4 tI #I~ I If AN p'1%%'9Ak M0 Po9*t% toI Nol!AMI't t | 'tfl TRIOO1S(StII a;) ,1

(In Analysis of" n-Dimensional

"Normal Probabilities

lsvepeiil hy I A (iGRA and R. H. (4IRISTEN

1 ýJUN D D C

AUMI 8 1970

Systens Ensinectirl$ Operid~lo

THE AEROSPACE CORPORATION

Pre'psed for SPACE. AND MISSILE SYSTEMS ORGANIZATION

AIR FORCE SYSTEMS COMMANI)LOS ANGELES AIR FORCE STATION

Los Angeles, California

TIIS DOCUMENT HAS BEEN APPROVED FOR PUBLICRELEASE AND SALE: ITS DISTRIBUTION IS UNLIMITED

Air r'orce litprrt No, A rtialout, Hir Nio,RA •gO +TK w7Tu-Z1 TII .0n¢,(,f(41I.+ .0I) .

ON ANALYSIS Of' n4DIMENSIONALNORMAL PROVlABILITTKS

Prepared byI. A, Oura and R. H. Gersten

Engineering Science Operations

70 JUN 02

Systems Engineering OperationsTHE AEROSPACE CORPORATION

Prepared for

SPACE AND MISSILE SYSTEMS ORGA:,., 'IAIR FORCE SYSTEMS COM4AN'

LOS ANGELES AIR FORCE STATIOLos Angeles, California

Contract No. F04701-69-C-ot)o•(

This document has been approved tfr publLrelease and sale; its distribution is unlimitecd

FOREWORD

The report is published by The Aerospace Corporation, El Segundo,California under Air Force Contract No. F04701-69-C.0066.

This report, which documents research carried out from May 1969

through March 1970 by R. H. Gersetn and 1. A. Gur&, Satellite Navigation

Department, Guidance and Navigation Subdivision, Electronics Division,

Engineering Science Operations, for the Ground Data Systems Office, Group

III Programs Directorate, Satellite Systems Division, Systems EngineeringOperations, was submitted for review and approval to SAMSO(SMUE) on

19 June 1970,

Approved

D.'F. Meronek, Director .......Guidance and Navigation Subdivision Groud Data ystems OfficeElectronics Division Group III Programs DirectorateEngineering Science Operations Satellite Systems Division

Systems Engineering Operations

Publication of this report does not constitute Air Force approval of the

report's findings or conclusions. It is published only for the stimulation and

exchange of ideas.

Thom-as M. -Sumn-er,-Col, USAF-

Director of EngineeringDefense Support Program

-ii-

ABSTRACT

Although Gaussian probability densities are extremely useful inengineering analyses, they ar'e frequently misinterpreted in this context.

This report In specifically designed to clarify this situation. While the

material presented may be well known among statisticians, tho engineering

community appears to require such exposition.

The report begins by developing, from first principles, the contoursof constant probability associated with n-dimensional normal density

* !functions. Analytic expressions are then derived for the probability that

the random variables under study will be contained within these contours.

The results obtained are fully discussed from an engineering viewpoint.

-iii-

Although error analyses constitute one of the more frequently

encountered types of engineering problems, the topic is fraught with fallacies,

misconceptions, and distortions. One basic difficulty occurs in attempting to

interpret covariance matrices. While the usual assumption of Gaussianness is

reasonable, there is widespread tendency to assign erroneous probability con-

fidence levels to the associated error ellipsoids. For example, it is not well-

known in the engineering community that a Z-sigma ellipsoid carries a different

probability confidence level than a 2-sigma ellipse or, for that matter, than a2-sigma line segment.

In this report, an attempt will be made to clarify this situation,Although the material to be presented is supposedly "well-known" among a

miniscule clique of theorists, it is often the subject of much debate and thecause of considerable confusion among engineers. For this reason, the

development will begin with a tutorial review of n-dimensional Gaussian prob-ability densities and their associated contours of constant probability in var-ious coordinate systems. The results obtained will then be used to develop

expressions for the probability associated with error contours in arbitrarydimensional spaces. The practical engineering aspects of the theory with

an example will be considered in the final section.

NOTATIONAL CONVENTIONS

I The symbols A, C, T, A, 1Z I and represent matrices.

