Fat-Tailed Distributions and Levy Processes

Louis Mello Fat-Tailed Distributions and Lévy Processes The notion that natural disasters can be controlled is, of course, fa rcical; history is permeated w ith examples of countless failed attem pts at this pointless task; it is synonymous with trying to build a perpetual motion machine. Nonetheless, there are w ays to reduce their impact on human communities, particularly by looking away from the normal hypothesis. 8/16/2008 Louis Mello Page 2 Introduction By definition natural catastrophes are extreme events, i.e. , they are events that possess, at times, an infinitesimal likel ihood of occurrence. In statistical te rms this means that they usually occur on the tails of a probability distribution. This , in t urn, elevates the importance of a correct measurement of the distribution’s tails , not to mention the accurate identification of the distribution to which the dat a observed from nature belong s. When we refer to extreme events we are, in fact, speaking of their placement in time, or, the fact that they occur extremely infrequently . As a result, the need to be able to predict such epi sodes with proper precision is of overwhelm ing importance to human communities that live under the constant threat of a major natural disaster such as the t sunam is and hurricanes that have shaken the world over the past few years. Obviously, we cannot con trol t hese events; however, we can better prepare ourselves for them through the d iscriminating use of f at-tailed statistical methods which come under broad scop e of Lév y processes. For m any reasons that a re w idely known, the scienti fic comm unity has com e to rely over- abundantly on t he Gaussian or Normal distribution. Ease of com putation and a vast array of analyses are just the more self-evident reasons. The computer and the ability to generate rapid simulations have helped bridge the gap b etween the analytica l and the exper im ental. Also notewor thy i s t he g reater presence o f more adv anced studies of these complex proces ses, which result from the use of m odern technology . It is clear, regardless of how enamored one is with the Gaussian distribution and i t s attendant ease of use, limitat ions notwith standing, that there is an urgent necessity to explore the no tion that othe r processes m ay well be bette r suited for de termining pro bability est imates for ext rem e events. This m onograph does not venture experime ntation due t o the autho r’s limited experience with natural disaster s. However , t he mathem atics show that there is a clear and proven way t o obtain the parameters tha t would allow one to proceed w ith such expe rimentation. It is in t his spirit, as well as in the hope that other scientists will take on the task of per form ing detailed testing on na tural p henomena, that this non-trivial recom mendation i s made. Louis Mello Page 3 Stochastic Lév y Processes – A Basic Def in ition Let us define a probability spa ce as   ,, P  F where  is the sam ple s pace of all conceivab le results, F is a algeb ra   of t he subsets of  and P is the positive measure of t otal mass 1 on   ,  F , which is essentially a probability. A stochastic process i s t hen defined as a related group of random variables     ,0 X t t  defined on   ,, P  F and taking values from a space   , E E . We will, f rom now on, refer to E as a state spac e. It is now easy to se e that every   Xt is represented by a corresponding metric mapping   F , E from  to E and, hence, is rea lly nothing m ore than an aleatory observ ation made on E at some t . We will also allow that E is a Euclid ean space . d  Let, for now, d E   . A stochastic Lévy process     ,0 X X t t  must satisfy: 1. X possesses independent and sta tionary increm ents, 2.   0 0 1 PX    3. X is a continu ous process, h ence, for al l 0 a  and for all 0 s  ,       lim | | 0 ts P X t X s a     . The Lévy- Khintchine equation pe rmits the g eneralizati on of a m apping : d t    which yields:         d iu X t iu y tt t e e p dy      E  (1.1) Where t p is the distribut ion of   Xt , i.e.,   1 t p P X t    and E is the mathematical expectation. t  must be continuous and positive definite. We rem ind the reader: Bochner’s theorem states that any and al l continuous positiv e definite mapping s from to d  are Fourier transform s of finite m easures on . d  Then, given axiom 1, we can state that each   Xt is infinitely divisible. For each n   there exists a probability m easure , o n d tn p  with characteristic function , tn  such that       , for each . n d t t n u u u    This l eads us to the fundamental t heorem of infinitely divisible probability measures . T he Lévy- Khintchine theor em can be w ritten: Louis Mello Page 