Solved – What does weighted cumulative frequency distribution mean

Question

I have two sets of data (temperature and catch) and I am following a proposed method in an article I am reading on the empirical cumulative function (ECDF) analysis. Firstly, I have derived the ECDF for my temperature data using the ecdf function in R. The second step was to get the catch-weighted cumulative distribution for temperature, which I honestly cannot understand the concept.

$$
W(t)=\frac1n\sum_{i=1}^n\frac{y_i}{y}I_{\{x_i\}}
$$

where $y_i$ is the catch for day $i$; $y$ mean is the average of catch for all days and $I_{\{x_i\}}$ is the indication function.

How do I compute for this curve?

Answer 1

Here are a couple of base R suggestions, one for where the weights are integers but not too large and the second for where the weights are simply positive

# example data
df <- data.frame(temp=c(50,20,10,40), weight=c(3,1,4,2))

# unweighted empirical CDF
plot.ecdf(df$temp,
  main="unweighted ecdf")

# weighted empirical CDF if weights are positive integers or counts
plot.ecdf(rep(df$temp, df$weight),
  main="weighted ecdf 1 - using counts")

# weighted empirical CDF if weights are positive 
dfsorted <- df[order(df$temp), ]
dfsorted$cumfreq <- cumsum(dfsorted$weight) / sum(dfsorted$weight)
dfsorted2 <- dfsorted[rep(1:nrow(df), each=2),]
dfsorted2$cumfreq <- c(0,dfsorted2$cumfreq[-2*nrow(df)])
plot(dfsorted2$temp, dfsorted2$cumfreq, type="l",
  main="weighted ecdf 2 - general weights", xlab="temp", ylab="cumfreq")

So the unweighted ecdf looks like

and the first weighted ecdf looks like

and the second weighted ecdf looks like