2. The unsubscripted symbols x, y, z are column vectors.

3. The asterisk ( *) is used to indicate matrix transposition.

4. The symbol 6denotes statistical expectation.5. All other symbols represent scalar quantities.

-1

GAUSSIAN DENSITY FUNCTIONS AND PROBABILITY CONTOURS

Let x be an n-dimensional Gaussian random vector with mean x and

covariance matrix C given by

x de(x)-*(1 )

C (x - Z)(x -=)*

In general, C is positive semidefinite. However, since the case of singular

C is of no practical interest, it will be assumed that C is strictly positive

definite.

The probability density function for x is

(())n/Z(Det C) 1 1/ exp [-i(x- )*C-l (x -x (2)

It follows from the above that contours of constant probability density are

defined by

(x -') *C (x -1 ) = k (3)

for arbitrary constant k. Geometrically, Equation (3) describes hyperellip-

soids in n-space.

Now, it is desirable to define a space in which the coordinate axes are

coincident with the princip-l axes of Equation (3). Toward this end, define

a new n-dimensional zero mean random vector y by

y = A(x- )

Since C is symmetric, A can be chosen so that

A* = A '

Det A 1 (5)

e(yy*) A A ACA*

with

a 2"1 0

a2a2

A = (6)

0 a 2

n

The elements of A are, of course, the eigenvalues of C, and the columns of Aare the eigenvectors of C. Thus A is an orthogonal matrix which trans -

forms the covariance C into a covariance A associated with the principal axes

of the hyperellipsoid given by Equation (3).

In general, if random vectors x and y are related by a one-to-one

(nonsingular) mapping, the associated density functions are related by(Ref. 1)

Py (y) Px(x)Det[L#] (7)

where

1x 1 'x 1

ax2 ax2 ax2

-T-J IV (y1 Z'n (8)

ax C)x axn n .. n

-3-

For the case at hand, the mapping in question is defined by Equation (4), so

that application of Equations (Z) and (5) to Equation (7) yields

P (y) = Det A- 1 1"

_ 12 exp (-~- I y)((ZnT) n2(Det C) lZY

(2Tr)n/Z(Det A)/ 2 exp y A- Iy)

where the last result follows from Equations (5). Finally, from

Equation (6)

I 1(a 1 Y2 YnP (y) = (Zn)nI•alZ'' n exp +[a(4 -4 ... + 2n (10)

It follows from Equation (10) that contours of constant probability density

in y-space are the hyperellipsoids given by

2 Z 2

+ +Y+ = ka 1 n

These contours are identical to those defined by Equation (3). However,

the principal axes are now coincident with the y-coordinate axes as desired,

ie Y ' Yz . ' Yn are independently distributed.

For ease of manipulation, it is convenient to further transform the y

variables to a space in which the constant probability contours are hyperspheres.

In line with this goal, define the new random vector z by

z Ty (12)

where

-4-

1/00

T A C1/ 2 = Ia2 (13)

01/lan

From a relationship analogous to Equation (7)

Syi

OZ) =- De exp [ z*TIA-IT"1 (14)

(2,,n)/ a 0 - n

which reduces to

pz(Z) ._ I exp [ z + z + + z (15)(2T,)n/ I 1 2 n

upon application of Equations (12) and (13). Thus, as desired, the contours

of constant probability density defined by Equation (15)

2 2+ Z k2 (16)zI + z + ... + z n -

are hyperspheres of radius k. Comparison of Equation (15) with the general

Gaussian density function, Equation (2), shows that , P, .12 . ., z are

independent Gaussian random variables with zero mcaji :,r..d unit variances.

From the basic concept of density functions, the pr - ihiWi th,,t

ZIP z 2 . n lies within the hypersphere of radius k ib Lj U>

Pn (k) =. dz I d cz2 f1dz n exp 1/ 2 z + + +

I -k '1 •I Z I "'2 " n-I

5(F,

From the precdinlg analys i, it ti cluar tl at l•(k) ti also the

probability that x lies within the hyperellipsoid of Mquation (4). Trhu, the

basi, problem of finding p n(k), tho probability conftdotice level of the "k-

sigma" hyperellipsoid in u-space, reduces to -ovaluation of &quaticni (17).

