Dear Spencer Graves,
Thank you for your comments and suggestions.
I knew that the true optimum values are about (0.69, 0.0, 0.0, -0.3, 0.3,
2.3, -3.0 ,-3.0). That's why I used these values as starting values.
Nonetheless, the last three estimates are so bad.
Regards,
Kathryn Lord
Spencer Graves wrote:
>
> Dear Katheryn:
>
> I'm confused by your claim that, "Even though I used the true
> parameter values as initial values, the results are not very good."
>
> When I ran it, 'optim' quit with $value = -35835, substantially
> less than fr2(theta0) = -0.3.
>
> Could you please review your question, because I believe this will
> answer it.
>
>
> MORE GENERAL OPTIM ISSUES
>
> It is known that 'optim' has problems. Perhaps the simplest thing
> to do is to call 'optim' with each of the 'methods' in sequence, using
> the 'optim' found by each 'method' as the starting value for the next.
> When I do this, I often skip 'SANN', because it typically takes so much
> more time than the other methods. However, if there might be multiple
> local minima, then SANN may be the best way to find a global minimum,
> though you may want to call 'optim' again with another method, starting
> from optimal solution returned by 'SANN'.
>
> Another relatively simple thing to do is to call optim with
> hessian = TRUE and then compute eigen(optim(..., hessian=TRUE)$hessian,
> symmetric=TRUE). If optim found an honest local minimum, all the
> eigenvalues will be positive. For your example, the eigenvalues were as
> follows:
>
> 19000, 11000,
> 0.06, 0.008, 0.003,
> -0.0002, -0.06,
> -6000
>
> This says that locally, the "minimum" identified was really a
> saddle point, being a parabola opening up in two dimensions (with
> curvatures of 19000 and 11000 respectively), a parabola opening down in
> one dimension (with curvature -6000), and relatively flat by comparison
> in the other 5 dimensions. In cases like this, it should be moderately
> easy and fast to explore the dimension with the largest negative
> eigenvalue with the other dimensions fixed until local minima in each
> direction were found. Then 'optim' could be restarted from both those
> local minima, and the minimum of those two solutions could be returned.
>
> If all eigenvalues were positive, it might then be wise to restart
> with parscale = the square roots of the diagonal of the hessian; I
> haven't tried this, but I believe it should work. Using 'parscale' can
> fix convergence problems -- or create some where they did not exist
> initially.
>
> I'm considering creating a package 'optimMLE' that would automate
> some of this and package it with common 'methods' that would assume that
> sum(fn(...)) was either a log(likelihood) or the negative of a
> log(likelihood), etc. However, before I do, I need to make more
> progress on some of my other commitments, review RSiteSearch("optim",
> "fun") to make sure I'm not duplicating something that already exists,
> etc. If anyone is interested in collaborating on this, please contact
> me off-line.
>
> Hope this helps.
> Spencer
>
> kathie wrote:
>> Dear R users,
>>
>> I used to "OPTIM" to minimize the obj. function below. Even though I
>> used
>> the true parameter values as initial values, the results are not very
>> good.
>>
>> How could I improve my results? Any suggestion will be greatly
>> appreciated.
>>
>> Regards,
>>
>> Kathryn Lord
>>
>> #------------------------------------------------------------------------------------------
>>
>> x = c(0.35938587, 0.34889725, 0.20577608, 0.15298888, -4.24741146,
>> -1.32943326,
>> 0.29082527, -2.38156942, -6.62584473, -0.22596237, 6.97893687,
>> 0.22001081,
>> -1.39729222, -5.17860124, 0.52456484, 0.46791660, 0.03065136,
>> 0.32155177,
>> -1.12530013, 0.02608057, -0.22323154, 3.30955460, 0.54653982,
>> 0.29403011,
>> 0.47769874, 2.42216260, 3.93518355, 0.23237890, 0.09647044,
>> -0.48768933,
>> 0.37736377, 0.43739341, -0.02416010, 4.02788119, 0.07320802,
>> -0.29393054,
>> 0.25184609, 0.76044448, -3.34121918, 1.16028677, -0.60352008,
>> -2.86454069,
>> -0.84411691, 0.24841071, -0.11764954, 5.92662106, 1.03932953,
>> -6.21987657,
>> -0.54763352, 0.20263192) # data
>>
>> theta0= c(log(2),0,c(0,-.3,.3),log(c(10,.05,.05))) # initial value(In
>> fact,
>> true parameter value)
>> n = length(x)
>>
>> fr2 = function(theta)
>> {
>> a1 = theta[1]; a2 = theta[2]
>> mu1 = theta[3]; mu2 = theta[4]; mu3 = theta[5]
>> g1 = theta[6]; g2 = theta[7]; g3 = theta[8]
>>
>> w1=exp(a1)/(1+exp(a1)+exp(a2))
>> w2=exp(a2)/(1+exp(a1)+exp(a2))
>> w3=1-w1-w2
>>
>> obj =((w1^2)/(2*sqrt(exp(g1)*pi))
>> + (w2^2)/(2*sqrt(exp(g2)*pi))
>> + (w3^2)/(2*sqrt(exp(g2)*pi))
>>
>> + 2*w1*w2*dnorm((mu1-mu2)/sqrt(exp(g1)+exp(g2)))/sqrt(exp(g1)+exp(g2))
>> +
>> 2*w1*w3*dnorm((mu1-mu3)/sqrt(exp(g1)+exp(g3)))/sqrt(exp(g1)+exp(g3))
>> +
>> 2*w2*w3*dnorm((mu2-mu3)/sqrt(exp(g2)+exp(g3)))/sqrt(exp(g2)+exp(g3))
>>
>> - (2/n)*(sum(w1*dnorm((x-mu1)/sqrt(exp(g1)))/sqrt(exp(g1))
>> + w2*dnorm((x-mu2)/sqrt(exp(g2)))/sqrt(exp(g2))
>> + w3*dnorm((x-mu3)/sqrt(exp(g3)))/sqrt(exp(g3)) )))
>>
>> return(obj)
>> }
>>
>> optim(theta0, fr2, method=BFGS", control = list (fnscale=10,
>> parscale=c(.01,.01,.01,.01,.01,1,1,1), maxit=100000))
>>
>>
>
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