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Some statistical characterizations of aircraft hijacking
Accid. Anal. & Prer.
V o l 6, pp. 115-123.
Pergamon Press.
Printed in Great Britain
SOME STATISTICAL CHARACTERIZATIONS OF AIRCRAFT HIJACKING* RICHARD E. Q U A N D T Department of Economics, Princeton University, Princeton, N.J. 08540, U.S.A.
(Received 21 September 1973)
1. I N T R O D U C T I O N
The frequent hijacking of aircraft is a relatively recent phenomenon. For operational purposes we take as the start of the series of intensive hijacking January 1, 1968: although some 12 hijackings occurred in the preceding 7 yr, there were 22 in 1968 alone. From early 1968 until the end of 1972 (at which time new and more stringent security regulations were brought into effect) we have witnessed an unprecedented stream of aircraft hijackings. Hijackers as a group, may have a variety of motivations: some hope to accomplish a political purpose, others may hope for monetary gain, and still others (as well as some of the former) may be psychologically-disturbed individuals for whom the accomplishment of such a deed is equivalent to rectifying some specific or general failure in life [Hubbard, 1973]. Whichever is the case, a hijacking whether successful or not, may well influence the occurrence of another. If this were the case the observed series of hijackings would not behave as do ordinary accident series in which the probability of the occurrence of an accident in a given time interval does not depend on when the last accident occurred: in other words the hijackings series could not be thought of as the realization of a Poisson process [Barnard, 1953; Miller and Quandt, 1967]. The basic purpose of this paper is to test the null hypothesis that the hijacking series was generated by a Poisson process. Section 2 contains a brief discussion of the data and some summary statistics derived from them. Section 3 is devoted to the test of the Poisson hypothesis. Section 4 contains a brief summary. 2. D A T A
AND
SUMMARY
CHARACTERISTICS
The data are taken from Chronology (~/ H!jacking of U.S. Registered Aircraft, prepared by the Office of Air Transportation Security, Federal Aviation Administration.'t The data provide, among others, the date of the incident, the type of flight (scheduled, chartered, private), the hijacker's boarding point and his destination. It is not immediately clear * I am indebted to Mrs. Elvira (Jiaimo for computer programming, to Mr. John G. Leyden of the Federal Aviation Administration for data on hijacking of aircraft, and to J. Durhin for valuable advice and to S. M. Goldfeld and the referees for a critical reading of the paper. Support from NSF grant GS-37519 is gratefully acknowledged. I am alone responsible for errors. "t" Departures from a Poisson process arc presumably caused bv a potential hijacker's awareness that another hijacking occurred, thus prompting him to take action himself. For'this reason we rejected using the more extensive series on hijacking in Hubbard [1973] since that series contains numerous events not widely reported in thc U.S. press. 115
116
RICHARD E. QUANDT
~hich series of incidents one wants to investigate. All the incidents in the Chronolog.l" pertain to U.S. registered aircraft but it is possible that the circumstances and motivations affecting the hijacking of scheduled flights is different from that of chartered and private flights and that these are different from flights originating abroad and those originating in the U.S. For this reason we shall treat in Section 3 four partially overlapping serics: Series 1: All hijackings Series 2: All hijackings except those affecting private aircraft Series 3: All hijackings except private and charter aircraft Series 4: All hijackings except private and charter aircraft and flights with foreign origin. It is obvious that many other series could have been generated and analyzed: in the interest of economy the analysis was restricted to the above four. The data comprising these series will reveal, it is hoped, the basic temporal characteristics of hijackings. Since there is no strong a priori reason to believe that hijackers exercise selectivity among types of flights, it is desirable to examine all four. However, Series 4 may well have the most homogeneous group of perpetrators. Table 1. Characteristics of hijackings by type of flight*
Total hijackings No. of hijackings of private aircraft No. of hijackings of chartered aircraft No. of hijackings of scheduled aircraft
1968
1969
1970
1971
1972
Total
22 4 ~ 16
40 0 0 40
28 0 1 27
27 0 2 25
31 0 2 29
14S 4 7 137
* The table includes four disputed incidents which are not regarded as hijackings by either the FAA or by the Department of Justice. Each of these incidents are regarded as hijackings by one o1" these agencies and we felt that we should not remove them from the series.
Tables 1 and 2 display some summary characteristics. In Table 1 we show the time series of hijackings by type of flight. In Table 2 the first three categories are overlapping but the overlap is extremely small: only three flights hijacked to Cuba originated abroad [Bahamas 1968, Mexico City 1969. Caracas 1971) and only one case of extortion had destination Cuba (1972). The total number of hijacked private and chartered aircraft is small as is the number of hijacked aircraft originating abroad;* hence one would not expect great differences among the properties of the various series analyzed. The overall pattern suggests that the desire to reach Cuba was the most important causative factor in 196~ and 1969; this factor declined in the years 1970 and 1971 accounting only for roughly one half of the hijackings; finally in 1972 extortion becomes the single most important causative factor. It is finally interesting to note that the number of airlines involved increases Table 2. Additional characteristics of hijackings
No. No. No. No.
of hijackings of aircraft originating abroad of hijackings with destination Cuba of hijackings involving extortion of airlines involved in hijackings
1968
1969
1970
1971
1972
Total
2 19 0 7
2 38 0 9
5 15 1 12
2 14 3 12
0 7 19 17
11 93 23
*All but two of the hijacked aircraft originating abroad were regularly scheduled flights.