It may appear that the most direct approinch to obtain the desired prob-

abilities is by integration of (17) uping n-dimenslonal polar coordinates,

While this technique readily yields the dusired results for n - 1|4, and

3, the integrals become extremely unwieldy for higher dimensions, This

difficulty can be overcome by recasting Equation (15) In terms of the

chi-square distribution and effectively collapsing n dimensions into one

dimension. Direct integration is then a simple task, yielding a recursion

formula for pn(k) valid for all n and k.

DLTERMINATION OF PROBABILITIES CORRESPONDINU TO "k-SIGMA"

CONTOURS.

Define the random vari-blbe u by the relation

u =z2 + z+ ... + aI zIt can be shown [Ref 3] that the probability density function for u, pu(u). is

chi-square; that is

P I un)/exp(-ul2) for u>0 19u(u) (1n)

ou(u) = 0 for uT0

where F( ) denotes the G.imina function.

In view of Equations (16) and (18) the interior of the k-sigma hyperellipsoid in

x-space corresponds to the line segment

0 < u k (20)in u-space. Thus,

-6-

a

Pn (k) - " (1.0 *u('l/ep (-u/1) du .lin P(nm I

Speviallaing Equation (I1) tor n i I a.d revallunM owt ri m a(1"0 yields

+.I k all"M

"aPForrn.no the subuit trton iw i that+ Equatton (•A) bmniomds

p (k) .7 1 /p(- 4df ti IrfUl ; 'k 1 (iA)0

F~or n a , not that Pl(I) + 1+ so that £~quattuti (e1) tmmUewdtly, yiuldv

p.} /•J -/ du I - ("~l 4)

A general recursion formula for all n can bc ,.gtibliihuad by

expressing Equation (i1) as

pn+ P (k) -•l•I~n•:~ u1/ 2 U|oxpki-l/) du 1•

n PI4+1r((n/4 4 1) uxtu/ u(S

Then, from the properties of the Gamma function,

2(n/n+l) n2nl"(n 1

j -~7 .

anti, (tir votiatatit m and a,

l i ,'x.p(av)dv a l (av)- mfvm1exp(av)dv M)

Thtus, N~quation ca•)vn We rowritten

Pn k n / (n.a)/Z exp(-u/Z)du (28)

k expt-ka/2)

Now, tl first term on the right side of Equation (28) in precisely pn(k) as

d'linatl by E(•tintion (41). Thus, Equation (Z8)can be expressed as the

rocurs|i-n formula

n(k)apW)-xpl-k /2)(29)pn+2ll pnl- na n-)/ rw 1

Application of Iquations (23) and (24) to Equation (29) yields, after some

manipulation,

p W - erf(k/\Z) - /\(T xp(-k 2 /Z) k + ( + + k (30)nI T I* G 130 5.' (n

for odd n

Pn(k) - I - expl-k l/2) 1 + k2 + k4 + ... + l(n-2 ) (31)n for even n T rI e4*e(nZ

These results were obtained by L. Schwartz[ Ref. 51 by an indirect method.

S.8-.

r

A graph of p n() vs k for solected values of n, as computed fromEquations (30) and (31), is presented in Figure 1. These same results arelisted in tabular form in Tables 1 through 8; Tables I through 4 displayingvalues of k corresponding to selected pn(k) and n, and Tables 5 through 8displaying values of pn(k) corresponding to selhcted k and n.

ENGINEERING INTERPRETATIONS

One of the conventional by-products of an error analysis is ann-dimensional covariance matrix. However, this array, though frequentlyencountered, is often misinterpreted. It should be clear from the precedingdiscussion that this matrix is related to the one-sigma hyperellipsoid inn-apace. Indeed, the probability that the random n-vector lies within itsboundaries is Pn(l) as given by Equation (30) or (31), not p,(l) (68%) as isoften incorrectly assumed. Specifically, pl(l) is the probability that anyone element of the random vector lies between the intercepts of the hyper-ellipsoid with the corresponding coordinate axis without regard to where theremaining elements lie. In short, for n > 1, Pn(l) assumes simultaneity,while pl(l) does not! Indeed, pn(1) is always smaller than pl(l) (SeeFigure 1).

