I Ie I I I I I I I I_ I I I I I I I .- I 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 composed of straight line segments with independent orientations, which have a continuous distribution, are discussed. formulas are obtained for the distance between initial and terminal points, and a new approximation discussed. The Rayleigh approximation is used to der- ive approximations to other characteristics of the random path. This reseach was supported by the Air Force Office of Scientific Research Grant No. AF-AFOSR-76o-65. DEPARTMENT OF STATISTICS UNIVERSITY OF NORTH CAROLINA Chapel Hill, N. C.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
I
IeIIIIIIII_IIIIIII.-I
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.
I
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
I
..IIIIIII
_IIIIIIII-.I
I
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
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-
I
~IIIIIIII_IIIIIII.eI
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
I
JIIIIIII
_IIIIIIII-.I
(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
I
\IIIIIIII_IIIIIII.-I
of
16
One further application may be noted here, though we shall not pursue
I
JIIIIIII
_IIIIIIII-.I
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
I
..IIIIIIII_IIIIIII.eI
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
I
JIIIIIII
_IIIIIIII-.I
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
I
'-IIIIIIII_IIIIIII.eI
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
I
JIIIIIII
_IIIIIIII-.I
I
~IIIIIIII_IIIIIII.eI
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.
[2] Chandrasekhar, S. (1943), "Stochastic problems in physics and astronomy,1IRev. Mod. Physics, 15, 1.89.
[3J Durand, D. and Greenwood, J. A. (1957), IIRandom unit vectors II: II Usefulness of Gram-Charlier series in approximating distributions," Ann. Math.Statist., 28, 978-986.
[4J Einstein, A. (1905), "Uber die von der molekularkinetischen Theorie derWarme geforderte ~wesung von in ruhenden Flussigkeiten suspendiertenTeilchen," Ann. d. Physik, 17, 549-560.
[5] Einstein, A. (1906), "Zur theorie der Brownschen Bewegung, II Ann. d. Physik,19, 371-381.
[6] Fisher, R. A. (1953), "Dispersion on a sphere,1I Proc. Roy. Soc. London,Series A, 217, 295-305.
[7] Greenwood, J. A. and Durand, D. (1953), "The distribution of length andcomponents of the sum of n random unit vectors," Ann. Math. Statist.,26, 233-246.
[8J Grosjean, C. S. (1953), IlSo1ution of the non-isotropic random flightproblem in the k-dimensional space," Physica, 19, 29-45.
[9J Gumbel, E. J., Greenwood, J. A. and Durand, D. (1953), "The circularnormal distribution: Theory and tables, II J. Amer. Statist. Assoc.,48, 131-152.
[10] Horner, F. (1946), IIA problem in the summation of simple harmonic functions of the same amplitude and frequency, but of random phase,1I Phil.Mag., 7th Series, 37, 145-162.
[11] Kluyver, J. C. (1905), "A local probability problem,1I Proc. Kon. Nederl.Akad. v. Wetensch., 8, 341-350.
[12J Kuhn, w. (1934), "Uber die Gestalt fadenformige Molekule in Losungen,1IKolloid-Zeit., 68, 2-15.
21
,electrons, " Ann. de Chim.
[13J Kuhn, W. (1946), "Dependence of the average transversal on the longitudinal dimension of statistical coils formed by chain molecules," J. PolymerSci., 1, 380-388.
[14J Kuhn, W. and Grim, F. (1942), "Beziehungen zwischen elastischen Konstantenund Dehnungsdoppelbrechung hochelastischer Stoffe," Kolloid-Zeit., 101,248-271.
[15J Kuhn, W. and Grun, F. (1946), "Statistical behavior of the single chainmolecule and its relation to the statistical behavior of assembliesconsisting of many chain molecules," J. Polymer Sci., 1, 183-199.
[16J Kuiper, N. H. (1960), "Tests concerning random points on a circle," Proc.Kon. Nederl. Akad. v. Wetensch, Series A, 63, 383-397.
[17J Langevin, P. (1905), "Magnetisme et theorie deset de Phys., 5, 70-127.
[18J Lord, R. D. (1948), "A problem on random vectors," Phil. Mag., 7th Series,39, 66-71.
[19J von Mises, R. (1918), "Uber die 'Ganzzahligkeit' der Atomgewichte undverwandte Fragen," Phys. Zeit., 19, 490-500.
[20J Pearson, E. S. (1963), "Some problems arising in apprOXimating to probability distributions, using moments," Biometrika, 50, 95-111.
[21J Pearson, K. (1905), "The problem of the random walk," Nature, 72, 294 and318.
[22J Pearson, K. (1906), "A mathematical theory of random migration," Math. Contr.Theory Evol. XV, Draper Co. Res. Mem., Biometric Series }.
[23J Rayleigh, Lord (Strutt, J. W.) (1880), "On the resultant of a large numberof vibrations of the same pitch and of arbitrary phase," Phil. Mag. 5thSeries, 10, 73-78.
[24J Rayleigh, Lord (1899), "On James Bernoulli's theorem in probabilities,"Phil. Mag., 5th Series, 47, 246-251.
[25J Rayleigh, Lord (1905), "The problem of the random walk," Nature 72, 318.
[26J Rayleigh, Lord (1919 a), "On the problem of random vibrations and randomflights in one, two and three dimensions," Phil. Mag., 6th Series, 37, 32131+7.
[27J Rayleigh, Lord (1919 b), "On the resultant of a number of unit vibrationswhose phases are at random over a range, not limited to an integralnumber of periods," Phil. Mag., 6th Series, 37, 498-515.
[28J Slack, Margaret (1946), "'The probability distribution of sinusoidaloscillations combined in random phase," J. !nst. Elect. Eng., 90(3),76-86.
22
I
JIIIIIII
_IIIIIIII-.I
I
..IIIIIIII_IIIIIII.eI
[29J Slack, Margaret (1947), Letter in Phil. Mag., 7th Series, 38, 297-298.
[30J von Smo1uchowski, M. (1906), "Zur kinetischen Theorie der BrownschenMo1eku1arbewegung und der Suspensionen," Ann d. Physik, 21, 756-780.
[31J Stephens, M. A. (1962 a), "Exact and approximate tests for directions, I,"Biometrika, 49, 463-477.
[32J Stephens, M. A. (1962 b), "Exact and approximate tests for directions, n"Biometrika, 49, 547-552.
[33J Watson, G. S. (1960), "More significance tests on the sphere,u Biometrika,47, 87-91.
[34J ~'latson, G. S. (1961), "Goodness-of-fit tests on a circle, I," Biometrika,1~8, 109-114.
[35J Watson, G. S. (1962), "Goodness-of-fit tests on a circle, II," Biometrika,1.1-9, 57-63.
[36J Watson, G. S. and Williams, E. J. (1956), "On the construction of significance tests on the circle and the sphere," Biometrika, 43, 344-352.
[37J Vogler, L. E., (1964), Percentage Points of the Beta Distribution, NationalBureau of Standards, Boulder, Colorado.