc.Dist1 value.
#' @param pWithin1 Numerical. Percentage of data points within acc.Dist of true
#' locations for point 1.
#' @param pWithin2 Numerical. Percentage of data points within acc.Dist of true
#' locations for point 2. If == NULL, defaults to pWithin1 value.
#' @param spTh Numerical. Pre-determined distance representing biological
#' threshold for contact.
#' @keywords contact location point
#' @references Farthing, T.S., Dawson, D.E., Sanderson, M.W., and Lanzas,
#' C. 2020. Accounting for space and uncertainty in real-time-location-
#' system-derived contact networks. Ecology and Evolution 10(11):4702-4715.
#' @import foreach
#' @export
#' @return Output is a list containing 5 named vectors. The first vector
#' describes summary statistics of the simulated distance distribution. The
#' second and third vectors describes varied confidence intervals (50-99%)
#' for the simulated distribiution. The fourth vector describes adjusted
#' spTh values that will capture approximately 84, 98, and 100% of true
#' contacts given the pre-determined spTh value (all calculated using the
#' Empirical rule). Finally, the fifth vector describes the actial observed
#' frequency of captured true contact given the spTh adjustments listed in
#' the fourth vector.
#' @examples
#' findDistThresh(n = 10, acc.Dist1 = 0.5, acc.Dist2 = NULL,
#' pWithin1 = 90, pWithin2 = NULL, spTh = 0.5)
#'
findDistThresh<-function(n = 1000, acc.Dist1 = 0.5, acc.Dist2 = NULL, pWithin1 = 90, pWithin2 = NULL, spTh = 0.666){
#bind the following variables to the global environment so that the CRAN check doesn't flag them as potential problems
i <- NULL
dist.distributionFunc<-function(x, simFPR, n.outOfContact, range.outOfContact){ #generate an expected-distance distribution from two points pulled from normal distributions with mean values at point1 and point2 x- and y-values, and standard deviations derived from the acc.Dist and pWithin values.
euc=function(x) {
point1 = x.cor=unlist(c(unname(unlist(x[1])),unname(unlist(x[2]))))
point2 = x.cor=unlist(c(unname(unlist(x[3])),unname(unlist(x[4]))))
euc.dis = raster::pointDistance(p1 = point1, p2 = point2, lonlat = FALSE)
return(euc.dis)
}
conf1.1 <- (1 - unname(unlist(x[4]))/100)/2 #calculate alpha/2 for the accuracy distribution
conf1.2<-unlist(ifelse(conf1.1 >0, conf1.1, ((1 - 99.99/100)/2))) #if conf = 0, it is replaced with this value which is extremely close to 0, so that qnorm doesn't return an inf value
if(conf1.1 == 0){ #Ensure that users know that users know if conf1.1 was changed
warning("pWithin1 == 100%. To prevent an error here, probability of true locations falling outside of an acc.Dist1 radius around reported point locations changed from 0 to 0.01 (i.e., pWithin1 = 99.99). As a result, spTh estimates may be inflated.", immediate. = TRUE)
}
zscore1<-abs(stats::qnorm(conf1.2)) #calculate z-score
conf2.1<- (1 - unname(unlist(x[5]))/100)/2
conf2.2<-unlist(ifelse(conf2.1 >0, conf2.1,((1 - 99.99/100)/2))) #if conf = 0, it is replaced with this value which is extremely close to 0, so that qnorm doesn't return an inf value
if(conf2.1 == 0){ #Ensure that users know that users know if conf1.1 was changed
warning("pWithin2 == 100%. To prevent an error here, probability of true locations falling outside of an acc.Dist2 radius around reported point locations changed from 0 to 0.01 (i.e., pWithin1 = 99.99). As a result, spTh estimates may be inflated.", immediate. = TRUE)
}
zscore2<-abs(stats::qnorm(conf2.2)) #calculate z-score
x1<-data.frame(p1.x = stats::rnorm(unname(unlist(x[1])), mean = 1, sd = unname(unlist(x[2]))/zscore1),p1.y = stats::rnorm(unname(unlist(x[1])), mean = 1, sd = unname(unlist(x[2]))/zscore1),p2.x = stats::rnorm(unname(unlist(x[1])), mean = 1, sd = unname(unlist(x[3]))/zscore2),p2.y = stats::rnorm(unname(unlist(x[1])), mean = 1+unname(unlist(x[6])), sd = unname(unlist(x[3]))/zscore2), stringsAsFactors = TRUE) #note that one of the points is spTh-m away from the other
dist.distr<-apply(x1,1,euc)
dist.mean<-mean(dist.distr)
dist.sd<-stats::sd(dist.distr)
dataFall84_98_100 <- c((dist.mean + dist.sd), (dist.mean + (dist.sd*2)), (dist.mean + (dist.sd*3))) #due to the empirical rule, approximately we can calculate where 68, 95, and 99% of data fall (i.e., the distances required to capture contacts in these cases).