Another common misconception is that the square roots of thediagonal elements of the covariance matrix represent the lengths of thesemi-axes of the one-sigma error hyperellipsoid. Actually, they bearno direct relationship to this contour. Note that the intercepts of theone-sigma hyperellipsoid with the coordinate axes are given by thereciprocal square roots of the diagonal elements of the inverse of thecovariance matrix as can be shown by examination of Equation (3).These quantities, however, only provide the coordinate interceptsof the one-sigma hyperellipsoid and do not, in general, define thesemi-axes of the hyperellipsoid. The distinction between thesedisappears only when the coordinate axes and principal axes coincide.

9-

Since the probability level corresponding to a one-sigma hyper-

ellipsoid in n-space varies with n and is small for large n, it is generally

more desirable to consider hyperellipsoids related to a specific probability

confidence level (such as 50%). The analog of the covariance matrix for such

a region can be found by multipliIng the covariance matrix by the k2 t cor-

responding to any given pn(k). The resulting matrix can be convenientlynnamed the "k-variance matrix" and the associated error contour called the

"k-sigma hyperellipsoid. " As with the covariance matrix, the square roots

of the reciprocals of the diagonal elements of the inverse of the k-variance

matrix represent the intercepts of this hyperellipsoid with the coordinate

axes.

It is evident in many cases that the diagonal elements of the inverse

of the covariance (or k-variance) matrix will not adequately describe the prob-

ability distribution of a given random vector. Now, in general, a hyper-

ellipsoid in n-space is not very useful since it cannot be readily visualized. In

addition, the enclosed region is meaningless if the various components of the

random vector do not represent the same physical quantities (i. e., if they

are not measured in commensurate units).

This is algebraically equivalent to diagonalizing the covariance matrix,2multiplying the resultant by k , and rotating back to the initial coordinates.

-10 -

In such situations, insight can be gleaned by projection into lower

dimensional subspaces, usually 2-or 3-space, in which each of the components

can be expressed in the same units. Two possibilities are available: (1) 2-

or 3-dimensional projections of a general n-dimensional hyperellipsoid which

constrains all random variables can be considered; (2) k-sigma ellipses

or ellipsoids can be found for compatible 2 or 3 element sets of the random

vector without regard to the behavior of the remaining elements. In the

former case, the desired contour is established by extracting the proper parti-

tion of the inverse of the k-variance matrix. In the latter case the desired

ellipse or ellipsoid can be obtained by partitioning the covariance matrix

first, multiplying the result by the appropriate k 2 for p,(k) or p 3 (k) and

then performing a matrix inverse. In either case, the magnitudes and direc -

tions of the principal axes of the final contours can be found by solving the

related eigenvalue-eigenvector problem.

Instead of considering the geometrical problems of displaying hyper-

ellipsoid error regions, a possible alternative is to present results in terms of

hyperspheres with the same probability level. Although this approach may

seem appealing at first, it has accompanying disadvantages. Unless the

principal axes of the given hyperellipsoid are nearly equal, use of the corre-

sponding hypersphere can lead to serious errors in engineering judgement

because all information regarding preferred directions will be lost. Further-more, even when the use of hyperspheres is justifiable, the desired radius isnot easily determined. For n=2 and nz.3 special algorithms exist for this com-putation [References 2, 4]. In these cases, when pn (k) .50, the results are

the well-known Circular Error Probability (CEP) and Spherical Error Probability(SEP), respectively.

- 11 -

In order to illustrate the above concepts, consider the various

50% probability contours in the xI - x2 plane associated with a zero

mean Gaussian random vector x = (x,, x 2 , x 3 ) whose covariance

matrix is

1 0. 95 0. 90]= 0 0.95 1 0.95 (32)

0. 90 0. 95 1

First establish the xI - x2 projection of the 3-dimensional

ellipsoid which encloses 50% of the values of xI. As indicated above,

the appropriate partition of the inverse of the k-variance matrix must

be found. For the case at hand, n = 3, P 3 (k) = 0. 50, so that Table 1

yields k = 1.5382. Thus

4338 -4.226 .112f

(k2P) -4 226 8.453 -4.226 33)

