Determination of mean, variance, covariance, and correlation estimates for axes rotation of bivariate normal elliptical distributio

Crutcher, H. L.

Falls, L. W.

English

NASA Technical Reports Server

NASA TECHNICAL 
MEMORANDUM 
NASA TM X-64595 
DETERM l NAT ION OF STAT ISTICS FOR ANY 
ROTAT ION OF AXES OF A B lVAR /ATE 
NORMAL ELLIPTICAL D ISTR I BUTION 
By Lee W. Falls and Harold L. Crutcher 
Aero-Astrodynamics Laboratory 
May 11, 1971 
NASA 
George C. Mdrsba t Center 
Manhdll Spdce Flight Center, 
MSFC - Form 3190 (September 1968) 
https://ntrs.nasa.gov/search.jsp?R=19710017502 2020-03-23T15:49:48+00:00Z
DETERMINATION OF STATISTICS FOR ANY ROTATION OF AXES OF A 
BIVARIATE NORMAL ELLIPTICAL DISTRIBUTION 
NASA TM X-64595 
4. T I T L E  AND SUBTITLE 
George C ,  Marshal l  Space F l i g h t  Center 
Marshall  Space F l i g h t  Center,  Alabama 35812 
5. REPORT DATE 
1 1 .  CONTRACT t-- 
NASA 
Washington, DC 20546 
TECHNICAL mMBMNDUM 
1 14. SPONSORBNG AGENCY CODE 
16. ABSTRACT P 
A method i s  presented f o r  the  de te rmina t ion  of the  es t imates  of means, varlanees,  
covar iances ,  and c o r r e l a t i o n s  f o r  any r o t a t i o n  of axes of a  b i v a r i a t e  normal e l l i p t i c a l  
d i s t r i b u t i o n .  This problem a r i s e s  i n  connect ion wi th  v e h i c l e  launches a t  Cape Kennedy, 
F l o r i d a ,  when winds normal and p a r a l l e l  t o  t h e  v e h i c l e  f l i g h t  pa th  a r e  launch coa- 
s t r a i n t s .  A vec to r  wind d a t a  sample i s  resolved i n t o  wind components i n  a rectangular 
coordina te  system, and the  b i v a r i a t e  normal e l l i p t i c a l  d i s t r i b u t i o n  theory is appl ied  
t o  t he  d a t a  sample of component winds. However, t h e o r e t i c a l  i n fe recces  are r e q u l r e d  
f o r  any a r b i t r a r y  f i r i n g  azimuth of t he  launch veh ic l e .  This r equ i r e s  a rotathsn of 
axes of t he  b i v a r i a t e  normal s u r f a c e  and the  computation of new s t a t i s t i c s  a f t e r  
r o t a t i o n .  
-t- 
"u. S. Dept. of commerce, NOAA, Environmental Data Serv ice ,  Nat ional  Climatic  Center ,  
Ashev i l l e ,  North Carol ina.  
18. DISTRIBUTION STATEMENT 
b i v a r i a t e  normal e l l i p t i c a l  d i s t r i b u t i o n  Unclass i f  ied-Unl imited 
Means, var iances ,  covar iances  and 
c o r r e l a t i o n  c o e f f i c i e n t s  4- ,' / - 
,C / / c L >  r ' - ( -  
E. D. G e i s s l e r  
D i rec to r ,  Aero-As t rodynaaics  Laboratcry 
UNCLASSIFIED I UNCLASSIFIED I 9 $3 -00  
B %lSFc - Form 3292  (May 1969) 
DETERMINATION OF STATISTICS FOR ANY ROTATION OF AXES 
OF A BIVARIATE NORMAL ELLIPTICAL DISTRIBUTION 
SUMMARY 
A method i s  presented f o r  t he  de te rmina t ion  of the  es t imates  of 
means, va r i ances ,  covar iances ,  and c o r r e l a t i o n s  f o r  any r o t a t i o n  of axes 
of a  b i v a r i a t e  normal e l l i p t i c a l  d i s t r i b u t i o n .  This problem a r i s e s  i n  
connect ion wi th  v e h i c l e  launches a t  Cape Kennedy, F lo r ida ,  when winds 
normal and p a r a l l e l  t o  the  veh ic l e  f l i g h t  pa th  a r e  launch c o n s t r a i n t s .  
A vec tor  wind d a t a  sample is  reso lved  i n t o  wind components i n  a rectangu- 
