estimating the angle between the mean directions of two spherical distributions

Austral. J. Statist. 37(2), 1995, 179-191

ESTIMATING THE ANGLE BETWEEN THE MEAN DIRECTIONS OF TWO SPHERICAL DISTRIBUTIONS

TOBY LEWIS’ AND NICHOLAS I. FISHER^ University of East Anglia and

CSIRO Division of Mathematics & Statistics

Summary

This paper considers the problem of calculating a confidence interval for the angular difference between the mean directions of two spherical random variables with rotationally symmetric unimodal distributions. For large sample sizes, it is shown that the asymptotic distribution of 1 - cos &, where & is the sample angular difference, is approximately exponential if the true difference is zero, and approximately normal for a ‘large’ true difference; a scaled beta approximation is determined for the general case. For small sample sizes, a bootstrap approach is recommended. The results are applied to two sets of palaeomagnetic data.

Key words: Bootstrap; confidence interval; directional data; beta distribution; Fisher distribution; spherical mean directions.

1. Introduction

G.S. Watson, in a talk entitled ‘Past, present and future of directional data analysis’, which he gave t o the Statistics Earth and Space Sciences Conference in Leuven in August 1989, discussed ‘a very old but neglected problem - estimating the angle between two Fisher distributions’. He suggested a confidence interval for this angle, cy say, based on an approximately normal sampling distribution for the angle & between the sample mean directions.

In this paper we consider the more general situation where two vectorial distributions are sampled, each having rotational symmetry about a uniquely defined mean direction, but otherwise arbitrary. The aim is to find a large- sample confidence interval for a, the angle between the true mean directions. In Watson’s paper, where the two distributions are specifically Fisher, the proposed confidence interval is of the form

[& - i(&) - z6(&) , & - i(&) + 26(&)], ~~

Received March 1993; revised January 1995; accepted March 1995. ‘Centre for Statistics, University of East Anglia, Norwich NR4 7TJ, England. 2Division of Mathematics k Statistics, CSIRO, Locked Bag 17, North Ryde, NSW 2113, Aus- tralia. Acknowledgements. The authors thank Michael Buckley and the referee. Their careful read- ings of the manuscript have enabled the authors to correct a number of errors.

180 TOBY LEWIS AND NICHOLAS I. FISHER

where ti is the angle between the two sample mean directions, B ( 6 ) is an estimate of its bias, +(&) is an estimate of its standard error, and z is the appropriate N ( 0 , l ) upper quantile.

In our paper the form of the proposed confidence interval depends on whether a is ‘small’ or ‘large’, and an interval of form (1) is appropriate when a differs ‘sufficiently’ from zero, but not otherwise; the above words ‘small’, ‘large’ and ‘sufficiently’ are quantified in the discussion below.

2. Notation and Basic Results

For a generic random three-dimensional unit vector with colatitude 0 and longitude @ and non-zero mean resultant length p, we denote its mean direction by ( p , ~ ) or, in direction cosines, by the unit vector t = ( ( , q , C ) . With this notation, let (O,, a1) and ( Q 2 , G2) be two independent random variables, each with a distribution rotationally symmetric about its mean direction. Then, given respective samples of sizes nl, n2, (Ojj,qbij) ( j = 1,. . . ,ni; i = 1,2), the usual sample estimates of ti and pi are

and Ri given by

iii sinki sin f i i = ( l / n i ) Cj = ( l / n J Cj2jj = 2; Rj cos f i i

Rjsinkicosci = ( l /nj) C j

where 2 . . ‘3 = sin9jjcos4ij7 yij = sinOjjsinr$ij, rij = coseij. (3)

We assume that nl and n2 are large enough for the distributions of 3j, fji, Zi to be taken as jointly normal by the central limit theorem.

We require a confidence interval for the angle a E [ O , T } between c1 and f 2 , given by

(4) T cos a = t1 [2.

Assume first that a # 0, (Y # T. Define two sets of coordinate axes with common Ox,

making an angle (i) (Oz,Oyl,Ozl) with Oz, in the direction tl, Oy, in the plane of (t1,t2)

(ii) ( O x , 0 y 2 , 0 z 2 ) with Oz2 in the direction t2 and 0 y 2 orthogonal to Ox and

These are illustrated in Figure l a (case 0 < a < +T), and Figure Ib (case

- a1 with t2, and Ox orthogonal to t1 and t2.

l a *

T T < N < T ) . 1

ESTIMATING THE ANGLE BETWEEN TWO SPHERICAL DISTRIBUTIONS 181

Y2 X Fig. 1-Definition of two sets of coordinate axes with common Ox, for two different

possible values of a. (a)-Case 0 < Q < $ T ; (b)-Case f 7 r < a < 7 r .

