[R] generate bi-variate normal data

[Liaw, Andy]

If (X, Y) ~ N(mx, my, sx^2, sy^2, r)  (r being the correlation, not
covariance), then 

Y|X ~ N(my + r * (sy / sx) * (X - mx), sy^2 * (1 - r^2))

You can use this to work out what you need to do.

Best,
Andy


________________________________

	From: Shin, David [mailto:david.shin at pearson.com] 
	Sent: Wednesday, July 05, 2006 1:14 PM
	To: Liaw, Andy; 'r-help at stat.math.ethz.ch'
	Subject: RE: [R] generate bi-variate normal data [Broadcast]
	
	

	Thanks for Andy's comment and help. I should have used a better
example for my question. Below is my exact question and I will appreciate a
lot for any insights.

	I generate a bi-variate normal distribution with mean = c(0, 0.2)
and variance covariance matrix = matrix(c(1, .025, .025, .0025), nrow = 2):

	> x <- rmvnorm(10, c(0, 0.2), matrix(c(1, .025, .025, .0025), nrow =
2))

	> x

	            [,1]      [,2]

	 [1,]  0.1595351 0.1715898

	 [2,] -0.5177577 0.1839222

	 [3,] -0.8794011 0.1896593

	 [4,]  1.0584185 0.2208470

	 [5,]  0.1960055 0.2199169

	 [6,]  0.6450406 0.1773001

	 [7,] -2.2160986 0.1810803

	 [8,]  0.2131569 0.1223121

	 [9,] -0.3598349 0.2402232

	[10,] -0.3905455 0.1787059

	> x.sum <- apply(x,1,sum)

	> x.sum

	 [1]  0.3311249 -0.3338355 -0.6897418  1.2792655  0.4159225
0.8223407 -2.0350183  0.3354690 -0.1196116 -0.2118395

	if I call x[,1] as theta.year1 and x[,2] as growth.year1 then the
mean of x.sum is 0+0.2 = -.2 and the standard deviation of x.sum is
sqrt(1+.0025+2*.5*1*.05) = 1.0259

	<<...OLE_Obj...>> 

	assume the correlation is again 0.5, I would like to generate
another bi-variate normal distribution with the fixed first column that
equals to x.sum. If the mean and SD of the second column is 0.2 and 0.05,
respectively, the mean of this bi-variate normal distribution is c(0.2, 0.2)
and the variance-covariance matrix is: matrix(c(1.0259^2, 0.5*1.0259*0.05,
0.5*1.0259*0.05, 0.0025), 2)

	Thanks again for helping me.

	David






		-----Original Message-----
	From: Liaw, Andy [mailto:andy_liaw at merck.com]
	Sent: Wednesday, July 05, 2006 11:50 AM
	To: Shin, David; 'r-help at stat.math.ethz.ch'
	Subject: RE: [R] generate bi-variate normal data

	From: Shin, David

	> 

	> Dear all,

	> 

	> I would like to generate bi-variate normal data given that 

	> the first column of the data is known. for example:

	> I first generate a set of data using the command, x <- 

	> rmvnorm(10, c(0, 0), matrix(c(1, 0, 0, 1), 2))

	> 

	> then I would like to sum up the two columns of x:

	> x.sum <- apply(x, 1, sum)

	> 

	> now with x.sum I would like to generate another column of 

	> data, say y, that makes cbind(x.sum, y) follow a bi-variate 

	> normal distribution with mean = c(0, 0) and sigma = 

	> matrix(c(1, 0, 0, 1),2)

	x.sum as you described would be distributed as normal with mean=0
and

	variance=2 (so you might as well just use x.sum <- rnorm(10, 0,
sqrt(2))),

	so I don't see how you can get to the second step where you want
x.sum to

	have variance=1.  Also, since the covariances are 0, you could just
generate

	the columns separately using rnorm() and cbind() them together.

	It might be helpful for you to get some basic understanding of math
stat.  I

	only say that because most likely there are other steps to whatever
task you

	are doing (people are unlikely to be generating random numbers just
for

	kicks), and there's no telling what other things you are doing

	inefficiently, or even erroneously.

	Andy

	 

	> I will appreciate for all insights.

	> 

	> David s.

	> 

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