Some statistical characterizations of aircraft hijacking
117
steadily. Alternatively one might look at the fraction of hijackings in each year by, say. the three most frequently hijacked airlines: this figure is 67 per cent in 1968 (Eastern, National, Delta), drops to 56 per cent in 1970 (TWA, Eastern, Pan American) and finally drops to 34 per cent in 1972 (Pacific Southwest, American, United). These figures suggest the increasing diffusion in motives, methods and destinations of hijackers. Finally one may examine whether hijackers exhibit any preference for certain days of the week. For Series 4 we display the fraction of hijackings by year and by day of week in Table 3. Except for Sundays which show a consistently low percentage, the variation from year to year is considerable. If hijackers had no preferences a m o n g flights, the hijacking frequencies by day of week would tend to show the same pattern as the number of flights by day of week. Since there appears to be no economical way of determining this figure, we hypothesized that the relative frequency of flights on M o n d a y through Friday is 16 per cent each and on Saturday and Sunday 10 per cent. Using a g 2 criterion of goodness-of-fit we cannot reject the hypothesis that no preference is given by hijackers to flights on certain days. According to this s u m m a r y characterization the risk of being hijacked did not vary significantly during the week on the average. Nevertheless, there were some sizeable year-to-year variations in the weekly patterns. In other words, although the weekly patterns are not steady, they do not on the average exhibit a bias in favor of some day or days. Table 3. H~ackings(percentage)by year and day ofweek*
Monday
Tuesday Wednesday
Thursday Friday Saturday Sunday
1968
1969
1970
1971
1972
Total
13 13 20 13 7 27 7
19 19 14 14 17 6 11
24 19 14 14 10 10 10
12 0 17 8 42 12 8
I1 4 25 11 36 4 10
16 11 18 12 23 10 10
* Percentages may not add up to 100 due to rounding.
3. TESTS OF THE POISSON PROCESS We first define the fundamental characteristics o f a Poisson process and then apply some tests of hypotheses.* Define N(t), t _>_0, as the number of events that have occurred in the interval (0, t). The quantity N ( t ) is said to have independent increments if the r a n d o m variables N ( t ~ + hat) - N ( t z + hat), h = 0, 1, 2... are mutually independent. The process N ( t ) is a Poisson process if it has stationary independent increments and if the number of events in the interval t 2 - - t I ----- At(At > 0) has Poisson distribution with mean 2At, i.e. if e - ;"a'0.At)~ P r o b IN(t_,) - N ( t t ) = i I = . i! Such a process is called a homogeneous Poisson process. If one requires that N ( t ) have * For the derivation of the properties of Poisson processes see Parzen, 1962.
118
RICHARD E. QUANDT
only independent increments, one obtains a nonhomogeneous Poisson process, i.e. one with intensity that is itself a function of time. 2(t).* Let ti be the length of time in days elapsed since January 1, 1968 to the ith event and let W~be the interarrival time I4,] = t / - ti_ 1, with W1 = tl. The following are well-known theorems: Theorem 1. The interarrival times W~of a Poisson process are independently distributed according to the exponential distribution ./'(W~t = 2e - ~"'. Theorem 2. Let T b e the length of the period under observation (from January 1, 1968 to November 11, 1972). Then the quantities rg = t i T are independently and uniformly distributed over the interval ((3,1). [Parzen, 1962, pp. 135-141 ].
Testino for the homogeneous Poisson process Since Theorem 2 gives the exact distribution of the z~ under the null hypothesis that hijackings represent a homogeneous Poisson process, the appropriate test in practice is to test whether the z; are uniformly distributed over the (0,1) interval. Two standard tests with good power are the Kolmogorov-Smirnov test and the Cram6r-von Mises test. The Kolmogorov-Smirnov statistic is D = sup I F(x i) - S ( x i ) I , where F(x) is the hypothesized and S(x) the sample cumulative distribution function. Exact significance levels have been tabulated by Birnbaum who observes that the asymptotic distribution is quite accurate for n = 80. The Cram6r-von Mises statistic is W2 = n~1 [S(y) - y]2dy where y = F(x). For computational purposes we use the equivalent expression 1
W2 =
i=I
Yi
;/
+ 121~
Significance points have been tabulated by Anderson and Darling [1952]. These two statistics are displayed in Table 4. Table 4. K o l m o g o r o v - S m i r n o v and C r a m e r - v o n M i s e s s t a t i s t i c s
D W2
Series l
Series 2
Series 3
Series 4
0.0726 0-1196
0.0965 0.1954
0.1072 0-2066
0.1083 0.2238
None of the D-values is significant at the 0.05 level and only Series 3 is significant at the 0.1 level. None of the W 2 figures is significant at the 0"2 level. On the basis of these statistics we cannot reject the hypothesis that the series represent a Poisson process. A more subtle examination is required, however, since we are interested in the possible bunching of incidents. The need for such examination may be illustrated with a simple example. Assume that 10 incidents have occurred and their ri values are 0.1, 0.2, 0.3 . . . . . 1.0 exactly. Then the Kolmogorov-Smirnov and Cram6r-von Mises statistics are both equal to zero and the data fit the null hypothesis perfectly. Now assume a slight bunching holding the total number of incidents constant. Let the ri series be slightly altered so that it is 0.101, 0.199, 0.301, 0.399 . . . . 0-901, 0.999. The Kolmogorov-Smirnov and Cram6r-von * If more than one event can happen in an interval, no matter how small, one is dealing with a generalized Poisson process. Since we are assuming that in an interval sufficiently short at most one event can occur, and since the measurements of time implicit in the data arc by days, we have assumed in those few instances (15) in which two events happened on the same day that they actually occurred 1/2 day apart.
Some statistical characterizations of aircraft hijacking
119
Mises statistics are still negligible in magnitude but the sample cumulative distribution function crosses the theoretical one after every observation, exhibiting a systematic lowamplitude oscillation around it. If oscillations of all frequencies are present, the null hypothesis is not contradicted. However, if some frequencies predominate in the oscillations, some systematic departure from the null hypothesis is indicated even if the amplitude of the oscillations is small. Such oscillations are therefore of interest and can be investigated by employing the components of the Cram6r-von Mises statistic [Durbin and Knott, 1972]. For a series of length n, denote the components by z. i (/' = 1, 2 . . . . ). Durbin and Knott show that the following decomposition holds: J
W,,2
_2
~=1 -nj i= .I -re-
where the z. i can be computed from 2 .L -hi
rl i= 1
cos(i~z vi) i = 1, 2 . . . .
Since the z,,j can also be represented as z,,i = \/(2n)jnS~ES(y) - y] sin(jrcy)dy, they can be shown to be the Fourier sine coefficients in the expansion of S ( y ) - y and are thus the proper frequency domain decomposition of a random function which vanishes at the ends of the range. The first 20 components for the four series are displayed in Table 5. A fair number of components is significant, suggesting the presence of bunching. N o intuitively clear bunching pattern emerges from the significant components.