Write Qi, Ui (i = 1,2) for the colatitude and longitude of ti with respect to (Oz,Oyj ,Ozi) . Also write 0: for the angle between (Oi ,Oj ) and tj, and .

Ai = E(cos20f), (5)

the second trigonometric moment of (O;, Oi).

mality of ( Z i , y i , Z i ) it follows that Then, as in Fisher & Lewis (1983), on the above assumption of joint nor-

0 Ui is independent of (Ri,Qi) with uniform distribution on [ 0 , 2 ~ ) which we write

ui N U[O,iT]

(we use N to denote ‘is distributed as’ and M to denote ‘is approximately distributed as’);

0 R: sin2 Qi N [(l - A,)/4ni] xi; 0 Ri cos Qi N N ( p , , [1+ Ai - 2pq]/2ni) independently of @ sin2 Qi.

( 6 ) (7)

It can be shown that, in the above notation, the angle ii betwen il and t2 is given by

C = cos ii = sin Q1 sin Q2(cos Ul cos U2 + sin Ul sin U2 cos a) + sin Q1 cos Q2 sin Ul sin (Y - cos Q1 sin Q 2 sin U2 sin ar + cos Q1 cos Q2 cos a. (8)

In (8), Q1, Q2, U,, U2 are independent random variables with U,, U2 uni- formly distributed on [0,27r). Omitting for the moment the subscript i, so that Q, n, 8, p etc. may refer either to Q1, nl, &, pl etc. or to Q2, n2, R2, p2 etc., write

L = tan2 Q, (9) M,, = sinPQ cossQ ( T , S = 0,1, . . .). (10)


From (6) and (7) we can write

where X, 2 are independent random variables with 2 N N(0, l ) and X exponential with mean 1, and P , 6 are parameters, necessarily positive, given by

Then sin2 Q = L/(1+ L ) , cos2 Q = 1/(1+ L ) ,

We require the values, to order n-4, of the 14 quantities

prs = E(Mr,) for 1 5 T + s 5 4. (15)

We have calculated these by expanding the right hand side of (14) as far as terms of order n-*, noting that p and 6 are both O(n-'), and then taking expectations term by term, noting that for any non-negative integer Is

Appro'kimate formulae for the moments of the distribution of C in (8) can now be deduced. To simplify the expressions involved, set

We can assume that the positive quantities pi and 6; are small, say less than 0.01 in most data situations, and that 6, is of the same order as Pi or smaller. For example, in the case of a Fisher distribution with concentration parameter R; >> 1, we have pi N l / n i K i , so pi < 0.01 for ni > 20, tcj > 5; and Si 21

l /ni6? << pi. Thus in (16), in terms of the small quantities P1 and P2 (or of no1 where no = min(nl, n,)), B is of the first order, P is of the second order, and R is of the third order.


We find, after some lengthy elementary algebra, that the first four cumulants of C are given by

p = (1 - B + P)c+0(B3), a2 = B2c2 (1 + O(B)) + Bs2 (1 + O(B)), rig = -2B3c3(l + O(B)) - 3B2s2c(1 + O(B)),

(17) (18) (19)

I C ~ = 6B4c4(1+O(B)) +12B3s2c2(1+O(B)) - 12Rs4(l+O(B)), (20)

where c = cosa, s = sincr, and that these results hold good even when the restriction a # 0, a # K is dropped. However, the form of the distribution depends critically on a, as we now show.

3. The Basic Parameter X

(a) Consider first the extreme case a = 0. Then the dominant terms in (18), (19), (20) are

2 N B 2 , Kg 21 -2B3, ~4 N 6B4,

respectively. Write T = 1 - cos & = 1 - C. Then the sampling distribution of T has skewness and peakedness coefficients

consistent with an exponential distribution with parameter B, and with origin -P which = 0 to order B from (16) and (17). (a’) Similarly, for the extreme case CY = K, T’ = 1 + cos 6 = 1 + C (= 2 - T ) has the same approximately exponential distribution. (b) Now consider the case when sin2 a >> B cos2 a (roughly speaking when a and K - a are ‘not small’). From (18)-(20) we get, for C,

a2 21 Bs2, K~ II -3B 2 2 s c, K~ N 12B3s2c2 - 12Rs4,

So in this case the sampling distribution of T can be taken as approximately normal,

