STUDY NUMBER C

Circular Error ProbabilityV4. 'of a

Quantity Affected by a Bias

D Iistribution of this dooiimontIs nilimited. .

June 1963 - \. ,:

USAF Aeronautical Chart and Information CenterChart Research )cineBion

Gaeophysica and Spac Sciences Branch

CIRCULAR ERROR PROBABILITY OF A QUANTITYAFFECTED BY A BIAS

Prepared by

Melvin E. Shultz

Distribution of this document

is U1limIted.

GECPHYSICAL STUDIES SECTIONGECPHYSICAL AND SPACE SCIENCES BRANCH

CHART RESEARCH DIVISIONPRODUCTION AND DISTRIBUTION PLANT

I

NOPICES

This publication was prepared for internal uselI in the Geophysical and Space Sciences Branch

and does not constitute an approved AeronauticalChart and Information Center publication.

ii

COMENTS

TITrZ PAGENOTICES

LIN OF TABLES ±11

ABSTRACT iv

1. Purpose and Scope 1

2. Circular Error 33. Elliptleal Error I

% - a. Differences 15

5. Conclusions 20

ii

LIST' 0? TABLES

TABLE NO. TITLE PAGE

2.xp -Re 21.1 R~~ax,Y, "0t '2

2 Rea cRaxx2

3 Roxay. Re 23

4 Re . Rxoxx 24

Lt

-!

A procedure for determinina the radius of the39.35# 50o A 90% probability circles for a biaseddistribution is presented. Both circular and ellip-tical, normal bivariate distributions are considered.The elliptical distribution is replaced by an equiv-alent circular distribution and an approxiate radiuobtained using the circular distribution procedure.Tables giving an indication of the error resultingfrom this replacement are included.

iv

1. Purpose and Scope

In error analysis It is assumed that all systematic errors have

been removed from the observational data. This is seldom true for it

is virtually impossible to eliminate all systematic errors. In the

analysis it is hoped that those systematic errors of consequence have

been eliminated leaving only numerous small systematic errors whose

combinations cannnt be dtstinguished from random errors. When a sys-

tematic error of consequence has not been removed, the value obtained

is said to be biased. That is, the value deviates from its true or

accepted value by some known or undetermined amount. Systeamtic errores

their origin, foriN and removal are discussed in ACIC RP-2 and ACIC TR-96.

This report is concerned with the effect these undeteated, biased

quantities have on the probability interpretation appiied in error

analysis. That its it is 4esired to determine the radius of a circle

which will include a certain portion of an error distribution effected

by a biased quantity. This may best be explained in the following

manner: consider a missile, with circular standard error w r, aimed

at a point T, (see Fig. 1). The missile is biased (or the point mis-

identified) by the amount d; therefore the distribution of impacts are

centered around the point A not T. It is desired to know how large

the R. circle (centered on T) must be to include a certain proportion

of the impacts distributed around A. Or, given limiting Values for

He and r, how large a value of d can be allowed? These are the ques-

tions with which this report is concerned. Only the normal bivariate

error distribution will be considered.

Fig. 1

In the preceding paragraph the error distribution is a special

case, that isj the distribution is considered circular. To have

a circular distribution the standard errors in the x and y directions

must be equal (ox a ay). When circular, the probability function

of a bivarilate normal distribution is simplified by reducing the two

parameters a. and a y to one parameter a.. Section 2 is concerned

with the determination of R. and probability values under the cir-

cular condition.

Unfortunately it is unusual in bivariate analysis to obtain a

circular distribution. The general case is an elliptical distri-

bution where ax ý ay. Since this complicates the solutions, the

elliptical distribution is replaced by an equivalent circular

distribution in Section 3 to obtain an approxim~te solution by the

method purposed in Section 2.

2_

2. Circular Error. If no bias error exists (d - 0), the center of

the error circle a. coincides with the center of the probability

circle Rc. Equation (1) expresses the probability of a circle of

radius R. with a circular standard error a 0 .