Table 5. Components of the Cram+r-yon Mises statistic Series Component j
1
2
3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0-4227 0.1871 1.7868" 2-1140" 1.3945 1.7517" 1.5491 0.4039 1-2531 0.4556 1.9109" 0.0134 0-8369 1"3903" 0.2322 0"6736 1"3085 1.9722' 1.3104 0.8410
0.8650 0-5286 2"0103" 2.1879" 1.3210 1.5882 1-7947" 0-2103 1.3928 0-4479 1"8358" 0-2128 1.1096 1.6877" 0.4865 0.8726 1.2174 1-9739" 1.3761 0-9545
0.7577 0'7538 2'3297t 2'2824t 1.6826" 1.4410 1-7513" 0'1227 1.1889 0.2568 2"0307* 0-4404 1.1495 1.8405" 0.4156 1"2379 1.2545 1-6132 1.4744 1.0622
0.9395 0'0730 2.3053t 2.4951t 1.4599 1"3211 1.5278 0-0503 1"4212 0.1576 1-5519 0-3734 0-6668 2-1045" 0.7737 1"5659 1.3918 1-5798 1.6567" 1.3485
* Significant at the 0-05 level. t Significant at the 0-01 level. A.^,,. 6 2
J3
120
RICHARD E. Q [ ANDT
It may be verified that the number of significant components is significantly greater than would be expected under the null hypothesis. For large n. it seems reasonable to regard the z,i as approximately normal with mean zero and unit variance. Then defining the I% =
f I:,,il
\/2./~ e 1'2 2 du
)
quantities V,i are uniformly distributed on (0.I) under the null hypothesis, Employing the one-sided K o l m o g o r o v - S m i r n o v test, to test the hypothesis that some of the 1,,i are too large, we reject it at the 0.01 level for all four series.* Although the standard K o l m o g o r o v Smirnov and C r a m 6 r - v o n Mises tests fail to reject the null hypothesis, the finer Durbin K n o t t procedure does detect a departure from it.
Testing for nonhonlogeneous Poisson processes Having determined that sufficient bunching is present in the data to reject the hypothesis of a homogeneous Poisson process for aircraft hijacking, it is of interest to determine whether the assumption of a nonhomogeneous process represents a significant improvement over the homogeneous one. There are. of course, numerous ways in which nonhomogeneity might be introduced into a Poisson process. A c o m m o n way is to assume that the parameter 2 of the exponential distribution of interarrival times is some simple function of ti, say a polynomial [Parzen, 1962]. This formulation is not adopted here since it does not appear plausible that the mean interarrival time changes in a polynomial of somc reasonably low order. If a change in the mean does occur over time. it is presumably because of the changing character of hijackers, with possible different mean rates of hijacking, due to the shift from Cuba-minded hijackers to extortion-minded ones. Assuming that the parameter ). shifts from 2! to ).2, the shift can be accomplished in one of two ways: (1) One may assume that at some unknown point in time there is a discontinuous shift from ).~ to ).2. (2) One may assume that the parameter ). is )., for some period, that this is followed by a transition period during which shifts from 2, to 2,. followed by a period during which it stays at ).~. Under the first specification the likelihood function is .
._~
e
5[ 14,]. i= I
5~_,
14~)
(3.1)
i=h ( 1
which is to be maximized with respect to 2~, ),~ and k. The likelihood function has similar form if it is posited that there are two shifts [and thus three parameters 2 ~, 22, 23 ) [Quandt, 1958]. Under the second specification we define
i
t,
di . . . .
1
, \,' 2~zo"
e
- ~1~-.
o-)-'
d~
(3.2)
where /x and a are unknown parameters. It is a property of di that for relatively, small r and as i varies from 1 to n, its value changes from 0 to 1. Hence it can properly be used to characterize the transition between two processes. Moreover, the smaller is a, the more sudden the transition and the more the second specification resembles the first [Goldfeld and Quandt, 1972]. * I am indebted
t o J. D u r b i n
for t h i s o b s e r w l t i o n
and result.
Some statistical characterizations of aircraft hijacking
121
Assume that the observed interarrival time I4;. is the convex combination of interarrival time W1,.and IV_,i from two homogeneous Poisson processes with weights dl and (1 - d;). Then I/Vii -~ diWli + (1
--
(3.3)
di)W2i
Since W1i, W_,~are exponentially distributed with parameters ).p).2. the pdf of the random variable W/can be obtained as the convolution of the pdfs of diWl~ and of (1 - d~)W_,,,and is h(Wi) =
~1J~2 (1
--
[e -~a''l-ai)wl`
--
e -~<''a°w2~]
(3.4)
di)2 2 + di21
The resulting log likelihood function l o g L = nlog(2,2_0 + i=~ log~(e-'~" e - ° '-'~"w~ ' =~d~) ' " " w ~Vl~~
;
(3.5)
where d~ is replaced by (3.2) is maximized with respect to 21, 22, # and a. We compare in Table 6 the estimate for the homogeneous Poisson process and the two specifications of the nonhomogeneous ones. The quality I is the likelihood ratio where the homogeneous process represents the null hypothesis. Under generalized regularity conditions - 2log/is approx Z2 distributed with degrees of freedom equal to the number of restrictions implied by the ull hypothesis [Kendall and Stuart, 1962]. Table 6. Comparison of estimates for various Poisson processes Series 1 Homogeneous Poisson process 2 0.0833 Specification 1 21 0.0276 22 0.0897 k 5 - 2log/ 9.872 Specification 2 2~ 0-0463 22 0-0886 ,u 260.160 a 28.921 - 2log/ 5.600 Implied k 10 Implied date Sept. 16. 1968
Series 2
Series 3
Series 4
0.0811
0-0771
0-0698
0.0211 0.0878 4 11.810
0.0293 0.0871 9 13.773
0-0261 0.0790 8 12.788
0.0353 0.0900 305.697 9.478 8.078 9 Nov. 1, 1968
0-0283 0.0868 307.000 0.042 13.482 9 Nov. 2, 1968
0.0291 0.0784 317-747 4.478 11.008 9 Nov. 13, 196S