( P 1122 -4.226 4.338 (

and the desired contour is given by the equation

x x] 4.338 -42261 [x] 3[4.226 8.453] Lx2

Equation (34) implicitly carries a constraint on x 3 . A 50%

contour in the xI - x2 plane which ignores the value of x 3 can also be

established. To accomplish this, first partition the P-matrix, then

multiply by the k2 satisfying P 2 (k) - 0. 50 (where from Table 1,

k - 1. 1774) and finally invert the result

-12 -

[I X21 2 1 0.95-I

L. 177 4 )Z 0 (35)

-0 95 1 x 2

or

-7. 029 7. 399 x

Observe that the reciprocal square roots of the diagonal

elements of the matrices in (34) and (36) give the coordinate axes inter-

cepts of the respective ellipses, while the solutions to the corresponding

eigenvalue-eigenvector problem yields the principal axes and orientation

angles. Note also that the square roots of the diagonal elements of the

P-matrix are the standard deviations of the individual elements x 1 , x 2 ,

and x 3 . Multiplying each by k = 0. 67449 (obtained from Table 1 for

pl(k) = 0.50) gives the 5016 boundaries on each element of x without regard

to the behavior of the remaining two elements.

The various contours and boundaries disci: ;,.d above are

illustrated in Figure 2. For completeness the CEP ,- l rrol.ction

of the SEP are also shown on the diagram. The exampo -Vi, t li

considerable variation possible for different contours ;i . , .% Ii,

the same covariance masrix. 1he analyst, therefore, mti.u. careuflly

consider the alternatives and their interpretations ic',r - 4 J I

choice. Whilv in general, there is no "best" geoum-trit :t rpreltititI

of a Gaiiisian probability distribution,error ellips.s iid vllipstids .are

usually preferabie since they contain more ln toi;tat ii tihan thehe ttlr fpormins.

-II

........... t............. .. .................. ........ ....................

- . ~.*.........-.... ..... .. . .

. . . .. . . .. . ..

0.0

-14-

2.0 PROJECTION OF SEP

500/ ELLIPSE COMPUTEDBY IGNORING X3 [Eq(36)]

PROJECTION OF 50% ELLIPSOID

.2.0 -1.6 .1.2 -0.8 -0.4 00. 12 16 20

-2.0

Fig. 2. Probability Contours for Exainiple

N CCOOO Vo&-&4 C N4CCO 0000-cmo OCOCOOCOOCCOCO04D 0OM

LaJkAWW W "W AJWWW W

4 - vm i .- m jQ.fy) n pZ 0- 04 0.4 'C.- P" , 0.4 0" Nf Oj r14 tI 4 -M (Mf'

0- C)aQ0Q :

0 Qo o oD o0 0 Q 0 l0 0 00 4.4 4 4* +4 44+*+4 *..+4*

4)A J W U ;U 4JJJ AA oA.) 4L: U) .U At) .J 4U J A) W "

.Fn C, 00 7~.f k pM N .4 - N J - 0 0 aN-

t~~~~ Vrl 0- X, 3% a- Jq *w Uwl N - rN 4

,L , iA. .4 j'. w. .-* L,. .,,, t

* -C 00 4 C C 0 0Y 4 a ..l - ' -3 A. C. V- '% !- -S P4 L; P- N .

7 C.;

~L %&- 4. L . 4 L . U. .'A ~ I. A j .