T M N(1 - c + Bc, Bs2). (22)

Clearly the form of the sampling distribution of T depends essentially on the relative magnitudes of s2 and Bc2, and thus on the basic parameter A , defined by

tan2 a s2 = A B C ~ , i.e. x = - B -


The normal approximation for T based on (21) requires X to be large (for a quantification of ‘large’, see Section 4). However, whatever the value of X we have, to within a factor 1 + O ( B ) ,

4. The Approximate Sampling Distribution of T

The actual sampling distribution of T = 1 - cos h for any a E [0, T] has the following properties: (i) its support is on the interval [0,2];

(ii) its mean, to order B, is pT = 1 - c + Bc, from (16) and (17); (iii) its variance, to within a factor 1 + O(B), is

(iv) its skewness, to within a factor 1 + O ( B ) , is

3X + 2 from (24);

(X + 1)3/2’ 7 J T ) =

( v ) its peakedness, to within a factor 1 + Q(B), is

from (24); 6(2X + 1)

%(TI = (A + 1)2 ’

(vi) its limiting form, to within a factor 1 + O(B) , is exponential with mean B as a -+ 0, and is 2 - (exponential with mean B ) as (Y 4 T ;

(vii) its form for parameter K - a is the reflection in T = 1 of its form for parameter a; in particular, for a = +T it is symmetrical about mean 1.

We now show that a good approximation to the distribution of T is the distribution of the scaled beta random variable Y ,

T z Y , Y = 2 H , (25 )

where H has the beta distribution with parameters u , v,

and the parameters u and v are suitably chosen.


Y clearly has property (i). Its mean, variance, skewness and peakedness are given by

2u PY = ; where w = u + v,

4212)

4 = w y w + 1) ' and for w >> 1, which is the case (see below),

For Y to satisfy properties (ii) and (iii),

u = $w[l - (1 - B ) c ] , v = f w [ l + (1 - B)c] , (31)

~ U V 1 - (1 - B ) 2 ~ 2 w + l = - -

w2a$ - B2c2 + Bs2 . Thus w is a large quantity whose order of magnitude is O ( l / B ) . In what follows, we give results correct to within factors 1 + O(B) and equivalently 1 t O(w- ' ) . To this order of approximation we have, from (32) and (31),

2 - B B lim u = 1, lim v = - = v*, say.

2 lim w = lim w = - a +O c+l B ' a+O a-+O

Thus, if K = Y / B with pdf f r c ( k ) , so that H = $ B K ,

whence

log lim f ~ ( k ) = -k(l+ O ( B ) ) + O ( B ) , lirn f ~ ( k ) = e - k ( l + O ( B ) ) , a+O Cr+O

with a similar calculation for a --+ x . Thus Y satisfies property (vi). If a is replaced by x - a, but c still denotes cos a, the expressions for u and

v in (31) are interchanged; in particular, when a = f?r , u = v. So Y satisfies property (vii).

As regards property (iv) we have, from (29), (31), (32) and (23), and omitting a factor 1 + O ( B ) ,


cf. 3X + 2

yl(T) = (X + 1)3/2

As X increases from 0 to 00, n ( Y ) and 7,(T) both decrease from 2 down to 0. The difference r l (Y) - y,(T) has maximum value 1.362 - 1.119 = 0.243 when X = 2( l+fi) = 5.464. We can regard Y as satifying property (iv) to a reasonable level of approximation.

The remaining property is (v). From (30), (31), (32) and (23) we have, to within a factor 1 + O ( B ) ,

cf. 6(2X + 1)

Y2(T) = ( A + 1)2

y2(Y) = ~ ~ ( 2 ' ) = 6 for X = 0; a,s X increases to (say) 4/B, they decrease from 6 down to O(B2) and 3B + O ( B 2 ) respectively, i.e. effectively to zero. In between, when X is o( 1/B) but non-zero, their values diverge to some extent; some typical comparative values, to within a factor 1 + O ( B ) , are as follows:

YAY) Y2(T) 0 6 6 1 5.33 4.50 4 3.33 2.16

16 1.26 0.69

The term XB in (35), negligible for X = o(l /B) , is only relevant for large A, of order 1/B or greater, in which case y2(Y) and 72(T) are both O(B) or smaller and, in effect, negligible.

To sum up, Y has the required properties (i), (ii), (iii), (vi) (vii) of T ; its performance as an approximation to T is satisfactory in relation to property (iv) and 'not bad' in relation to (v).

