An Analysis Approach: Chap 9

Reading Questions

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Fig RQ9- 1 shows three hypotheses about the time between so-called hundred-year storms. Of course, there is no fixed time; the time from the last big storm to the next big storm is uncertain. (“Hundred-year” storms are identified by the storm magnitude, not timing, based on statistical analysis of past storms and the intervals between them.)

We express the uncertainty with a probability distribution, as described in Chapter 7. The exponential distribution is the standard model for the time until the next event. So all three hypotheses in Fig RQ9- 1 have the exponential form. Where they differ is in the rate of the storms. Hypothesis N (for neutral) says the rate for hundred-year storms is 1 per 100 years, just what you would expect from the name “hundred-year storm.” Hypothesis S (for “sceptic”) is that the rate for hundred-year storms is actually one per one-hundred fifty years. Hypothesis C (for “climate”) is that, with climate change, the rate for hundred-year storms is 1 per 50 years.

Figure RQ9- 1: Three competing hypotheses for the interval until the next so-called 100-year storm.
  1. Fig RQ9- 1 is just about hypotheses. A likelihood is not a hypothesis, but rather something calculated in the context of a specific hypothesis. What is that “something”?
  1. Assign a likelihoood to each of the hypotheses in Fig RQ9- 1 in the context of having just observed the next “100-year”-magnitude storm, which arrived only 0 years after the previous one. (Hint: You can read the answer directly from Fig RQ9- 1.)
  1. Similarly to (2), assign a likelihood to each of the hypothesis in Fig RQ9- 1 in the context of having just observed the next “100-year”-magnitude storm, which arrived 150 years after the previous one. (Hint: Same as in (2).)
  1. Fig RQ9- 2 shows the likelihood function for a different observation: a “100-year” magnitude storm that comes 30 years after the last one. There is an exact correspondence between the colored curves in Fig RQ9- 1 and the colored points in Fig RQ9- 2. