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G O O D N E S S - O F - F I T A N D D I S C O R D A N C Y TESTS FOR SAMPLES FROM THE WATSON DISTRIBUTION O N THE SPHERE

D. J. BESTAND N. I. FISHER CSIRO Division of Mathematics and Staristics, Australia Summary The only parametric model in current use for axial data from a rotationally symmetric bipolar or girdle distribution on the sphere is the Watson distribution. This paper develops methods for evaluating the model as a fit t o data using graphical and formal goodness-of-fit tests, and tests of discordancy. Key words: Axial data; discordancy

tcsts;

outlicrs; tcst of

fit;

Watson's distribution.

1. Introduction Data which are measured as undirected lines, or axes, arise in a variety of areas. Much of Structural Geology is concerned with the orientations of joint or fracture planes, and with fold axes. In the former case, the orientation of the plane can be described by the normal or pole to the plane; and in the latter case, the axis is measured directly. In Rock Magnetism, it is of interest to determine the maximum susceptibility axis of magnetic fabric in rock samples. Thus, problems concerning axes and orientations of planes can be treated as problems concerning distributions of points on a spherical surface. The only distribution on the sphere in current use as a model for these data is the Watson distribution. The general form of the density function of a random axis X with direction cosines (X, Y , 2 ) is

( C ( K ) / ~ Jexp C ){ K ( X ' A ) ' }

(1)

where

and 1 is a fixed axis with direction cosines (a,/j,'y ) . If K > 0, the distribution is bipolar, and rotationally symmetric about the principal axis 31 with direction cosincs (A,p , 1,) say. If K < 0, the distribution is of girdle form, conctntratcd around a n equator which tics in ;i ptanc t i o r v f ~ ito l )r. I n particular. wtlcn thc distribution

14

D J BEST AND N. I FISHER

is referred to its polar axis as reference axis, (so that the reference axis becomes (0, 0, 1)) the probability element for the density has the simple form sin

e de d @ ,

(2)

0 S 6 < n, 0 S @ S 2n, where (8, 4) are the colatitude and longitude given, in terms of the direction cosines, by sin 8 cos C#I = x , sin 8 sin @ = y , cos 8 = z. The Watson distribution is a special case of the general Bingham distribution for axial data (Bingham 1964, 1974). It was first studied as a model for axial data by Watson (1965), although prior to that it had been suggested by a number of writers (see Fisher, Lewis & Embleton (1986) for details). Whilst the corresponding Fisher distribution for unimodal samples of vector data has been used a great deal in the literature, virtually no use seems to have been made of the Watson distribution as a convenient means of summarizing a sample, yet the reasons for fitting a parametric model to axial data seem just as compelling as they are for the vectorial case. The aim of this paper is t o provide a set of methods for evaluation of the Watson distribution as a suitable model for a single sample of data. Questions of statistical inference concerning the polar axis and concentration parameter have been largely dealt with by Watson (1965; 1983, Chapter 5). See Fisher, Lewis & Embleton (1986, Chapter 6) for a review of this material. Because our methods had to be evaluated initially using simulated data, there was a need to develop suitable generators of pseudorandom Watson deviates. These are described in Section 2. Section 3 gives basic estimation of parameters, including approximations required in later sections. Sections 4 and 5 are concerned with goodness-of-fit procedures and discordancy tests, respectively, for the Watson bipolar distribution, Sections 6 and 7 with corresponding results for the Watson girdle distribution. Each method is illustrated using samples of data from Structural Geology. The discussion parallels the material for the Fisher distribution in Lewis & Fisher (1982), Fisher, Lewis & Willcox (1981), and Fisher & Best (1984). The basis of the methods in this paper is that, if a random sample of n axes from the Watson bipolar (say) distribution is rotated so that the sample principal axis is aligned along the z-axis, then the resulting n axes can be treated for many practical purposes as n independent observations from the distribution with probability element given by (2) (for n , K not too small). Analogous approximations were used in the papers cited above, in developing methods for the Fisher distribution; in that case, the quality of the approximations was investigated in considerable detail (see also Fisher and Willcox, 1975).

