# [Numpy-discussion] please change mean to use dtype=float

Tim Hochberg tim.hochberg at ieee.org
Fri Sep 22 11:06:10 CDT 2006

```Sebastian Haase wrote:
> On Thursday 21 September 2006 15:28, Tim Hochberg wrote:
>
>> David M. Cooke wrote:
>>
>>> On Thu, 21 Sep 2006 11:34:42 -0700
>>>
>>> Tim Hochberg <tim.hochberg at ieee.org> wrote:
>>>
>>>> Tim Hochberg wrote:
>>>>
>>>>> Robert Kern wrote:
>>>>>
>>>>>> David M. Cooke wrote:
>>>>>>
>>>>>>> On Wed, Sep 20, 2006 at 03:01:18AM -0500, Robert Kern wrote:
>>>>>>>
>>>>>>>> Let me offer a third path: the algorithms used for .mean() and
>>>>>>>> .var() are substandard. There are much better incremental algorithms
>>>>>>>> that entirely avoid the need to accumulate such large (and therefore
>>>>>>>> precision-losing) intermediate values. The algorithms look like the
>>>>>>>> following for 1D arrays in Python:
>>>>>>>>
>>>>>>>> def mean(a):
>>>>>>>>      m = a[0]
>>>>>>>>      for i in range(1, len(a)):
>>>>>>>>          m += (a[i] - m) / (i + 1)
>>>>>>>>      return m
>>>>>>>>
>>>>>>> This isn't really going to be any better than using a simple sum.
>>>>>>> It'll also be slower (a division per iteration).
>>>>>>>
>>>>>> With one exception, every test that I've thrown at it shows that it's
>>>>>> better for float32. That exception is uniformly spaced arrays, like
>>>>>> linspace().
>>>>>>
>>>>>>  > You do avoid
>>>>>>  > accumulating large sums, but then doing the division a[i]/len(a)
>>>>>>  > and adding that will do the same.
>>>>>>
>>>>>> Okay, this is true.
>>>>>>
>>>>>>
>>>>>>> Now, if you want to avoid losing precision, you want to use a better
>>>>>>> summation technique, like compensated (or Kahan) summation:
>>>>>>>
>>>>>>> def mean(a):
>>>>>>>     s = e = a.dtype.type(0)
>>>>>>>     for i in range(0, len(a)):
>>>>>>>         temp = s
>>>>>>>         y = a[i] + e
>>>>>>>         s = temp + y
>>>>>>>         e = (temp - s) + y
>>>>>>>     return s / len(a)
>>>>>>>
>>>>>>>
>>>>>>>> def var(a):
>>>>>>>>      m = a[0]
>>>>>>>>      t = a.dtype.type(0)
>>>>>>>>      for i in range(1, len(a)):
>>>>>>>>          q = a[i] - m
>>>>>>>>          r = q / (i+1)
>>>>>>>>          m += r
>>>>>>>>          t += i * q * r
>>>>>>>>      t /= len(a)
>>>>>>>>      return t
>>>>>>>>
>>>>>>>> Alternatively, from Knuth:
>>>>>>>>
>>>>>>>> def var_knuth(a):
>>>>>>>>      m = a.dtype.type(0)
>>>>>>>>      variance = a.dtype.type(0)
>>>>>>>>      for i in range(len(a)):
>>>>>>>>          delta = a[i] - m
>>>>>>>>          m += delta / (i+1)
>>>>>>>>          variance += delta * (a[i] - m)
>>>>>>>>      variance /= len(a)
>>>>>>>>      return variance
>>>>>>>>
>>>> I'm going to go ahead and attach a module containing the versions of
>>>> mean, var, etc that I've been playing with in case someone wants to mess
>>>> with them. Some were stolen from traffic on this list, for others I
>>>> grabbed the algorithms from wikipedia or equivalent.
>>>>
>>> I looked into this a bit more. I checked float32 (single precision) and
>>> float64 (double precision), using long doubles (float96) for the "exact"
>>> results. This is based on your code. Results are compared using
>>> abs(exact_stat - computed_stat) / max(abs(values)), with 10000 values in
>>> the range of [-100, 900]
>>>
>>> First, the mean. In float32, the Kahan summation in single precision is
>>> better by about 2 orders of magnitude than simple summation. However,
>>> accumulating the sum in double precision is better by about 9 orders of
>>> magnitude than simple summation (7 orders more than Kahan).
>>>
>>> In float64, Kahan summation is the way to go, by 2 orders of magnitude.
>>>
>>> For the variance, in float32, Knuth's method is *no better* than the
>>> two-pass method. Tim's code does an implicit conversion of intermediate
>>> results to float64, which is why he saw a much better result.
>>>
>> Doh! And I fixed that same problem in the mean implementation earlier
>> too. I was astounded by how good knuth was doing, but not astounded
>> enough apparently.
>>
>> Does it seem weird to anyone else that in:
>>     numpy_scalar <op> python_scalar
>> the precision ends up being controlled by the python scalar? I would
>> expect the numpy_scalar to control the resulting precision just like
>> numpy arrays do in similar circumstances. Perhaps the egg on my face is
>> just clouding my vision though.
>>
>>
>>> The two-pass method using
>>> Kahan summation (again, in single precision), is better by about 2 orders
>>> of magnitude. There is practically no difference when using a
>>> double-precision accumulator amongst the techniques: they're all about 9
>>> orders of magnitude better than single-precision two-pass.
>>>
>>> In float64, Kahan summation is again better than the rest, by about 2
>>> orders of magnitude.
>>>
>>> I've put my adaptation of Tim's code, and box-and-whisker plots of the
>>> results, at http://arbutus.mcmaster.ca/dmc/numpy/variance/
>>>
>>> Conclusions:
>>>
>>> - If you're going to calculate everything in single precision, use Kahan
>>> summation. Using it in double-precision also helps.
>>> - If you can use a double-precision accumulator, it's much better than
>>> any of the techniques in single-precision only.
>>>
>>> - for speed+precision in the variance, either use Kahan summation in
>>> single precision with the two-pass method, or use double precision with
>>> simple summation with the two-pass method. Knuth buys you nothing, except
>>> slower code :-)
>>>
>> The two pass methods are definitely more accurate. I won't be convinced
>> on the speed front till I see comparable C implementations slug it out.
>> That may well mean never in practice. However, I expect that somewhere
>> around 10,000 items, the cache will overflow and memory bandwidth will
>> become the bottleneck. At that point the extra operations of Knuth won't
>> matter as much as making two passes through the array and Knuth will win
>> on speed. Of course the accuracy is pretty bad at single precision, so
>> the possible, theoretical speed advantage at large sizes probably
>> doesn't matter.
>>
>> -tim
>>
>>
>>> After 1.0 is out, we should look at doing one of the above.
>>>
>> +1
>>
>>
> Hi,
> I don't understand much of these "fancy algorithms", but does the above mean
> that
> after 1.0 is released the float32 functions will catch up in terms of
> precision precision ("to some extend") with using dtype=float64 ?
>
It sounds like there is some consensus to do something to improve the
precision after 1.0 comes out. I don't think the details are entirely
nailed down, but it sounds like some combination of using Kahan
summation and increasing the minimum precision of the accumulator is
likely for sum, mean, var and std.

-tim

```