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 mathematical models of real life phenomena. The first 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 find 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 confirmed 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

−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 aﬃne 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) k≤n at the points of continuity of the limit distribution for some translation constants bn and scale constants an > 0, then the three types (excluding the trivial constant) are the distributions of the following random variables:

β Ψβ, negative Weibull: Y = −V , −β Φβ, Fr´echet, inverse Weibull: Y = V (2) Λ, Gumbel, log-Weibull: Y = ln V , where β > 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 β − 1 ln V = lim . β→0 β

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 ±∞. The condition

F (ct) β lim = c (F ∈ Va(β) in short) t→a F (t) describes, accordingly:

β = 0: slow variation (like a logarithm); ﬁnite β 6= 0 regular variation (like a power); |β| = ∞ rapid variation (like an exponential).

By simple algebra:

1 ∈ V (−β) ⇐⇒ F (t) ∈ V (β) ⇐⇒ F (1/s) ∈ V (β−1) F (t) ∞ ∞ 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 speciﬁc 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 elementary probability and analysis.

7 All random variables are transforms of a uniform random variable

Let U have the uniform distribution on [0, 1].

1/β Example: V = − ln U is exponential, Wβ = V is Weibull, etc.

For the survival function G(x) = P(X > x), its right-continuous inverse:

G−1(u) =def inf { a : G(a) ≤ u } =⇒ X =D G−1(U)

Example. For a Weibull random variable X = W = V 1/β, β > 0, © ª G(w) = exp −wβ , w ≥ 0,G−1(x) = (− ln x)1/β, 0 < x ≤ 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 ﬁnite 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 φ(V ), where g(v/n) − b φ(v) = lim n . (5) n→0 an

Deﬁnition. Quantities Φ and φ are of the same type if

Φ = φ α + β, α > 0, β,

Step 1. “Types Lemma”. The type is preserved in the limit.

Let φn 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 , φ (v) − b φ (v) − A φ(v) = lim n n , Φ(v) = lim n n . n an n Bn Then a b − B there exist numbers α = lim n , β = lim n n (6) n An n An Conversely, if (6) holds and either φ or Φ exists, so exists the other. Under both assumptions, Φ(v) = φ(v)α + β, v ∈ W.

Step 2. Let φ(v) exist everywhere, be right-continuous and non-constant.

Then for every u > 0 there exist numbers βu, αu > 0 such that

φ(uv) = αu φ(v) + βu, v > 0.

10 Aﬃne stability

The multiplicative vs additive notation:

f(x) = g(e−x), x = − ln u. (7)

Let T ⊂ R be unbounded from above.

Let f : (0, ∞) → R be nonconstant & nondecreasing. f is aﬃnely 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→∞,y→∞ a(y + t)

The function φ(e−x) (5) is aﬃnely stable by Step 2.

For example, the integer value function f(x) = [x] would be stable in this weak sense for T = { n }, 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 aﬃne transformations: x0 7→ 0, x1 7→ 1,L(0) 7→ 0,L(1) 7→ 1)

Step 5. The function is self-stabilizing, e.g., if L(c) 6= 0

B(t) = f(t),Ac(t) = f(c + t) − f(t).

(HS Algebra)

(a,b) Then, write simply Lc instead of L and check that L(u) L = L, L (0) = 0,L (c) = 1, and L (u) = . (9) 1 c c c L(c)

12 When f is aﬃnely stable

Step 6. The key equation.

³ ´ Lc(u + v) = Lc(u) Lc(v + c) − Lc(v) + Lc(v), c > c0. (10)

Proof. Let t → ∞: f(u + v + t) − f(t) f(c + t) − f(t) · ¸ f(u + v + t) − f(v + t) f(v + c + t) − f(t) f(v + t) − f(t) = − f(c + v + t) − f(v + t) f(c + t) − f(t) f(c + t) − f(t) f(v + t) − f(t) + . f(c + t) − f(t)

Step 7. Lc is right-continuous. (Basic calculus)

Step 8. L has no intervals of constancy. “0” and “1” in Step 5 are unique. (Elementary analysis.)

