Advice on calculating a function to describe upper bound of data

I ha开发者_开发百科ve a scatter plot of a dataset and I am interested in calculating the upper bound of the data. I don't know if this is a standard statistical approach so what I was considering doing was splitting the X-axis data into small ranges, calculating the max for these ranges and then trying to identify a function to describe these points. Is there a function already in R to do this?

If it's relevant there are 92611 points.

You might like to look into quantile regression, which is available in the quantreg package. Whether this is useful will depend on whether you want the absolute maximum within your "windows" are whether some extreme quantile, say 95th or 99th, is acceptable? If you are not familiar with quantile regression, then consider the linear regression which fits a model for the expectation or mean response, conditional upon the model covariates. Quantile regression for the middle quantile (0.5) would fit a model to the median response, conditional upon the model covariates.

Here is an example using the quantreg package, to show you what I mean. First, generate some dummy data similar to the data you show:

set.seed(1)
N <- 5000
DF <- data.frame(Y = rev(sort(rlnorm(N, -0.9))) + rnorm(N),
                 X = seq_len(N))
plot(Y ~ X, data = DF)

Next, fit the model to the 99th percentile (or the 0.99 quantile):

mod <- rq(Y ~ log(X), data = DF, tau = .99)

To generate the "fitted line", we predict from the model at 100 equally spaced values in X

pDF <- data.frame(X = seq(1, 5000, length = 100))
pDF <- within(pDF, Y <- predict(mod, newdata = pDF))

and add the fitted model to the plot:

lines(Y ~ X, data = pDF, col = "red", lwd = 2)

This should give you this:

I would second Gavin's nomination for using quantile regression. Your data might be simulated with your X and Y each log-normally distributed. You can see what a plot of the joint distribution of two independent (no imposed correlation, but not necessarily cor(x,y)==0) log-normal variates looks like if you run:

x <- rlnorm(1000, log(300), sdlog=1)
y<- rlnorm(1000, log(7), sdlog=1)
plot(x,y, cex=0.3)

You might consider looking at their individual distributions with qqplot (in the base plotting functions) remembering that the tails of such distrubutions can behave in surprising manner. You should be more interested in how well the bulk of the values fit a particular distribution than the extremes ... unless of course your applications are in finance or insurance. Don't want another global financial crisis because of poor modeling assumptions about tail behavior, now do we?

qqplot(x, rlnorm(10000, log(300), sdlog=1) )