THE WATSON DISTRIBUTION ON THE SPHERE

15

2. Simulation of Data from the Watson Distribution The problem of generating pseudo-random variables from the Watson distribution is not straightforward because: (i) the distribution function does not have a closed form, (ii) there is no standard form independent of K (i.e. unlike say the normal distribution N ( p , a2) it is not possible to concentrate on the standard form N ( 0 , 1) and simply transform to N ( p , a')), (iii) there are no known characterizations or convenient transformations of other distributions which can be employed. To overcome these difficulties we propose algorithms based on the envelope-rejection technique which is a modification of the classical acceptance-rejection method described, for example, in Abramowitz & Stegun (1965, p. 952). The envelope-rejection technique may be defined as follows. Let f ( x ) be the probability density function (pdf) of a random variable X which is to be sampled. Let Y be a random variable with pdf proportional to g ( x ) , an upper envelope forf(x) (i.e. g ( x ) B f ( x ) , all x ) , and let (Ibe a u(0, 1) random variable. If (y, u) is a realization of (Y,U), y is accepted as a realization of X if f(y)/g(y) > u. The distribution of the accepted values is then exactly the required distribution. However, for the method to be efficient it must (a) be easy to generate a realization, y , (b) have an acceptance ratio = {Ig(y) d y ) } - ' , not too near zero, and (c) be easy to evaluate f 01)/go11.

Bipolar Case The simplified form of the probability density element for the Watson distribution is given by (2) with K >O. Put x = cos 8 and consider only 0s 8 < n/2. Take g ( y ) proportional to CeKY where C = K(eY- 1)-'. Then, if u 1 and u2, are pseudo-random values from U(0, l), the envelope-rejection algorithm is: 0. Set c = K(eK- I)-' 1. Ser y = K-' log ( u , / c + 1) 2. If u2 < exp ( ~ ( y ' y ) ) go to step 4 3. Return to step 1 4. Set 8 = COS-' (y). In steps 1 and 2, u 1 and u2 are to be taken as. new pseudo-random values on each call to them. The speed of the algorithm could be improved a little on many computers by adding

If

ci2

< 1+

KC^' - y ) go to step 4

16

D. J . BEST AND N . I . FISHER

between steps 1 and 2. This so-called “squeeze” depends on the bound e“ 11 + x and avoids, at least some of the time, the use of t h e

exponential function at step 2. Clearly criteria (a) and (c) above are satisfied. The acceptance ratio, R ( K ) say, can be expressed in the ) D l ( x )= alternative forms (1 - e-K)-1K1’2D1(K112) or D 2 ( ~ 1 / 2where e-”’ I$e’ dt and D2(x)= xe-”’ I: e” dt are forms for Dawson’s Integral used in Table 7.5 of Abromowitz & Stegun (1965, p. 319). Thus R(0) = 1, R(1) = 0.93, R(1) = 0.85, R ( 4 ) = 0.62 and R ( m ) = 0.5. This means criterion (b) is also satisfied.

Girdle Case For this case the simplified probability density element of Watson’s distribution is given by (2) with K < 0. Again put x = cos 6 for 0 S 6 S x / 2 but now take g ( y ) proportional to C(1+ E2x2)-’ where = ( - K ) ” ~ and C = E(tan-’ (E))-’. Then, if u 1 and u2 are pseudorandom values from U(0, l), the envelope-rejection algorithm is: 0. Set 5- = ( - K ) ‘ ~ . Set C = E(tan-’ (E))-’ I. Set y = E-’[tan { u l tan-’ ( E ) } ] 2. If u2 < (I - E2y2)exp (E’y’) go to step 4 3 . Return to step 1 4. Set 8 = cos-’ (y).