Step 9. Let qc = L(1 + c) − L(c) > 0, c > 0. Then, Lc(u + c) = 1 + qc Lc(u).

(Proof. Interchanging u and v in (10), ³ ´ ³ ´ Lc(u) Lc(v + 1) − Lc(v) + Lc(v) = Lc(v) Lc(u + c) − Lc(u) + Lc(u).

This yields a function whose values at arbitrary u and v are equal. So it must be constant:

Lc(v + c) − Lc(v) − 1 Lc(u + c) − Lc(u) − 1 = = qc − 1, Lc(v) Lc(u) for some constant qc. But qc = L(1 + c) − L(c), choosing u = 1 and using (9).)

13 Almost there...

c Step 10. If qc 6= 1, denoting q = q1, then qc = q . That is,

L(1 + c) − L(c) = qc.

Proof. Iterating the formula in Step 9,

L (u + nc) = (1 + q + ··· + qn−1) + qnL (u) c µ c ¶ c c n 1 1 (11) = qc + Lc(u) − . qc − 1 qc − 1

For a rational c, let Nc = { n : nc are integers }. Whence nc n q − 1 qc − 1 = L(nc) = L(c)Lc(nc) = L(c) . q − 1 qc − 1

By Types Lemma , the functions Lc, L, 1 are linearly dependent, so are the discrete functions

n n n nc c qc , q , 1 on Nc. This is possible only when qc = q , i.e., qc = q . Since qc is right-continuous, the equality will hold for all real c.

Summary

1. L should be either linear or exponential;

2. The Extreme Values Theorem follows by returning to the multiplicative notation. ¡ ¢ f(x) 7→ g(x) = G−1 1 − e−x

q 6= 1: the Weibull distributions q = 1: the Gumbel distribution

3. An aﬃne function f is self-stabilizing: translation B(t) = f(t) scale A(t) = f(t + c) − f(t)

14 Step 11.

Theorem 1

Let c > 0 and f be aﬃnely, where T = { tn }, tn % ∞ and the limit function

(8) satisﬁes L(0) = 0,L(1) = 1. Then, there exists q > 0 such that qc = L(1 + c) − L(c) = qc for every c > 0. Further,

(A) If q = 1, then L(u) = u;

(B) if q 6= 1, then there is λ > 0 such that

eλu − 1 (B ) for q > 1, L(u) = . + eλ − 1 1 − e−λu (B ) for q < 1, L(u) = − 1 − e−λ ( q < 1 if and only if f is bounded from above).

15 Proof. (A): Let qc = q = 1. The formula in Step 9 yields Lc(cn) = n, and from Step 5 we infer this implies L(cn) = L(c)n. With c = 1/n, we obtain L(1/n) = 1/n. With c = 1/k, we obtain L(n/k) = n/k. Thus L(u) = u, since L is right-continuous.

(B): Let q 6= 1 and c be rational. Let Nc = { n : nc are integers }. We rewrite (11) with c u = 0, once for Lc using qc = q and (A), and then again for L: qcnL(c) L(c) qcn 1 − = L(nc) = − . qc − 1 qc − 1 q − 1 q − 1

When q < 1, we let n → ∞, n ∈ Nc. When q > 1, we subtract the expressions for L(nc). In both cases L(c) 1 qc − 1 = , i.e., L(c) = , qc − 1 q − 1 q − 1 which extends to real c, by the right continuity (i). To ﬁnish, we put λ = ln q.

To prove the second statement in (B−), let qc = L(1 + c) − L(c) < δ < 1. Hence, there exists an integer N such that for every t ≥ N, ³ ´ f(1 + c + t) − f(c + t) < δ f(1 + t) − f(t) .