4 Theorem 1. If     ,0 X X t t  is a Lévy process , then     tu t ue    for each 0, d tu   where         1 0 1 11 2 d iu y y u ib u u a u e iu y y dy                 (1.2) for some d b   , a non-negative d efinite symmetr ic dd  matrix a and a Borel mea sure  on   0 d   for which       2 0 1. d y dy        Conversely, g iven a mapping if t he form (1.2) it i s always possibl e to construct a Lé vy process for wh ich     . tu t ue    We call attention to the fact that the mapping : d    is, in fact, t he characteristic exponent  of X . It is conditionally positive definite in that   1 ,1 0 , f or all , , , n i j i j n ij c c u u n c c           with 1 0. n i i c    Finding  A strictly stable Lévy process is one f or which a change in the time scale will have a si milar effect on the spatial scale. Definition 1. Let  be an i nfinit ely divisible probability measure on d  . We will consider it stable if, for any 0 a  there are 0 b  and d c   such that     , ˆˆ a i c z z bz e   (1.3) and we will cons ider it strictly sta ble if, for any 0 a  there is 0 b  such that     ˆˆ a z bz   (1.4) A semi-stable process is defined as some 0 a  where 1 a  and there are 0 b  and d c   satisfying (1.3) . We will cal l it strictly se mi-stable if, for some 0 a  with 1 a  , there is 0 b  satisfying (1.4) . Louis Mello Page 5 Definition 2. Let   :0 t Xt  be a Lévy process on d  . It is called a stable, stri ctly stable, semi-s table or strictly semi-stable proce ss if the distribution of t X at 1 t  is respectively, stable, strictly stable, sem i- stable or strictly s emi-stable. Example 1. I f  is Gaussian on d  , then   , / 2 , ˆ z Az i z ze     and  is stable, as it satisfies (1.3) with 11 22 and . b a c a a         Thus, i f  is Gaussian with a mean = 0, then i t is strictly stab le. If  is Cauchy on d  with parameter c > 0 and d    , then   , ˆ c z i z ze     and  is strictly stable, sa tisfying (1.4) with . ba  If 0 n n bx n b         (1.5) with b > 1, 02   , and   0 \ 0 , d x   then     2 1 x dx      and the i nfinitely divisibl e distribution with generating triplet   0 , , 0  is semi-stable wit h ab   in (1.3), but it is not stable. The semi-stability arises from       0 , 00 ˆ exp 1 , 1 n i z b x n n n D z b e i z b x b x             Definition 3. Let   :0 t Xt  be a stochastic process on d  . We shall call it self-similar if, for any a > 0, there is b > 0 such that     d : 0 : 0 at t X t bX t    (1.6) or it is called latu sensu self -similar if, for any a > 0, there are b > 0 and a function   ct from   0,  to d  such that       d : 0 : 0 . at t X t bX c t t     (1.7) Louis Mello Page 6 It is called semi self- si milar if, f or some a > 0 with 1 a  , there i s b > 0 satisfying(1.6). It is called latu sensu semi self-si milar if, for some a > 0 with 1 a  , there are b > 0 and a function   ct satisfying(1.7). The link between th ese concept s and stable pro cesses, etc. can be form ed as follo ws: Proposition 1. L et   :0 t Xt  be a Lévy process on d  . Then it is s elf-similar, latu sensu s elf-similar, semi self-similar or l atu sensu semi self-similar if and only if it is, re spectivel y, strictly stable, stable, strictly semi-stab le or semi-stable. P R O O F : Let 1 . X P   Let us fu rther assum e that   t X is latu sen su semi-stab le. By definition there is a positive 1 a  for which (1.3)holds with some b and c. The Lévy processes   at X and   t bX tc  correspond to the distribu tions with cha racteris tic functions   ˆ a z  and   , ˆ i c z bz e  , respective ly. Hence, i f   t X and   ' t X are Lévy processes in law on d  such that ' 1 1 X X PP  , then   t X and   ' t X are i dentical in law and we can write     d at t X b X tc  (1.8) and hence   t X is latu sensu semi self-similar. Conversely, i f   t X is latu sensu semi self-similar, then it follows from (1.7)that   1 1 a X bX c PP   , i.e.       1, ˆˆ a i c z z bz e   and   t X is semi stable. Simultaneously, we h ave sh own that     1 c t tc  .The other assertions ar e proved simi larly. Q.E.D. Theorem 2. Let   :0 t Xt  be a broad-sense self- si milar or semi self-similar stochastically continuous, non-trivial process on d  with 0 const X  a.s. Denote by  the set of all a > 0 such that there are b > 0 and   ct satisfying       d : 0 : 0 at t X t bX c t t     . Then: i. There is H > 0 such that, for ev ery , H a b a    ii. The set   1,    is non empty. Le t 0 a be the infimum o f this set. If     00 1 , the n = : and n t a a n X     is not broad-sense s elf-similar. If     0 1 , t h en 0 , and t aX     is broad-sense sel f-similar. Louis Mello Page 7 Consider a self-sim ilar function   yt . The difference between the maximum and m inimum values of y in a time interval t  defines a range for that interval,   . Rt  Given that y is