Sorry! It was a bad idea to run through more than 250 letters in two hours at the very end of a week! In my answer I wrote: ======================================== Dear Mark: In his answer to your question Tim Berryhill showed a program that will generate autocorrelated numbers. Unfortunately, he did not said how to choose coefficient 'a' and 'b' in the main equation x(i+1)=a*x(i)+b*variate for getting predefined correlation 'r'. The answer is: a=r/(1+r**2) b=1/(1+r**2) ; !!!!!!!!!!!!!!!!!!!!!!!!!THIS IS WRONG!!!!!!!!!!!!!!!!!!!!!!!!!!!!! THE CORRECT ANSWER IS: if 'x(i)' and 'variate' are independent standard Gauss , then a=r, b=sqrt(1-r**2) .

Thus for generating two series with predefined auto- and crosscorrelations you can use the following .

/* problem: generate two sequences {x(i)} and {y(i)} so, that Var(x)=Var(y)=1, Cov(x(i),x(i+1))=a, Cov(y(i),y(i+1))=b, Cov(x(i),y(i))=c

Solution: +++++++++++++++++++++++++++++++++++ */ data a (replace=yes keep= x y ); /* N, a,b,c - just for example */ N=20000 ; a=0.2 ; b=0.4 ; c=0.7 ; /*.............................. */ ka=sqrt(1-a**2) ; kb=sqrt(1-b**2) ; kc=sqrt(1-c**2) ; m=c*(1-a*b)/ka/kb ; if abs(m)>=1 then abort return ; km=sqrt(1-m**2) ; /* the first values */ x=rannor(789054) ; y0=rannor(123456) ; y=a*x+ka*y0 ; output ; do i=2 to N ; addx=rannor(123456) ; addy0=rannor(123456) ; addy=m*addx+km*addy0 ; x=a*x+ka*addx ; y=b*y+kb*addy ; output ; end ; run ; data b; set a ; retain x2 y2 ; output ; x2=x ; y2=y ; run; proc corr pearson ; var x2 x y2 y ; run; ========================= Execution of this program will give:

CORRELATION ANALYSIS

4 'VAR' Variables: X2 X Y2 Y

Simple Statistics

Variable N Mean Std Dev Sum

X2 19999 0.0003744 1.00461 7.48734 X 20000 0.0004076 1.00460 8.15103 Y2 19999 0.01347 1.00340 269.32089 Y 20000 0.01343 1.00339 268.66632

Simple Statistics

Variable Minimum Maximum

X2 -4.07777 3.98698 X -4.07777 3.98698 Y2 -3.74807 4.17530 Y -3.74807 4.17530

CORRELATION ANALYSIS

Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / Number of Observations

X2 X Y2 Y

X2 1.00000 0.19924 0.69619 0.28383 0.0 0.0001 0.0 0.0 19999 19999 19999 19999

X 0.19924 1.00000 0.13096 0.69615 0.0001 0.0 0.0001 0.0 19999 20000 19999 20000

Y2 0.69619 0.13096 1.00000 0.40048 0.0 0.0001 0.0 0.0 19999 19999 19999 19999

Y 0.28383 0.69615 0.40048 1.00000 0.0 0.0 0.0 0.0 19999 20000 19999 20000 ===========================================================

Sorry for a misleading answer!

Ilya Novikov Ph.D. Senior Statistician fax : 972-3-5348360 Dept. of Clinical Epidemiology voice: 972-3-5303256 The Chaim Sheba Medical Center mail : epid04@post.tau.ac.il Tel Hashomer Israel 52621

On Fri, 19 Jul 1996 TWB2%Rates%FAR@GO50.COMP.PGE.COM wrote:

> Mark, Generating pseudo random numbers is not for people who have to post here > for suggestions. Further, posting C questions here is a lot like posting > construction questions to a medical list "because, after all, lots of you guys > work in buildings." That said, assuming you have a good random number generator > you can simulate first order autocorrelation in SAS as follows: > DATA AUTOCORR(KEEP=STEP RESULT); > VARIATE=RANNOR(123456); > BIAS=.2*VARIATE; > DO STEP=1 TO 100; > VARIATE=RANNOR(123456); > RESULT=(.8*VARIATE)+BIAS; > OUTPUT; > BIAS=.2*VARIATE; > END; > STOP; > RUN; > > You can use a different random number function if you prefer. You can change > the degree of correlation by changing the values .2 and .8. You could simulate > second order correlation by keeping a few more values around. This code is a > little odd--the _N_ counter will always be 1, rather than running from 1 to 100. > Because the datastep only iterates once, there is no need to RETAIN values. > > Tim Berryhill - Contract Programmer and General Wizard > TWB2@PGE.COM > Frequently at Pacific Gas & Electric Co., San Francisco > The correlation coefficient between their views and > my postings is slightly less than 0 > ----------------------[Reply - Original Message]---------------------- > > Sent by:Mark Sale <Marksale@AOL.COM> > Dear users, > Can anyone help me generate autocorrelated random numbers? That is if the > mean of the numbers is 1 and the number at time x is 1.5234 the probability > of the number at time x+1 being greater than 1 is > 0.5. I'd actually like a > reference to a method, since I want to write (or download) some C code to do > this. > Thanks > Mark Sale M.D. > Assistant Professor of Medicine > Georgetown Univerity > > ===================================================================== >