hampel

hampel(x; spread=mad(x), threshold=2)

Identify outliers using the Hampel criterion.

Given vector x, identify elements xₖ such that

|xₖ - m| > t S,

where m is the median of the elements, the dispersion scale S is provided by the function spread, and the parameter t is given by threshold. The return value is a Bool vector.

By default, spread is mad and threshold is 2.

hampel(x, K; spread=mad, threshold=2, boundary=:truncate)

Apply a windowed Hampel filter to a time series.

Given vector x and half-width K, apply a Hampel criterion within a sliding window of width 2K+1. The median m of the window replaces the element xₖ at the center of the window if it satisfies

|xₖ - m| > t S,

where the dispersion scale S is provided by the function spread and the parameter t is given by threshold. The window shortens near the beginning and end of the vector to avoid referencing fictitious elements. Larger values of t make the filter less agressive, while t=0 is the standard median filter.

For recursive filtering, see hampel!

The value of boundary determines how the filter handles the boundaries of the vector:

• :truncate (default): the window is shortened at the boundaries

• :reflect: values are reflected across the boundaries

• :repeat: end values are repeated as necessary

hampel(x, weights; ...)

Apply a weighted Hampel filter to a time series.

Given vector x and a vector weights of positive intgers, before computing the criterion each element in the window is repeated by the number of times given by its corresponding weight. This is typically used to make the central element more influential than the others.

CREDITS

This function is adapted from HampelOutliers.jl and you should consult the original source for more details and examples. The differences with respect to original HampelOutliers.jl functions is that here we created different methods for Hampel.identify and Hampel.filter and called them collectively just hampel and let the multi-dispatch do the work.

Returns

• A vector of Boolean saying if points were considered regular or outliers.

or

• A filtered version of x

Example

using GMT

t = (1:50) / 10;
x = [1:2:40; 5t + (@. 6cos(t + 0.5(t)^2)); fill(40,20)];
x[12] = -10; x[50:52] .= -12; x[79:82] .= [-5, 50, 55, 0];
m = hampel(x, 4, threshold=0);
y = hampel(x, 4);

plot(x, marker=:point, mc=:blue, lc=:blue, label="Original", xlabel="k", ylabel="x_k")
scatter!(m, ms="2p", mc=:red, MarkerEdgeColor=true, label="Median filter")
scatter!(y, ms="2p", mc=:green, MarkerEdgeColor=true, label="Hampel filter", show=true)