Let c = 1, dn = f(1 + tn) − f(tn). Since dn+1 < δ dn for n ≥ N, the telescopic series P n(f(1 + tn) − f(tn)) converges, i.e., f is bounded from above. That f is unbounded when qc > K > 1 follows analogously.

16 Corollaries.

(C1) When is a given f attracted to a speciﬁc extreme type? f is aﬃnely stable if and only if f(x + t) − f(x + a) L(x) − L(a) lim = , (12) T f(y + t) − f(y + b) L(y) − L(b) where at least three of four nonnegative parameters x, y, a, b are distinct, and L(u) = u or L(u) = eαu.

Switching from the additive to the multiplicative notation, and denoting s = sn = 1/n, we translate the above as follows:

(C10) X →aﬀ Y , where Y is one of the nontrivial extreme variables, if and only if g(x) = G−1(1 − e−x) satisﬁes g(vs) − g(ps) Φ(v) − Φ(p) lim = , (13) s→0 g(us) − g(qs) Φ(u) − Φ(q) where Φ(v) = ln v or Φ(v) = vβ, β 6= 0.

17 Streamlining

The illegal limit substitution of the argument in the stabilizing functions 1 − e−ps 1 − e−ps = · s 7→ p · s s can now be justiﬁed. The Extreme Values Theorem itself authorizes the replacement of the cumbersome G−1 (1 − e−x) by the simple G−1(x).

(C2) The choice of simple quantile stabilizers.

If X →aﬀ Y , where Y is one of the nontrivial extreme variables, then g(sy) − G−1(sp) Φ(y) − Φ(p) lim = (14) s→0 G−1(sx) − G−1(sq) Φ(x) − Φ(q)

Proof. It suﬃces to take p = q, and diﬀerent distinct x and y. For δ > 0 consider s < δ and p0, x0 such that

0 0 1 − e−ps < ps < 1 − e−p s < p0s, 1 − e−xs < xs < 1 − e−x s < x0s.

Then, use the monotonicity and (right-)continuity.

By the same token: (C3) The attraction to an extreme type is described:

X →aﬀ Y , where Y is one of the nontrivial extreme variables, if and only G−1(sy) − G−1(sp) Φ(y) − Φ(p) lim = . (15) s→0 G−1(sx) − G−1(sq) Φ(x) − Φ(q) where at least three parameters are distinct, and Φ(x) = xβ or Φ(x) = ln x.

18 Example: Weibull distribution

For a Weibull random variable X = W = V 1/β with the shape parameter β: © ª G(w) = exp −wβ , w ≥ 0,G−1(x) = (− ln x)1/β, 0 < x ≤ 1.

The ﬁrst stabilizers are unpleasant, e.g., the translation is

−p/n 1/β bn = (− ln(1 − e )) .

Previously, the replacement of the minimum U (n) by 1 − U (n) was necessary to avoid the forbidden substitution V U (n) ∼D n

In contrast, the stabilization of the minimum follows trivially, and the minimum of Weibull variables is attracted to the Weibull distribution itself. µ ¶ − ln U 1/β n1/βW (n) =D n1/β =D W. n After (C2), the maximum of Weibull variables can be handled easily.

The Weibull distribution is attracted to the Gumbel extreme type.

Since (x − a)c − xc ∼ a c xc−1 as x → ∞,

W − (ln n)1/β (ln n − ln V )1/β − (ln n)1/β ln V ln ln U −1 (n) ∼D ∼D =D . (ln n)1/β−1 (ln n)1/β−1 β β

19 That was easy. So, what is diﬃcult?

The characterization condition in the Karamata language describes the variation of increments of inverse tails at 0:

(C4) Let X →aﬀ Y , where Y is one of the nontrivial extreme variables. Con- sider the scaled increment

φ(s) = G−1(sy) − G−1(sp)

(the choice of y, p is arbitrary). Then

β 1. Weibull: Φ(v) = v if and only if φ ∈ V0(β).