-3 L

*'.Pag 164

0 000 coo 00 Coc 0000000D"c~0 0 0 000 c o oc o0 a00000;00

0 04DO & W OM m m~oP-4 ram M4M PMko" M 0o

MI *.sj NOo 0* 00zNON 1 ;r01 AA 0;f

C 0 jWM4 &0 vtk-z 40 O~Q - NM4 ~0 v

0- a u.M 0 4' M M a I- *I- a, N P. N 0 M 0%

OOOOOM#OAOOOOOOOC0-OM4 4

4 *y ** 4 4,. 44, 4,,A

93~~.~

0*******.*.. ****000-4 - NNNNNN )v

o C, 0 C 0 0 a 000000c 001 C C,

4' # 40 *' 0 000 00 00 0' C

V J'04 , V X1*n\*****

.t 'tC' l -- -4 * -D 'r

0' C r. C Aj Q. C, J) C I .A

itj -

Page 17

o 4 0 4 A 0 A 4 0 A 01 SO S6 S o

e* 4 4 4 4 4 4.. 4 4•,J .AJW•6, J IJb'J •.i ~ta

M A ) # A ev0p -0

* S • c• * 0 N** M2. a-No V

W~& Iii W~ kiiW WW

f'~0- a 0 4 ON a Nf I~

M . q m 'IV mJ 'I Fvm 4m 4

C. Z- t . Z J 0 a 0 C aL 6 L M ". e. .1 kAL ge6s.,.

* * o 0 mp ee 0 4 -Ie 4

4 4 4, • m •, '4 • i tf 40 N 4

U. • "' L , a&. L € ~ .. r,; • .w , tj La. L.7 4 .4 . L 0)*~~~. I;-.. Mege *g . *

"a I- N" It ýL V T . 4. P .-, al k, l 'r r n I•*I j , .L., - r ,4

* CU'Q *C * C J, C reJ C tC

CU Page 18

lipli~

f" 4% 06 a ag N 0 4A i t a 1 00 14

9--- * ,:- p:!p.SA.0

44 , (W 4 1 4 0 e. * 4v,,

"" C ., -O Ca , V 0 • " Q'

iJ

aý A

P

l'aU

Poll4 1S

*0 0s 4S O O g go"soal 0 I!se0

""OP *go No*1091"" 0 W 0 0 0

* ~ eq~~*sees* seeo

lo N N p-4N 00*00c06000ONO0950 6600000000C OO dSO S

a 0

P-41-W ~e..- WdW ow ; yw ;V go go Po o - 0 0000 00 000

.- fN tv c'-J3 Q. Q c a it, a c--~

_ 3 NN N \ 2k 31 .1 'r J, 4% l V" OR PA f P4 V-1

VA 14 1 fooot " 0-0" " 0-a 0 0 0 oo 0 410

cC Q C.,. cM;~ Pag C.c200000

4 W %wwwwwww

0N 00r m 0000.00PO N A. *0P M4 0000000

0 ii0000000000004bS...~k ..... * .S . ...... **

SOI W, O 03 0*000000

V-* Aprt s"*li O

m~ 404A MMNNNFQmWe4S0 @@cc@coo

*0 @0.. 00 W o@0** W40000 0=000

91l a050 NOock e 000000610440*'

U~ J1Ja. M &L IAI LL' J.I&I& WIW~UI %.LJ IW W&WJ4W

* S S S 0 S S S ~SN 0 a 0 %V a 0 0 0 S 0 0 5 0 0l 0b

MC4 m 4 rM N N m~ * - # a -in t- X 34 C. c c 0

N 4 C 0 @C 0 @0 000 N4N 0N4. O r 0U)0 lC 0#1PC, CMN*t) e(g,*,,,, OhS Sleeg Lilo C100. c.. .

Pag 1

M C.~ W I* %v 0 M~ f4 N~ 0-0- woo V-40 04 U-40-v4 0000o 00P-0-00000000000co0 00000.o00 00000000 00

I* loco*8N~ mmeo manicO A~ 00000

*4S 4mPJh4hdj 40hJ j.:~o 2 @000.0N001 :4 4 0 ;;44

* ~ ~ MM00 NNWC 0 . 004004464 .; . 0000

-- ~ ~ ~ 0-ou 00 0 0 0 0 0 0 0 0 0 0 0400000

* 0 01"o mtoinfl-*

N M 1 ph- N@my iw 0 ty - 0fUI 0 NO-PO4400000* 0 0 ~ 0 0 0 6 ~0 0 9.40 00 04 WON 0 v 0 0 0 0 0 000 0 0

