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    <p>I use these incremental/recursive mean and median estimators, which both use constant storage:</p> <pre><code>mean += eta * (sample - mean) median += eta * sgn(sample - median) </code></pre> <p>where <em>eta</em> is a small learning rate parameter (e.g. 0.001), and <em>sgn</em>() is the signum function which returns one of {-1, 0, 1}. (Use a constant <em>eta</em> if the data is non-stationary and you want to track changes over time; otherwise, for stationary sources you can use something like <em>eta</em>=1/n for the mean estimator, where n is the number of samples seen so far... unfortunately, this does not appear to work for the median estimator.)</p> <p>This type of incremental mean estimator seems to be used all over the place, e.g. in unsupervised neural network learning rules, but the median version seems much less common, despite its benefits (robustness to outliers). It seems that the median version could be used as a replacement for the mean estimator in many applications.</p> <p>I would love to see an incremental mode estimator of a similar form...</p> <p><strong>UPDATE</strong></p> <p>I just modified the incremental median estimator to estimate arbitrary quantiles. In general, a quantile function (<a href="http://en.wikipedia.org/wiki/Quantile_function" rel="noreferrer">http://en.wikipedia.org/wiki/Quantile_function</a>) tells you the value that divides the data into two fractions: p and 1-p. The following estimates this value incrementally:</p> <pre><code>quantile += eta * (sgn(sample - quantile) + 2.0 * p - 1.0) </code></pre> <p>The value p should be within [0,1]. This essentially shifts the <em>sgn</em>() function's symmetrical output {-1,0,1} to lean toward one side, partitioning the data samples into two unequally-sized bins (fractions p and 1-p of the data are less than/greater than the quantile estimate, respectively). Note that for p=0.5, this reduces to the median estimator.</p>
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    1. PO"On-line" (iterator) algorithms for estimating statistical median, mode, skewness, kurtosis?
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    1. USTyler Streeter
    1. USTyler Streeter
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    1. COThis median estimator is great. Do you know if there are similar estimators for 0.25/0.75 quantiles?
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      1. USGacek
    2. CO@Gacek, sure: split the input stream into Lohalf < median and Hihalf > median, and use running-median on each half.
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      1. USdenis
    3. CO@Gacek, yes, see http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=614292&tag=1
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      1. USIan Goodfellow
 

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