Under the first specification one of the parameters, k, is discrete and it has been found that the Zz distribution is a poor approximation to - 2 l o g / i n this case. Under the second specification the Z2 approximation may be employed. At a 0"05 level of significance, Series 2-4 lead to rejection of the null hypothesis; at a 0.01 level, only Series 3 does. Thus the assumption that two regimes were at work represents a considerable improvement over the hypothesis of a homogeneous Poisson process. The estimate for/t is the value of ti at which rate of transition from the first regime to the second was fastest. According to Series 2-4 this occurred during the first two weeks of November 1968. The
122
RICHARD E. QUANI)T
t r a n s i t i o n was rapid in a n y event: the +_2a range was longest for Series 2 (38 daysl but is estimated to as short as less t h a n a day for Series 3. The result for Series 3 is therefore essentially identical between specifications 1 a n d 2. The difference between the two regimes is in the m e a n as has been suggested in the previous Section from visual inspection of the c u m u l a t i v e sample distribution. 4. C O N C L U D I N G COMMENTS There is little d o u b t that the series o n aircraft hijacking exhibit at least two types of departure from what one w o u l d expect in a h o m o g e n e o u s P o i s s o n process. First, there is a p r o n o u n c e d shift in the m e a n or intensity of the process near the end of the first year of the period selected for observation. This allows us to classify the epochs of hijacking as follows: (11 Prior to 1968, the period of occasional hijackings: (2) F r o m J a n u a r y 19(,~ to N o v e m b e r 1968, the period of m o v i n g into high gear: (3) F r o m 1968 to the end of 1972. the peak period. So far we are fortunate to be able to classify 1973 as (4) the period of essentially no hijackings. M o r e interesting p e r h a p s is the discovery of b u n c h i n g via the D u r b i n K n o t t analysis of the c o m p o n e n t s of the C r a m 6 r - v o n Mises statistic. This does not allow us to characterize the b u n c h i n g in detail [say, the hijackings occur in threes as bishops' deaths (according to a n c i e n t lore)]. It does not, therefore, allow us directly to q u a n t i f y the c h a n g e d risk of being hijacked, given that one has recently occurred. But it does suggest that it is w r o n g to treat hijackings as i n d e p e n d e n t events: thus extra vigilance in a p p l y i n g preventive measures is likely to be most useful in periods i m m e d i a t e l y following a hijacking. REFERENCES Anderson T. W. and Darling D. A. [1952] Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. ,4ml.qfmath. Star. 23, 193 212. Barnard G. A. [1953] Time intervals between accidents: A note on Maguire. Pearson and Wynn's paper. Biometrika, 40, 212 213. Birnbaum A. W. [1952] Numerical tabulation of the distribution of Kolmogorov's statistic for finite sample size. J. Am. Stat. Ass. 47, 425-441. Durbin J. and Knott M. [1972] Components of Cramer-von Mises statistics. J. Ro). Star. Soc. Series B. 34, 290 307. Goldfeld S. M. and Quandt R. E. [ 1972] Xonlinear Methods in Ecommwtrics. North Holland Publishing. Hubbard D. G. [1973] The Sk£jacker. Collier Books. Kendall M. G. and Stuart A. [1961] The Adraneed Theory Statistics. Vol. 2. Harper. Miller R. E. and Quandt R. E. [1967] An anahsis of the randomness of air accidents. J. Roy. Aer. Soc. 71, 23 27. Parzen E. [1962] Stochastic Processes. Holden Day. Quandt R. E. [1958] The estimation of the parameters of a linear regression system obeying two differentregimcs. J. Am. Star. Ass., 53, 873 880. Abstract--The paper examines hijackings of U.S. registered aircraft since 1968. After some summary characterizations of the series of hijackings, the paper tests the null hypothesis that the) represent a homogeneous Poisson process. Using the Durbin Knott decomposition of the Cram6r yon Mises statistic, this hypothesis is rejected. The paper further investigates whether formulating the model as a nonhomogeneous Poisson process represents a significant improvement over the assumption of homogeneity. This question is answered in the affirmative.The major conclusion of the paper, of practical significance,is that hijackings are not independent of one another and security precautions may need to be intensified in periods following a hijacking. ROsum6--Ce texte examine le detournement d'avions am~ricains depuis 1968. Apr6s quelques caract6risations sommaires des s6ries de detournements, le texte met /~ l'essai l'hypoth6se nulle qu'ils repr6sentent un proc6d6 homogene de Poisson. Utilisant la d6composition Durbin-Knott de la statistique Bram6r-von-Mises cette hypoth/~seest rejet+e. Le texte 6tudie ensuite si la formulation
Some statistical characterizations of aircraft hijacking du mod/~le en tant que proc6d6 de Poisson non-homog+ne represente une amelioration notable sur la supposition d'homog6n6it6. Cette question est repondue par l'affirmative. La conclusion principale de cet ouvrage d'importance pratique est que les d6tournements ne sont pas independants l'un de l'autre et les mesures de securit6 devront 6tre intensifi6es dans les p6riodes suivant un d6tournement.
Zusammenfassung--Der Bericht untersucht Flugzeugentfiihrungen seit 1968 von Flugze~gen, die in den U.S. eingetragen waren. Nach einigen zusammenfassenden Charakterisierungen der Reihen der Entfiihrungen priift der Bericht die Nullhypothese, dab sie einen gleichartigen Poisson Vorgang darstellen. Mittels Zerlegung der Cram~r-von Mises Statistik nach Durbin-Knott wird die Hypothese verworfen. Der Bericht untersucht weiterhin, ob die Formulierung des Modells als ein nicht gleichartiger Poisson Vorgang eine bedeutende Verbesserung zu der Annahme der Gleichartigkeit darstellt. Diese Frage wird bejah6nd beantwortet. Der HauptschluB des Berichts von praktischer Bedeutung ist die Feststellung, dab Flugzeugentfiihrungen nicht voneinander unabhS.ngig sind, und dab SicherheitsmaBnahmen m6glicherweise in Zeitr~iumen nach einer Entfiihrung verst/irkt werden miissen.