Each new use of steps 1 and 2 requires new values of u I and u2. Again, if speed is important a “squeeze” could be applied between steps 1 and 2. Also, as before, criteria (a) and (c) are satisfied. Let @ ( x ) denote the cumulative distribution function of the - 2 ~ ) ” ~-) standard normal distribution; then R ( K )= n112{@(( O.S}(tan-’ ( - K ) ’ ~ ) . Thus R(O)= 1, R(3) =0-98, R(1) = 0.94, R(10) = 0.70 and R ( m ) = 0.56 which verifies that criterion (b) is satisfied. 3. Parameter Estimation, Approximations and Definitions Suppose XI,. . . , X, constitute a random sample from the Watson distribution, where Xi has direction cosines ( x I , yi,zI), i = 1, . . . , tz. Define the orientation matrix T to be T = Ci XiXl‘. Let the eigenvalues of T be t,, j = 1, 2, 3 where 0 d t1S f2 S t3S n, C t j = n and t, = t,/n. Let hl, ii2, ii3 be the corresponding eigenvectors (with u,’u, = 1). Then, as noted, for example, in Watson (1983), the maximum likelihood estimator of 1 is 8, in the bipolar case and ill in the girdle case. Further, the maximum likelihood estimator, R, for K is given by f3 = D ( k ) in the bipolar case and by t l = D ( k ) in the girdle case where

17


On integrating the numerator by parts

and then using the exponential series and integrating by parts D(K)=

(113 + K / 5 . . .)(I f K13. . .)-' = 113 + (4145)K. . . ,

+

and thus D ( K )= (1 4 ~ / 1 5 ) / 3for small K. Further, for large negative K it is easy to show that D ( K )= - 1 / ( 2 ~ ) . (The relative values of the normalised eigenvalues f l , t2,t3 give an indication of the shape of the distribution. Since t1+ t2 t3= 1 and TI 5 i2S i,, we must have 0 5 tlS 1/3, 1/3 5 f 35 1. For uniform distributions, tl, t2and t3 are all approximately 113. For bipolar distributions, there is one large eigenvalue, i3> 1/3, the other two being small, tl< 1/3, f2 < 1/3. For girdle distributions, there are two large eigenvalues, t3> 1/3, f 2> 1/3, and one small one, tl < 1/3. See Fisher, Lewis & Embleton (1986, 93.4) for further details and references.) These approximations combined with regression analysis on tabulated values of D - ' ( t 1 ) give the following approximation for the girdle case

+

0 5 71 5 0.06 (2t1)-5 i? = D - ' ( f l ) = 0.961 - 7.08f1 + 0*466/f1 0.06 < 71 S 0.32 3.75(1 - 3i1), 0.32 < ti I1/3.

{

The approximation yields results less than 1% in error. Similarly, an approximation for the bipolar case is: i? = D - ' ( t 3 )

={

3-75(313- l), 113 5 33 5 0.34 -5.95 + 14*9?3+ 1*48(1- i3)-' - 11*05t:, 0.34 < T 3 5 0 - 6 4 -7.96 + 2.15t3+ (1 - t3)-' - 13*25tg, f 3 > 0-64.

In the following sections distributional approximations for large K are needed. Watson (1965) notes that for the girdle case and when the data are referred to ( A , p, v ) as reference axis so that (2) applies, cos 8 has a truncated normal distribution with variance ( 2 ~ ) ~When '. K is large, truncation is unimportant and thus for large K , 2~ cos2 8 will be distributed as x:; Watson (1965) indicates that K 1 5 is large enough. For the bipolar case, consider equation (2) and the approximation that sin 8 = 8, which is reasonable for large K ; the probability density is approximately proportional to e K ( ' - " ) , so that putting t = O2 gives a density approximately proportional to e-"'. This suggests that, for large K , 2 ~ ( 1 -cos28) is approximately distributed as a xf variate.

18

D . I . BEST AND N. 1. FISHER

[Mardia (1972, p. 281) gives the approximation as x:, which is incorrect; degrees of freedom for F subsequent variates in equations (9.7.22) and (9.7.23) are also incorrect, as a consequence]. Simulations indicate that K should be at least 10 for the x: approximation to be reasonable. In the following sections we will also have occasion to rotate the data so that the sample principal or polar axis is aligned along the corresponding to (i, p, +) and z-axis. To do this find the angles (8, then apply the rotation

4)

(;)=(

cos 8 cos -sin+ sin 8 cos

+ cos 8 sin + cos fj + sin 6 sin +

s-:

