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Extreme Events: the Amazing Weibull Distribution

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Extreme Events: the Amazing Weibull Distribution Extreme events: the amazing Weibull distribution Jerzy Szulga Department of Mathematics and Statistics, Auburn University, AL 36849 Summary 1. Extreme probability distributions in eleven easy steps 2. Weibull distribution 3. The earthquakes: waiting for the Big One http://www.math.auburn.edu/~szulgje/talks/csu.pdf 1 There are two competing yet complementary avenues to develop mathemat- ical models of real life phenomena. The ¯rst approach refers to the laws of science, physics, logic, and deduction. In the second \black box approach", in spite of ungrasped causality one can still seek and ¯nd patterns in the data. In time, the discovered patterns will become rules, then laws, giving rise to a theory. Next, its laws will be either con¯rmed or contradicted by new observations, and the process will continue. We live in the extreme world, struggling with extreme forces of nature, atop of which come extremities of our own. Sometimes, their partial understanding is possible. We will focus on a probabilistic model that emerged in 1930s. The model may apply to a structure under the action of many random forces. Perhaps, the forces are too complicated to be described or simply unknown. The breaking point is reached when the maximum of these forces exceeds a certain level. Surprisingly, it can be proven that the behavior of the maximum is no longer arbitrary, for it will be constrained by just three types of probability distributions. One of the cases encompasses the Weibull distribution. On the other hand, the Weibull distribution is detected in many natural processes, time and time again, although the causes and reasons behind its presence remain foggy. I will present a mathematical outline of the extreme probabilities, and explain the role and properties of the Weibull distribution. Worldwide earthquake data will illustrate its connection to the real world. 2 From chaos to order A typical random noise 4 3 2 1 0 −1 −2 −3 0 100 200 300 400 500 600 700 800 900 1000 Figure 1: Independent random variables. 2 χ (6) binomial, sixes in 100 rolls of a die 25 30 20 15 20 10 10 5 0 0 0 200 400 600 800 1000 0 200 400 600 800 1000 geometric, waiting for the first six on a die Cauchy 200 30 150 20 100 50 10 0 0 −50 0 200 400 600 800 1000 0 200 400 600 800 1000 Figure 2: At the top of the chaotic behavior, maxima behave rather orderly. 3 History 1928 Ronald Fisher and Leonard Tippet distinguished only three types of limit distributions of the maxima of a sequence of i.i.d. random variables, subject to shift and scale. Two random variables X and Y are of the same type, if, within some a±ne transformation, they have the same distribution X =D aY + b; a > 0 1943 Boris Gnedenko proves that, if µ ¶ P max Xk · anx + bn ! P(Y · x); n ! 1; (1) k·n at the points of continuity of the limit distribution for some translation con- stants bn and scale constants an > 0, then the three types (excluding the trivial constant) are the distributions of the following random variables: ¯ ª¯, negative Weibull: Y = ¡V , ¡¯ ©¯, Fr¶echet, inverse Weibull: Y = V (2) ¤, Gumbel, log-Weibull: Y = ln V , where ¯ > 0 and V is an exponential random variable with the unit intensity. Gumbel's variable is but a cluster point of the Weibull variables: V ¯ ¡ 1 ln V = lim : ¯!0 ¯ 4 Weibull variables Frechet variables Gumbel variables 100 20 15 80 10 15 60 5 10 40 0 5 20 −5 0 0 −10 0 1 0 1 0 1 Weibull densities Frechet densities Gumbel densities 0 0 0 Figure 3: Three types of growth, three types of probability densities. 5 Karamata theory, the bliss and curse 1933 Jovan Karamata's concepts of variations of functions has had a great impact on the probability theory (see Feller's book). Consider a function F (t) in a vicinity of a point a which could be §1. The condition F (ct) ¯ lim = c (F 2 Va(¯) in short) t!a F (t) describes, accordingly: ¯ = 0: slow variation (like a logarithm); ¯nite ¯ 6= 0 regular variation (like a power); j¯j = 1 rapid variation (like an exponential). By simple algebra: 1 2 V (¡¯) () F (t) 2 V (¯) () F (1=s) 2 V (¯¡1) F (t) 1 1 0 The concepts are simple but many techniques are hard, cumbersome, and deep. One needs a great deal of real and complex analysis, function theory, measure theory, probability, etc. A typical presentation of the extreme value theory begins with Karamata. 