1 (2 x+ y2)]dxd1y (1

Fig. 2 illustiates equation (1):

y

Fia. 2

When the center of the ac circle does not coincide with the Re

circle, equation (1) takes the form:

Pr Ix2 + Y2 Ij R2] exp (2a)2na 2oJ

f f 2~3va c

y

Fia. 3

From Fig. 3 it can be seen that, if the coordinate axes are

rotated by the angle ct so that d lies along the x-axise equation

(2s) can be expressed:

PrEX2 + y2 R3 .( F..l exp [. ((x - d)2+ Y231dxdy (2b)f f2sg'u 2 2

vhere: d ) (y .-)2 Lor all cacao.

Fig. I illustrates equation (2b).

y

K

0onverting equation (2b) into polar coordinates:

x a roo 0

y a r min

Pr 1 an 3[ 1 r2 -2rd coose+ A rdrd

"i'o ONO

P• [ro- J) 21oJ. 2E MM-.20 (r2 _- oO r (3a)S2a4 2s off , 2,O0,

iwo 000

vbzen d 0 eaustion (A&a haA the. frnvm

2 2I

[2@

hm A p o, equiamo (31) canot be I&tepted In losea tom.

ViUtlUs (1956) o•a0ped his table of Oircular WAil. Probablities

bY the Polvamaia1:

-r OM- d- a (3b).

TabWlated alues of probability awe given for l v- velues. The

catation, of Pr for varous values of Lo a A relatively easy

6

vith a machine calculator but very tim conmsming. The desired

degree of accuracy in Pr In obtained by uundn

(d2 )n2 until the desired correct digit is unaffected.

Inmnle 1:

LtA-. , 1c

20 0(p. -. 0.60653

. ,r o.6o653[(l) (1 -0.60653) + 0.5 (1 - 0.6653 . 1.5)

+ o..25 (1 - 0.6o653 1.62s5) + 0.020833 (1 - 0.60653 . 1.645833)

+ o.oo26o1 Ci. 0.60653 • 1.648437) + 0.0oo06 (1 - 0.60653. 1.6187)

+ 0.0ooo0 (i - o.60653 , 1.6.8719)]

Pr - o.6o653 [o.10,06 a 0.26722 or 26.719%

The table vas utilized to develop a nmoaram and graph, Figs. 5

and 6,o s an uncomplicated mthod for obtaining values of H0 &d

probability. (The probability curves of 39.35v 50, and 90 per cent

were graphed because they are the probability levels of circular

standard, CEP, and C1MS errors.) The folloving steps should be fol-

loved in using the nm and probability curves: enter Pig. 5 With

dd value; place a straight edge at the origin 0 to that it passes over

dthe . value; obtain numerical value from the nomopmm A. With the A

value, enter rig. 6 and determine an .• value from the intersection

7

of the A value vith the dmsolpiA Tnwn1'414+%r nii,. ,qh #4v.1Il

is the r•stue of the probability circle resulting from a normal

Circzlar distribution displaced fkam the origin tr distance d.

NXatple 2: Given A missile weaegm system with a circular standard

error (a.) of 15Cm. The xisul2a Is aimed at a point 200m in the

x Airection, and WaOsm in the y direction from the tarset. vhat to

the readius of the 50% probability eircle centered an the taemrtt

Asmas the origi of coordinate eystem) (X) Y) (0j, 0) as thetarget.

Solutio:

"" 2V,0000 " ý e40 a

4 _ __* 9-- 2 - .99we- .150

enter Fig. 5 with

A a 9.96

enter Fai. 6 with A - 9.96

at 50%, !a- a 3.15Go

Be a 3.15. 150

The result (472m) Is the radius of a target centered circle

vhich has a 50% probablilty of containing the missile impact.

8

10.0

9.0

3.0 8.0

2.5 7.0

2.0 6.0

a .5A 5i.0

1.0 4.0

0.5 3.0

0.0 2.0

1.0

E-0.0

ma

2 3

S~0

.- 9-

Tidorder marpmVnR~Manv mmsma44#%w'a fn,." 20 2t: iz .-. A Ov~i -- I-,

ability levels are:

O). -0.098 (ro• ) + 0.3885 () . o.06fr () + 1.o2o

039.35 -0oo

(4) -. .,0535 ( ) o + 0.3952 ( -)2 0.o453 () + 1.3823To0

050 *0.0011

'd 9 .. 03623 (n-) +01 + 2122

U * *0.01il

90IThe () values have a standard deiio n f*.0 ih da

deviation of 0.010 occurring at -*0. ValUesof( R hav a standard

deviAtioU of *0.004 MAd MAXIMM deviation of 0.008 ocourring at

d a 3.00, azd at 90% the values have utandad deviation of *0.011

with a mAXIma deviation of 0.020 ooeueii4 at at - 0..