Conclusion. The sampling distribution of T can be well approximated by the scaled beta distribution with pdf

where u and v are given by (31), (32).

for sufficiently large A: As noted earlier at (21) and (22), the distribution of T is effectively normal

T M N(l - c + Bc. Bs').


How large must X be in order that a normal distribution for T can reasonably be assumed? To answer this, we note from (24) that the skewness y1 and peakedness y2 of T are

3x + 2 6(2X t 1) 'l (A + 1)3/2 ' 72 = ( X + 1 ) 2 -

A normal distribution for T can reasonably be assumed when these are small. If we take 'small' to mean less than 0.1, say, then 71 < 0.1 implies X > 900 and y, < 0.013; similarly yl < 0.2 implies X > 225 and y2 < 0.053. So for X > 300, say, we can reasonably use the normal approximation (22). Higher values of X may commonly be encountered in real data sets; for instance, see Example 2 below, where X II 74000. It is essentially a question of the size of a. For example, X will exceed 300 for values of a between 60" and 120" if B = 0.01, and for values of Q between 29" and 151" if B = 0.001.

Note that in the particular case where the two samples have been drawn from Fisher distributions with respective concentration parameters IC, and ri2 , B = ( n , p , ~ , ) - l + ( n , p z ~ 2 ) - ' (see e.g. Fisher & Lewis, 1983 Example 3).

5. Calculation of a Confidence Interval for a: Two Examples

The approximate sampling distribution (36) involves, through (31) and (32), the quantities Q and B = x:=l(l - Aj)/4nipq, and therefore the mean resultant lengths p,, p2 and the second cosine moments A , , A 2 . In applying this distribution to interval estimation for a, we use for B the estimate

1 - A, B=C-. 2

4ni Rq i= l (37)

Here the unknown quantities p l , p2 are estimated by R, and R,; and the unknown quantities A, , A , are estimated either directly by the sample second cosine moments or, if the form of the distribution of (O i , Sj) is known, by using the functional relationship of A; to pi and hence of the estimate of A j to Bj.

We give two examples. In the first, ii is so small that the confidence interval extends down to a = 0. In the second example, the confidence interval covers a = $T, illustrating the fact that no difficulties arise if both positive and negative values of cos a are consistent with the observed cos ii.

For these purposes, we use two pairs of samples: (a) data sets given in Fisher, Lewis & Embleton (1987, 1993 Table B21 p.304) as sets B and C, but which we call I1 and I11 to avoid confusion (see also loc. cit. Example 7.11, p.213); and (b) data sets given in Table B23 p.306, Zoc. cit., as sets A and B, but which we call I and 11. For each of these pairs, we calculate an approximate 95% confidence interval for Q. Two of the samples are rather small, so that the examples are really for illustration only; see Section 6.

188

Set 1

TOBY LEWIS AND NICHOLAS I. FISHER

Set 2

270 270

90 90

Fig. 2-Two samples of directions of remanent magnetisation in red silts and claystones, from two sites in Eastern New South Wales. The coordinate system is

(Declination, Inclination), and the data are plotted in equal-area projection.

TABLE 1 Summary statistics for the data of Example I

Quantity Set 1 (11) Set 2 (111) i 1 2

39 16 n i (Pi 7 3 , ) . (12.8', 359.6') (14.9', 349.0') Ri 0.9940 0.9807

= n; C cos 2e* 0.9760 0.9248

6, = ( 1 - A i ) / ( 4 n i R ? ) 1.56 x 1 0 - ~ 1.222 x 1 0 - ~

Example 1. Figure 2 shows the data sets from (a) in equal-area projection. The samples are directions of remanent magnetisation in red silts and claystones, made at two locations in Eastern New South Wales (Australia). Summary statistics are given in Table 1. We get B = 0.001378, d = 2.14", cosd = 0.999304, and t = 1 - cosd = 0.000696.

For a = 0, T has, approximately, an exponential distribution with mean h = 0.00138. The observed value t = 0.00070 is in the main mass of this distribution (approximately at the lower 40% quantile), so the required 95% confidence interval extends down to a = 0 and is, say, [O,aU]. The upper confidence limit ayu is the value of a for which Pr{T < t I a} = 0.05, and is obtained from the approximating scaled beta distribution (36) by calculating Pr{Y < 0.000696 I a} for a few suitable values of a. This can, for example, be done easily on MINITAB by using the CDF command with BETA suhcommand.