"( 5)

cos 0

to each data point and so obtain xf = (O;, #:) for i = 1, . . . , n, relative to (6, +), where x f = sin Of cos #;, y f = sin 0: sin cpf, z: = cos 6;. Note that, for bipolar data, t3= C:=, zt2, and that for girdle data, f l = zf'. One of the formal goodness-of-fit procedures we investigate is based on the Kolmogorov-Smirnov statistic 0,. If t : , . . . , t,' is a sample of data from a population with distribution function is F ( x ) then

c7=l

D, = max (D;,D;) where 0,'= max {iln - F ( t f i , } , 1 liSn

Dl = max {F(tfi,) - (i - l)/n}, 1LiSn

and f:l)$. . . Sf?,, are the ordered values of t : , . . . , t,'. For the bipolar case we have tf = 22(1- zf') and F ( t ) = 1 - e-"2 while for the girdle case t; = -2Rzf' and F ( t ) = ( 2 ~ r ) -f'i ~s - l R e - m ds. In sections 6 and 7 discordancy tests using the eigenvalues of the orientation matrix will be considered for samples of size n - 1, n - 2 and n - 3 which are obtained by deleting, say, the ith point, the ith and jth points and the ith, jth and kth points. In an obvious notation we define T(i,j ) = T - xix! - xjx; T(i) and T(i, j , k) can be defined similarly. Suppose T(i,j) has eigenvalues Zl(i, j ) S t2(i,j ) . S Z 3 ( i , j ) and likewise T(i) has Zl(i) 5 Z 2 ( i ) S Z 3 ( i ) , T(i, j , k) has Z(i, j , k) Ie2(i,j , k) 5 t3(i,j , k). 4. Goodness-of-fit Tests for the Bipolar case

a. Graphical Tests Let Qf = 1 - z t 2 , i = 1, . . . , n denote by Q , , , S .. . IQ,,,the ordered Q,'s. If the data { ( x f ,y,, z,)} are a random sample from a


19

Watson bipolar distribution, then a plot of Q(,) against log { n / (n - i $)}, i = 1, . . . , n should be well approximated by a straight line passing through the origin. The slope of this line gives an estimate of 1 / ~ yielding , a graphical estimate of K. A plot of the ordered @: values against 2n(i- 0*5)/nshould lie approximately along a 45" line through the origin. These methods are reasonable provided K Z 10. These provide tests on the marginal distributions of colatitude and longitude variates when measured relative to the principal axis of the distribution. Lewis & Fisher (1982) describe a probability plot for assessing the joint distribution of colatitude and longitude. This can be done here by checking the independence of the rotated variates O* and a*,using the X-plot technique of Fisher & Switzer (1985) on the data (QI, 4?),. . ( e n , 4):)-

+

b. Formal Tests

Formal goodness-of-fit tests can be applied to the sets S,= . . . , r x } where rt = (1 - z:*)28 and Su = {@:, . . . , @:}. To test S, use can be made of the modified Kolmogorov-Srnirnov statistic ME(&) = (0, -0*2/n)(f+ i 0.26 + 0 . 5 / G ) . Stephens (1974) has tabulated the distribution of ME(&). The distribution of M E is independent of the parameter values of the underlying Watson distribution but dependent on the fact that r:, . . . , r: are (approximately) exponential, since the parameters have had to be estimated. The adequacy of using M,(D,,) to test for the bipolar Watson distribution was checked by a simulation study. For various (n,K) pairs 10,000 pseudo-random samples of size n were simulated using the method of section 2, the corresponding 10,000 values of ME(D,,)found, and the upper lo%, 5% and 1% points compared with Stephens's tabulated values. Agreement was good for K S 10 and Table l(a) shows some typical results. Note that any of the goodness-of-fit tests provided by Stephens (1974) for testing exponentiality could be used to test S,. To test Su, Kuiper's V statistic (see, for example, Stephens, 1974) or the Prentice statistic P,, where {t:,

can be used. Approximate powers found by means of a simulation study indicated there was little difference in the power performance of V and P,,. Table l(b) shows some typical powers based on 500 samples of size n with a=O.lO. The alternative used was a von Mises distribution with concentration parameter K* = 1, 2 and 5. Pseudorandom von Mises variates were generated as in Best & Fisher (1979). The critical values used were estimates as above from 10,000 samples of size n and showed little variation with K for K 2 10. The percentage points of P, can be approximated by using those of x: for n 2 30

