On Aug 22, 2012 6:55 AM, "Owen Marschall" <[email protected]> wrote: Anyone know of an easy way to create a random symmetric matrix (more specifically, a matrix whose entires are each picked from a standardGaussian distribution)?
Note that 'A random symmetric matrix' is a contradiction in terms. A symmetric matrix is not random, and a matrix whose entries are each picked from a standard Gaussian distribution is almost never symmetric. Rather than accept the problem formulation you need to think about what you really want, and why. Perhaps you want a matrix with random real eigenvalues and with random eigenvector basis? - Bo. >________________________________ > Fra: Ric Sherlock <[email protected]> >Til: [email protected] >Sendt: 7:30 onsdag den 22. august 2012 >Emne: Re: [Jprogramming] Creating random symmetric matrices > >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 ><[email protected]> 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" <[email protected]> >>>>> 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 >>>>>> ---------------------------------------------------------------------- >>>>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>>>> >>>>> ---------------------------------------------------------------------- >>>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>> >>>> ---------------------------------------------------------------------- >>>> For information about J forums see http://www.jsoftware.com/forums.htm >>>> >>> ---------------------------------------------------------------------- >>> For information about J forums see http://www.jsoftware.com/forums.htm >> >> ---------------------------------------------------------------------- >> For information about J forums see http://www.jsoftware.com/forums.htm >---------------------------------------------------------------------- >For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