l a r  coord ina te  system, and the  b i v a r i a t e  normal e l l i p t i c a l  d i s t r i b u t i o n  
theory i s  appl ied  t o  the  d a t a  sample of component winds. However, 
t h e o r e t i c a l  in ferences  a r e  requi red  f o r  any a r b i t r a r y  f i r i n g  azimuth of 
t he  launch veh ic l e .  This r equ i r e s  a  r o t a t i o n  of axes of the  b i v a r i a t e  
normal s u r f a c e  and the  computation of new s t a t i s t i c s  a f t e r  r o t a t i o n ,  
I. INTRODUCTION 
I n  1846, Bravais [I] presented the  theory  of normal frequency dis- 
t r i b u t i o n s  i n  two and more v a r i a b l e s .  Most modern s t a t i s t i c a l  textbooks 
inc lude  some d i scuss ion  of such d i s t r i b u t i o n s .  Outstanding con t r ibu t ions  
a r e  no tab le  i n  t he  works of Bertrand [ 2 ] ,  Czuber [ 3 ] ,  Pearson [ 4 ] ,  
Kluyver 151, and Student  161. 
Fur ther  developments and some p r a c t i c a l  a p p l i c a t i o n s  of these  d i s -  
t r i b u t i o n s  w i t h  p a r t i c u l a r  regard t o  geophysical  problems a r e  presented 
by Crutcher  and Baer [ 7 ] ,  Crutcher and Moses [81, and Groenewoud, Hoaglin, 
V i t a l i s ,  and Crutcher [ 9 ] .  Anderson [ l o ]  and Hald [ l l ]  t r e a t  t he  subject 
i n  genera l  form. This r e p o r t  p re sen t s  a  method f o r  t he  de te rmina t ion  of 
t he  es t imates  of means, va r i ances ,  covar iances ,  and c o r r e l a t i o n s  f o r  any 
r o t a t i o n  of axes of a  b i v a r i a t e  normal e l l i p t i c a l  d i s t r i b u t i o n ,  Although 
t he  procedure can be extended t o  the  case  f o r  more than two v a r i a b l e s ,  
t h i s  c a s e  is  no t  discussed here ,  
This problem a r i s e s  i n  t he  determina.tion of p r o b a b i l i t i e s  of success-  
f u l  launches a t  Cape Kennedy, F lo r ida .  For example, winds exceeding some 
given va lue  normal and p a r a l l e l  t o  t he  f l i g h t  pa th  may be defined as 
launch c o n s t r a i n t s .  Those winds normal t o  the  f l i g h t  pa th  a r e  designated 
a s  r i g h t  and l e f t  c ros s  winds, while  those  p a r a l l e l  t o  the  f l i g h t  pa th  
a r e  called head and t a i l  winds. A vec to r  wind d a t a  sample i s  reso lved  
i n t o  wind components i n  a  r ec t angu la r  coord ina te  system and the b i v a r i a t e  
normal e l l i p t i c a l  d i s t r i b u t i o n  theory is appl ied  t o  the d a t a  sample of 
component winds. However, t h e o r e t i c a l  in ferences  a r e  requi red  f o r  any 
a r b i t r a r y  f i r i n g  azimuth of the  launch veh ic l e .  Making such inferences  
requires a  r o t a t i o n  of t he  axes of the  b i v a r i a t e  normal s u r f a c e  and the  
computation of new s t a t i s t i c s  a f t e r  r o t a t i o n .  
II. THE BIVARIATE NORMAL ELLIPTICAL DISTRIBUTION 
Let  X r  and x2 be normally d i s t r i b u t e d  v a r i a b l e s  w i t h  parameters 
pol) and (529a2) .  51 and 52 a r e  t he  r e s p e c t i v e  means, 
le op and a2 a r e  t he  r e s p e c t i v e  s tandard  dev ia t ions .  Le t  p be the  
correlation c o e f f i c i e n t ,  which i s  a  measure of t he  dependence between 