Considering exam~ple 2 vith A. a 2.99s the () value Will be

- -0.0535 (26.7306) + 0.3952 (8.910) -0.0453 (.o9900 +e 1.1813)

= 3.149 ± 0.004

3. Elliptioal wrror. As stated In section 1, a bivariate error

analysis generally re sits in an elliptical error distribution. With

ox jI op. the probability intepgal takes the foam of equation (4).

11

The gecmtrical expresaion of equation (4.) Is shovn by Fig. 7.

Fla. 7

12

Equalivu, (4) i not integraple In closed form, but a solution in

obtained by a method developed by Rosenthall and Rodden (1961)1.

Evaluation of (4)j for all possible values of (x - i), (y -

a• and i-, vould result in a table of monstrous proportions,

therefore, a method of approximation will be introduced so that, in

many Instances, the table will not be required.

If d is zero, i and j are zero and equation 4 Is expressed:

prEX2 + y2 SRI 1J~;-x [1 (2 + 2* (5)

Equation (5) determines the radius of a circle, (x 2 + Y2 a )p vhich

has a probability density equivalent to that of the normal elliptical

distribution. ACIC TR-96 discusses the use of equation (5) for re.

placing a normal standard elliptical distribution with a more convenient

circular distribution. 7R-96 ougeests that:

V a o.50o O(a + GY) for o.2 s i * 1.o

vill produce a standard error circle (a.) which will effectively replace

the standard error ellipse. By replacing the standard error ellipse,

the evaluation of equation (4) is reduced to an approximate solution

obtained through the nomogram and probability curves or equations of

section 2.

1. Ref. 1

13

Examule 3: Given a in iomffl wiwn~ Yav-tc= wit r..

F-- I - .'%LSUIA ~ U.U

(oX) of 1.0 units and a downrange error (0y) of 0.6 units, The

missile Is aimed at a point 2.0 units in the x-direction and 1.0

unit in y-direction ftr the trae position of the target. What

Is the radius of the 39.35%, 50% and 90% probability circle (OR)

centered on the target? Fig. 8.

(x- ) - 2.0; (y- ) w 1.0; Z- o.6Ox 1.0

a 0 o.5 ([.o + o.6) - o.8

d *1 + . .. 2.24I

d . 2.24 * 2.80

enter Fig. 5 with A - 9.33

enter Fig. 6 H for 39.35% " 2.72 R0 a 2.1.8

Re

~for 50% -2.98 Rew0 -3

Re fo 0 .23 He a 3.38ac

ar00

x(x-~m. Fig. 8

"14

4. RE - Differences. An error in either probability or

results from replacing the error ellipse by the correspoiling

circular form. This error is the difference between the true value

of the elliptical error distribution minus the approximation value.

A br!ef discussion of these differences is presented in order to

qualify the approximation method proposed in section 3. The

notation in this discussion will be as follows: The R. value i1

the radius obtained from a circular distribution by equation (3b).

The RE value refers to the radius obtained from an elliptical

distribution. RE will be used in general cases where the orien-

tation of the ellipse is not in question. R 0 xx is used to signify

the radius obtained by equation (4) in place of RE when orientation

is considered. The first subscript denotes the direction of the

larger standard error value and the second the larger biased

components or RVx, means that ox >cy and (x - i)>(y - j). The

values of R. are those obtained from Vitalis' table and the RE

(Roxlv aete) values are obtained from the tables of Rosenthal and

Rodden.

