[R] The effect of tolerance in all.equal()

[Ashim Kapoor]

Dear All,

This answer is very clear. Many thanks.

I am now confused about how str*ucture works. Where can I read more about
when does it  return language / logical / chr ? I would want to read that
so I can interpret the result of structure. I don't think ?str contains
this.To me, logical and chr make sense, what does language mean? I think I
need to read some more.

Many thanks,
Ashim

On Tue, Apr 25, 2017 at 3:14 PM, Martin Maechler <maechler at stat.math.ethz.ch
> wrote:

> >>>>> Ashim Kapoor <ashimkapoor at gmail.com>
> >>>>>     on Tue, 25 Apr 2017 14:02:18 +0530 writes:
>
>     > Dear all,
>     > I am not able to understand the interplay of absolute vs relative and
>     > tolerance in the use of all.equal
>
>     > If I want to find out if absolute differences between 2
> numbers/vectors are
>     > bigger than a given tolerance I would do:
>
>     > all.equal(1,1.1,scale=1,tol= .1)
>
>     > If I want to find out if relative differences between 2
> numbers/vectors are
>     > bigger than a given tolerance I would do :
>
>     > all.equal(1,1.1,tol=.1)
>
>     > ############################################################
> ######################################################################
>
>     > I can also do :
>
>     > all.equal(1,3,tol=1)
>
>     > to find out if the absolute difference is bigger than 1.But here I
> won't be
>     > able to detect absolute differences smaller than 1 in this case,so I
> don't
>     > think that this is a good way.
>
>     > My query is: what is the reasoning behind all.equal returning the
> absolute
>     > difference if the tolerance >= target and relative difference if
> tolerance
>     > < target?
> (above, it is    tol  >/<=  |target|  ie. absolute value)
>
>
> The following are desiderata / restrictions :
>
> 1) Relative tolerance is needed to keep things scale-invariant
>    i.e.,  all.equal(x, y)  and  all.equal(1000 * x, 1000 * y)
>    should typically be identical for (almost) all (x,y).
>
>    ==> "the typical behavior should use relative error tolerance"
>
> 2) when x or y (and typically both!) are very close to zero it
>    is typically undesirable to keep relative tolerances (in the
>    boundary case, they _are_ zero exactly, and "relative error" is
> undefined).
>    E.g., for most purposes, 3.45e-15 and 1.23e-17 should be counted as
>    equal to zero and hence to themselves.
>
> 1) and 2) are typically reconciled by switching from relative to absolute
> when the arguments are close to zero (*).
>
> The exact cutoff at which to switch from relative to absolute
> (or a combination of the two) is somewhat arbitrary(*2) and for
> all.equal() has been made in the 1980's (or even slightly
> earlier?) when all.equal() was introduced into the S language at
> Bell labs AFAIK. Maybe John Chambers (or Rick Becker or ...,
> but they may not read R-help) knows more.
> *2) Then, the choice for all.equal() is in some way "least arbitrary",
>     using c = 1 in the more general   tolerance >= c*|target|  framework.
>
> *) There have been alternatives in "the (applied numerical
>  analysis / algorithm) literature" seen in published algorithms,
>  but I don't have any example ready.
>  Notably some of these alternatives are _symmetric_ in (x,y)
>  where all.equal() was designed to be asymmetric using names
>  'target' and 'current'.
>
> The alternative idea is along the following thoughts:
>
> Assume that for "equality" we want _both_ relative and
> absolute (e := tolerance) "equality"
>
>    |x - y| < e (|x|+|y|)/2  (where you could use |y| or |x|
>                              instead of their mean; all.equal()
>                              uses |target|)
>    |x - y| < e * e1          (where e1 = 1, or e1 = 10^-7..)
>
> If you add the two inequalities you get
>
>    |x - y| < e (e1 + |x+y|/2)
>
> as check which is a "mixture" of relative and absolute tolerance.
>
> With a somewhat long history, my gut feeling would nowadays
> actually prefer this (I think with a default of e1 = e) - which
> does treat x and y symmetrically.
>
> Note that convergence checks in good algorithms typically check
> for _both_ relative and absolute difference (each with its
> tolerance providable by the user), and the really good ones for
> minimization do  check for (approximate) gradients also being
> close to zero - as old timers among us should have learned from
> Doug Bates ... but now I'm really diverging.
>
> Last but not least some  R  code at the end,  showing that the *asymmetric*
> nature of all.equal() may lead to somewhat astonishing (but very
> logical and as documented!) behavior.
>
> Martin
>
>     > Best Regards,
>     > Ashim
>
>
> > ## The "data" to use:
> > epsQ <- lapply(seq(12,18,by=1/2), function(P) bquote(10^-.(P)));
> names(epsQ) <- sapply(epsQ, deparse); str(epsQ)
> List of 13
>  $ 10^-12  : language 10^-12
>  $ 10^-12.5: language 10^-12.5
>  $ 10^-13  : language 10^-13
>  $ 10^-13.5: language 10^-13.5
>  $ 10^-14  : language 10^-14
>  $ 10^-14.5: language 10^-14.5
>  $ 10^-15  : language 10^-15
>  $ 10^-15.5: language 10^-15.5
>  $ 10^-16  : language 10^-16
>  $ 10^-16.5: language 10^-16.5
>  $ 10^-17  : language 10^-17
>  $ 10^-17.5: language 10^-17.5
>  $ 10^-18  : language 10^-18
>
> > str(lapply(epsQ, function(tl) all.equal(3.45e-15, 1.23e-17, tol =
> eval(tl))))
> List of 13
>  $ 10^-12  : logi TRUE
>  $ 10^-12.5: logi TRUE
>  $ 10^-13  : logi TRUE
>  $ 10^-13.5: logi TRUE
>  $ 10^-14  : logi TRUE
>  $ 10^-14.5: chr "Mean relative difference: 0.9964348"
>  $ 10^-15  : chr "Mean relative difference: 0.9964348"
>  $ 10^-15.5: chr "Mean relative difference: 0.9964348"
>  $ 10^-16  : chr "Mean relative difference: 0.9964348"
>  $ 10^-16.5: chr "Mean relative difference: 0.9964348"
>  $ 10^-17  : chr "Mean relative difference: 0.9964348"
>  $ 10^-17.5: chr "Mean relative difference: 0.9964348"
>  $ 10^-18  : chr "Mean relative difference: 0.9964348"
>
> > ## Now swap `target` and `current` :
> > str(lapply(epsQ, function(tl) all.equal(1.23e-17, 3.45e-15, tol =
> eval(tl))))
> List of 13
>  $ 10^-12  : logi TRUE
>  $ 10^-12.5: logi TRUE
>  $ 10^-13  : logi TRUE
>  $ 10^-13.5: logi TRUE
>  $ 10^-14  : logi TRUE
>  $ 10^-14.5: chr "Mean absolute difference: 3.4377e-15"
>  $ 10^-15  : chr "Mean absolute difference: 3.4377e-15"
>  $ 10^-15.5: chr "Mean absolute difference: 3.4377e-15"
>  $ 10^-16  : chr "Mean absolute difference: 3.4377e-15"
>  $ 10^-16.5: chr "Mean absolute difference: 3.4377e-15"
>  $ 10^-17  : chr "Mean relative difference: 279.4878"
>  $ 10^-17.5: chr "Mean relative difference: 279.4878"
>  $ 10^-18  : chr "Mean relative difference: 279.4878"
>
> >
>

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