-4k-(pv~ @00000090Oak00 00 Oka000 043k 40000000

Oh a,. ag~t -Do ooo 000 af4f Ca NiJ 0 6 ftg mo". .0O0a000000

0 a*c 40.0 000woo 0000 C00 000000 co . 0 0c),

* S &SS S S * *l m0, , e * 0 .0* 0 .a

0, '0 0 -& & &~ 04 0 0 0 0 NN 0 0 0 Oh 0 0&&0 00 00 0If t&Avl P. IOO 0laa

u-QG C00 0C00 0QC 0000CC@ 000 00

*Sl~l~lgggg llgl~g SPage.**22

~~~~c c o c~-~ee4~O Q o@o a 00 ac0c

14 (Up- uk- cyl- e- mo-0 9 404 000 QN 4eJ Now a4 cook wo-

* 000000000000000000 0 o@@,o 000QCC

* NF-I- NP. 01- moo '0E'hfl4n 00*@04 @4 r.W a okr %D P- 04o-N.P.O'N-O@0af a'Omoovo

*I** * I* *** 010000'00a** 000 00 00 e.0 **1** 00

4 444 44mm~ *am I- *AD a a-..0@-a a a C 000OCO04000 00a0a0C00a000

O - o m r- m ONNN 0'Pu.u"N £ 0&s )Op u-.1M Z 240'a 000

M ND 00 4- N D 0%~ N 4 *, V~ X P4 P- NM3 N F-o d- M~ N 0- P-~ %D P- -0 -0Q 4

o 4l 4V-O z cz0a a 4' v.-.r-4-C lpOca4004P 00

N 4 NoA #I O f-~ N- .0 40 't 4..@ P4 C. M No 0C N0 M- T 4 r- Or 0

o" X .M 1 ()iuCJ,1 DO j ky * )L .130 71 ~- .r

O-d p-. VW p-W 0i 1 r- ':L NEOI l N. ~00- -O f C

Nag 23

RE FERENCES

[I] Davenport: W. B. and W. L. Root, "An Introduction to the Theoryof Random Signals and Noise, " McGraw-Hill, 1958, pp. 37-38.

[2) Greenwalt, C. R. and M. E. Shultz, "'Principles of Error Theory andCartographic Applications, " Report No. 96, Aeronautical Chartand Information Center, St. Louis, Mo. , Feb., 1962.

[3) Parzan, E., "Modern Probability Theory and Its Applications,"John Wiley and Sons, 1960, p. 181 and p. 325.

[4) Schulte, R. J. , "Four Methods of Solving the Spherical ErrorProbable Associated with a Three-Dimensional Normal Distribution,"AD 666646, Holloman Air Force Base, New Mexico, Jan., 1968.

[5) Schwartz, L., "Multivariate-Gaussian Error Regions," HughesAircraft Co. Memo 2281. 1-60, 1967.

- 24 -

UNCLASSIFIEDSecurity Classification

DOCUMENT CONTROL DATA. R & D(Security cloejilcetlse of title, how of ebstract and indoxing an•rotetion must be entered when the overall report I& clavelfled)

1. ORIGINATING ACTIVITY (CSOpf~t* ethet) 2e REPORT SECUAITY CLASSlWICATION

The Aerospace Corporation Unclassified

El Segundo, California 2b aRouP

S. REPORT TITLE

ON ANALYSIS OF n-DIMENSIONAL NORMAL PROBABILITIES

4 O[UCRIPTIVE NOTES (Type of report and Incluelve do#e*)

5 AU THORIS) (F-eXt nmow. widle Initlel, lost nome)

Gersten, Robert H.Gura, Ira A.

6 REPORT DATE 7e. TOTAL NO. OF PAGEO | 7b. NO. Or ners

70 JUN 02 28 ,Ie CONTRACT ON GRANT NO. te. ORIGINATORN9 REPORT NUMaER(S)

F04701-69-C-0066 TR-0066(5129-0i)-26. PROJEC T No.

it rb. "THRPOIRT NO(S) (Any other numbers that may be as ia~e'

d. SAMSO-TR-70-28310. DISTRIBUTION STATEMENT

This document has been approved for public release and sale; itsdistribution is unlimited.

e. SUPPLEMENTARnY NOTES 1S.60PONSORING MILITARY ACTIVITY

Space and Missile Systems Organization

Air Force Systems CommandLos Angeles Air Force Station

-I3.AUSTNAC T

Although Gaussian probability densities are extremely useful in engineeringanalyses, they are frequently misinterpreted in this context. This reportis specifically designed to clarify this situation. While the material pre-sented may be well known among statisticians, the engineering communityappears to require such exposition.

"-The report begins by developing, from first principles, the contours ofconstant probability associated with n-dimensional normal density functions.Analytic expressions are then derived for the probability that the randomvariables under study will be contained within these contours. The resultsobtained are fully discussed from an engineering viewpoint.

Do AO•m 141 UNCLASSIFIEDSecurity C lai•J •"acJn

UNCLASSIFIED

Security Classification14

KEY WORDS

Probability Theory

Covariance Matrices

Error Analysis

Gaussian Probability

Probability Confidence Levels

Error Contours

Distribution Statement (Continued)

Abstract (Continued)

UNCLASSIFIED

-ecurity Cla1 Nificat-in