20

D. J . BEST AND N. 1. FISHER

TABLE 1 ( a ) Estimated percentage points of M,(D,) in the bipolar c u e for K = 20, (Y = {O.Ol, 0.05,0.1) and n = (10, 15,

20, 30, SO). LY

0.1

0.05

0.01

1.005 1.005

1-334 1.312 1.312 1.312 1-306 1.308

n

10 15 20 30 50

0.997 0.996

1.113 1.108 1.105 1.100 1.100

Stephens

0.990

1.094

1.005

( b ) Estimafed powers of P,, and V (bipolar care) for von Mises aLernatiues with concentration parameter 'K = (1, 2, 5 } and with a =

0.10 n

P"

V

10 20 30

0.12 0.14 0.14

0.10

1

0-24

2

10 20 30

0.31 0.38

0-19 0.34 0.41

5

10 20 30

0.65 0.90 0.97

0.60 0.90 0.99

' K

0.15

0.15

(Prentice, 1984). For n = 10 and LY = 0.10, 0.05 and 0.01 approximate values are 4.55, 5.52 and 7.49, and for n = 20 and the same LY, approximate values are 4.60, 5-82 and 8.19, from simulation. The percentage points of V agreed well with those of Stephens (1974) for K B 10. However, P,, appears more powerful than V for the girdle case and so we would recommend its use. Example 1. Figure l a is an equal-area projection of 35 measurements of LA axes in Ordovician turbidites (Powell ef al. 1985), in polar coordinates. Figure l b shows a rotation of the data to centralize the sample principal axis. We see that the data set exhibits some degree of rotational symmetry, except for two outlying points in the lower hemisphere.


21

90

210 Fig. la.-Equal-area projection of 35 measurements of L:,axes in Ordovician turbiditcs, in polar coordinates.

Fig. 1b.-Rotation

of Figure l a data.

Suppose we wish to evaluate the Watson bipolar distribution as a model for these data. The corresponding probability plots are shown in Figures 2a and 2b, with the outliers clearly evident in Figure 2a, and with Figure 2b suggesting some departure from rotational symmetry. If we set aside the two outliers for the moment, and consider fitting the model to the reduced data set, we obtain the probability plots in Figures 3a and 3b. These appear reasonably acceptable, and Figure 3:I yields the graphical estimate RGn = 20, in good agreement with the maximum likelihood estimate R = 20.5. (A X-plot for assessing dependence of the variates shows a modest degree of positive association, due possibly to a small amount of sampling bias). For formal testing of the reduced data set, we obtain for the colatitude tcst,

D. J. BEST AND N. I. FISHER

22 0.35

0.3

0.2s

w

0. I

0.01

EXPONENTIAL OUANTILE

Fig. 2a.-Watson

bipolar distribution probability plot for Figure 1 data.

I

0.7s

W

i t z a

3 O

0.6

W A

n 5

a v)

0.2s

0

-

I

i

1

UNIFORM OUANTILE

Fig. 2b.-Uniform

probability plot for Figure 1 data.

23

THE WATSON DlSTRIBUTlON ON THE SPHERE

/

0

l

.*

j

0

I

1

I

I

I

I

z

3

4

S

EXPONENTIAL QUANTILE

Fig. 3a.-Watson

0

bipolar distribution probability plot with 2 outliers removed.

0.2s

0.5

0.75

1

UNIFORM QUANTILE

Fig. 3b.-Uniforrn

probability plot of Figure 1 data with 2 outliers rernovcd.

24

D. J . BEST AND N. 1. FISHER

DT = 0-0800, D ; = 0.1293 and the modified value M,(D,) = 0.7610, which is quite an acceptable value. To test for rotational symmetry, P,, = 2.73, corresponding to a significance probability of 0.065. We conclude that the Watson bipolar distribution is a reasonable model for the reduced data set. We consider testing the two outliers for discordancy, in 6.