6 Main issues 1. The proof of the extreme types theorem; 2. The choice of stabilizing constants; 3. The domains of attraction: which distributions will be attracted to the speci¯c type? The language: A. in terms of quantiles or inverse tails; B. in terms of tails or survival functions. Claim: \A" is easy. \B" is harder. The connection \A"$\B" is the hardest. We will outline 1{2{3, using the language\A". No external reference is needed, except for basic algebra, calculus, and ele- mentary probability and analysis. 7 All random variables are transforms of a uniform random variable Let U have the uniform distribution on [0; 1]. 1=¯ Example: V = ¡ ln U is exponential, W¯ = V is Weibull, etc. For the survival function G(x) = P(X > x), its right-continuous inverse: G¡1(u) =def inf f a : G(a) · u g =) X =D G¡1(U) Example. For a Weibull random variable X = W = V 1=¯; ¯ > 0, © ª G(w) = exp ¡w¯ ; w ¸ 0;G¡1(x) = (¡ ln x)1=¯; 0 < x · 1: 90 1 G(x) 0.5 G−1(x) 40 0 0 20 40 90 20 0 0 0.5 1 Figure 4: The survival (tail) function, and its inverse (quantile) function. Convergence of distributions D Xn ! X: P(Xn > x) ! P(X > x) at the continuity points of the limit distribution. Now: D ¡1 ¡1 ¡1 Xn ! X () Gn (u) ! G (u) at the continuity points of G : 8 Order statistics (k) Sort a ¯nite sequence (xk) to the ascending x(k) % or descending x &: (n) (1) x(1) = x = min xk; x = x(n) = max xk: k k For a sequence U1;:::;Un of independent uniform variables on [0; 1]: D 1=n (n) D D 1=n D ¡V=n U(n) = U ;U = 1 ¡ U(n) = 1 ¡ U = 1 ¡ e : (3) Indeed: µ ¶ h in ¡ ¢ n 1=n P U(n) · x = P max Uk · x = P(U · x) = x = P(U · x): k·n So, for X with the tail G(x) and its independent copies (Xk), we have ¡ ¢ ¡ ¢ (n) D ¡1 D ¡1 1=n X = G U(n) = G U ; ¡ ¢ ¡ ¢ (4) D ¡1 (n) D ¡1 ¡V=n X(n) = G U = G 1 ¡ e ; (U - uniform, V - exponential). The tempting simple replacement µ ¶ V V U (n) =D ; hence X =D G¡1 ; n (n) n is illegal! We have only a seemingly useless limit property: µ ¶ ³ ´ ³ ´n (n) x x x ¡x P U > = P min Uk > = 1 ¡ ! e = P(V > x): n k·n n n 9 Functions instead of probabilities We will examine the function g(x) = G¡1(1 ¡ e¡x) when x ! 0. Translation: Nondegenerate limit distributions in the the Extreme Values Theorem appear as distributions of Á(V ), where g(v=n) ¡ b Á(v) = lim n : (5) n!0 an De¯nition. Quantities © and Á are of the same type if © = Á ® + ¯; ® > 0; ¯; Step 1. \Types Lemma". The type is preserved in the limit. Let Án be a sequence of real functions on a set W that has at least two points and (an; bn) and (An;Bn) be two pairs of numerical sequences such that an > 0;An > 0. Suppose that there are non-constant functions on W , Á (v) ¡ b Á (v) ¡ A Á(v) = lim n n ; ©(v) = lim n n : n an n Bn Then a b ¡ B there exist numbers ® = lim n ; ¯ = lim n n (6) n An n An Conversely, if (6) holds and either Á or © exists, so exists the other. Under both assumptions, ©(v) = Á(v)® + ¯; v 2 W: Step 2. Let Á(v) exist everywhere, be right-continuous and non-constant. Then for every u > 0 there exist numbers ¯u; ®u > 0 such that Á(uv) = ®u Á(v) + ¯u; v > 0: 10 A±ne stability The multiplicative vs additive notation: f(x) = g(e¡x); x = ¡ ln u: (7) Let T ½ R be unbounded from above. Let f : (0; 1) ! R be nonconstant & nondecreasing. f is a±nely stable if for some functions a(t) > 0 and b(t) the limit exists f(x + y + t) ¡ b(t) L(a;b)(x) = lim (8) T 3t!1;y!1 a(y + t) The function Á(e¡x) (5) is a±nely stable by Step 2. © ª 1 Without this step, we would have only a poor set T = n . The limit along such T would yield a much weaker notion of a±ne stability, insu±cient for our purposes. For example, the integer value function f(x) = [x] would be stable in this weak sense for T = f n g, but only with bn = n; an = 1. 11 Immediate properties Step 3. All limit functions L(a;b) are of the same type. (Step 1, Types Lemma.) Step 4. There is a main form L = L(a0;b0) such that L(0) = 0;L(1) = 1: (use a±ne transformations: x0 7! 0; x1 7! 1;L(0) 7! 0;L(1) 7! 1) Step 5.
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