robfilter (version 3.0)

adore.filter: A Robust Adaptive Online Repeated Median Filter for Univariate Time Series

Description

Procedure for robust online extraction of low frequency components (the signal) from a univariate time series by a moving window technique with adaptive window width selection (ADaptive Online REpeated median FILTER).

Usage

adore.filter(y, p.test = 15, minNonNAs = 5,
                min.width = 11, max.width = 100, width.search="geometric",
                rtr=2, extrapolate=TRUE, calc.qn = FALSE)

Value

adore.filter returns an object of class adore.filter. An object of class adore.filter is a list containing the following components:levela numeric vector containing the signal level extracted by the RM filter with adaptive window width.slopea numeric vector containing the corresponding slope within each time window.widtha numeric vector containing the corresponding window width used for the level and slope estimations.level.lista list which contains with as many elements as the length of the input time series. If at time t, the window width was not reduced, the entry level.list[[t]] simply corresponds to level[t]. However, if more than one iteration took place, level.list[[t]] is a vector which contains all level estimations which were evaluated until the final estimate mu[t] passed the goodness of fit test and was stored.slope.lista list containing the slope estimations corresponding to the values in level.list.width.lista list containing the window widths used for the estimations in level.list and slope.list.sigmaa numeric vector containing the corresponding scale within each time window estimated by the robust Qn estimator (only calculated if calc.qn = TRUE, else sigma does not exist).In addition, the original input time series is returned as list member y, and the settings used for the analysis are returned as the list members min.width, max.width, width.search, p.test, minNonNAs, rtr, extrapolate, and calc.qn. Application of the function plot to an object of class aoRM returns a plot showing the original time series with the filtered output.

Details

The adore.filter works by applying Repeated Median (RM) regression (Siegel, 1982) to a moving time window with a length varying between min.width and max.width. For each point in time, the window width is adapted to the current data situation by a goodness of fit test for the most recent signal level estimation. The test uses the absolute value of the sum of the RM residuals in the subset specified by p.test. The critical value for the test decision corresponds to a slightly modified 0.95-quantile of the distribution of the test statistic and is stored in the data set critvals. A more detailed description of the filter can be found in Schettlinger, Fried, Gather (2008).

References

Schettlinger, K., Fried, R., Gather, U. (2008) Real Time Signal Processing by Adaptive Repeated Median Filters, International Journal of Adaptive Control and Signal Processing, submitted. Siegel, A.F. (1982) Robust Regression Using Repeated Medians, Biometrika 69 (1), 242-244. Rousseeuw, P. J., Croux, C. (1993) Alternatives to the Median Absolute Deviation, Journal of the American Statistical Association 88, 1273-1283.

See Also

robreg.filter, wrm.filter, madore.filter.

Examples

# # # # # # # # # #
# Short and noise-free time series
series <- c(rep(0,30),rep(10,30),seq(10,5,length=20),seq(5,15,length=20))

# Adaptive online signal extraction without & with 'restrict to range' rule
t.without.rtr <- adore.filter(series, rtr=0)
plot(t.without.rtr)
t.with.rtr1 <- adore.filter(series, rtr=1)
lines(t.with.rtr1$level, col="blue")
t.with.rtr2 <- adore.filter(series)
lines(t.with.rtr2$level, col="green3",lty=2)
legend("top",c("Signal with rtr=1","Signal with rtr=2"),col=c("blue","green3"),lty=c(1,2),bty="n")

# # # # # # # # # #
# Short and noise-free time series + 1 outlier
ol.series <- series
ol.series[63] <- 3

# Adaptive online signal extraction without & with 'restrict to range' rule
t.without.rtr <- adore.filter(ol.series, rtr=0)
plot(t.without.rtr)
t.with.rtr1 <- adore.filter(ol.series, rtr=1)
lines(t.with.rtr1$level, col="blue")
t.with.rtr2 <- adore.filter(ol.series)
lines(t.with.rtr2$level, col="green3",lty=2)
legend("top",c("Signal with rtr=1","Signal with rtr=2"),col=c("blue","green3"),lty=c(1,2),bty="n")

# # # # # # # # # #
# Noisy time series with level shifts, trend changes and shifts in the scale of the error term
true.signal  <- c(rep(0,150),rep(10,150),seq(10,5,length=100),seq(5,15,length=100))
series2      <- true.signal + c(rnorm(250,sd=1), rnorm(200,sd=3), rnorm(50,sd=1))

# Adaptive online signal extraction with additional Qn scale estimation
s2 <- adore.filter(series2, calc.qn=TRUE)
par(mfrow=c(3,1))
plot(s2)
plot(s2$sigma,type="l",main="Corresponding Qn Scale Estimation",ylab="sigma",xlab="time")
lines(c(rep(1,250),rep(3,200),rep(1,150)),col="grey")
legend("topleft",c("True scale","Qn"),lty=c(1,1),col=c("grey","black"),bty="n")
plot(s2$width,type="l",main="Corresponding Window Width",ylab="width",xlab="time")