2. Gumbel: Φ(v) = ln v if and only if φ ∈ V0(0).

Statements simple to state but hard to prove, in general:

• replace “increments of a function” by the function itself;

−1 • G ∈ V0(−β) ⇐⇒ G ∈ V∞(−1/β);

−1 • M − G ∈ V0(β) ⇐⇒ G ∈ VM (1/β);

−1 −1 • For a non-decreasing F , F ∈ V∞(β) ⇐⇒ F ∈ V0(β )

20 Tails

Tails with distinct asymptotic structure are easy to classify.

1. Let G(x) = P(X > x) ≈ x−α as x → ∞. Then X →aﬀ Fr´echet.

Computations: G−1(x) ≈ x−1α as x → 0. Hence, putting β = 1/α and letting s → 0, (sy)−β − (sp)−β (y)−β − (p)−β ≈ , (sx)−β − (sq)−β (x)−β − (q)−β i.e., Φ(x) = x−β, pointing to the Fr´echet distribution.

2. Let X be bounded from above by a constant M. Let G(x) ≈ (M − x)α, as x → M. Then G−1(x) ≈ M − xβ with β = 1/α, and clearly Φ(x) = xβ, pointing to the Weibull distribution.

3. If G(x) ≈ e−xβ , X →aﬀ Gumbel (as Weibull X does).

4. Superlight tails such as G(x) ≈ e−ex or slowly decreasing ones such as G(x) ≈ (log x)−1 are not attracted to any of the three types. They may be attracted to the constant. Typically, there is no translation but there might be scaling stabilizers.

5. Gnedenko provided detailed characterizations beyond the most common situations described above.

21 Weibull Models

Waloddi Weibull, a material engineer, introduced the class of distributions, later named after him, in a 1939 paper on the strength of materials.

3.5 Weibull densities with various shape parameters β>0 3 β<1 2.5

2 β=1 1.5 β>1

0.5

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

β β Figure 5: f(x) = xβ−1e−(x/α) is the density of αV 1/β αβ

Various applications: strength of steel, fiber, concrete, laminate composites, ... fineness of particles in sprays, grinds, mixtures,... lasting of devices, inventories, and organisms, ... failures of components, software, sales, traffic flows, ... precipitation, ocean wave heights, floods, earthquakes, winds, temperature, ...

......

As the Weibull variable is a transformation of the exponential, new transformations such as the translation lead to a variety of multiparameter models.

22 Poltergeist

On 30 January 1986, I and my wife watched the “Poltergeist” movie. I had scary dreams that night. Next day we had lunch at home, a rented upper ﬂoor in an old two story house that was build by our landlord in 1920s. Suddenly glasses and plates trembled, jumped, and felt to the ﬂoor. Books were thrown from the shelves against the walls and the glass of pictures broke.

No, in spite of we thought ﬁrst it wasn’t a Poltergeist.

On 31 January 1986, at 11:46 EST, an earthquake of magnitude 5 on the Richter’s scale occurred about 40 km east of Cleveland, and about 17 km south of the Perry Nuclear Power Plant. The earthquake was felt in 11 states, in the District of Columbia and parts of Ontario, Canada. Thirteen aftershocks were detected as of 15 April, with six occurring within the ﬁrst 8 days.

(Bulletin of the Seismological Society of America; 1988; v. 78; pp. 188-217.)

23 Point process of events

Example. The sum of independent exponential r.vs. yields the Gamma r.v.:

Sn = V1 + ··· + Vn.

Sn’s interpreted as incoming signals or events induce the classical Poisson process, deﬁned as the time-dependent counter:

N(t) = # { n : Sn ≤ t }

The earthquakes as “signals” on the time axis serve often as an illustration of the Poisson process. Even in layman’s eyes, this is not the best example:

• A Poisson process has no clusters but earthquakes come in packs, pre- shocks, shocks, and aftershocks.

