# How to calculate prediction intervals for LOESS?

I have some data that I fitted using a LOESS model in R, giving me this:

The data has one predictor and one response, and it is heteroscedastic.

I also added confidence intervals. The problem is that the intervals are confidence intervals for the line, whereas I am interested in the prediction intervals. For example, the bottom panel is more variable then the top panel, but this is not captured in the intervals.

This question is slightly related: Understanding the confidence band from a polynomial regression, especially the answer by @AndyW, however in his example he uses the relatively straightforward interval="predict" argument that exists in predict.lm, but it is absent from predict.loess.

So I have two very related questions:

1. How do I get the pointwise prediction intervals for LOESS?
2. How can I predict values that will capture that interval, i.e. generate a bunch of random numbers that will eventually look somewhat like the original data?

It is possible that I don't need LOESS and should use something else, but I am unfamiliar with my options. Basically it should fit the line using local regression or multiple linear regression, giving me error estimates for the lines, and in addition also different variances for different explanatory variables, so I can predict the distribution of the response variable (y) at certain x values.

• Is this a pointwise prediction interval? Mar 13, 2015 at 5:31
• What do you mean by "this"? And I'm not sure if it's pointwise or not. My question 2 is what I'm looking for - unfortunately I am not familiar with the nomenclature. Mar 13, 2015 at 5:34
• By 'this' I mean "the thing the question is asking about in the title" Mar 13, 2015 at 5:38
• The spread may be variable (that's why I opted for local regression in the first place). Single predictor. Mar 13, 2015 at 7:52
• For anyone else wondering about point wise vs simultaneous intervals: en.wikipedia.org/wiki/Confidence_and_prediction_bands Jun 10, 2022 at 8:09

I don't know how to do prediction bands with the original loess function but there is a function loess.sd in the msir package that does just that! Almost verbatim from the msir documentation:

library(msir)
data(cars)
# Calculates and plots a 1.96 * SD prediction band, that is,
# a 95% prediction band
l <- loess.sd(cars, nsigma = 1.96)
plot(cars, main = "loess.sd(cars)", col="red", pch=19)
lines(l$x, l$y)
lines(l$x, l$upper, lty=2)
lines(l$x, l$lower, lty=2)


Your second question is a bit trickier since loess.sd doesn't come with a prediction function, but you can hack it together by linearly interpolating the predicted means and SDs you get out of loess.sd (using approx). These can, in turn, be used to simulate data using a normal distribution with the predicted means and SDs:

# Simulate x data uniformly and y data acording to the loess fit
sim_x <- runif(100, min(cars[,1]), max(cars[,1]))
pred_mean <- approx(l$x, l$y, xout = sim_x)$y pred_sd <- approx(l$x, l$sd, xout = sim_x)$y
sim_y <- rnorm(100, pred_mean, pred_sd)

# Plots 95% prediction bands with simulated data
plot(cars, main = "loess.sd(cars)", col="red", pch=19)
points(sim_x, sim_y, col="blue")
lines(l$x, l$y)
lines(l$x, l$upper, lty=2)
lines(l$x, l$lower, lty=2)


• Exactly what I was looking for. When looking at the method he used by seeing the code with loess.sd, it's not too different from what @rnso suggested in a comment to another question of mine. Thanks! Apr 9, 2015 at 12:30
• Bootstrap to generate the intervals? Jan 30, 2017 at 14:22