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PATHS AND CHAINS OF RANDOM STRAIGHT-LINE SEGMENT

by

N. L. Johnson

University of North Carolina

Institute of Statistics Mimeo Series No. 424

March 1965

Classical problems associated with paths composedof straight line segments with independent orientations,which have a continuous distribution, are discussed.~ment formulas are obtained for the distance betweeninitial and terminal points, and a new approximationdiscussed. The Rayleigh approximation is used to der­ive approximations to other characteristics of therandom path.

This reseach was supported by the Air Force Office ofScientific Research Grant No. AF-AFOSR-76o-65.

DEPARTMENT OF STATISTICS

UNIVERSITY OF NORTH CAROLINA

Chapel Hill, N. C.

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IeIIIIIIII_IIIIIII.-I

Paths and Chains of Random Straight-Line Segments

By

N. L. Johnson

University of North Carolina

1. Introduction and Historical Outline

The motion of a point traveling along a succession of straight-line

segments of equal length (see Figure 1, section 2), but randomly and inde­

pendently oriented, has been studied, in various contexts, by a considerable

number of workers over a considerable period of time. Rayleigh (1880, 1899,

1905, 1919 a,b) studied such motions, firstly in a plane and later in three

dimensions (and also in one dimension). In the 1880 paper, he obtained appro­

priate formulas for the distribution of the distance between the initial

and terminal points of a path of N segments. In 1919 he gave exact explicit

distributions (for the three dimensional problem) for some small values of N.

The two-dimensional problem was posed, in connection with the study of random

migration, by K. Pearson (1905) who also noted the relevance of Rayleigh's

solution (K. Pearson (1905, 1906) also Rayleigh (1905». In response to this

statement of the problem, Rayleigh (1905) gave an outline of his analysis, and

an exact solution, in the form of an integral, was obtained by Kluyver (1905).

Numerical evaluation of this integral was not effected, however, until the

work of Greenwood and Durand (1955). An exact explicit general solution for

the three-dimensional case ,~s given by R. A. Fisher (1953). This expresses

the probability integral of the distribution of distance in terms of a set of

polynomials, each polynomial corresponding to a different interVal of values

of the distance.

At about the same time that Pearson and Kluyver 1fere working on

the t,'lo-di:mensional problem, Smoluchowski (1906) was working on the three­

dimensional problem, in connection vuth the study of Brownian motion (with

finite time between charges of path direction). He also considered the

rather more difficult problenl in which the angle between successive segments

is fixed (though orientation in three dimensions is assumed random, subject

to this constraint). Smoluchowski stated that his attention was drawn to the

problem by two papers by Einstein (1905, 1906). However, the methods used by

the latter were of a different nature, being based on continuous variation

with each coordinate of the traveling point varying accordingly to a Wiener

process (as it would now be called).

Kuhn (1934), apparently independently, initiated stUdy of the three­

dimensional problem, regarding the "path" as a chain cOITq>osed of links of

fixed length, but independent random orientations, which he proposed asa

model for certain kinds of cl1ain molecules. Kuhn I s paper was followed by

later papers by Kuhn and GrUn (1942, 1946) and Kuhn (1946). In these papers

a number of problems were solved approximately, including the problem studied

by Sn~luchowski, in which the angle between successive links (here regarded

as tile "valence angle") is fixed.

The accuracy of approximate solutions has been discussed, for the

two-dimensional case, by Horner (1946), Slack (1946, 1947) and, in consider­

able detail, by Greenwood and Durand (1957). Lord (1948) commented generally

on accuracy of approximation, pointing out that accuracy might be expected to

increase with number of dimensions. Stephens (1962 a,b) has studied the

accuracy of approximations for both two- and three-dimensional cases.

The case of fixed unequal segment lengt"hS was discussed by Rayleigh

(1919 a). A formal system of analysis (based on a method ascribed to

2

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IeIIIIIIII_IIIIIII•eI

A. A. l'ifarkov) ,applicable to random varying (though independent) segment

lenGrtlls "laS developed by Chandrasekl1ar (1943). This Ims found. to lead to

si~le explicit results in certain special cases, such as tl~ case of segment

vectors '\'lith independent, id.entically normally distributed components. GroG;)ean

(1960) allo'\'lcd for random variation in segment length, and for a general

orientation distribution, thougll he retained. the asst~tions of independence

bet-;ieen sevnents, and betlreen length and orientation of the same segment.