5. Goodness-of-fit Tests for the Girdle Case a. Graphical Tests

The tests are similar to those described for the bipolar case except that Qi = z f 2 , i = 1, . . . ,n and the ordered value Q C i , is plotted against F - * ( ( i - $ ) / n ) ,where F is the distribution function of x:. The slope of the best fitting straight line estimates 1/12~l.To check on the z:~), . . . , (Q,, 2:') can be joint distribution, a %-plot of (Ql, performed.

b. Formal Tests To test Sc = {t?, . . . , t,*} where now tf = -2k(z,*)' it is suggested that the statistic Mc(D,,) = On($ - 0.04 + 0.7fi) be used. The form of Mc(D,) and the actual constants were determined using standard regression techniques on a table of percentage points derived for various (n, K ) pairs from simulation of 10,000 samples of size n. The algorithm for generating pseudo-random Watson variates of girdle form which is given in section 2 was used. The approximate percentiles of Mc(D,,) are 10%: 1-04, 5%: 1.15, 1%: 1.36 and are quite satisfactory for --K 2 5. (As with the bipolar case, Mc does not depend on the true parameter values, but is not distribution-free). TABLE 2 Esrimared powers of P,,and V (girdle case) for von Mires alterriatives with concentration parameter K * = { l , 2, 5 ) and wirh a = 0.10

1

10 20 30

0.14 0.14 0.16

2

10 20 30

0.30 0.54 0.65

5

10 20 30

0.94 0.99 1.oo

0.14 0.14 0.15 '

0.21 0.33 0.44 0.67 0.92 0.99


25

As for the bipolar case S , = {@;, . . . , @:} can be tested by either Kuiper's V or Prentice's P,, statistics. However, as indicated in Table 2, P,, seems preferable to V in terms of power. Again as before, for n 2 30 the percentage points of P,, can be approximated by those of x: while for n = 10 and 1y = 0.10, 0.05 and 0.01 approximate values are 4.48, 5.63, 8.30 and for n = 20 and the same LY values approximate values are 4-58, 5.90 and 8-85.

Example 2. Figure 4 is an equal-area projection of 40 poles to axial-plane cleavage surfaces of fi folds in Ordovician turbidites (Powell et al. 1985), in polar coordinates. It appears from the plot that the data may be drawn from a rotationally symmetric girdle distribution; however, there is an outlying point (122", 177") somewhat distant from the equatorial plane along which the bulk of the data fall. To investigate whether the Watson girdle distribution is a suitable model for these data, consider first using graphical methods. The probability plots are shown in Figure 5 , with the outlier clearly visible in Figure 5a. However, if the outlier is disregarded and the probability plots re-calculated, we obtain the pictures in Figure 6. The colatitude plot in Figure 6a is now reasonably linear, and yields the graphical estimate RGR = -40.0 (whereas R = -38.0). The longitude plot in. Figure 5b checking on rotational symmetry also appears reasonable. (A X-plot revealed a small degree of positive .association, as occurred in Example 1, and probably for the same reason). However, as a formal check, we calculate the goodness-of-fit statistics. For the colatitude distribution, D; = 0.1010, 0, = 0.0925, and the modified statistic MG(D,,)= 0.631, which is much smaller than the upper 10% 90

210

Fig. .l.--Equal-area projection of 30 poles to ;rxi;tl-plane clc;ivnge surfaces of F, folds in Ordovician turbidites, in polar coorrlinatcs.

D. J. BEST A N D N. I. FISHER 0.2-

0.15-

W

ez 4

0.1-

W 2 P

t 4

u) 0.05

-

,... . . ..a

I

0

0

I

.

I

2

0.25

I

1

I

3

1

I

4

0.5

I

0

5

0.7s

UNIFORM QUANTILE

Fig. 5b.-Uniform

probability plot of Figure 4 data.

1

6

,

I 7

I

27

THE WATSON DISTRIBUTION ON THE SPHERE 0.1

o.oe

/

W

d

C

z

0.06

4: 3

0

4 n

5

0.04

In

2

I

0

3

4

x: Fig. 6a.-Watson

s

6

7

QUANTILE

girdle distribution probability plot for Figure 4 data with the outlier removed.

0

0.25

0.5

0.75

I

UNIFORM OUANTILE

Fig. 6b.-Uniform

probability plot for Figure 4 data with the outlier removed.

