Issue/655: Standard deviation and variance

[allmywatts]

This PR implements std and var methods (see: #655) using the same method as used for var_axis, std_axis.
The variance is computed by flattening the array of N dimensions into a 1D array of the length .len() of the original array.

The return type is scalar A instead of Array<A, D:Smaller>

An attempt was made to refactor var_axis to use var, but this resulted in a performance regression so original implementation is kept and we accept some code duplication. 😛

pub fn var_axis(&self, axis: Axis, ddof: A) -> Array<A, D::Smaller>
where
    A: Float + FromPrimitive,
    D: RemoveAxis,
{
    let mut sum_sq = Array::<A, D::Smaller>::zeros(self.dim.remove_axis(axis));

    for (lane, sum) in self.lanes(axis).into_iter().zip(sum_sq.iter_mut()) {
        *sum = lane.var(ddof);
    }
    sum_sq
}

Thanks @LukeMathWalker for the mentoring. 😀

[LukeMathWalker]

The main issue we had here was that Zip does not allow to iterate over two n-dimensional producers that yield items with different dimensionality (0 for the output array and 1 for the lanes that we want to reduce).

[bluss]

I don't quite see the issue with using Zip here

[LukeMathWalker]

Sorry, it took me a while to get back to this - @bluss is indeed right, I was able to put it together using Zip. If you still have the attempts we made together @allmywatts at RustFest I'd be curious to understand what I was doing wrong there 🤔

I opened a PR against your branch with the Zip version @allmywatts - allmywatts#1
We might want to have a var_unchecked, as we did at the beginning, that we use from both var and var_axis to avoid redundant input checks in var_axis.

[hameerabbasi]

@allmywatts If you don't mind can I take over this PR and open a new one?

[allmywatts]

@hameerabbasi hey, sorry for the late reply, yes please go ahead.

[bluss]

This looks ready to merge, right?

[bluss]

Just to note, iter is not an efficient way to iterate a "flattened" array, so this should potentially use unary Zip instead, for example, or ArrayBase::fold. In the high level, this is documented in the module documentation, about array traversals.

can of course be fixed as a later change.

[bluss]

Super minor thing, but as long as there is a division and a checked type conversion inside the actual loop, I don't think var_unchecked with the purpose of saving input checks saves that much.

[bluss]

"This must not fail" does not add information, just use .unwrap(). I'm just afraid that the user will conclude some kinds of into_shape() (to 1D) always succeed, and they don't.

[bluss]

Superseded by #790