Apart from the distribution of distance between the initial and

ter;:rl.nal points, the distribution of relative orientation of seGments of the

paUl :nas been the object of study. Kuhn and GrUn (1942) considered the

distribution (in three dimensions) of the angle between a randomly chosen

linl: and tIle "terminal axis" joininG the initial and terminal points of the

chain. It is interesting to note that the approximate distribution they

obtained had been introduced (is the study of magnetism) by Langevin (1905)

about the time that Pearson, IQ.uyver and Smoluchowski were working on the

distribution of distance between initial and terminal points. Statistical

properties of the Langevin distribution were studied by R. A. Fisher (1953).

Recently, a number of statistical procedures based on this distribution have

been developed by G. S. Watson and co-workers (Watson and Williams (1956),

Watson (1960), Stephens (1962 b».

The corresponding two-dimensional distribution, described by Mises

(1918), has been discussed and named the "circular normal" distribution by

Gumbel et al. (1953). It also haS recently been used as the basis for

statistical procedures developed by Watson and Williams (1956), Watson (1961,

1962) and Stephens (1962 a). Breitenberger (1963) has discussed ways of

describing the genesis of both the two- and three-dimensional distributions•

3

4

In tnis paper, relatively simple metnods will be used to study

and other statistical procedures connected with these problems.

points, but some other, related problems will be discussed later. We will

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when Segment Lengths are Fixed

may be fixed or may be represented by a

that the ,J,th segment is of length J., ~michJ

In Figure 1, Po represents the

initial position of the point (or beginning

random variable.

of the chain) and P. lP, represents theJ- J

,J,th straight line segment. It is supposed

Figure 1 is applicable, whatever the number (v > 1) of dimensions,

Distribution of Terminal Distance

P~1"+1J" ,d/ L1 p.-1

(7 'PI ~ , i-2., ,/ ,

/ ,/

'/

"p~I/"o P.

'1

11FrGl1RE 1

2 2 - 2rj _l .e. cos Bj

+ l:r. = rj

_lJ J J

or, equivalently

(1)2 2

+ r. 1 J j+i:r

j = r. 1 u.J- J- J J

though the distribution of the angle ej(=po Pj

- l Pj

) depends on v. The

distance POPj

between the initial point and the end of the Jth segment will

be represented by rj

•

From the triangle Po Pj - l Pj

not, here, be directly concerned with the development of tests of significance,

larly concerned with the distribution of distance between initial and terminal

fashion, to cases where the segments (or links) have lengths which are

randomly distributed (and even correlated). We will, at first, be particu-

random paths and chains. These methods can be adapted, in a straightforward

2.

5

When v = 3, it is

Adding together (N-l) equations like (1) (with j = 2,3, ••• ,N), and

N= 1: l:

j=l JE(r~ I [.t} )

N~ (r~) I(.t)) = E( [1: rj_l.tj uj ]21[.t}) •s j=2

z2_ N N_2N - 1: r. 1 .t. u. + 1: r:j •

j=2 J- J J j=l

where the .t j , sand u j •s are nmtually independent random variables, and r j

is independent of .ti and ui for i > j, but not for i S j. Each uj

has the

same distribution, taking values in the interval -2 < u. < 2. The probability- J-

density function of uj when v = 2 (two dimensions) is

In each case the fornnlla applies for the interval -2 S uj S 2 only. Whatever

the value of v, uj is distributed symmetrically about zero, so E(U;) = 0 if

s be odd.

noting that r l equals .tl

, we obtain

where [.t} denotes the set of fixed values .tl,.t2' ••• '~. The sth conditional

f2 .central mment 0 rN 1S

We now proceed to evaluate the mments of r;, conditional on the

.tj's having fixed values. Since E(U~) = 0 if s be odd,

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and so

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(since E(u.) = 0)1

(using (j»

2 2 N L~ 2 2 4 N 4 4+ 6 [E (u )] E E E thr:l. + E(u ) E U . .e.