self- similar, the ensemble- averaged value of R will sca le with t  . We can write:   H R t c t    (1.9) where c and H a re constants; H defines t he Hurst exponent 1 . For data t hat are o nly approximately self- similar, we use this relation to check their proximit y to self -sim ilarity, and also to obtain an effective value for H . We proceed as follows: create a mov ing window t  one point at a ti me through the raw data; an array of values   Rt  is created from which the mean R is found, t hus reducing the effects of uneven sampling. This is repeated for a range of t  within the length of the data set. A plot of   log against log R t t  will reveal any deviations from se lf -sim ilarity, while the slope will yield the best estim ate of H . Linear reg ression is u tilized to calcula te the 95% conf idence interv al for H . Trivially, a function that is exactly constant over time has H = 0. At the other extreme, H = 1 indicates a function whose range increases l inearly wit h time (for a positive c in (1.9). Interm ediate values of H are g enerated by f ractal functions: Random Gaussian noise possesse s 0 .2 H  while Gaussian Random Walks (whose next value in tim e is 1 t y    , where  is a random Gaussian increment) will yield 0 .5 H  . The value of H does not uniquely establish correlation; however, uncorrelated series may present signific ant probabil ities of observing greater va lues as the time-scale increases. We now define 1 H   as a di mension in the probability space defined for the Generalized Lévy Characteristic Fun ction (GLCF):     lo g 1 tan 2 t f t i t t i t                (1.10) where: t = a constant of in tegration .  = the location par ameter of the mean.  = is the scale par ameter to adjust differenc es in tim e frequency of dat a.  = is the measure o f skewness w ith  ranging between - 1 and +1.  = the kurtosis and the fatness of the tails. Only when  = 2 does the distribution becom e equal to the Gaussian distr ibution. 1 Hurst, H. E. “ The Long Term Storage of Reservoirs ”, Transactions of the American Society of Engineers, 116, 1951 Louis Mello Page 8 Proof of the Rela tionship between b D and H The fractal dim ension is rel ated to the Hurs t exponent by means of: 2 B DH  (1.11) Proof: Reca ll that to calculate the box counting dimension B D of a f ractal ob ject, w e cov er the ob ject with N boxes of s ide length  and then com pute B D using: * log ( ) lo g ( ) 1 log B NV D       (1.12) Where N = the number of boxes of length  used to cover a line segment and * V  is the minimum count of one dimension al boxes n eeded to cov er said line segm ent. For a fractional Brownian motion trace, suppose that we isolate a time series of T time steps that sp ans 1 unit of time. During each time step of length 1 T the averag e vertical range of the f unction is 1 H T    due to the scaling pr opertie s of any self-similar process. In order to cover the plot of t he f unction during a single tim e step, a rec tangle of width 1 T and heig ht 1 H T    is required. The area of this rect angle is 1 1 H T     , so the number of squa res wit h side leng th 1 T needed to cover it is 1 1 H T     . For all T of the t ime steps, the total number of squares needed to cover the plot of the function is 2 1 H T     . If we let N = 2 1 H T     and 1 T   , then the box counting dimension is given by:   2 1 log 2 log H B T DH T        (1.13) Louis Mello Page 9 The time series s pans 1 unit of tim e, so * 11 B D V  . This relationship between the fractal dimension and the Hurst exponent aligns perfectly with the notion of fractal dimension as a measure of the roughness of an object. As H increases and the fractional Brownian motion displays greater persistence, the plot of the function becomes smoother and B D decreases accordingly . Conversely, as H decr eases and the fractional Brownian motion is more anti - persistent, the p lot of the fu nction becom es more ja gged and B D increases. The literature demonstrates the existence of a parametric function in which each coordinate’s function i s a Brownian motion trace. Similarly , we can construct fractional Brownian motion paths from traces. T o calculate the box counting dimension of a fractional Brownian motion path with two coordinates, we examine a section of the path that results from T time st eps spanning 1 unit of time. During each tim e step of length 1 T , each of the two tra ces has range 1 H T    , so we can cov er the path during a single time step with a square of side length 1 H T    . For all T time steps, the path requires T such squares to cover it. Letting N = T and 1 H T      , we have:   log 1 1 log B H T D H T         (1.14) Since H can fall between 0 and 1, 1 H can assume values greate r than 2. However, the