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 4th 5th 40th %
sumInt [0..] 6.064e3 1.746e3 1.560e3 1.540e3 1.626e3 1.544e3 cycles
sumDouble [0..] 7.835e5 3.032e5 3.104e5 2.837e5 3.051e5 8.957e4 cycles
sumPoly [0..] 1.139e5 7.660e4 7.638e4 7.648e4 7.636e4 7.674e4 cycles
sum Int 1.601e4 1.186e4 1.167e4 1.158e4 1.176e4 1.168e4 cycles
sum Double 2.756e4 1.189e4 1.158e4 1.163e4 1.158e4 1.159e4 cycles
sum Poly 1.170e4 1.172e4 1.163e4 1.166e4 1.158e4 1.164e4 cycles
fold sum 1.177e4 1.174e4 1.178e4 1.185e4 1.177e4 1.175e4 cycles
fold av 2.935e4 1.190e4 1.181e4 1.182e4 1.177e4 1.181e4 cycles
fold ma 1.289e4 1.202e4 1.205e4 1.203e4 1.206e4 1.201e4 cycles
fold std 2.052e5 1.210e5 7.236e5 1.364e5 1.348e5 1.324e5 cycles
fold maL1 8.218e4 1.192e5 1.052e5 2.567e5 1.168e5 7.856e4 cycles
fold absmaL1 3.405e5 5.966e4 5.976e4 6.025e4 5.939e4 5.975e4 cycles
```

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