STL decomposition of time series with missing values for anomaly detection

Question

I am trying to detect anomalous values in a time series of climatic data with some missing observations. Searching the web I found many available approaches. Of those, stl decomposition seems appealing, in the sense of removing trend and seasonal components and studying the remainder. Reading STL: A Seasonal-Trend Decomposition Procedure Based on Loess, stl appears to be flexible in determining the settings for assigning variability, unaffected by outliers and possible to apply despite missing values. However, trying to apply it in R, with four years of observations and defining all the parameters according to http://stat.ethz.ch/R-manual/R-patched/library/stats/html/stl.html , I encounter error:

time series contains internal NAs

when na.action = na.omit, and

series is not periodic or has less than two periods

when na.action = na.exclude.

I have double checked that the frequency is correctly defined. I have seen relevant questions in blogs, but didn't find any suggestion that could solve this. Is it not possible to apply stl in a series with missing values? I am very reluctant to interpolate them, as I do not want to be introducing (and consequently detecting...) artifacts. For the same reason, I do not know how advisable it would be to use ARIMA approaches instead (and if missing values would still be a problem).

Please share if you know a way to apply stl in a series with missing values, or if you believe my choices are methodologically not sound, or if you have any better suggestion. I am quite new in the field and overwhelmed by the heaps of (seemingly...) relevant information.

Answer 1

In the beginning of stl we find

x <- na.action(as.ts(x))

and soon after that

period <- frequency(x)
if (period < 2 || n <= 2 * period) 
    stop("series is not periodic or has less than two periods")

That is, stl expects x to be ts object after na.action(as.ts(x)) (otherwise period == 1). Let us check na.omit and na.exclude first.

Clearly, at the end of getAnywhere("na.omit.ts") we find

if (any(is.na(object))) 
    stop("time series contains internal NAs")

which is straightforward and nothing can be done (na.omit does not exclude NAs from ts objects). Now getAnywhere("na.exclude.default") excludes NA observations, but returns an object of class exclude:

    attr(omit, "class") <- "exclude"

and this is a different situation. As mentioned above, stl expects na.action(as.ts(x)) to be ts, but na.exclude(as.ts(x)) is of class exclude.

Hence if one is satisfied with NAs exclusion then e.g.

nottem[3] <- NA
frequency(nottem)
# [1] 12
na.new <- function(x) ts(na.exclude(x), frequency = 12)
stl(nottem, na.action = na.new, s.window = "per")

works. In general, stl does not work with NA values (i.e. with na.action = na.pass), it crashes deeper in Fortran (see full source code here):

z <- .Fortran(C_stl, ...

Alternatives to na.new are not delightful:

• na.contaguous - finds the longest consecutive stretch of non-missing values in a time series object.
• na.approx, na.locf from zoo or some other interpolation function.
• Not sure about this one, but another one Fortran implementation can be found for Python here. One could use Python of possibly install R from source after some modifications, in case this module really allows missing values.

As we can see in the paper, there is no some simple procedure for missing values (like approximating them in the very beginning) which could be applied to the time series before calling stl. So considering the fact that original implementation is quite lengthy I would think about some other alternatives than whole new implementation.

Update: a quite optimal in many aspects choice when having NAs could be na.approx from zoo, so let us check its performance, i.e. compare results of stl with full data set and results when having some number of NAs, using na.approx. I am using MAPE as a measure of accuracy, but only for trend, because seasonal component and remainder crosses zero and it would distort the result. Positions for NAs are chosen at random.

library(zoo)
library(plyr)
library(reshape)
library(ggplot2)
mape <- function(f, x) colMeans(abs(1 - f / x) * 100)

stlCheck <- function(data, p = 3, ...){
  set.seed(20130201)
  pos <- lapply(3^(0:p), function(x) sample(1:length(data), x))
  datasetsNA <- lapply(pos, function(x) {data[x] <- NA; data})
  original <- data.frame(stl(data, ...)$time.series, stringsAsFactors = FALSE)
  original$id <- "Original"
  datasetsNA <- lapply(datasetsNA, function(x) 
    data.frame(stl(x, na.action = na.approx, ...)$time.series, 
               id = paste(sum(is.na(x)), "NAs"), 
               stringsAsFactors = FALSE))
  stlAll <- rbind.fill(c(list(original), datasetsNA))
  stlAll$Date <- time(data)
  stlAll <- melt(stlAll, id.var = c("id", "Date"))
  results <- data.frame(trend = sapply(lapply(datasetsNA, '[', i = "trend"), mape, original[, "trend"]))
  results$id <- paste(3^(0:p), "NAs")
  results <- melt(results, id.var = "id")
  results$x <- min(stlAll$Date) + diff(range(stlAll$Date)) / 4
  results$y <- min(original[, "trend"]) + diff(range(original[, "trend"])) / (4 * p) * (0:p)
  results$value <- round(results$value, 2)
  ggplot(stlAll, aes(x = Date, y = value, colour = id, group = id)) + geom_line() + 
    facet_wrap(~ variable, scales = "free_y") + theme_bw() +
    theme(legend.title = element_blank(), strip.background = element_rect(fill = "white")) + 
    labs(x = NULL, y = NULL) + scale_colour_brewer(palette = "Set1") +
    lapply(unique(results$id), function(z)
      geom_text(data = results, colour = "black", size = 3,
                aes(x = x, y = y, label = paste0("MAPE (", id, "): ", value, "%"))))
}

nottem, 240 observations

stlCheck(nottem, s.window = 4, t.window = 50, t.jump = 1)

co2, 468 observations

stlCheck(log(co2), s.window = 21)

mdeaths, 72 observations

stlCheck(mdeaths, s.window = "per")

Visually we do see some differences in trend in cases 1 and 3. But these differences are pretty small in 1 and also satisfactory in 3 considering sample size (72).