28

D. J. BEST AND N. 1. FISHER

critical value. For the longitude distribution, P,, = 2.14, corresponding to a significance probability of 0-117. So we can accept the Watson girdle distribution as a suitable model for these data.

6. Discordancy Tests for the Bipolar Case For K 2 10, 2 ~ ( 1 -(~15)’)2& as in Section 3 and so 2~ c;=~(1 ( ~ i b )4~xk. ) (The justification for these approximations, apart from that given in Section 3, is that the distributions of the KolmogorovSmirnov statistics for the marginal colatitude and longitude variables in simulated samples match those suggested by the approximate theory). When 5 is estimated by c3, n

2K

(1 - (X;L)*)

2K(n - t 3 )

Xgn-2,

i= 1

and similarly, 2 ~ ( n 1 - f 3 ( i ) ) ~ X L -If ~it . can be assumed that - 1 - Z,(i) and (n - Z,) - ( n - 1 - Z,(i)) = 1 Z , ( i ) - Z3 are statistically independent, it follows that 1 f,(i) - f, is approximately distributed as x: and that (n - 2)(1 f,(i) - f,)/(n - 1 - Z,(i)) is approximately distributed as an F-variate with 2 and 2n - 4 degrees of freedom. (The analogous approximation for the Fisher distribution was investigated in great detail by Fisher & Willcox (1978); for similar reasons, it is satisfactory here). Define Hkl)= maxi (n - 2)(1 f,(i) Z,)/(n - 1- Z3(i)). The statistic Hkl) is then analogous to En of Fisher, Lewis & Willcox (1981) and is appropriate for testing a single outlier for discordancy. H y ) , like En, has the advantage of easy generalization to the case of more than one outlier and simulations show that, for n

+ +

+

+

K

2 10,

where I[A] denotes the indicator function of event A . This result, partly based on properties of F variates, is derived for En in Fisher, Lewis & Willcox (1981) and, as there, use of only the first term, not involving the indicator function, is often sufficient. A generalization of H;’), H:) say, which is suitable for testing r outliers en bloc is H:) = max [(n - r - l)/t](t I,.

..

+ Z,(i,, . . . , i,)

-1,

- f , ) / ( n - t - T 3 ( i , , . . . , i,)).


29

TABLE 3 Entries are values w(a, n , t ) such that P(H!'> w ) approximates the percentiles shown, where H!' = block-test statistic for I outtiers, I = (2. 3). ~

