I'm trying to calculate a covariance matrix using weighted data in a single pass, and I'm not sure that I'm doing it correctly. I found a wikipedia article which gave the following python* code:

def online_covariance(data1, data2):
    mean1 = mean2 = 0.
    M12 = 0.
    n = len(data1)
    for i in range(n):
        delta1 = (data1[i] - mean1) / (i + 1)
        mean1 += delta1
        delta2 = (data2[i] - mean2) / (i + 1)
        mean2 += delta2
        M12 += i * delta1 * delta2 - M12 / (i + 1)
    return n / (n - 1.) * M12

I think I should be able to adapt this to include weights by sticking in weights in a few places, something like this:

def online_covariance(data1, data2, weights):
    mean1 = mean2 = 0.
    M12 = 0.
    n = len(data1)
    for i in range(n):
        wt = weights[i]
        delta1 = (data1[i] - mean1) * wt / (i + wt)
        mean1 += delta1
        delta2 = (data2[i] - mean2) * wt / (i + wt)
        mean2 += delta2
        M12 += i / wt * delta1 * delta2 - M12 * wt / (i + wt)
    return n / (n - 1.) * M12

I'm aware that there may be some problems with the n / (n - 1.) factor in the last line, but I'm not overly concerned with that since I have large statistics and weights are all small compared to the sum weight.

I've run some tests and the output looks reasonable, but my "derivation" here is pretty crude, is it valid? Is there some better reference on this than the wikipedia article I found?

*(note that I don't care about it being python, and in reality I'm calculating the full matrix, not just one element)

  • $\begingroup$ You can always just write a few test cases and check that you get the same result as with the full weighted covariance. Julia has some online and streaming packages under MIT license. I'm not aware of any Python package, although it might be used inside several packages. (The denominator might need sum of previous weights instead of i.) $\endgroup$
    – Josef
    Commented Jan 30, 2016 at 18:04
  • $\begingroup$ Did you ever find out if your algorithm is correct? $\endgroup$
    – Azmisov
    Commented Apr 21, 2017 at 3:14
  • $\begingroup$ No, but it seemed "about right" in a few limiting cases (very big and small weights, and weights of 1), so I just used it and moved on. $\endgroup$
    – Shep
    Commented Apr 23, 2017 at 9:55

1 Answer 1


I dug into the derivation for the incremental, weighted variance algorithm. I came up with this, which gives the correct results. It turned out to be just a really simple modification of the variance algorithm. I've updated the wikipedia article with the code sample:

def online_weighted_covariance(data1, data2, data3):
    meanx = meany = 0
    wsum = wsum2 = 0
    C = 0
    for x, y, w in zip(data1, data2, data3):
        wsum += w
        wsum2 += w*w
        dx = x - meanx
        meanx += (w / wsum) * dx
        meany += (w / wsum) * (y - meany)
        C += w * dx * (y - meany)

    population_covar = C / wsum
    # Bessel's correction for sample variance
    # Frequency weights
    sample_frequency_covar = C / (wsum - 1)
    # Reliability weights
    sample_reliability_covar = C / (wsum - wsum2 / wsum)

Note we use dx when incrementing C; you can't do (x - meanx), since you need to use the old meanx.


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