For the 39.35 and 50% probability levels it was found that

if the ci direction and the direction of maximum bias, (x - 9)

or (y - g)', coincides the R. value is too large. This problem is

shown in Fig. 9a where o.*ay and (x - i)-(y - 9). In this

particular instance, see ex. 3, for 50% probability R. - 2.38

while ROXx w 2.35. If the amax direction and the direction

15

of maximum bias do not coincide (Fig. 9b), the Rc value is too

small. .cne ox in this situation is 2.41 units and the Rc value

is the scme for both cases 2.38 units. However, for some of the 90%

probability level values R y< Rc< Rox*x.

The next question: with the same error ellipse as Fig. 9a and

9b, what are the R and probability values when (x - i) - 1, and

(y - j) - 2? Since d is the same as ex. 3, the Re value for 50%

probability level is again 2.38 units, Fig. 10. By rotating Fig. 10a

through an angle of 90* to the left it is a reflection of Fig. 9a,

thus R cry m " R OXx. Similarly, Fig. lOb rotated to the left 90O

is a reflection of 9b and Ra, y R X

If d makes an angle of 450 with the coordinate axis (Fig. 13a

and b) and (x - )- (y - ), the direction of a. is inmaterial.

Both Fig. lla and lib have equal R and probability values. By

rotating lla through an angle of 90" to the left it becomes a

reflection of l1b. Computing R0 values at C - 450 produces

the values of rinim•m. difference (RK - Re). As d progresses in

either direction away from a n 45* the difference can be ex-

pected to increase to maximws at 0 and 900. Tables 1 and 2

give values of (Roxy - Rc) and (Rc - Rcyxx) respectively, to

indicate a possible maximum magnitude for the differences. The

tables are computed for the O ratios of 0.8, 0.6, 0.4 and 0.2Umax

and for probability levels of 39.35, 50 and 90%Y. Tables 3 and 4

give the probability difference for each of the (RE - nc) terms

in tables 1 and 2.

16

94

9b

Fig. 9

17

1a0

10b

Fig. 10

1 18

fig

IV

l1b

Fig. U1

19

The R values for example.(3) are:

Re 39.35% - 2.18 RaxX - 2.09

He 50 - 2.38 BOxX 2.33

Re 90 - 3.38 x - 3.51

From table 2ap b, and c for (x - ) 2.0 and (y - 0) - 0, the

maxinmu differences to be expected are 0.118, 0.66, and -0.189

respectively. The values in the example all fall within the ex-

pected differences.

5. Conclusions. The simplified computation suggested is feasible

in most cases. Where the error In the R value created by the trans-

formation from elliptical to circular forms is great enough) such

treatment cannot be used, Tables I 2j, 3, and 4 are designed to

assist In determining the manitude of such errors. Whether to

apply the less complicated method or not is tht. choice of the user.

The Rosenthal and Rodden method must be used where the error is

considered decisive.

20

R -R

Por 39.35% Probability

(- ),(yR - 5.) (o.o, 0.5) (0.0, 1.0) (o.o, 1-5) (0.0, 2.0)

0.8 0.012 0.039 0.053 0.059

0.6 o.o26 0.o66 0.413. 0.1(,70.4 0057 0.128 04150 o.146

0.2 o.o67 0.147 o.166

(a)Par 50$ Probability(. ,(y. .) .(0.0,0 .5)) (o.o, 1.0) (0.o0 .15) (o.o, 2.0)

0.8 0.012 0.033 0.037 0.038

0.6 o.oe6 0.068 O.077 o0o66

0.4 o.o44 0.090 o.o94 0.082

0.2 o.o0 8 0.090 0.089

(b)

par 90% Probabilit(x.-);(Y.. ) (o.0, 0.5) (o0., 1.0) (o., 1.5) (o.o, 2.0)

0.8 0.009 -0.021 -0.052 -0.074

0.6 0.041 -0.034 -0.088 -0.129

0.4 0.117 .0-0.25 -0.120 -0.1850,2 0.250 -0.047 -0.099

(c)

Table 1

21

For' 39.35% Prbblity(x- i),(y- •) (o.5, 0.0) (1.o, 0.0) (1.5, 0.0) (2.0, 0.0)

%~inam0.8 - 0.039 0.062 0.059

0.6 - 0.089 0.119 0.118

0.4 o.149 0176 0..

