[Rd] [R] choose(n, k) as n approaches k

[John Mount]

At the risk of throwing oil on a fire.  If we are talking about fractional values of choose() doesn't it make sense to look to the gamma function for the correct analytic continuation?  In particular k<0 may not imply the function should evaluate to zero until we get k<=-1.

Example:

``` r
choose(5, 4)
#> [1] 5

gchoose <- function(n, k) { 
  gamma(n+1)/(gamma(n+1-k) * gamma(k+1))
}

gchoose(5, 4)
#> [1] 5
gchoose(5, 0)
#> [1] 1
gchoose(5, -0.5)
#> [1] 0.2351727
```

> On Jan 14, 2020, at 10:20 AM, peter dalgaard <pdalgd using gmail.com> wrote:
> 
> OK, I see what you mean. But in those cases, we don't get the catastrophic failures from the 
> 
>        if (k <  0) return 0.;
>        if (k == 0) return 1.;
>        /* else: k >= 1 */
> 
> part, because at that point k is sure to be integer, possibly after rounding. 
> 
> It is when n-k is approximately but not exactly zero and we should return 1, that we either return 0 (negative case) or n (positive case; because the n(n-1)(n-2)... product has at least one factor). In the other cases, we get 1 or n(n-1)(n-2)...(n-k+1) which if n is near-integer gets rounded to produce an integer, due to the
> 
>        return R_IS_INT(n) ? R_forceint(r) : r;
> 
> part.
> 
> -pd
> 
> 
> 
>> On 14 Jan 2020, at 17:02 , Duncan Murdoch <murdoch.duncan using gmail.com> wrote:
>> 
>> On 14/01/2020 10:50 a.m., peter dalgaard wrote:
>>>> On 14 Jan 2020, at 16:21 , Duncan Murdoch <murdoch.duncan using gmail.com> wrote:
>>>> 
>>>> On 14/01/2020 10:07 a.m., peter dalgaard wrote:
>>>>> Yep, that looks wrong (probably want to continue discussion over on R-devel)
>>>>> I think the culprit is here (in src/nmath/choose.c)
>>>>>     if (k < k_small_max) {
>>>>>        int j;
>>>>>        if(n-k < k && n >= 0 && R_IS_INT(n)) k = n-k; /* <- Symmetry */
>>>>>        if (k <  0) return 0.;
>>>>>        if (k == 0) return 1.;
>>>>>        /* else: k >= 1 */
>>>>> if n is a near-integer, then k can become non-integer and negative. In your case,
>>>>> n == 4 - 1e-7
>>>>> k == 4
>>>>> n - k == -1e-7 < 4
>>>>> n >= 0
>>>>> R_IS_INT(n) = TRUE (relative diff < 1e-7 is allowed)
>>>>> so k gets set to
>>>>> n - k == -1e-7
>>>>> which is less than 0, so we return 0. However, as you point out, 1 would be more reasonable and in accordance with the limit as n -> 4, e.g.
>>>>>> factorial(4 - 1e-10)/factorial(1e-10)/factorial(4) -1
>>>>> [1] -9.289025e-11
>>>>> I guess that the fix could be as simple as replacing n by R_forceint(n) in the k = n - k step.
>>>> 
>>>> I think that would break symmetry:  you want choose(n, k) to equal choose(n, n-k) when n is very close to an integer.  So I'd suggest the replacement whenever R_IS_INT(n) is true.
>>>> 
>>> But choose() very deliberately ensures that k is integer, so choose(n, n-k) is ill-defined for non-integer n.
>> 
>> That's only true if there's a big difference.  I'd be worried about cases where n and k are close to integers (within 1e-7).  In those cases, k is silently rounded to integer.  As I read your suggestion, n would only be rounded to integer if k > n-k.  I think both n and k should be rounded to integer in this near-integer situation, regardless of the value of k.
>> 
>> I believe that lchoose(n, k) already does this.
>> 
>> Duncan Murdoch
>> 
>>>    double r, k0 = k;
>>>    k = R_forceint(k);
>>> ...
>>>    if (fabs(k - k0) > 1e-7)
>>>        MATHLIB_WARNING2(_("'k' (%.2f) must be integer, rounded to %.0f"), k0, k);
>>> 
>>>> Duncan Murdoch
>>>> 
>>>>> -pd
>>>>>> On 14 Jan 2020, at 00:33 , Wright, Erik Scott <ESWRIGHT using pitt.edu> wrote:
>>>>>> 
>>>>>> This struck me as incorrect:
>>>>>> 
>>>>>>> choose(3.999999, 4)
>>>>>> [1] 0.9999979
>>>>>>> choose(3.9999999, 4)
>>>>>> [1] 0
>>>>>>> choose(4, 4)
>>>>>> [1] 1
>>>>>>> choose(4.0000001, 4)
>>>>>> [1] 4
>>>>>>> choose(4.000001, 4)
>>>>>> [1] 1.000002
>>>>>> 
>>>>>> Should base::choose(n, k) check whether n is within machine precision of k and return 1?
>>>>>> 
>>>>>> Thanks,
>>>>>> Erik
>>>>>> 
>>>>>> ***
>>>>>> sessionInfo()
>>>>>> R version 3.6.0 beta (2019-04-15 r76395)
>>>>>> Platform: x86_64-apple-darwin15.6.0 (64-bit)
>>>>>> Running under: macOS High Sierra 10.13.6
>>>>>> 
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>>>>>> 
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> 
> -- 
> Peter Dalgaard, Professor,
> Center for Statistics, Copenhagen Business School
> Solbjerg Plads 3, 2000 Frederiksberg, Denmark
> Phone: (+45)38153501
> Office: A 4.23
> Email: pd.mes using cbs.dk  Priv: PDalgd using gmail.com
> 
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---------------
John Mount
http://www.win-vector.com/ <http://www.win-vector.com/> 
Our book: Practical Data Science with R
http://practicaldatascience.com <http://practicaldatascience.com/> 






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