• The Poisson counts of events in disjoint time intervals are independent. In contrast, a today’s earthquake makes tomorrow’s one more likely.

Interevent times. Southern CA earthquakes, Me > 4, 1968−2006 150

100

0 0 200 400 600 800 1000 1200 1400 1600 1800 2000

Figure 6: Interevent times between almost 2000 Southern California earthquakes of magnitude ≥ 4, between 1968 and 2006. Clusters (low values) are clearly visible. Any period, any location, and cutting magnitude shows the same pattern.

24 Weibull interquake times

To account for clusters and correlation between disjoint intervals consider Weibull interevent times, W = αV 1/β:

Sn = W1 + ...Wn.

To verify the model, a sample of 2000 might be too small. With the help of Auburn applied math major, Kristin Seamon, 65,536 (the magic Microsoft number) moments of magnitude 4 or higher worldwide earthquakes between 1973 and 1986 were examined.

Earthquakes 1973−1986 90

−15

−30

−45

−60

−75

−90 −180 −150 −120 −90 −60 −30 0 30 60 90 120 150 180 The magnitudes: 4+ (cyan), 5+ (green), 6+ (red), 7+ (blue), 8+ (black o). The locations of earthquakes draw the boundaries of tectonic plates

Figure 7: Self explanatory

The two parameters, scale α and shape β were estimated from the sample using the “Method of Moments”. µ ¶ µ ¶ 1 2 E W = βΓ 1 + , E W 2 = β2Γ 1 + α α Both parameters are functions of E W and E W 2. In lieu of the ﬁrst and the second moment we use the sample mean and the sample quadratic mean.

25 We generated an assortment of synthetic Weibull samples using Matlab’s random number generator.

data histogram, 50 bins log−log plot: data histogram 5 20 10 =0.9137 κ 0 10 10

−5 0 10 0 0.5 1 1.5 −2 0 2 =0.073862, 10 10 10 λ histogram, simulated Weibull log−log plot: histogram, simulated 5 20 10

0 10 10

−5 0 10 −2 0 2 sample parameters 0 0.5 1 1.5 10 10 10 red: simulated, blue: data blue: data, black: ideal 20 20

10 10

0 0

0 0.5 1 1.5 bin size=0.020542 0 0.5 1 1.5

Figure 8: A perfect ﬁt of empirical and synthetic densities. Too good to be true?

Franz Kafka, Aphorisms:

”All human errors are impatience, a premature breaking oﬀ of methodical procedure, an apparent fencing-in of what only seems to be.”

The graphs are standard histograms built upon pooling the numerical data into equal length bins. What’s wrong with equal bins? Does it aﬀect the shape and ﬁt?

It does when the data is unbalanced. So, we changed the number of bins and zoomed on both tails. We examined the uniform binning of the logarithm (exponential binning of the original).

26 Log-log plots

Consider a function p = f(t) > 0, t > 0. Change the variables:

x = ln t (or, t = ex), q = ln p (or, p = eq)

Examples of the log-log transformation:

density f(t) exponential λe−λt 7→ q = c − λex, Weibull (λ = 1/α1/β) λβtβ−1e−λtβ 7→ q = c + (β − 1)x − λeβx, λα α−1 −λt x Gamma Γ(α)t e 7→ q = c + (α − 1)x − λe , β κ−1 −(t/θ)β βx generalized Gamma θΓ(α) (t/θ) e 7→ q = c + (κ − 1)x − λe ,

data (blue stars), synthetic (red dots), ideal (black line) 30 bins 60 40 =0.9137

κ 40

20 20 sample size 65534 =0.073862,

λ 0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8

zoom on the right tail (40%) zoom on the left tail (33%) .001 60

0 40 .002

.001 20

0 0 0.7 0.8 0.9 1 0 .00002 .00004 .00006

log−log plots (shifted) log−log plots (shifted)

0 0 10 10

−5 −5 10 10

−1 0 −6 −4 −2 10 10 10 10 10 uniform bins exponential bins

Figure 9: The left column: uniform bins, the right column: exponential bins.

