online turns a statistic (in haskell this can usually be thought of as a fold of a foldable) into an online algorithm.

Imagine a data stream, like an ordered indexed container or a time-series of measurements. An exponential moving average can be calculated as a repeated iteration over a stream of xs:

The 0.1 is akin to the learning rate in machine learning, or 0.9 can be thought of as a decaying or a rate of forgetting. An exponential moving average learns about what the value of x has been lately, where lately is, on average, about 1/0.1 = 10 x’s ago. All very neat.

The first bit of neat is speed. There’s 2 times and a plus. The next is space: an ema is representing the recent xs in a size as big as a single x. Compare that with a simple moving average where you have to keep the history of the last n xs around to keep up (just try it).

It’s so neat, it’s probably a viable monoidal category all by itself.

Haskell allows us to abstract the compound ideas in an ema and create polymorphic routines over a wide variety of statistics, so that they all retain these properties of speed, space and rigour.

```
av xs = L.fold (online identity (.* 0.9)) xs
-- av [0..10] == 6.030559401413827
-- av [0..100] == 91.00241448887785
```

`online identity (.* 0.9)`

is how you express an ema with a decay rate
of 0.9.

online works for any statistic. Here’s the construction of standard deviation using applicative style:

```
std :: Double -> L.Fold Double Double
std r = (\s ss -> sqrt (ss - s**2)) <$> ma r <*> sqma r
where
ma r = online identity (.*r)
sqma r = online (**2) (.*r)
```

1 cycle = 0.4 nanoseconds.

```
sum to 1,000
run first 2nd 3rd median av.
rSumInt' 9.72e3 1.68e3 1.56e3 1.63e3 1.71e3
rSumDouble' 1.41e6 2.94e5 3.04e5 9.19e4 1.84e5
rSumPoly' 9.17e4 9.15e4 9.13e4 7.90e4 1.10e5
rSumInt 1.58e4 1.18e4 1.17e4 1.17e4 1.17e4
rSumDouble 2.62e4 1.18e4 1.16e4 1.16e4 1.18e4
rSumPoly 1.17e4 1.17e4 1.16e4 1.16e4 1.16e4
rSumSum 1.16e4 1.16e4 1.17e4 1.16e4 1.16e4
rAvTestMain 2.85e4 1.19e4 1.19e4 1.19e4 1.20e4
rMaTest 1.26e4 1.20e4 1.20e4 1.20e4 1.27e4
rStdTest 2.14e5 1.26e5 8.05e5 1.16e5 2.03e5
rMaL1Test 1.73e5 8.30e4 1.14e5 7.61e4 1.21e5
rabsmaL1Test 3.34e5 5.89e4 5.93e4 5.93e4 1.06e5
```

```
stack build --test --exec "$(stack path --local-install-root)/bin/online-bench" --exec "$(stack path --local-bin)/pandoc -f markdown -i other/readme_.md -t markdown -o readme.md --filter pandoc-include --mathjax"
```