jointseg (version 1.0.2)

estimateSd: Robust standard deviation estimator

Description

Estimate standard deviation of an unimodal signal with possible changes in mean

Usage

estimateSd(y, method = c("Hall", "von Neumann"))

Arguments

y

A numeric vector

method

Method used to estimate standard deviation

Details

von Neumann's estimator is proportional to the mean absolute deviation (mad) of the first-order differences of the original signals: mad(diff(y). By construction this estimator is robust to 1) changes in the mean of the signal (through the use of differences) and 2) outliers (through the use of mad instead of mean).

The proportionality constant \(1/\sqrt 2 \times 1/\Phi^{-1}(3/4)\) ensures that the resulting estimator is consistent for the estimation of the standard deviation in the case of Gaussian signals.

Hall's estimator is a weigthed sum of squared elements of y. Let m=3. \(sigma^2 = (\sum_{k=1}^{n-m}\sum_{j=1}^{m+1}(\code{wei[i]}\code{y}[i+k])^2)/(n-m)\)

References

Von Neumann, J., Kent, R. H., Bellinson, H. R., & Hart, B. T. (1941). The mean square successive difference. The Annals of Mathematical Statistics, 153-162.

Peter Hall, J. W. Kay and D. M. Titterington (1990). Asymptotically Optimal Difference-Based Estimation of Variance in Nonparametric Regression Biometrika,521-528

Examples

# NOT RUN {
n <- 1e4
y <- rnorm(n)  ## a signal with no change in mean
estimateSd(y)
estimateSd(y, method="von Neumann")
sd(y)
mad(y)

z <- y + rep(c(0,2), each=n/2)  ## a signal with *a single* change in mean
estimateSd(z)
estimateSd(z, method="von Neumann")
sd(z)
mad(z)

z <- y + rep(c(0,2), each=100)  ## a signal with many changes in mean
estimateSd(z)
estimateSd(z, method="von Neumann")
sd(z)
mad(z)  

# }