Charles R Harris charlesr.harris at gmail.com
Sat Oct 18 22:37:55 EDT 2014

```On Sat, Oct 18, 2014 at 7:17 PM, Nathaniel Smith <njs at pobox.com> wrote:

> Okay! I think I now actually understand what was going on with
> np.gradient! The discussion has been pretty confused, and I'm worried
> that what's in master right now is not the right solution, which is a
> problem because we're supposed to cut 1.9.1 tomorrow.
>
> Background:
>
> Wikipedia has a good explanation of how this works:
> https://en.wikipedia.org/wiki/Numerical_differentiation
> The key point is that there are multiple formulas one can use that all
> converge to the derivative, e.g.
>
>    (f(x + h) - f(x)) / h
>
> or
>
>    (f(x + h) - f(x - h)) / 2h
>
> The first is the textbook definition of the derivative. As h -> 0, the
> error in that approximation shrinks like O(h). The second formula also
> converges to the derivative as h -> 0, but it converges like O(h^2),
> i.e., much faster. And there's are many many formulas like this with
>    https://en.wikipedia.org/wiki/Finite_difference_coefficient
> In practice, given a grid of values of f(x) and a grid stepsize of h,
> all of these formulas come down to doing a simple convolution of the
> data with certain coefficients (e.g. [-1/h, 1/h] or [-1/2h, 0, 1/2h]
> for the two formulas above).
>
> quadratically diminishing error term, for interior points in the grid.
> For points on the boundary of the grid, though, this formula has a
> problem, b/c it requires you look at points to both the left and the
> right of the current point -- but for boundary points one of these
> the first formula instead (the "forward (or backward) finite
> difference approximation with linear error", where
> "forward"/"backward" means that it works on the boundary). As the web
> page linked above shows, though, there's an easy alternative formula
> that works on
>
> The change in numpy 1.9.0 that has caused all the fuss, is that we
> the linearly accurate approximation for boundary points.
>
>
> 1) it produced (hopefully small) changes in the calculation for the
> boundaries. In general these should have made the calculations more
> accurate, but apparently this did cause problems for some people who
> wanted perfectly reproducible output.
>
> 2) it tickled a nasty bug in how gradient and masked arrays work
> together. It turns out gradient applied to masked arrays has always
> been seriously buggy; there's some horribleness in the masked array
> implementation that means the output array that np.gradient is writing
> into ends up sharing a mask array with the input. So as we go along
> calculating the output, we also mutate the input. This can definitely
> corrupt the input, for example here:
>     https://gist.github.com/njsmith/551738469b74d175e039
> ...and I strongly suspect it's possible to find cases where it means
> you get the wrong answer.
>
> For mysterious reasons involving the order in which values were
> calculated or something, the version of np.gradient in 1.9.0 tickled
> this bug in a much worse manner, with early mutations to the input
> array ended up affecting later calculations using the same input
> array, causing cascading errors. This is the cause of the massive
> problems that matplotlib encountered in their test suite:
>
> https://github.com/matplotlib/matplotlib/issues/3598#issuecomment-58859663
>
> There's a proposed fix for the masked array issues at:
>     https://github.com/numpy/numpy/pull/5203
> For the reasons described above, I would be very wary about using
> workaround would be to make a local copy of the definition of
> np.gradient, and immediately after the 'out' array is allocated do
>
> Once this bug is fixed, the "new" gradient code produces results which
> are identical to the "old" gradient code on the interior; only the
> boundary is different.
>
> ------
>
> So here are my concerns:
>
> - We decided to revert the changes to np.gradient in 1.9.1 (at least
> by default). I'm not sure how much of that decision was based on the
> issues with masked arrays, though, which turns out to be a different
> issue entirely. The previous discussion conflated the two issues
> above.
>

My concern was reproducibility. The old behavior wasn't a bug, so we should
be careful that old results are reproducible.

>
> - We decided to gate the more-accurate boundary calculation behind a
> kwarg called "edge_order=2". AFAICT now that I actually understand
> what this code is doing, this is a terrible name -- we're using a
> 3-coefficient kernel to compute a quadratically accurate approximation
> to a first-order derivative.

Accuracy is a different problem and depends on the function being
interpolated. As you point out above, the order refers to the *rate* of
convergence. Normally that is illustrated with a loglog plot of the
absolute value of the error against h, resulting in a straight line once
the function is sufficiently smooth over the range of h. The end points are
a bit special because of the lack of bracketing. Relative to the interior,
one is effectively extrapolating rather than interpolating and things can
become a bit unstable. Hence it is useful to have a safe option, here
linear extrapolation, as an alternative to a higher order method.

> There probably exist other kernels that
> are also quadratically accurate. "Order 2" simply doesn't refer to
> this in any unique or meaningful way. And it will be even more
> confusing if we ever add the option to select which kernel to use on
> the interior, where "quadratically accurate" is definitely not enough
> information to uniquely define a kernel. Maybe we want something like
> edge_accuracy=2 or edge_accuracy="quadratic" or something? I'm not
> sure.
>
> - If edge_order=2 escapes into a release tomorrow then we'll be stuck with
> it.
>

Order has two common meanings in the numerical context, either degree, or,
for polynomials, the number of coefficients (degree + 1). I've noticed that
the meaning has evolved over the years, and these days an equivalence with
degree seems to be pretty common. In the present context, it refers to the
power of the h term in the error term of the Taylor's series approximation
of the derivative. The precise meaning needs to be elucidated in the notes,
as the order doesn't map one-to-one into methods.

> Some possible options for 1.9.1:
> - Just revert the entire np.gradient code to match 1.8, and put off a
> real fix until 1.9. This at least avoids getting stuck with a poor
> API.
>
> - Put the masked array bug fix into 1.9.1, and leave the gradient code
> the same as 1.9.0; put off designing a proper API for selecting the
> old behavior for 1.9.2. (Or 1.10 I guess.) This doesn't solve the pure
> reproduceability problems, but it might be enough to make matplotlib
> work again at least?
>

The only reason I see to keep the current default is to maintain behavior
from 1.9.0 to 1.9.1. I don't think 1.9.0 will last long in the wild. Anyone
working on the cutting edge will likely update almost immediately, and
distros releases in the near future will probably have 1.8.2.

> - Delay 1.9.1 to rehash the issues and make a decision on what we want
> to support long-term.
>

Might want to delay a few days in any case to settle the Mac/Windows issues.

Chuck
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