hfi,hfj,i<j 1 J i<j 1 J

(6)

(4)

6

The calculations for s=3 and s=4 follO"l similar lines. Thus

Sirular calculations lead to

Putting s = 2,

? N 2J.L. (~ I{.t}) = 3E [E E r. .t.u. (1'. ll.u . ) I{.e} ]. ~ i < j 1-1 1 1 J- J J

2 N 2__ 2= 3E (u ) EEl. l--:E ( l' • lU •E ( r. 1 11'. l' U • , [l} ) [ .e} ) •

i < j 1 J 1- 1 J- 1- 1

2 .2 j-l 2No'" E(r. 1 11'. l' u., Ce}) = r. 1 + r. l.t·u. + E lh'J- 1- 1 1- 1- 1 1 h=i

N N-l N"There E(u

2) = E(U~) (for any j), and r.I: means E E. In later fornmlas,

1<.1 i=l j=i+lhigher order summations will be introduced, using similar conventions.

7

In two dimensions

In three dimensions

~4(r~,/) • (t[E(u2)]2[1+E(u2)JN(N-l)(N-2)(N-3)

+ 3[E(u2)]2N(N_l)(N_2) + ~ E(u4) N(N-1»)18 •

212 4var(rNI/) • 2 E(u ) N(N-l)!

2 l[ 2 J2 6~3(rN'/) • 2 E(u) N(N-l) (N-2)/

(8)

The second term on the right-hand side of (6) is actually

N N N(6[E(u2)J2 (E E E + E E E) + E(u4)[2+E(u2)]E E E) I~!~!~

h<i<j i<h<j i<j<h J

but, for our purposes, it can be written in the form shown, since E(u2) =4v- l

and E(u4) =48[v(V+2) ]-1 and so 6[E(u2)J2 • E(u4)[2+E(u2)J.

If each straight-line segment is of the same fixed length I, then

(using I in place of (llto denote 11 =12 • ••• • IN • / ).

(10)

(11) var(rill) • N(N-l)14

(12) ~3(ri I) =2N(N-1) (N_2)/6

(13) ~4 (ril f) • 3N(N-l)(3~-1lN+ll)f8 •

21/ 2 4(14) var(rN ). '3 N(N-1)f

(15) ~3(r~ll) • ~ N(N-1) (N_2)/6

(16) ~4(r~ll) • ~ N(N-1)(35~-115N+108)/8

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2Approximations to Distribution of r N

For the case of constant segment length the asymptotic distributions

of (V/(NJ2»r~, as N increases, is that of x2 with v degrees of freedom.

(Rayleigh (1880), Pearson (1905), Kluyver (1906), Smoluchowski (1906), Kuhn

(1934) etc.) This result can be obtained (see Stephens (1962 a, pp. 473-4)

by introducing Cartesian coordinates xtN

(t ~ 1,2, ..• ,v) for PN, with xtO

= 0

for all t, so that Po is at the origin of coordinates, and noting that

(i) xtN is the sum of independent, identically distributed random

variables v3t ) (the projections of links Pj - l Pj

on the xt axis) which each

have a finite range of variation, expected value zero and variance f2 jv ,

and (ii) xlN' x2N' ••• , xVN are uncorrelated (though not independent).

It may be noted that a similar argument applies also when the fj's

are independent random variables with identical distributions with finite

expected value and standard deviation. The only modification is the replace­

ment of f2 by E(f2).

Horner (1946), Slack (1946,1947) and Greenwood and Durand (1955)

found that the asymptotic distribution did not give very good results for N

less than 10 when v = 2 (i.e., in two dimensions). However, as Lord (1948)

pointed out, the approximation would be expected to be better as v increases.

Rayleigh (1919 a,b), Pearson (1906) and Kuhn (1934) investigated possible

improvements of the X2 approximation, and various methods of approximation (in

the case v=2) were compared by Durand and Greenwood (1957).

For relatively small values of N, a natural method of approximation

2is to use a Pearson Type I curve with the same first two moments as r N, and

2the same range of variation (from 0 to (Nf)). This gives exactly correct

results when N = 2.

8

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(for N > 2).

