[Jprogramming] Creating random symmetric matrices

bill lam bbill.lam at gmail.com
Thu Aug 23 10:09:48 UTC 2012 

machine translation is [:-:]*>: while that from Shaw is -:@(*>:)

the main difference in this case 13 : tends use [: to form a train
of fork but human tends to form compound verb using @  
That said, certainly some prefer [: over @.

IMO 13 : is similar to google translate, it is useful but the result
could be aweful.

Чтв, 23 Авг 2012, Linda Alvord писал(а):
> If you compare these two tacit versions for the creation of the matrix, I
> think my version is optimal because it is "read" from right to left.  I
> don't care much if it or most any function is fast, as it always beats me in
> completing whatever it is doing.
> 
> Maybe someone will write an optimizer to turn my code into optimal code and
> then the computer can speed along in the fast lane without ever explaining
> it to me.
> 
> It could do all the final comparisons and choosing of optimal timings. It
> could do it on its own and just keep the results to itself. It, notice how I
> have humanized it since it one of my best mathematical buddies, reads from
> left to right and right to left with equal easy.  However, most humans
> probably don't it makes it difficult for us to communicate and become
> readers of both styles.
> 
> So....  how do you judge which of these two versions is optimal?
> 
> Linda
> 
> 
> -----Original Message-----ow
> From: programming-bounces at forums.jsoftware.com
> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Ric Sherlock
> Sent: Thursday, August 23, 2012 3:18 AM
> To: programming at jsoftware.com
> Subject: Re: [Jprogramming] Creating random symmetric matrices
> 
> Hi Linda,
> Basically tacit definitions contain no reference to their arguments.
> http://en.wikipedia.org/wiki/Tacit_programming
> 
> By that definition the following assignment isn't tacit because the "y"
> refers to the right argument.
> 
>    trian=: 13 :'-:y*>:y'
> 
> However the resulting definition for trian is tacit:
> 
>    trian
> [: -: ] * >:
> 
> As Bill suggests, 13 : '' is a useful tool for understanding how to convert
> explicit to tacit, but it won't necessarily generate "optimal"
> tacit definitions.
> 
> On Thu, Aug 23, 2012 at 3:29 PM, Linda Alvord <lindaalvord at verizon.net>
> wrote:
> > If you are able to use  5!:4  and get a tree, is that a way to tell if 
> > the definition is tacit?
> >
> > Thanks for letting me know that I was not clear about tacit.
> >
> > Linda
> >
> > -----Original Message-----
> > From: programming-bounces at forums.jsoftware.com
> > [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Ric 
> > Sherlock
> > Sent: Wednesday, August 22, 2012 9:36 PM
> > To: programming at jsoftware.com
> > Subject: Re: [Jprogramming] Creating random symmetric matrices
> >
> > Just a pedantic nitpick with regard to terminology here.
> > Ewart's verbs were already tacit. I think the key thing that your 
> > versions show is how that tacit translates to explicit and then how 13
> > : retranslates them to another tacit version without hooks.
> >
> > On Thu, Aug 23, 2012 at 12:45 PM, Linda Alvord 
> > <lindaalvord at verizon.net>
> > wrote:
> >> Ewart's function in tacit version:
> >>
> >>    trian=: 13 :'-:y*>:y'
> >>    indsy=: 13 :'(>.|:i.y)+/tri i.y'
> >>
> >>    indsy 4
> >> 0 1 3 6
> >> 1 2 4 7
> >> 2 3 5 8
> >> 3 4 6 9
> >>    trian
> >> [: -: ] * >:
> >>    indsy
> >> ([: >. [: |: i.) +/ [: tri i.
> >>
> >> Linda
> >>
> >>
> >> -----Original Message-----
> >> From: programming-bounces at forums.jsoftware.com
> >> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of Henry 
> >> Rich
> >> Sent: Wednesday, August 22, 2012 6:46 PM
> >> To: programming at jsoftware.com
> >> Subject: Re: [Jprogramming] Creating random symmetric matrices
> >>
> >> Why yes, that's much better.  Very clever way of filling the triangle.
> >>
> >> Henry Rich
> >>
> >> On 8/22/2012 4:28 PM, Ric Sherlock wrote:
> >>> Note that Ewart Shaw has responded to this on the Wiki with a 
> >>> solution that only generates the required number of random numbers:
> >>> http://www.jsoftware.com/jwiki/EwartShaw/RandomSymmetricMatrix
> >>>
> >>> On Wed, Aug 22, 2012 at 5:30 PM, Ric Sherlock <tikkanz at gmail.com> wrote:
> >>>> The other option is not to add them in the first place?
