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1 Author(s): Kerby Shedden, Ph.D., 2010 License: Unless otherwise noted, this material is made available under the terms of the Creative Commons Attribution Share Alike 3.0 License: We have reviewed this material in accordance with U.S. Copyright Law and have tried to maximize your ability to use, share, and adapt it. The citation key on the following slide provides information about how you may share and adapt this material. Copyright holders of content included in this material should contact with any questions, corrections, or clarification regarding the use of content. For more information about how to cite these materials visit Any medical information in this material is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. Please speak to your physician if you have questions about your medical condition. Viewer discretion is advised: Some medical content is graphic and may not be suitable for all viewers. 1 / 13

2 Statistical analysis of rare events: collisions between Earth and Near Earth Objects Kerby Shedden Department of Statistics, University of Michigan Monday 18 th February, / 13

3 Distributions of rare events and the rate/intensity relationship The rate at which an event occurs is the expected number of occurrences per year. Let N denote a random variable describing the number of events per year, and let F denote the cumulative distribution function of N. Thus P(N k) = F (k), and the rate is E[N]. Most hazardous events occur at varying levels of intensity: Earthquakes have a moment magnitude. Hurricanes have a rating on the Beaufort scale. Tornadoes have a rating on the Enhanced Fujita scale. The rates of these hazardous events are decreasing functions of their intensities. 3 / 13

4 Power law relationships Let N w be a random variable giving the number of annual occurrences of events of intensity w or greater, and let λ w = E[N w ] be the corresponding rate. Many events have the property that their rate and intensity follow a power law rate/intensity relationship. This means that log λ w is approximately a linear function of log w: Equivalently, log λ w = α + β log w. λ w = e α w β. 4 / 13

5 Power law relationships If we compare the event rates for intensity w and 10w, we get: e α w β e α (10w) β = (1/10)β. In other words, if you consider an even that is 10 times more severe than a given event, the frequency drops by a factor of 10 β. 5 / 13

6 Simulation from power law distributions The power law only describes the mean structure. If we want to simulate data, we need to specify the complete distribution. An easy choice is the Poisson distribution. For example, suppose the power law is as follows: log λ w = log w, and we are interested in w = 5, 10, 15, 20, 30, 40, 50. The corresponding annual rates are , , , , , , / 13

7 Simulation from power law distributions A draw from the Poisson distribution with rate λ is approximately equal (in distribution) to the number of successes in n Bernoulli trials with success probability λ/n. The approximation is better if n is large. Conceptually, there are n opportunities for the rare event to occur, with each opportunity having probability p of actually occurring. Choosing n = 10000, we get λ = , , , , , , and / 13

8 Simulation from power law distributions Suppose we want to simulate N w for a single value of w. We can achieve this as follows: 1. Set p = λ/n (where λ is obtained from w using the power law). 2. Simulate u 1,..., u n as iid uniform random variables on (0, 1). 3. Set I i = I(u i < p). Note that each I i is a Bernoulli trial with success probability p. 4. Set N w = i I i. Note that there are other (better) ways to simulate a single Poisson value, but as we will see next, this approach is useful for us. 8 / 13

9 Simulation from power law distributions If w > w, then N w and N w are statistically dependent, since every event of intensity at least w is also an event of intensity at least w (thus N w N w ). We can handle this by coupling the simulation of N w for multiple values of w. 9 / 13

10 Simulation from power law distributions Suppose w 1,..., w d are values of w for which we want to simulate the corresponding N w values. This can be achieved as follows: 1. λ 1,..., λ d are the rates corresponding to the w j, determined from the power law. 2. p j = λ j /n are the success probabilities corresponding to the λ j. 3. Let u 1,..., u n be iid uniform random variables on (0, 1). 4. Set I ij = I(u i < p j ). 5. Set N wj = i I ij 10 / 13

11 Power law extrapolation We are especially interested in events that have only occurred a few times, if ever, in recorded history: Category 5 hurricanes striking a major populated area. Earthquakes with magnitude 7 striking a major populated area. Large meteoroid impacts on Earth. How to estimate the frequency of such events? If there is a power law relationship, we can estimate the power law parameters (α and β) from low-intensity events (e.g. minor earthquakes, small meteoroid strikes), and extrapolate to the rare events. 11 / 13

12 Meteor strikes Some frequencies: 4 meters around 1/year 7 meters around 1/5 years 1 km around 1/500,000 years 5 km around 1/20 million years Cretaceous-Paleogene extinction event 66 million years ago: > 10 km diameter Tunguska event in 1908: 100 meters 2013 Chelyabinsk event: 20 meters 12 / 13

13 Data for estimating meteor strike frequency Craters: not very useful due to weathering and erosion; most meteor events are air bursts not impacts. Observations of fireballs from ground need to infer size from brightness, sound, radar, etc. Observations of cratering on the moon. Observations from satellites or space telescopes. 13 / 13