1 -1 () (1 -1ex • 2 \I-N j 13 = N-l 2 \I-N ) •

~4 (ri l{)~4(r;I{)

ii3(r;1 {)

~3(ril {)

This method is equivalent to approximating the probability density

function of YN = (rN/(N{))2 by

with

9

2The third and fourth central moments of r N corresponding to the approximating

distribution (1'7) are

The ratios of approximating to exact values of the third and fourth central

moments are

Some values of these ratios are shown in Table 1. In assessing these ratios

as indices to the accuracy of approximation it should be borne in mind (see

Pearson (1963)) that variation in high order moments may have little effect on

computed probabilities. (It may be more informative to consider the "stand­

ardized" ratios ::Jfi3/~3 and ~ft4/~4 .)

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10

Application of this approximation requires the evaluation of the incon~lete

approximation, but the approximation improves for larger values of R.

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Type ,I approx.

0.1450.4250.7040.8950.9992

Table 2

0.1250.4180.7140.9000.9979

Probability Integral of r7

(V=2)

Pr[r7 :s R]

~ approx.

0.1330.4350.7240.8980.9942

Table 1

Ratios of Approximate to Exact 2Central Mo~nts__~! r N

Third ltt>ment Fourth Moment

N ~ v=3 v=2 v=33 0.750 0.818 0.900 0.9334 0.800 0.857 0.818 0.8965 0.833 0.882 0.806 0.8887 0.875 0.913 0.823 0.895

10 0.909 0.937 0.856 0.91520 0.952 0.968 0.910 0.950

R/ (Nt)

1.02'.03'.0!~.o

6.0

v = 2 given by these authors

beta function ratio I (CX,f3) with x::= (r/(Nl)2. Lower percentage points can bex

obtained (using linear harmonic interpolation) from the recently published

comparisons with exact values of the probability integral of r, for N ::= 7 and

tables of Vogler (1964). This is, unfortunately, not so for the upper percentage

points, so no direct comparison has been made with the upper 5%and 1% points

given by Greenwood and Durand (1955). Special calculations gave the follovring

For small values of R the present approximation is not even as good as the i

4. Variable segment Length

true. )

N N~2'(rN2) =E(E(u2)E E 1:/2

j+ ( E 12

j)2)

i<j ~ j=l

N N=E([2+E(u2)]E E/~I~ + E 14

j )i<j j=l

2 N 2~l'(rN) = E( E I j ) = N~'

j=l 1

11

Thus

The x2 approximation may also be used in obtaining approximations to

other properties of random paths and chains (see, e.g., Stephens (1962 a,b)).

mation on which they are based; they improve as v and/or N increase.

If the I's are not constants, but random variables, the moments about

zero of r; can be obtained by finding the expected values of the corresponding

conditional moments,

In section 5 of this paper there will be found some applications of this kind.

2The accuracy of these approximations depends closely on that of the X approxi-

If the 12 ,s are mutually independent and identically distributed,

with finite moments (about zero) ~~, then the moments of r; can be expressed

in terms of these u"s"s •

(The corresponding relationship for central moments is, of course, not generally

(18)

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etIIIIIII-.I

(26)

l[ 4 2] 2 2+ 2' E(u ) + l2E(u ) + 6 N(N-l) 112 + 6E(u )N(N-l)1l3Ili + Nlll~ •

12

(22)

(26)

(24)

(20)

So, in two dimensions

Expressing equations (19)-(21) in terms of central moments (for both

2 ,2rN

and ,. ), we find

In a similar fashion we obtain

and

(21)

and in three dimensions

~4(r~) ~ ~N(N-1)(35~-115N+ 108)~i4 + ~ N(N-l)(25N-36)~2~i2

+ ~N(N-1)~~ + 8N(N-1)~3~1 + N~4 .

13

lations among them are

(28)

then

while

It can be seen that, for sufficiently large N, the most important

term in ~s(r~) is that containing ~ls. This term can be obtained from the

corresponding formula for fixed 11 = t2

= ... = IN = I, by replacing fS by

~'~. It is to be expected that ri will tend to its asymptotic distributionN~l 2

(that of (-V-) x (X with v degrees of freedom)) more rapidly when ~2' ~3' ~4

are small compared with ~l.

