Peak Detection in Time Series

Question

I'm currently working on a little project in which I want to compare two time-series. The similarity measure is really vague, they are considered to be similar if the two time series roughly have the same shape.

So I thought to myself "Well if they only need to have the same shape, I just compare the peaks of the two time-series, if the peaks are at the same position, then surely the time-series will be similar"

My problem now is to find a good algorithm for the peak detection. I used google, but I only came up with the paper Simple Algorithms for Peak Detection in Time-Series. The problem is, the algorithms described in this paper work well with really extreme and thin peaks, but in the most cases, my time-series have rather flat peaks so they will not be detected.

Does anybody know where I could find or search for an algorithm which would detect the peaks shown in the following image?

Answer 1

You seem to simply look for slope inversion (from positive to negative and vice versa). A rough java algo could be (not tested):

List<Point> points = ... //all the points in your curve
List<Point> extremes = new ArrayList<Point> ();
double previous = null;
double previousSlope = 0;

for (Point p : points) {
    if (previous == null) { previous = p; continue; }
    double slope = p.getValue() - previous.getValue();
    if (slope * previousSlope < 0) { //look for sign changes
        extremes.add(previous);
    }
    previousSlope = slope;
    previous = p;
}

Finally, a good way to measure similarity is correlation. In your case, I would look at % move correlation (in other words, you want your 2 series to go up or down at the same time) - that's typically what is done in finance where you calculate the correlation between 2 assets returns for example:

• create 2 new series with the % move for each point of the 2 series
• calculate the correlation between those 2 series

You can read more about returns correlations here for example. In summary, if your values are:

Series 1  Series 2
 100        50
 98         49
 100        52
 102        54

The "returns" series will be:

Series 1  Series 2
 -2.00%     -2.00%
 +2.04%     +6.12%
 +2.00%     +3.85%

And you calculate the correlation of those 2 returns series (in this example: 0.96) to get a measure of how much the 2 curves look alike. You might want to adjust the result for variance (i.e. if one shape has a much wider range than the other).

Answer 2

You can use a very simple local extremes detector:

// those are your points:
double[] f = {1, 2, 3, 4, 5, 6, 5, 4, 7, 8, 9, 3, 1, 4, 6, 8, 9, 7, 4, 1};
List<Integer> ext = new ArrayList<Integer> ();
for (int i = 0; i<f.length-2; i++) {
  if ((f[i+1]-f[i])*(f[i+2]-f[i+1]) <= 0) { // changed sign?
    ext.add(i+1);
  }
}
// now you have the indices of the extremes in your list `ext`

This will work nice with smooth series. If you have a certain variation in your data, you should put it through a low pass filter first. A very simple implementation of a low pass filter would be the moving average (every point is replaced by the average of the nearest k values, with k being the window size).