ESTIMATING T H E ANGLE BETWEEN T W O SPHERICAL DISTRIBUTIONS 189

Set 1 270 270

'-7-

't, L

90

A -30 90

Set 2

270

60

30

d

90

Fig. 3-Two samples of directions of natural remanent magnetisation in Old Red Sand- stone rocks from sites in Pembrokeshire, Wales. The coordinate system is (Declination, Inclination), and the data are plotted in equal-area projection. Set 1 is shown with the lower hemisphere on the right and reversed, so the sphere is 'hinged' at (OD, -90'). The data for Set 2 are rotated so the sample mean direction is ( O D , 0').

We get the following results

a 0" 2" 4" 4.3" 4.4" 4.5" Pr{Y < 0.000696 I a} 0.396 0.275 0.074 0.055 0.049 0.044

The approximate 95% confidence interval for a is thus [0,4.4"]. (Note: write 770.975 for the upper 2+% point of the exponential distribution with mean i. If ti had been sufficiently large for the value of t to exceed 770.975, a two-sided 95% confidence interval for Q would be available.) Example 2. Figure 3 shows the data sets from (b) in equal-area projection, using the display method described in Fisher et al. (1987, 1993 $1). The samples


TABLE 2 Summary statistics for the data of Example 2

Quantity Set 1 (I) Set 2 (11)

i 1 2 n i 35 13

R, 0.9522 0.9785 ( b i , C i ) (86.2', 163.1') ( 0 ° , 0.)

0.8317 1.326 X

0.9158 1.692 x loe3

are directions of remanent magnetisation in Old Red Sandstone rocks from two sites in Pembrokeshire, Wales (UK). The second set has been rotated to (O,O), so that the difference between the sample mean directions is approximately 90". Table 2 gives summary statistics for the data. We get 3 = 0.003018, cos8 = 0.0668 and ii = 86.2". X is large (.i 73900) so (22) can be used; the sampling distribution of cos ti is then approximately normal with mean (1 - fl) cos a and variance B sin2 a. Approximate 95% confidence limits [aL, au] are then given hY

0.99698 cos cy k 1.96 x 0.05493 sin cr = 0.0668,

whence a L = 80.0", au = 92.3".

6. A Bootstrap Approach for Small Samples

One way of seeking to construct a bootstrap confidence interval for a when either n1 or n2 is small is to base a resampling scheme on an asymptotically pivotal quantity (as was done by Fisher & Hall (1989) in estimating a confidence cone for a mean direction). However, the complex nature of the asymptotic distribution in some situations (see the Appendix, Zoc. cit.) suggests that it would be difficult to make this workable, particularly in terms of obtaining accurate coverage p r oh abilities .

A simpler, if computationally more expensive, approach is to use the 'naive' bootstrap (based on a resampling scheme which incorporates the assumption of rotational symmetry - see Fisher & Hall, 1989) to obtain an initial bootstrap confidence interval for a of nominal size say 95%, and then use the iterated bootstrap to bring the actual coverage closer to the nominal coverage. See Hall & Martin (1988) and Graham et al. (1990) for general introductions to the iterated bootstrap, and comments by Fisher and Hall in the discussion of Efron (1991) for an intuitive explanation of how bootstrap iteration effects this calibration.

References

EFRON, B. (1991). Jackknife-after-bootstrap standard errors and influence functions (with discussion). J . Roy. Statist. SOC. Ser. B 53, 83-127.

ESTIMATING T H E ANGLE BETWEEN T W O SPHERICAL DISTRIBUTIONS 191

FISHER, N.I. & HALL, P.G. (1989). Bootstrap confidence regions for directional data . J. Amer. Statist. Assoc. 84, 996-1002. Correction (1990) J . Amer. Statist. Assoc. 85, 608.

- & LEWIS, T. (1983). Estimating the common mean direction of several circular or spherical distributions with differing dispersions. Biometrika 70, 333-341. Correction (1984) Biometrika 71, 655.

-, - & EMBLETON, B.J.J. (1987). Statistical Analysis of Spherical Data. 1st paperback edition (with corrections), 1993. Cambridge: Cambridge University Press.

GRAHAM, R.L., HINKLEY, D.V., JOHN, P.W.M. & SHI, S. (1990). Balanced design of boot- s t rap simulations. J. Roy. Statist. SOC. Ser. B 52, 185-202.

HALL, P. & MARTIN, M.A. (1988). O n bootstrap resampling and iteration. Biometrika 7 5 , 66 1-67 1.

estimating the angle between the mean directions of two spherical distributions

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