Wolfgang Resch - Notes

Smoothing data with Savitzky-Golay filters

June 26, 2014

Savitzky-Golay filters can be used to smooth data by fitting a low degree polynomial to the values in a sliding window using least squares. For equally spaced data points a set of coefficients can be calculated that can be applied to a subset of the data in a sliding window to estimate the smoothed signal at the center of the window and obtain results equivalent to fitting each window. These filters are good at preserving line shape while removing higher frequency noise. For more details see the Wikipedia entry.

In addition to creating smoothed data, smoothed versions of the mth derivative of the data can also be generated by Savitzky-Golay filters.

The sgolay function in the signal package calculates these coefficients for us and the filter (from signal, not base R) can be used to apply these coefficients to data. Rather than returning a single set of coefficients that estimates the signal at the center of the sliding window based on values before and after the center of the window, sgolay also returns coefficients that only use values after the point to be estimated, coefficients that only use values preceeding the point to be estimated, and everything in between. That way, the filter can be used to estimate smoothed signal at all points of the signal, including the end points where there are no preceeding/following data points.

Some example data:

n <- 100
x <- (1:n) / n
clean.data <- ((exp(1.2*x)+1.5*sin(7*x))-1)/3
df <- data.frame(
    x = x,
    clean = clean.data,
    noisy = clean.data + rnorm(n, 0, 0.12)

The filter coefficients (1nd order polynomials, 0th derivative, length 11):

sg <- sgolay(p=1, n=13, m=0)
## [1] 13 13

Applied to the data

df$sg <- filter(sg, df$noisy)

Plotting the results

p <- ggplot(df) +
    geom_point(aes(x, noisy), size = 2) +
    geom_line(aes(x, clean), linetype = "longdash") +
    geom_line(aes(x, sg), col = "red2", size = 1) +
    ylab("y") +
## R version 3.1.0 (2014-04-10)
## Platform: x86_64-apple-darwin10.8.0 (64-bit)
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
## attached base packages:
## [1] stats     graphics  grDevices utils     datasets  methods   base     
## other attached packages:
## [1] knitr_1.6     ggplot2_1.0.0 signal_0.7-4 
## loaded via a namespace (and not attached):
##  [1] colorspace_1.2-4 digest_0.6.4     evaluate_0.5.5   formatR_0.10    
##  [5] grid_3.1.0       gtable_0.1.2     labeling_0.2     MASS_7.3-31     
##  [9] munsell_0.4.2    plyr_1.8.1       proto_0.3-10     Rcpp_0.11.2     
## [13] reshape2_1.4     scales_0.2.4     stringr_0.6.2    tools_3.1.0