Gaussian smoothing of time series

Question

Venkatessh 2013-5-16

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回答： Andreas Pagel 2021-8-5

I have a time series with measurements taken at time t along with measurement uncertainties. I would like to smooth this data with a Gaussian function using for example, 10 day smoothing time.

How could this be done?

Thank you

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Answer 1

Wayne King 2013-5-16

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You do not tell us how many samples represents 10 days in your t variable. That is an important piece of missing information. Here, I'll just assume that t is in days and you have 1 sample per day. You'll have to adjust accordingly if that is not accurate.

If you have the Signal Processing Toolbox, you can use gausswin

    x = randn(1000,1);
    w = gausswin(10);
    y = filter(w,1,x);

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Xen 2018-4-25

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Please correct me if I am wrong, but the accepted answer has a problem. The gaussian window is not normalized, thus your filtered vector will have larger values than expected. I created a window of length 5 and this essentially doubled the amplitude of my vector. w must have a unit sum:

w = w/sum(w);

Chinmayee L M 2021-8-1

I ran to the same problem. It is not normalised. But, can you please explain the normalisation that you have suggested? How is that a normalisation?

This doesn't take into account the length and width of the window. I want a normalisation factor that accounts for the length and width of the window.

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Answer 2

Image Analyst 2013-5-16

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If you don't have the Signal Processing Toolbox, make up your weighted window, then use conv():

filteredSignal = conv(originalSignal, gaussianWindow);

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Ashraful Haque 2020-5-18

Hey I know this comment is from a long time ago. But Hopefully you see my reply. My question was how do I create a gaussian window function without the signal processing toolbox? What are the input(s) and output(s)?

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Answer 3

Andreas Pagel 2021-8-5

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I had a similar issue with some 'random' noise spikes on the signal which I wanted to eliminate. The usual smoothing and moving avg approaches I found when searching for solutions did not match my expectations as the noise would still distord the results.

So, I created some kind of of a gauss filter, using a scatter recording approach.:

the code is still a bit rough but does it's job.

I get 10 readings, record them in an array thay counts how ofter a given value is found and pull then the value that got most hits;

#define sampleSize 10
int sampleArray[sampleSize + 1][2] = {{0}}; // initialise array

Sorry for being lazy...

I know it's not optimal but I like the easy way of using real references for adressing the array, ie 1st record sits in array[1]

push () records the values and increments the counter for a given value:

void push (int val) {
  short i = 0;
  for (i = 1; i <= sampleSize; i++) {
    if (debug) Serial.printf("%i-%i: %i #%i\n", i, val, sampleArray[i][0], sampleArray[i][1]);
    if (sampleArray[i][0] == val) {
      ++sampleArray[i][1];
      return;
    }
    else if (sampleArray[i][0] == 0) {
      sampleArray[i][0] = val;
      ++sampleArray[i][1];
      return;
    }
    else if (i == sampleSize) Serial.printf("ERROR - too many values: %i\n", i);
  }
}

and pull() returns the value with most hits:

int pull() {
  int maxCnt = 0;
  int maxVal = 0;
  short i = 0;
  for (i = 1; i <= sampleSize; i++) {
    if (sampleArray[i][1] > maxCnt) {
      maxCnt = sampleArray[i][1];
      maxVal = sampleArray[i][0];
    }
  }
  return maxVal;
}

since I wanted to simplify the tests, I did not even use a data source but simply used random numbers.

int readValue() {
  short i = 0;
  // reinit array
  for (i = 1; i <= sampleSize; i++) {
    sampleArray[i][0] = 0; 
    sampleArray[i][1] = 0;
  }
  for (i = 1; i <= sampleSize; i++) {
    //push(AnalogRead(ADC));
    push(random(20, 35));
  }
  return pull();
}