123
V o l 6, pp. 115-123.
Pergamon Press.
Printed in Great Britain
SOME STATISTICAL CHARACTERIZATIONS OF AIRCRAFT HIJACKING* RICHARD E. Q U A N D T Department of Economics, Princeton University, Princeton, N.J. 08540, U.S.A.
(Received 21 September 1973)
1. I N T R O D U C T I O N
The frequent hijacking of aircraft is a relatively recent phenomenon. For operational purposes we take as the start of the series of intensive hijacking January 1, 1968: although some 12 hijackings occurred in the preceding 7 yr, there were 22 in 1968 alone. From early 1968 until the end of 1972 (at which time new and more stringent security regulations were brought into effect) we have witnessed an unprecedented stream of aircraft hijackings. Hijackers as a group, may have a variety of motivations: some hope to accomplish a political purpose, others may hope for monetary gain, and still others (as well as some of the former) may be psychologically-disturbed individuals for whom the accomplishment of such a deed is equivalent to rectifying some specific or general failure in life [Hubbard, 1973]. Whichever is the case, a hijacking whether successful or not, may well influence the occurrence of another. If this were the case the observed series of hijackings would not behave as do ordinary accident series in which the probability of the occurrence of an accident in a given time interval does not depend on when the last accident occurred: in other words the hijackings series could not be thought of as the realization of a Poisson process [Barnard, 1953; Miller and Quandt, 1967]. The basic purpose of this paper is to test the null hypothesis that the hijacking series was generated by a Poisson process. Section 2 contains a brief discussion of the data and some summary statistics derived from them. Section 3 is devoted to the test of the Poisson hypothesis. Section 4 contains a brief summary. 2. D A T A
AND
SUMMARY
CHARACTERISTICS
The data are taken from Chronology (~/ H!jacking of U.S. Registered Aircraft, prepared by the Office of Air Transportation Security, Federal Aviation Administration.'t The data provide, among others, the date of the incident, the type of flight (scheduled, chartered, private), the hijacker's boarding point and his destination. It is not immediately clear * I am indebted to Mrs. Elvira (Jiaimo for computer programming, to Mr. John G. Leyden of the Federal Aviation Administration for data on hijacking of aircraft, and to J. Durhin for valuable advice and to S. M. Goldfeld and the referees for a critical reading of the paper. Support from NSF grant GS-37519 is gratefully acknowledged. I am alone responsible for errors. "t" Departures from a Poisson process arc presumably caused bv a potential hijacker's awareness that another hijacking occurred, thus prompting him to take action himself. For'this reason we rejected using the more extensive series on hijacking in Hubbard [1973] since that series contains numerous events not widely reported in thc U.S. press. 115
116
RICHARD E. QUANDT
~hich series of incidents one wants to investigate. All the incidents in the Chronolog.l" pertain to U.S. registered aircraft but it is possible that the circumstances and motivations affecting the hijacking of scheduled flights is different from that of chartered and private flights and that these are different from flights originating abroad and those originating in the U.S. For this reason we shall treat in Section 3 four partially overlapping serics: Series 1: All hijackings Series 2: All hijackings except those affecting private aircraft Series 3: All hijackings except private and charter aircraft Series 4: All hijackings except private and charter aircraft and flights with foreign origin. It is obvious that many other series could have been generated and analyzed: in the interest of economy the analysis was restricted to the above four. The data comprising these series will reveal, it is hoped, the basic temporal characteristics of hijackings. Since there is no strong a priori reason to believe that hijackers exercise selectivity among types of flights, it is desirable to examine all four. However, Series 4 may well have the most homogeneous group of perpetrators. Table 1. Characteristics of hijackings by type of flight*
Total hijackings No. of hijackings of private aircraft No. of hijackings of chartered aircraft No. of hijackings of scheduled aircraft
1968
1969
1970
1971
1972
Total
22 4 ~ 16
40 0 0 40
28 0 1 27
27 0 2 25
31 0 2 29
14S 4 7 137
* The table includes four disputed incidents which are not regarded as hijackings by either the FAA or by the Department of Justice. Each of these incidents are regarded as hijackings by one o1" these agencies and we felt that we should not remove them from the series.
Tables 1 and 2 display some summary characteristics. In Table 1 we show the time series of hijackings by type of flight. In Table 2 the first three categories are overlapping but the overlap is extremely small: only three flights hijacked to Cuba originated abroad [Bahamas 1968, Mexico City 1969. Caracas 1971) and only one case of extortion had destination Cuba (1972). The total number of hijacked private and chartered aircraft is small as is the number of hijacked aircraft originating abroad;* hence one would not expect great differences among the properties of the various series analyzed. The overall pattern suggests that the desire to reach Cuba was the most important causative factor in 196~ and 1969; this factor declined in the years 1970 and 1971 accounting only for roughly one half of the hijackings; finally in 1972 extortion becomes the single most important causative factor. It is finally interesting to note that the number of airlines involved increases Table 2. Additional characteristics of hijackings
No. No. No. No.
of hijackings of aircraft originating abroad of hijackings with destination Cuba of hijackings involving extortion of airlines involved in hijackings
1968
1969
1970
1971
1972
Total
2 19 0 7
2 38 0 9
5 15 1 12
2 14 3 12
0 7 19 17
11 93 23
*All but two of the hijacked aircraft originating abroad were regularly scheduled flights.