27 Alvaro Corral: Universal local versus united global scaling laws in the statistics of seismicity, 2004. The solid lines show the alleged Weibull densities. The data display distinc- tive ”upward bends”, or inﬂections, deviations from the Weibull distribution for small interevent times.

Figure 10: Corral log-log plots of the histograms of interevent times.

28 Alexander Saichev and Didier Sornette:

“Universal” Distribution of Inter-Earthquake Times Explained (2006) Theory of Earthquake Recurrence Times (2007)

Explanation:

1. The process of events might be a “mixture” or “superposition” or pooling of two independent processes, one Poisson with exponential interevent (a trend) times V and another process with Weibull times W (clusters).

2. The physical mixture yields no regularity (exception: Poisson processes).

3. Nevertheless, suppose that the interevent times T

a) are independent and identically distributed,

b) have a tail G(x) = P(T > x) and the mean µ.

29 4. Suppose that the process is stationary: the event count depends only on the duration of time intervals.

An observer at time t waits a random time At until the next event: Z 1 ∞ H(x) = P(At > x) = G(u) du. µ x Conversely:

G(x) = −µH0(x) =⇒ g = −G0 = µH00 = µh00

(well deﬁned when h is convex).

5. Deﬁne the model:

H(x) =

P(At > x) = P(no events in [t, t + x]) = P(no type I events AND no type II events) = P(V > x)P(W > x) = e−λx−(x/α)β .

6. The second derivative yields a multi-parameter density of the form

−λx−(x/α)β cα,β,λ(x)e that can be perfectly ﬁt to the bent up modiﬁed Weibull densities.

This is the ﬁnal explanation...

Or, is it?

30 5 synthetic Weibull samples (size 10000) for the chosen λ=1, κ=0.8 5 synthetic Weibull samples (size 10000) for the chosen λ=1, κ=0.8 3 2 10 10

2 10 1 10

1 10 0 10

0 10

−1 10

−1

14 exp−bins 10 14 exp−bins

−2 10 −2 10

−3 −3 10 10

−4 −4 10 10 −7 −6 −5 −4 −3 −2 −1 0 1 −8 −6 −4 −2 0 2 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 red stars: the pooled sample (size 50,000) vs. ideal pdf (shifted) with estimated λ‘ =0.9959, κ‘=0.7952 red stars: the pooled sample (size 50,000) vs. ideal pdf (shifted) with estimated λ‘ =0.9985, κ‘=0.7961

(a) Yes, we’ve got a bend! (b) Inconclusive

5 synthetic Weibull samples (size 10000) for the chosen λ=1, κ=0.8 5 synthetic Weibull samples (size 10000) for the chosen λ=1, κ=0.8 2 2 10 10

1 1 10 10

0 0 10 10

−1 −1 10 10 14 exp−bins 14 exp−bins

−2 −2 10 10

−3 −3 10 10

−4 −4 10 10 −7 −6 −5 −4 −3 −2 −1 0 1 −6 −5 −4 −3 −2 −1 0 1 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 red stars: the pooled sample (size 50,000) vs. ideal pdf (shifted) with estimated λ‘ =0.9971, κ‘=0.797 red stars: the pooled sample (size 50,000) vs. ideal pdf (shifted) with estimated λ‘ =1.003, κ‘=0.8025

Figure 11: The right tail is very stable. The left tail is somewhat wild, but rather up-wild than down-wild.

We experimented with many synthetic Weibull samples to mimic the Corral’s graph (eventually pooled together as an average). Some of them did, some did not. We ran also a“psychological test” on ourselves and our colleagues. We created 40 displays and let them be judged according to the “degree of bending”. The results:

50% inconclusive 40% an upward bend (20% strong, 20% weak) 10% a downward bend