names(dataFall84_98_100 ) <- c("spTh_84%Capture", "spTh_98%Capture", "spTh_100%Capture") #names
TruePositive <- length(which(dist.distr <= as.numeric(x[6]))) #number of contacts correctly identified.
TPR<- TruePositive/as.integer(x[1]) #True positive rate (i.e., sensitivity)
distr.summary <- c(dist.mean, dist.sd, min(dist.distr), max(dist.distr), TPR)
names(distr.summary) <- c("mean", "sd", "min", "max", "TPR")
dataFall.noise <- (unlist(foreach::foreach(i = 1:length(dataFall84_98_100)) %do% length(which(dist.distr <= as.numeric(dataFall84_98_100[i])))) / as.integer(x[1])) #this gives you the proportion of observed contact durations for each sd interval, relative to the SpTh value. It's a measure of noise, as we expect the TPR to be constant, but observed positive-contact rates reported here inflate this value.
names(dataFall.noise) <- paste("freq_",c("84%Capture", "98%Capture", "100%Capture"), sep ="")
#now we calculate various confidence intervals (50-99%) to report.
CIseq <- seq(50, 95, 5)
CIseq <- c(CIseq, 99)
CIvec_upper <- NULL
CIvec_lwr <- NULL
for (i in CIseq) {
alphaOver2 <- (1 - i/100)/2
marginOfError <- abs(stats::qnorm(alphaOver2)) *
(dist.sd/sqrt(unname(unlist(x[1]))))
CIvec_upper <- c(CIvec_upper, dist.mean + marginOfError)
CIvec_lwr <- c(CIvec_lwr, dist.mean - marginOfError)
}
names(CIvec_upper) <- paste(CIseq, "%-CI", sep = "")
names(CIvec_lwr) <- paste(CIseq, "%-CI", sep = "")
out.list <- list(distr.summary, CIvec_upper, CIvec_lwr, dataFall84_98_100 , dataFall.noise)
names(out.list) <- c("distribution.summary", "CI_upper", "CI_lwr", "spTh.adjustments", "contact.frequency")
return(out.list)
}
if(is.null(pWithin2) == TRUE){pWithin2 = pWithin1} #If == NULL, defaults to pWithin1 value.
if(is.null(acc.Dist2) == TRUE){acc.Dist2 = acc.Dist1} #If == NULL, defaults to acc.Dist1 value.
inputFrame<-data.frame(matrix(ncol =6, nrow = 1), stringsAsFactors = TRUE)
inputFrame[,1]<-n
inputFrame[,2]<-acc.Dist1
inputFrame[,3]<-acc.Dist2
inputFrame[,4]<-pWithin1
inputFrame[,5]<-pWithin2
inputFrame[,6]<-spTh
distributions<-apply(inputFrame,1,dist.distributionFunc) #creates a matrix of distance-distribution CI values
return(distributions[[1]]) #for some reason distributions is a nested list, so we have to designate that we want the further nested objects.
}
```

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