Some statistical characterizations of aircraft hijacking
117
steadily. Alternatively one might look at the fraction of hijackings in each year by, say. the three most frequently hijacked airlines: this figure is 67 per cent in 1968 (Eastern, National, Delta), drops to 56 per cent in 1970 (TWA, Eastern, Pan American) and finally drops to 34 per cent in 1972 (Pacific Southwest, American, United). These figures suggest the increasing diffusion in motives, methods and destinations of hijackers. Finally one may examine whether hijackers exhibit any preference for certain days of the week. For Series 4 we display the fraction of hijackings by year and by day of week in Table 3. Except for Sundays which show a consistently low percentage, the variation from year to year is considerable. If hijackers had no preferences a m o n g flights, the hijacking frequencies by day of week would tend to show the same pattern as the number of flights by day of week. Since there appears to be no economical way of determining this figure, we hypothesized that the relative frequency of flights on M o n d a y through Friday is 16 per cent each and on Saturday and Sunday 10 per cent. Using a g 2 criterion of goodness-of-fit we cannot reject the hypothesis that no preference is given by hijackers to flights on certain days. According to this s u m m a r y characterization the risk of being hijacked did not vary significantly during the week on the average. Nevertheless, there were some sizeable year-to-year variations in the weekly patterns. In other words, although the weekly patterns are not steady, they do not on the average exhibit a bias in favor of some day or days. Table 3. H~ackings(percentage)by year and day ofweek*
Monday
Tuesday Wednesday
Thursday Friday Saturday Sunday
1968
1969
1970
1971
1972
Total
13 13 20 13 7 27 7
19 19 14 14 17 6 11
24 19 14 14 10 10 10
12 0 17 8 42 12 8
I1 4 25 11 36 4 10
16 11 18 12 23 10 10
* Percentages may not add up to 100 due to rounding.
3. TESTS OF THE POISSON PROCESS We first define the fundamental characteristics o f a Poisson process and then apply some tests of hypotheses.* Define N(t), t _>_0, as the number of events that have occurred in the interval (0, t). The quantity N ( t ) is said to have independent increments if the r a n d o m variables N ( t ~ + hat) - N ( t z + hat), h = 0, 1, 2... are mutually independent. The process N ( t ) is a Poisson process if it has stationary independent increments and if the number of events in the interval t 2 - - t I ----- At(At > 0) has Poisson distribution with mean 2At, i.e. if e - ;"a'0.At)~ P r o b IN(t_,) - N ( t t ) = i I = . i! Such a process is called a homogeneous Poisson process. If one requires that N ( t ) have * For the derivation of the properties of Poisson processes see Parzen, 1962.
118
RICHARD E. QUANDT
only independent increments, one obtains a nonhomogeneous Poisson process, i.e. one with intensity that is itself a function of time. 2(t).* Let ti be the length of time in days elapsed since January 1, 1968 to the ith event and let W~be the interarrival time I4,] = t / - ti_ 1, with W1 = tl. The following are well-known theorems: Theorem 1. The interarrival times W~of a Poisson process are independently distributed according to the exponential distribution ./'(W~t = 2e - ~"'. Theorem 2. Let T b e the length of the period under observation (from January 1, 1968 to November 11, 1972). Then the quantities rg = t i T are independently and uniformly distributed over the interval ((3,1). [Parzen, 1962, pp. 135-141 ].
Testino for the homogeneous Poisson process Since Theorem 2 gives the exact distribution of the z~ under the null hypothesis that hijackings represent a homogeneous Poisson process, the appropriate test in practice is to test whether the z; are uniformly distributed over the (0,1) interval. Two standard tests with good power are the Kolmogorov-Smirnov test and the Cram6r-von Mises test. The Kolmogorov-Smirnov statistic is D = sup I F(x i) - S ( x i ) I , where F(x) is the hypothesized and S(x) the sample cumulative distribution function. Exact significance levels have been tabulated by Birnbaum who observes that the asymptotic distribution is quite accurate for n = 80. The Cram6r-von Mises statistic is W2 = n~1 [S(y) - y]2dy where y = F(x). For computational purposes we use the equivalent expression 1
W2 =
i=I
Yi
;/
+ 121~
Significance points have been tabulated by Anderson and Darling [1952]. These two statistics are displayed in Table 4. Table 4. K o l m o g o r o v - S m i r n o v and C r a m e r - v o n M i s e s s t a t i s t i c s
D W2
Series l
Series 2
Series 3
Series 4
0.0726 0-1196
0.0965 0.1954
0.1072 0-2066
0.1083 0.2238
None of the D-values is significant at the 0.05 level and only Series 3 is significant at the 0.1 level. None of the W 2 figures is significant at the 0"2 level. On the basis of these statistics we cannot reject the hypothesis that the series represent a Poisson process. A more subtle examination is required, however, since we are interested in the possible bunching of incidents. The need for such examination may be illustrated with a simple example. Assume that 10 incidents have occurred and their ri values are 0.1, 0.2, 0.3 . . . . . 1.0 exactly. Then the Kolmogorov-Smirnov and Cram6r-von Mises statistics are both equal to zero and the data fit the null hypothesis perfectly. Now assume a slight bunching holding the total number of incidents constant. Let the ri series be slightly altered so that it is 0.101, 0.199, 0.301, 0.399 . . . . 0-901, 0.999. The Kolmogorov-Smirnov and Cram6r-von * If more than one event can happen in an interval, no matter how small, one is dealing with a generalized Poisson process. Since we are assuming that in an interval sufficiently short at most one event can occur, and since the measurements of time implicit in the data arc by days, we have assumed in those few instances (15) in which two events happened on the same day that they actually occurred 1/2 day apart.
Some statistical characterizations of aircraft hijacking
119
Mises statistics are still negligible in magnitude but the sample cumulative distribution function crosses the theoretical one after every observation, exhibiting a systematic lowamplitude oscillation around it. If oscillations of all frequencies are present, the null hypothesis is not contradicted. However, if some frequencies predominate in the oscillations, some systematic departure from the null hypothesis is indicated even if the amplitude of the oscillations is small. Such oscillations are therefore of interest and can be investigated by employing the components of the Cram6r-von Mises statistic [Durbin and Knott, 1972]. For a series of length n, denote the components by z. i (/' = 1, 2 . . . . ). Durbin and Knott show that the following decomposition holds: J
W,,2
_2
~=1 -nj i= .I -re-
where the z. i can be computed from 2 .L -hi
rl i= 1
cos(i~z vi) i = 1, 2 . . . .
Since the z,,j can also be represented as z,,i = \/(2n)jnS~ES(y) - y] sin(jrcy)dy, they can be shown to be the Fourier sine coefficients in the expansion of S ( y ) - y and are thus the proper frequency domain decomposition of a random function which vanishes at the ends of the range. The first 20 components for the four series are displayed in Table 5. A fair number of components is significant, suggesting the presence of bunching. N o intuitively clear bunching pattern emerges from the significant components.