The moments of ri can be evaluated in a similar way when the 12,s

are not independent. If the 12 ,s all have the same moments, and the corre-

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In these two cases the limiting values of the moment-ratios as the same (as N

u2increases) as when there is no correlation. If, howeve~ all ~'s have the

same correlation, that is p(li, I~) = pfor all i ~ j.

and the limiting value of the variance of V;/N is increased in the ratio.

The other moments are affected also.

2It is possible to investigate cases where individual Ij's have

different distributions, using methods of the kind described above.

5. Uses of the Chi-Square Approximation to Distribution of Length

It can be seen from the results in the last section that the moment­

ratios of the distribution of r~ tend to those of X2 with v degrees of freedom

as N increases. Although the work of Greenwood and Durand (1955) indicated

that this approximation is not completely satisfactory for values of N even

as large as 20, it can be used with confidence for larger values of N (say

40-50 or more) and can provide useful insight into the nature of the distri-

bution of the end-points of segments of the path or chain.

As an example, suppose we wish to consider the distribution of the

distance dn

of Pn from the "terminal axis" POPN joining the extremities of

the (path or) chain (n < N). Assuming that there is no dependence between

lengths of segments, and that each f~ has the same distribution we haveJ

14

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(for D < min(r ,rN )n -n

jr~ _D2

2 2-n p(r

N)dr

N)

r N -n-n-n

The distribution of d can be obtained by direct analyses. Then

r r N' sin2¢2 n-n

d =---------n r -2r r ' N cos¢+r

N'

n n -n -n

15

where ¢ (the angle POPnPN), r n (poPn ) and rN_n(PnPN) are mutue.lly independent

random variables: -2 cos ¢ has the same distribution as u, and r~, rN~n are

distributed as r~ with N replaced by n, N-n respectively. We will now drop

the prime in r N' •-n

following demonstration, for ~ = 3 dimensions, is particularly simple.

provided D < min(r ,rN

). Otherwise the probability is zero.n -n

Since (for ~s 3) cos ¢ is uniformly distributed between -1 and +1

and

(since r 2 and rN2 are mutually independent). Now introducing the approximate

n -n

probability density function (corresponding to the asymptotic x2distribution)

1 22'2 3r

N(rN) exp[-2N~' ]

1

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of

16

One further application may be noted here, though we shall not pursue

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the

formuladown an approximate

2••• , r N ' where

slarge enough for

3ND2

2n(N-n)J..L' ):3 1

1 -K~L-2 r-=- )= 2e -K v 1t' •

We first consider the conditional distribution of r~ , given

XVN • Using the random variables u~t) defined in s:ction 4, we1

... ,

~nJ..Li x (x2 with two degrees of freedom)

In(l-~)J..Li x (x2 with two degrees of freedom), as compared with

we obtain from (35)

Nl

< N2

< •.• < Ns and Nl , N2-Nl ,

x2 approximation to be useful.

through Po'

The approximate expected value of d2, calculated from (37), isn

2n(~;n) • J..Li' If N is even and n =~ N, so that Pn is the mid-point of the

path or chain, the approximate expected value is gNJ..Li' or i rf2 if each link

is of length [, agreeing with Kuhn and GrUn (1942) and Kuhn (1946).

The conditional distribution of r~ given xlN ' ••• , xVN is approximately that211

e2From (37) we see that dn is approximately distributed as

xIX '1

have

for the distribution of the square of the distance of P from an arbitrary linen

the matter in great detail. It is possible to write

for the joint probability density function of r~ , r 21 N2 ,

••• , N -N 1 are eachs s-

Pr[d2 > D

2J =exp(-n .00 1

J ( '2 -Kz(Note that z-e) e dz

The right-hand side of (39) can be written as a weighted sum of terms

-1a :: (N -N 1) •s s s-

17

V-l(N2-Nl)~i x (Non-central x2 with v degrees of freedom and non­

centrality parameter vri [(N2-Nl)~iJ-l).1

Hence this is also the conditional distribution of r; given r; , and using2 1

the x2 approximation to the distribution of ri '1

222The approximate joint distribution of rN ' r N ' and r N is obtained by multi-1 2 3

plying (38) by the approximate conditional probability density function

p(r~ Ir; ) and so on. When v= 3 (three dimensions) the relationship3 2

leads to considerable simplification, giving

where

of the form

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distributed as

18

using the methods of this section may be mentioned the distribution of the

So far, we have been mainly concerned with the distance between the

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exp[quadratic function of r N ' "" r N J,1 s

Since any ordering of the N segments, given a set of N orientations,

using an obvious notation.