xi and x2, Now, the  equat ion  of the  two-dimensional normal d e n s i t y  
fuac t ion takes t he  form 
Given t he  v a r i a b l e s  xl and x2, we c a l c u l a t e  t he  e s t ima te s  of t he  
b i v a r i a t s  normal parameters.  Denote t h e  es t imates  a s  El, Z2, Sxl, Sx2, 
and rXlX,, A s p e c i a l  ca se  of t he  problem described here  i s  t o  
determine t he  es t imates  of t he  parameters where a  r o t a t i o n  of t h e  axes 
through an  angle  a reduces the  c o r r e l a t i o n  of components t o  zero. This 
case is  discussed i n  Crutcher and Baer [ 7 ] ,  and Groenewoud, Koaglin, 
Vitalis, and Crutcher [ 9 ] ,  From the  s t a t i s t i c s  i n  t h e  (x1,x2) space,  
we must determine the  s t a t i s t i c s  f o r  any r o t a t i o n  of t he  axes through 
an a r b i t r a r y  angle  a i n  t h e  fol lowing way. 
L e t  (yl,y2) be the  new space a f t e r  r o t a t i o n  wi th  or thogonal  axes 
Yl and Y2. 
Y1 = X1 cos a + X2 s i n  a 
Y2 = X2 cos a - Xi s i n  a, 
where a i s  t h e  angle  of r o t a t i o n ,  and xl and x2 a r e  t he  o r i g i n a l  axes 
along which the  values of X1 and x2 a r e  measured. Denote t h e  sample 
var iances  of (xl,x2) as  s2 and s2 r e s p e c t i v e l y .  The covariance sf 
( ~ 1 3 ~ 2 )  is X1 x2' 
S = r S S .  
X l X 2  x1x2 x 1  x2 
For t he  s p e c i a l  c a s e  when S o r  r i s  equal  t o  zero ,  the  var iance-  
x1x2 x1x2 
covar iance  ma t r ix  i s  
S e t t i n g  the  determinant  of t h i s  ma t r ix  equal  t o  zero  and so lv ing  
f o r  t he  two r o o t s  of t he  quadra t i c  equat ion is e s s e n t i a l l y  the  rotation 
of axes through an angle  a t o  where components along the  new axes are 
uncorre la ted .  Thus, the  r o o t s  Al and h2 a r e  the  var iances  along the 
major and minor axes,  r e s p e c t i v e l y ,  of t he  b i v a r i a t e  d i s t r i b u t i o n  and 
the  covariances ( c o r r e l a t i o n s )  a r e  zero. From these  d a t a ,  var iances  
and covariances ( c o r r e l a t i o n s )  a long any s e t  of new axes can be deter- 
mined by r o t a t i n g  the  axes from t h i s  b a s i c  pos i t i on .  Then the ma t r i x  
w i l l  be 
w h e r e  the new space i s  (y1,y2) and yl and y2 a r e  t he  axes.  
The s t a t i s t i c s  i n  t he  (y,, y2) space may be derived a s  fol lows:  Let 
$I, Y 2 )  S~ , s2 9 S be the  means, va r i ances ,  covariance,  and 
~1 y2 ~ 1 ~ 2 '  '~1~2 
correlation c o e f f i c i e n t  i n  the  (yl,y2) space. Taking mathematical 
expectations of equat ions (1) and (2), we have f o r  t he  new means 
;i, = ii, cos a + Z2 s i n  a 
- - - 
y2 = x2 cos a - X, s i n  a. 
Subt rac t ing  equat ion  (3)  from equat ion  ( I ) ,  we have 
- - - $V1 - 'Y1) = (X1 - X1) cos a + (X2 - X2) s i n  a, 
and s ~ h t r a c t i n g  equat ion (4) from equat ion  (2), we have 
- ( Y ~  - k2> = (x2 - k2) cos a - (x, - x,) s i n  a. 
Let 
b e  the dev ia t ions  from the  r e s p e c t i v e  means. Now, equat ions (5) and 
(63 become 
yl = xl cos a + x2 s i n  a 
y2 = x2 cos a - xl s i n  a. 
Squaring equat ions (7) and (8) g ives  
2 2 2 2 yl = xl cos a + x2 s i n 2 a  + 2x1x2 cos a s i n  a (9 1 