Table 5. Components of the Cram+r-yon Mises statistic Series Component j
1
2
3
4
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
0-4227 0.1871 1.7868" 2-1140" 1.3945 1.7517" 1.5491 0.4039 1-2531 0.4556 1.9109" 0.0134 0-8369 1"3903" 0.2322 0"6736 1"3085 1.9722' 1.3104 0.8410
0.8650 0-5286 2"0103" 2.1879" 1.3210 1.5882 1-7947" 0-2103 1.3928 0-4479 1"8358" 0-2128 1.1096 1.6877" 0.4865 0.8726 1.2174 1-9739" 1.3761 0-9545
0.7577 0'7538 2'3297t 2'2824t 1.6826" 1.4410 1-7513" 0'1227 1.1889 0.2568 2"0307* 0-4404 1.1495 1.8405" 0.4156 1"2379 1.2545 1-6132 1.4744 1.0622
0.9395 0'0730 2.3053t 2.4951t 1.4599 1"3211 1.5278 0-0503 1"4212 0.1576 1-5519 0-3734 0-6668 2-1045" 0.7737 1"5659 1.3918 1-5798 1.6567" 1.3485
* Significant at the 0-05 level. t Significant at the 0-01 level. A.^,,. 6 2
J3
120
RICHARD E. Q [ ANDT
It may be verified that the number of significant components is significantly greater than would be expected under the null hypothesis. For large n. it seems reasonable to regard the z,i as approximately normal with mean zero and unit variance. Then defining the I% =
f I:,,il
\/2./~ e 1'2 2 du
)
quantities V,i are uniformly distributed on (0.I) under the null hypothesis, Employing the one-sided K o l m o g o r o v - S m i r n o v test, to test the hypothesis that some of the 1,,i are too large, we reject it at the 0.01 level for all four series.* Although the standard K o l m o g o r o v Smirnov and C r a m 6 r - v o n Mises tests fail to reject the null hypothesis, the finer Durbin K n o t t procedure does detect a departure from it.
Testing for nonhonlogeneous Poisson processes Having determined that sufficient bunching is present in the data to reject the hypothesis of a homogeneous Poisson process for aircraft hijacking, it is of interest to determine whether the assumption of a nonhomogeneous process represents a significant improvement over the homogeneous one. There are. of course, numerous ways in which nonhomogeneity might be introduced into a Poisson process. A c o m m o n way is to assume that the parameter 2 of the exponential distribution of interarrival times is some simple function of ti, say a polynomial [Parzen, 1962]. This formulation is not adopted here since it does not appear plausible that the mean interarrival time changes in a polynomial of somc reasonably low order. If a change in the mean does occur over time. it is presumably because of the changing character of hijackers, with possible different mean rates of hijacking, due to the shift from Cuba-minded hijackers to extortion-minded ones. Assuming that the parameter ). shifts from 2! to ).2, the shift can be accomplished in one of two ways: (1) One may assume that at some unknown point in time there is a discontinuous shift from ).~ to ).2. (2) One may assume that the parameter ). is )., for some period, that this is followed by a transition period during which shifts from 2, to 2,. followed by a period during which it stays at ).~. Under the first specification the likelihood function is .
._~
e
5[ 14,]. i= I
5~_,
14~)
(3.1)
i=h ( 1
which is to be maximized with respect to 2~, ),~ and k. The likelihood function has similar form if it is posited that there are two shifts [and thus three parameters 2 ~, 22, 23 ) [Quandt, 1958]. Under the second specification we define
i
t,
di . . . .
1
, \,' 2~zo"
e
- ~1~-.
o-)-'
d~
(3.2)
where /x and a are unknown parameters. It is a property of di that for relatively, small r and as i varies from 1 to n, its value changes from 0 to 1. Hence it can properly be used to characterize the transition between two processes. Moreover, the smaller is a, the more sudden the transition and the more the second specification resembles the first [Goldfeld and Quandt, 1972]. * I am indebted
t o J. D u r b i n
for t h i s o b s e r w l t i o n
and result.
Some statistical characterizations of aircraft hijacking
121
Assume that the observed interarrival time I4;. is the convex combination of interarrival time W1,.and IV_,i from two homogeneous Poisson processes with weights dl and (1 - d;). Then I/Vii -~ diWli + (1
--
(3.3)
di)W2i
Since W1i, W_,~are exponentially distributed with parameters ).p).2. the pdf of the random variable W/can be obtained as the convolution of the pdfs of diWl~ and of (1 - d~)W_,,,and is h(Wi) =
~1J~2 (1
--
[e -~a''l-ai)wl`
--
e -~<''a°w2~]
(3.4)
di)2 2 + di21
The resulting log likelihood function l o g L = nlog(2,2_0 + i=~ log~(e-'~" e - ° '-'~"w~ ' =~d~) ' " " w ~Vl~~
;
(3.5)
where d~ is replaced by (3.2) is maximized with respect to 21, 22, # and a. We compare in Table 6 the estimate for the homogeneous Poisson process and the two specifications of the nonhomogeneous ones. The quality I is the likelihood ratio where the homogeneous process represents the null hypothesis. Under generalized regularity conditions - 2log/is approx Z2 distributed with degrees of freedom equal to the number of restrictions implied by the ull hypothesis [Kendall and Stuart, 1962]. Table 6. Comparison of estimates for various Poisson processes Series 1 Homogeneous Poisson process 2 0.0833 Specification 1 21 0.0276 22 0.0897 k 5 - 2log/ 9.872 Specification 2 2~ 0-0463 22 0-0886 ,u 260.160 a 28.921 - 2log/ 5.600 Implied k 10 Implied date Sept. 16. 1968
Series 2
Series 3
Series 4
0.0811
0-0771
0-0698
0.0211 0.0878 4 11.810
0.0293 0.0871 9 13.773
0-0261 0.0790 8 12.788
0.0353 0.0900 305.697 9.478 8.078 9 Nov. 1, 1968
0-0283 0.0868 307.000 0.042 13.482 9 Nov. 2, 1968
0.0291 0.0784 317-747 4.478 11.008 9 Nov. 13, 196S