fractal dimension of the pa th ca nnot exceed its Eu clidean dimension E D , so w e modify the box counting dimension to be: 1 m in , BE DD H     (1.15) Thus, the fractal di mensio n of regular Brownian motion and anti-persistent fractional Brownian motion paths is 2, a nd the fractal dim ension of per sistent paths lies b etween 1 and 2. Louis Mello Page 10 Conclusion The above demonstration s provide us with the necessary mathem atical support to state , unequivocally , t hat the f ractal dimension of a time series allows us to estimate the cha racteristic exponent of a GLCF through its relationship with the Hurst exponent. One could, then, estimate either H or b D and thus obtain  . Comparison of Tai l Prob abilities between a S tandard N ormal and a Standar dized Stable-Lévy Distribution (Tabl e 1) x P[Norm al] P[Levy] Ratio Levy/Norm al –10 7. 619853E-24 9. 861133E-05 1 2 ,9 4 1 ,36 8 ,74 2 ,5 2 2 ,80 0 ,00 0 –9 1. 128588E-19 0. 000192105 1 ,70 2 ,17 4 ,7 8 1 ,95 5 ,01 0 –8 6. 220961E-16 0. 000363951 5 8 5 ,03 9 ,30 8 ,5 3 3 –7 1. 280000E-12 0. 000670638 52 3 ,93 5 ,7 6 3 –6 9. 865900E-10 0. 001202084 1 ,21 8 ,42 3 –5 2. 866516E-07 0. 00209627 7 ,31 3 –4 3. 167124E-05 0. 003557187 112 –3 0. 001349898 0. 006697816 4 .96 1 7 1 9 9 0 –2 0. 022750132 0. 020136505 0 .88 5 1 1 5 9 5 –1 0. 158655254 0. 117935844 0 .74 3 3 4 6 6 0 The table above p rovides ample proof of the massive m agnitude o f difference between the Gaussian a nd fat - tailed distributions. This mea ns that the correct specification of the under lying distrib ution at the onset of the m odeling effort is essential to the correct estim ation of the intrinsic pro babilities. An example of a curve w ith "fat tails" is the Cauchy distribution (a member of the stable -Pa reto-Lévy class), shown ab ove. In the ca se of the norm al curve (on the lef t), the tails app roach zero a t -3.5 and 3.5 standard deviations. In the case o f the Cauchy distribution the cu rve is s till not even close to zero at - 5 and 5 standard deviations. This illustrate s the higher prob abilit y at the tail ends. Louis Mello Page 11 The si gnificanc e of fitting the appropriate distributio n to the problem at hand cannot be over - emphasized. As can be seen from Table 1, an error in the estimate of the probability associated with an extreme event can be colossal. One of the ma jor drawbacks with respect to fat-tailed dis tributions has always been t he absence of a closed -form for parameterization. Also, depending on the value of  there may also be the issue of infini te variance. There are several software i mplem entations of recently defined algorithm s based on the work of: Zolotarev (1986), Uchaikin and Zolotarev (1999), Christoph and Wolf (1992), Samorodnitsky and T aqqu (1994), Janicki and Weron (1994), and Nikias and Shao (1995) as well as th e related topic of modeling with the extremes of data and heav y tai led d istributions which is d iscussed in Embrechts e t al. (1997), Adler et al. ( 1998), and in Reiss and Thomas (2001). These implementations allow t he researcher to fit time series data to fat tailed stable Lévy class distri butions yield ing t he parameters and the necessary modules for g enerating pse udo-random numbers from the obse rved distribu tion. When the wel l-being and/or sa fety of who le populations lies i n the ba lance, there should be no parsimony of effort in determining the best possible solution to a problem, as catastrophes li ke Katrina have so poignantly poi nted out. There is no field of human endeavor that cannot benefi t from st udies predicated on the principles outlin ed in this paper, either directly or indirectly . It is in the i nterest of scientists and policym akers alike t o pursue this course of research i n order to better understand and integrate with th e world wh ich we all share. Louis Mello Page 12 References: 1. Bert oin, J. “Lév y Processes”, Cam bridge Tracts in Mathematics , C ambridge Univ ersity Press, (1996 ). 2. Barndorff-Nielsen, O.E. et al , “ Lévy Processes: Theory and Applic ations ,” Birkhäuse r, 2001. 3. Hurst , H.E. “Long - term Storage of Reservoirs, ” Transactions of the American So ciety of Civi l Engineers , (1951), 1 16. 4. J ones, R. H., “Mu ltivariate Autoregress ion Estimation Using Residual s,” Applied Time Series Analysis , Edited by D. F. Findley, N ew York: Acad emic Press, ( 1978), 139-162. 5. Malamud, B. D., Morein, G., and Turcotte, D . L., “ Forest Fires: An Exam ple of Self-org anized Critical Behav ior ,” Science , (1998 ), 281, 1840- 1842. 6. Shannon, C.E., and Weaver, W., The Mathematical T heory o f Communication, (U rbana: Un iversity of Illinois, 1963). 7. Steven M. Kay, Modern Spectral Estimation , (New York: Prentice Hall, 1988), 153-270.