2 2 2 y2 = x2 C O S ~ ~  + x1 s i n 2 a  - 2x1x2 cos a s i n  a. 1181 
Now, tak ing  expected values of equat ions (9)  and ( l o ) ,  we have 
f o r  the new var iances  i n  the  (yl, y2) space 
S" = s2 cos2a  + s2 s i n 2 a  + 2s cos a s i n  a 1[11) 
Y 1  X l  x2 x1x2 
and 
To d e r i v e  the  covar iance ,  SYlY2, i n  t he  (yl,y2) space ,  we m u l t i p l y  
equat ion  (7) by equat ion (8) and o b t a i n  
2 2 yly2 = x,x2(cos2a - s in2a )  + cos a s i n  a (x2  - xl) .  (1 3 )  
Taking the  mathematical expec ta t ion  of equat ion (13) g ives  the  
covariance i n  t h e  (yl, y2) space 
S = S (cos2a - s in2a) + cos a s i n  a ( s 2  - 2 
Y l Y 2  X l X 2  X 2  SXl)e ( 1 4 )  
F i n a l l y ,  t he  c o r r e l a t i o n  c o e f f i c i e n t  i n  the  (yl, yZ) space i s  d e f i n e d  
as  
Equa t ions  ( 3 ) ,  (4 ) ,  ( l l ) ,  ( 1 2 ) ,  (14) ,  and (15) g i v e  t h e  r e q u i r e d  
e s t i m a t e s  of t h e  s t a t i s t i c s  i n  t h e  (y1,y2) s p a c e .  
A computer program whose i n p u t  i s  t h e  s t a t i s t i c s  i n  t h e  (xl,x2) 
space and  any a r b i t r a r y  a n g l e  of r o t a t i o n  of  t h e  b i v a r i a t e  normal s u r -  
f a~ce  i s  a v a i l a b l e  upon r e q u e s t .  The o u t p u t  of  t h e  program i n c l u d e s  
t h e  r e q u i r e d  s t a t i s t i c s  i n  t h e  (y1,y2) s p a c e .  
REFERENCES 
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Acad. S c i .  P a r i s ,  Mem. Sav. E t rang .  2, pp. 255-332. 
B e r t r a n d ,  J . ,  1888: C a l c u l  des  p r o b a b i l i t e s :  Note s u r  l a  
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du t i r  a la c i b l e ,  Compt. Rend. a, pp. 387-391; pp, 521-522 ,  
Czuber,  Emanuel, 1891: T h e o r i e  d e r  Beobach tungs feh le r ,  B, 6, 
Teubner ,  L e i p z i g ,  pp. 400-411. 
Pea rson ,  K., 1900: On t h e  c r i t e r i o n  t h a t  a  g i v e n  s y s  tern of devia- 
t i o n s  from t h e  p r o b a b l e  i n  t h e  c a s e  of  a  c o r r e l a t e d  sys tem sf 
v a r i a b l e s  i s  s u c h  t h a t  it can  r e a s o n a b l y  b e  supposed t o  have 
a r i s e n  from a  random sampl ing ,  P h i l .  Mag. 50, pp. 157-175, 
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Meteorology,  VoL. 1, No. 4 ,  pp. 522-530. 
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R a d i o l o g i c a l  P h y s i c s  D i v i s i o n  Summary R e p o r t ,  ANL-6769, pp, 181-196, 
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APPROVAL 
DETERMINATION OF STATISTICS FOR ANY ROTATION OF AXES 
OF A BIVARIATE N O W L  ELLIPTICAL DISTRIBUTION 
The information in this report has been reviewed for security 
classification. Review of any information concerning Department of 
Defense or Atomic Energy Commission programs has been made by the MSFC 
Security Classification Officer. This report, in its entirety, has 
been determined to be unclassified. 
This document has also been reviewed and approved for technical 
accuracy, 
Chief, Vaugv Aer pace Environment Divis i:tn 
E, D, Geissler 
Director, .Aer o-As trodynamics Laboratory 
MSFC-RSA, Ala 


Determination of statistics for any rotation of axes of a bivariate normal elliptical distribution

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