Mismatch between documentation and code for the computation of non parametric variables IS and IV

[achey2016]

The documentation for IS and IV use formula with uncorrected variance
but the code use pandas.Series.var without specifying ddof=0, which by default correct for bias (using ddof=1).

For long recordings the results should be almost the same but for shorter recordings it could lead to slight differences with other tools.

in the documentation for IS

This variable is defined in [1]:

$$IS = \frac{d^{24h}}{d^{1h}}$$

with:

$$d^{1h} = \sum_{i}^{n}\frac{\left(x_{i}-\bar{x}\right)^{2}}{n}$$

where $x_{i}$ is the number of active (counts higher than a
predefined threshold) minutes during the $i^{th}$ period,
$\bar{x}$ is the mean of all data and $n$ is the number of
periods covered by the actigraphy data and with:

$$d^{24h} = \sum_{i}^{p} \frac{ \left( \bar{x}_{h,i} - \bar{x} \right)^{2} }{p}$$

What the current implementation does

$$IS = \frac{d^{24h}}{d^{1h}}$$

with:

$$d^{1h} = \sum_{i}^{n}\frac{\left(x_{i}-\bar{x}\right)^{2}}{n-1} = \mathrm{data.var()}$$

and:

$$d^{24h} = \sum_{i}^{p} \frac{ \left( \bar{x}_{h,i} - \bar{x} \right)^{2} }{p-1} = \mathrm{data.groupby([ data.index.hour, data.index.minute, data.index.second] ).mean().var()}$$

[viranovskaya]

Thanks for flagging this. I checked the current master branch, and both _interdaily_stability() and _intradaily_variability() still use Series.var() with the default ddof=1, while the equations in the documentation use population variance.

For the default 1-hour IS, the difference is not necessarily negligible even for long recordings: with complete hourly data, the correction for the 24-bin daily profile tends to 24/23, or about 4.35%, relative to ddof=0.

I can prepare a small synthetic regression test and a patch using ddof=0. Since this would change existing results, would you prefer a direct correction with a changelog note, or a compatibility/deprecation path?