percentile

~

~~~~

HL21

n

10%

5%

2t%

1%

0.1%

10

6.59 5.92 5.84

7.86 6.79 6.54 6.46 6.41

9.16 7.73 7.29 7.06 6.86

11.18 8.95 8.42 7.95 7.53

15.85 12.90 10.84 10.74 (8.39)

7.88 7.14 6.70 6.43

9.07 8-16 7.47

12.58 10.33 9.35 (7.47)

15

20 30 50

5.80

5.75

Hi"

n

15 20 30 50

6.10 5.79 5.59 5.55

7-03 6.42 6.17 6.01

746

Calculations based on 10,OOO simulations for n = 10, 15, 20, 30 and 1,ooO for n = 50.

Following the derivation for Hh') we see that, for K 2 10, as a rough approximation one can use the first-order Bonferroni bound and take

Upper critical values for H t ) , f = 2, 3 for K 2 10 are essentially the same as those of E t ) given in Table 5 of Fisher, Lewis & Willcox (1981) although the values given there for f = 3 are in error and should all be multiplied by (213). We give here a slightly improved (for t = 2) and corrected (for f = 3) table suitable for both H:) and E;).

Example 3. Returning to the data set of Example 1, we found that the reduced set with the two outliers omitted could reasonably be modelled by a Watson bipolar distribution. To test the two outliers jointly for discordancy, we calculate HL2)= 6-22, corresponding to a significance probability between 0.1 and 0-05. So there is not strong evidence that the points are discordant; however, we may wish to downweight their contribution when estimating K (see Fisher, Lewis & Embleton, 1986). 7. Discordancy Tests for the Girdle Case

Following the same reasoning as in the bipolar case and remembering that now the x z approximations are good for ~ > = 5it is

30

D. J. BEST A N D N.

1. FISHER

suggested that J:) = m q [ ( n - - 2 ) / t ] ( Z , - Zl(il, i,, ...,It

...

it))/tl(il, . . . if)

be used for testing f outliers en bloc. As a rough approximation one can again use the Bonferroni bound and take

Table 4 gives some upper critical values of J i ) for t = 1, 2, 3 for selected sample sizes. These values are based on 50,000 samples of the size shown and should be good for K 2 5. TABLE 4 Entries are values w ( a , n, t ) such that P(J:)Z w ) approximates the percentiles shown where JC) = block-test statistic for t outliers, t = (1, 2, 3). percentile

t=l

n

10%

5%

24%

1%

0-1

10 15 20 30 50

12.10 10.72 10.42 10.37 10-36

16.16 13.26 12-57 12.31 12.28

21.17 15.49 14.76 14.08 13.53

29.43 20.95 18-34 16-81 15.20

61.74 33-76 30.42 25.01 21.67

29.50 16.83 14-48 13.00 12.43

43.00 21.07 18.08 14.69 13.27

105.80 32.99 26.69 20.01 17.17

20.84 15-59 12.87 12.40

25.79 19-32 14.71 11.26

48.28 28.91 20.72 14.42

t=2

10 15 20 30 50

16.51 11.63 10.56 9.98 9.93

22.02 14.15 12.61 11.42 11.31 1=3

15 20 30 50

13-55 11.28 9.91 9.17

16.52 13.46 11.35 10.10

Example 4. For the data of Example, 2, suppose we test the outlier (122", 177") for discordancy. We obtain JA1)= 13.35, corresponding to a significance probability between 0-05 and 0.025 so there is some evidence that this point is discordant and hence requires special treatment (or alternatively, the model for the whole sample is incorrect).


31

References ABRAMOWITZ, M. & STEGUN,I. G. (1965). Handbook of Mathematical Functions, New York: Dover Publications. BEST, D. J. & FISHER,N. I. (1979). Efficient simulation of the von Mises distribution. Appl. Statist. 28, 152-7. BINGHAM,C. (1964). Distributions on the sphere and on the projective plane. Ph.D. Dissertation, Yale University. BINGHAM, C. (1974). An antipodally symmetric distribution on the sphere. Ann. Statist. 2, 1201-1225.

FISHER,N . I. & BEST,D. J. (1984). Goodness-of-fit tests for Fisher’s distribution on the sphere. Austral. 1. Sratisf. 26, 142-150. FISHER, N. I., LEWIS,T.& EMBLETON, B. J . J. (1986). Stat&id Analysis of Spherical Data. Cambridge: Cambridge University Press. In Press. FISHER,N. I., LEWIS,T. & WILLCOX,M.E. (1981). Tests of discordancy for samples from Fisher’s distribution on the sphere. Appl. Stutisr. 30,230-237. FISHER,N. I. & S W n R , P. (1985). Chi-plots for assessing dependence. Biometrika 72, 253-265.

FISHER,N. I. & WILLCOX, M. E. (1978). A useful decomposition of the resultant length for samples from von Mises-Fisher distributions. Commun. Statist.-Sirnula. Compula.

B 7(3), 257-267.

LEWIS,T. & FISHER,N. I. (1982). Graphical methods for investigating the fit of a Fisher distribution to spherical data. Geophys. J.R. usrr. SOC.69, 1-13. MARDIA,K. V. (1972). Statistics of Directional Data. London: Academic Press. T. J. (1985). Megakinking in the Lachian POWELL,C. MCA., COLE,J. P. & CUDAHY, Fold Belt. 1. Struct. Ceol. 7 , 281-300. PRENTICE, M. L. (1984). A distribution-free method of interval estimation for unsigned directional data. Biometrika 71, 147-154. STEPHENS,M. A. (1974). EDF tests for goodness-of-fit and some comparisons. 1. Amer. Sratist. Assoc. 69, 730-737. WATSON,G . S . (1965). Equatorial distributions on the sphere. Biometrika 52, 193-201. WATSON,G. S. (1983). Stutisticr on Spheres. University of Arkansas Lecture Notes in the Mathematical Sciences, Volume 6. New York: John Wiley. Received March 1985; revised September 1985