> >>>>
> >>>>     load 'stats/distribs'
> >>>>     (|: + ~: zeroTri ) >: zeroTri rnorm 5 5
> >>>>   0.346799 _1.22161    0.57274   0.556122 _0.329658
> >>>>   _1.22161 0.149955   _1.77435   _1.76668  0.831557
> >>>>    0.57274 _1.77435    0.77674 _0.0690683 _0.967551
> >>>>   0.556122 _1.76668 _0.0690683   0.720588 _0.195658
> >>>> _0.329658 0.831557  _0.967551  _0.195658  _0.12314
> >>>>
> >>>> In other words zero the items above the diagonal then transpose and 
> >>>> add to the non-diagonal items.
> >>>>
> >>>> Where zeroTri is an adverb defined as below:
> >>>>
> >>>> NB.*zeroTri a Zeros triangular items of matrix determined by verb 
> >>>> to
> > left
> >>>> NB. EG: < zeroTri mat   NB. zeros lower-tri items of mat
> >>>> NB. EG: <: zeroTri mat  NB. zeros lower-tri, off-diag items of mat
> >>>> NB. EG: > zeroTri mat   NB. zeros upper tri items of mat
> >>>> NB. EG: = zeroTri mat   NB. zeros all off-diag items of mat
> >>>> NB. EG: ~: zeroTri mat  NB. zeros diag items of mat
> >>>> zeroTri=: 1 :'([: u/~ i.@#) * ]'
> >>>>
> >>>> This is somewhat wasteful in that you generate a bunch of random 
> >>>> numbers that you then discard, but if that was an issue you could 
> >>>> work around it.
> >>>>
> >>>>
> >>>> On Wed, Aug 22, 2012 at 5:07 PM, Owen Marschall 
> >>>> <omarschall15 at amherst.edu> wrote:
> >>>>> You read my mind. I was thinking about this same problem tonight 
> >>>>> while
> >> at the opera, and I couldn't think of any way to only divide the 
> >> diagonals by sqrt(2) a second time--without loops, of course. I don't 
> >> quite yet understand why your solution works, but I'm sure with 
> >> enough staring and dictionary help I'll get it.
> >>>>>
> >>>>> Thanks,
> >>>>> Owen
> >>>>>
> >>>>> On Aug 21, 2012, at 8:25 PM, Henry Rich wrote:
> >>>>>
> >>>>>> If you have a matrix a of standard normal deviates, you can make 
> >>>>>> it symmetric with
> >>>>>>
> >>>>>> asymm =: (+ |:) a
> >>>>>>
> >>>>>> but what is the variance of the items of a?
> >>>>>>
> >>>>>> The variance of values off the principal diagonal will be the sum 
> >>>>>> of
> >> the variance of two independent standard normal deviates. i.e. 2.
> >>>>>>
> >>>>>> To return these values to variance 1 you need to divide by sqrt(2).
> >>>>>>
> >>>>>> But the variance of values ON the principal diagonal will be the 
> >>>>>> sum of
> >> two perfectly correlated random variables, i. e. 4.
> >>>>>>
> >>>>>> So you need to treat the principal diagonal differently.  You can
> >> reduce its variance by scaling it differently after the conversion to 
> >> symmetric, dividing the diagonal by sqrt(4) and the rest by sqrt(2):
> >>>>>>
> >>>>>> asymmgood =: asymm % %: +: >: e. i. # asymm
> >>>>>>
> >>>>>> (The e. i. bit is a standard idiom for making an identity matrix.
> >> Another one you see around is   = i.  but I avoid that because I think
> > monad
> >> = was wrongly defined and should be assigned for other purposes)
> >>>>>>
> >>>>>> If I've made a statistical blunder I'm sure someone will tell me.
> >>>>>>
> >>>>>> Henry Rich
> >>>>>>
> >>>>>> On 8/21/2012 3:10 PM, Owen Marschall wrote:
> >>>>>>> Ah, just what I needed. Thanks!
> >>>>>>>
> >>>>>>> On Aug 21, 2012, at 1:06 PM, Ric Sherlock wrote:
> >>>>>>>
> >>>>>>>> The primitive ( |: ) is transpose. E.g. :
> >>>>>>>>
> >>>>>>>>    |: i. 3 4
> >>>>>>>> 0 4  8
> >>>>>>>> 1 5  9
> >>>>>>>> 2 6 10
> >>>>>>>> 3 7 11
> >>>>>>>> On Aug 22, 2012 6:55 AM, "Owen Marschall"
> >>>>>>>> <omarschall15 at amherst.edu>
> >> wrote:
> >>>>>>>>
> >>>>>>>>> Anyone know of an easy way to create a random symmetric matrix 
> >>>>>>>>> (more specifically, a matrix whose entires are each picked 
> >>>>>>>>> from a standard Gaussian distribution)? I can start by doing
> >>>>>>>>>
> >>>>>>>>> load 'stats'
> >>>>>>>>> R=:normalrand N N
> >>>>>>>>>
> >>>>>>>>> but this is not symmetric, and I don't know of any way to 
> >>>>>>>>> symmetrize it without thinking in loops, which I'm training 
> >>>>>>>>> myself not to. If I could somehow take a transpose, that would 
> >>>>>>>>> solve the problem, but I don't know how to do that either.
> >>>>>>>>>
> >>>>>>>>> Thanks,
> >>>>>>>>> Owen
> >>>>>>>>> --------------------------------------------------------------
> >>>>>>>>> -
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