Under the first specification one of the parameters, k, is discrete and it has been found that the Zz distribution is a poor approximation to - 2 l o g / i n this case. Under the second specification the Z2 approximation may be employed. At a 0"05 level of significance, Series 2-4 lead to rejection of the null hypothesis; at a 0.01 level, only Series 3 does. Thus the assumption that two regimes were at work represents a considerable improvement over the hypothesis of a homogeneous Poisson process. The estimate for/t is the value of ti at which rate of transition from the first regime to the second was fastest. According to Series 2-4 this occurred during the first two weeks of November 1968. The
122
RICHARD E. QUANI)T
t r a n s i t i o n was rapid in a n y event: the +_2a range was longest for Series 2 (38 daysl but is estimated to as short as less t h a n a day for Series 3. The result for Series 3 is therefore essentially identical between specifications 1 a n d 2. The difference between the two regimes is in the m e a n as has been suggested in the previous Section from visual inspection of the c u m u l a t i v e sample distribution. 4. C O N C L U D I N G COMMENTS There is little d o u b t that the series o n aircraft hijacking exhibit at least two types of departure from what one w o u l d expect in a h o m o g e n e o u s P o i s s o n process. First, there is a p r o n o u n c e d shift in the m e a n or intensity of the process near the end of the first year of the period selected for observation. This allows us to classify the epochs of hijacking as follows: (11 Prior to 1968, the period of occasional hijackings: (2) F r o m J a n u a r y 19(,~ to N o v e m b e r 1968, the period of m o v i n g into high gear: (3) F r o m 1968 to the end of 1972. the peak period. So far we are fortunate to be able to classify 1973 as (4) the period of essentially no hijackings. M o r e interesting p e r h a p s is the discovery of b u n c h i n g via the D u r b i n K n o t t analysis of the c o m p o n e n t s of the C r a m 6 r - v o n Mises statistic. This does not allow us to characterize the b u n c h i n g in detail [say, the hijackings occur in threes as bishops' deaths (according to a n c i e n t lore)]. It does not, therefore, allow us directly to q u a n t i f y the c h a n g e d risk of being hijacked, given that one has recently occurred. But it does suggest that it is w r o n g to treat hijackings as i n d e p e n d e n t events: thus extra vigilance in a p p l y i n g preventive measures is likely to be most useful in periods i m m e d i a t e l y following a hijacking. REFERENCES Anderson T. W. and Darling D. A. [1952] Asymptotic theory of certain "goodness of fit" criteria based on stochastic processes. ,4ml.qfmath. Star. 23, 193 212. Barnard G. A. [1953] Time intervals between accidents: A note on Maguire. Pearson and Wynn's paper. Biometrika, 40, 212 213. Birnbaum A. W. [1952] Numerical tabulation of the distribution of Kolmogorov's statistic for finite sample size. J. Am. Stat. Ass. 47, 425-441. Durbin J. and Knott M. [1972] Components of Cramer-von Mises statistics. J. Ro). Star. Soc. Series B. 34, 290 307. Goldfeld S. M. and Quandt R. E. [ 1972] Xonlinear Methods in Ecommwtrics. North Holland Publishing. Hubbard D. G. [1973] The Sk£jacker. Collier Books. Kendall M. G. and Stuart A. [1961] The Adraneed Theory Statistics. Vol. 2. Harper. Miller R. E. and Quandt R. E. [1967] An anahsis of the randomness of air accidents. J. Roy. Aer. Soc. 71, 23 27. Parzen E. [1962] Stochastic Processes. Holden Day. Quandt R. E. [1958] The estimation of the parameters of a linear regression system obeying two differentregimcs. J. Am. Star. Ass., 53, 873 880. Abstract--The paper examines hijackings of U.S. registered aircraft since 1968. After some summary characterizations of the series of hijackings, the paper tests the null hypothesis that the) represent a homogeneous Poisson process. Using the Durbin Knott decomposition of the Cram6r yon Mises statistic, this hypothesis is rejected. The paper further investigates whether formulating the model as a nonhomogeneous Poisson process represents a significant improvement over the assumption of homogeneity. This question is answered in the affirmative.The major conclusion of the paper, of practical significance,is that hijackings are not independent of one another and security precautions may need to be intensified in periods following a hijacking. ROsum6--Ce texte examine le detournement d'avions am~ricains depuis 1968. Apr6s quelques caract6risations sommaires des s6ries de detournements, le texte met /~ l'essai l'hypoth6se nulle qu'ils repr6sentent un proc6d6 homogene de Poisson. Utilisant la d6composition Durbin-Knott de la statistique Bram6r-von-Mises cette hypoth/~seest rejet+e. Le texte 6tudie ensuite si la formulation
Some statistical characterizations of aircraft hijacking du mod/~le en tant que proc6d6 de Poisson non-homog+ne represente une amelioration notable sur la supposition d'homog6n6it6. Cette question est repondue par l'affirmative. La conclusion principale de cet ouvrage d'importance pratique est que les d6tournements ne sont pas independants l'un de l'autre et les mesures de securit6 devront 6tre intensifi6es dans les p6riodes suivant un d6tournement.
Zusammenfassung--Der Bericht untersucht Flugzeugentfiihrungen seit 1968 von Flugze~gen, die in den U.S. eingetragen waren. Nach einigen zusammenfassenden Charakterisierungen der Reihen der Entfiihrungen priift der Bericht die Nullhypothese, dab sie einen gleichartigen Poisson Vorgang darstellen. Mittels Zerlegung der Cram~r-von Mises Statistik nach Durbin-Knott wird die Hypothese verworfen. Der Bericht untersucht weiterhin, ob die Formulierung des Modells als ein nicht gleichartiger Poisson Vorgang eine bedeutende Verbesserung zu der Annahme der Gleichartigkeit darstellt. Diese Frage wird bejah6nd beantwortet. Der HauptschluB des Berichts von praktischer Bedeutung ist die Feststellung, dab Flugzeugentfiihrungen nicht voneinander unabhS.ngig sind, und dab SicherheitsmaBnahmen m6glicherweise in Zeitr~iumen nach einer Entfiihrung verst/irkt werden miissen.
123