0.2 0.187 0.209(a)

For 50 Probabi01).( - ), (y- ) (o. 5, 0.0) (1.0, o.0) (1.5,.0) (2.0, 0.0)

aux

0.8 - 0.033 o.oi4 0.036

0.6 a 0.072 0.076 0.066

0.4 a 0.107 0.101 0.082

0.2 . 0.109 0.103

(b)For 90% Pmotbatb.lity(z - f),(y- ) (o.5, 0.0) (1.o, 0.0) (1.5, 0.0) (2.0, 0.0)

0.8 - -0.0o5 -O.O66 -0.076

0.6 . -0.131 -0.170 -0.189

0.4. . -0.255 .o.02 -0.3

0.2 -o.388 -.0423

(a)

Table 2

22

I&rolun-ill Iy DIJ.Lrances

For 39.35$ ProbablIty(z -i),(y - ) (0.0, 0.5) (o.o, 1.0) (0.0, 1.5) (0.0, 2.0)

Oldnl

max0.8 -0.73$ -2.14$ -2.68% -2.91%

o.6 -1.79 -5.43 -6.86 ..6.61.

0.4 -3.76 -20.20 -12.19 -14.52

0.2 -7.46 -17.58 -21.07 a(a)

For 0%Probability(z -TAY,- 9) 10-0, 0.5) (0-0, 1.O) (o.o, 1.3),(o.o, a._o)

0.8 -0.77% -1.84% -1.93% -1.96%

0.6 -1.74 4.39 -4.g8 -4.27

0.4 -3.43 -7.49 .8.17,.

0.2 -5.23 -9,07 -11.99(b)

For 90%Probabi tfr(Z .0,0 0.5) (0.0o, 1.0) (0.0, 1.5) (0.0, 2.0)

0.8 -0.21% +0.49% ,1.18% +1.90%

o.6 -1.01 .0.83 +2,14 +3.2.0

OA -3.01 .o.61 +3.09 +4.8o

0.2 .2.85 -1.24 +2.51(a)

Table 3

23

P'o'ahl 1'1 tv Tn.i ,-,.

Ra - Rax

For _32350 Probability

(x X ),(y- y) (o.p, 0.0) (1.o0 0.0) (1.5, 0.0) (2.0, 0.0)

0.8 1.6o% 2.711% 2.43%

0.6 - 4.83 5.23 2.71

o- 7.73 7.39 6.56

0.2 8.84 8.48

(a)

Foz r0% Probabilityx. •)z,(Y -.,. " (o.5, 0.0) (1.0, 0.0) (1.5, 0.0) (2.0, 0.0)

%ain%Aax

0.8 a 1.65% 1.99% 1.51%

0.6 3.65 3.31 2.75

o.4 - 5.10 4.21 3.35

0.2 - 4.86 4.35 -

(b)Ior go0% Probambilty

X- , (0.5, 0.0) (1.0, 0.0) (1.5, 0.0) (2.0, 0.0)

GainOmaz

0.8 . -0.90% -1.24% -1.4.%

0.6 - -2.63 -3.42 -3.79

o.4 - -5.26 -6.13 -54

0.2 - 8.417 -9.59(a)

Table 4

24

REWINCES

1. Aeronautical Chart and Information Center Technical Report No. 96,"Prinoiples of ror Theory and Cartographio Applications,"Fekvay 1962.

2. Rosenthalo G.W., and Rodden, J.J., "Tables of the Integral ofthe 1Iliptical Blvariate Normal Distribution," May 1961, LockheedMissile and Space Division, Smnnyvale, Calif., (Prepared underbureau of Naval Weapons) Contract Ord 17017.

3. Vitalis, John A., p MTable of Circular Norml Probabilitles,"Jue 1956, .B1u Aineraft Corp.

25

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Circular Error Probability of a Quantity Affected by a Bias

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Scimlts, Melvin E.

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13. ABSTRACT

A procedure for determining the radius of the 39. 35. 50, and 90per cent probability circles for a biasod dlstributlio is presented.Both circular and elliptical normal bivariate distributions are con.siderod. The elliptical distribution it replaced by an equivalentcircular distribution and an approximate radius obtained, using thecircular distribution procedure. Tables giving an indication of theerror resulting from this displacement are Included.

DD I3 JAN _41473

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