[R] The gradient of a multivariate normal density with respect toits parameters

[Ravi Varadhan]

Karl,

You may want to look at the paper by Dwyer on "Some applications of matrix
derivatives in multivariate analysis" (JASA 1967), especially the Table 2 on
p. 617. 

Ravi.

----------------------------------------------------------------------------
-------

Ravi Varadhan, Ph.D.

Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology 

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

Email: rvaradhan at jhmi.edu

Webpage:
http://www.jhsph.edu/agingandhealth/People/Faculty_personal_pages/Varadhan.h
tml

 

----------------------------------------------------------------------------
--------


-----Original Message-----
From: Karl Ove Hufthammer [mailto:Karl.Hufthammer at math.uib.no] 
Sent: Monday, June 22, 2009 12:12 PM
To: r-help at stat.math.ethz.ch
Cc: Ravi Varadhan
Subject: Re: [R] The gradient of a multivariate normal density with respect
toits parameters

Ravi Varadhan skreiv:
> I am not aware of any.  May I ask what your purpose is?  You don't 
> really need this if you are going to use it in optimization, since 
> most optimizers use a simple finite-difference approximation if you 
> don't provide the gradient.  Using the numerical approximation from 
> "numDeriv" will be quite time-consuming in an optimization routine, 
> since numDeriv uses a high-order Richardosn extrapolation to compute 
> an accurate approximation of the gradient.
>   

No, I don't use it in an optimisation. The expression is part of a more
complicated formula used for calculating some estimates in a special
nonparametric model.

I won't use the numerical approximation; the alternative would be to
calculate the analytical expressions myself. It's not too difficult, but
tedious, and the expressions I end up with may not be the fastest or most
numerically accurate, so if there was a package implementing them in a good
way, it would be nice. :)

> Regardless of your purpose, there is a small bug in your function.  You
> should change   `dmvnorm(cbind(x,y),mu,sig)'  to
> `dmvnorm(cbind(xx,yy),mu,sig)'.

Yes, of course. I originally used x and y when creating the example, but
then discovered that the jacobian() function already used x as an argument
for something else, so I renamed them to xx and yy (though obviously not
everywhere!). I really should have tested it in a completely clean
environment before posting.

-- 
Karl Ove Hufthammer