V-l(Nf2+N'f,2) x (non-central x2 with v degrees of freedom and

non-centrality parameter VR2(Nf2+N'f,2)-1)

Approximate evaluation of the probability of a given class of configurations

distance between the terminal points PN, PN, od two independent chains with

initial points a distance R apart, The square of this distance is approximately

of the set of values (rN ' "" r N ) can be effected by term-by-term inte-I s 2 2

gration of expressions of type (40) with respect to r N ' "" r N •1 s

Among other interesting approximate results which can be obtained

leads to the same position for the terminal point; and since each such ordering

the angle between Pj-lPj and the terminal axis is the same for all values of j.

It is convenient to consider the angle between the final segment, PN-IPN, and

the terminal axis POPN• Denoting this angle by ~, we have from the triangle

is, under our assumptions, equally likely, it follows that the distribution of

initial and terminal point~ Po' PN respectively. We now consider the distri­

bution of the angle between a randomly chosen segment P. IP" and the terminalJ- J

axis POPN, This distribution will eVidently depend on the length of POPN(rN),

and it is this conditional distribution that we will consider. We will confine

6. Distribution of Angle of Inclination to Terminal Axis

our attention to the case of segments of constant length, f, in three dimensions.

(40)

(-loS cos ~ oS 1)

cos(41)

(42)

it can be seen that (in three dimensions)

Kuhn and Grun (1942) studied this problem, by dividing the range of

19

2Hence the conditional distribution of cos ~, given r N, is directly derivable

from that of r~_l' given r~. From the relation (see (1))

Hence

Hence, from (41)

and

Now introducing the x2 approximation for the distribution of r;_l we find

approximately.

variation of ~ into a large number of small intervals and max:lmizing the

way were found to lead to a distribution of the same type as (lJ.3) but with

probability, among sets of WI s constrained to give a specified value of r N•

The limiting relative frequencies of different values of V, obtained in this

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3rN[(N-l)fJ-l replaced by the inverse Langevin function of rN/(NI); i.e., the

value of ~ satisfying.

(44)

F',r small values of rN/(Nf), (43) implies t3 .;. 3rN/(Nf). Since the

expected value and standard deviation of r N are both of order VN, the ratio

rN/(Nf) will usually be small when N is large. Our result is thus in qUite

good agreement with that of Kuhn and Griin.

7. Continually Varying Orientation as a Limiting Case

It would appear natural to try to approach the situation in which

the point moves with a continuously varying orientation by considering the

limit of the case considered in section 2 (With corrnnon fixed segment length f)

as N tends to infinity and t to zero, Nt remaining constant. (If the point be

imagined to move with a fixed velocity, Nt is proportional to the time the

point is in motion.) However, from (7( and (8) we have

and in fact lim ~s(r~lt) = 0 for all s > 1. As the limit is approached the

probability of the terminal point being in any neighborhood, however small,

of the initial point tends to one.

A non-degenerate limit can be obtained by assuming (see Chandrasekhar

(1943, p. 19), also Einstein (1905)) that Nt2 remains constant--equal to ~ say.

Then the limiting values of the moments of r~ are

lim E(r~lt) = ~j lim ~2(r~lt) = ~ E(u2

)¢2;

. 21· 1 [ 2 J2 311m ~3(rN t) = 2 E(u) ¢

lim ~4(rilt) =~[E(u2)J2[1+E(u2)J¢4

20

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and the limiting distribution of r~ is that of (~/v) x (x2 with v degrees of

freedom). If the Ij's are independent identically distributed random variables

and ~ tj

has a proper limiting distribution ri also has a proper limiting

2distribution, but it need not be of X form.

REFERENCES

[1] Breitenberger, E. (1963)Ii-Analogues of the normal distribution on thecircle and the sphere,' Biometrika, 50, 81-88.

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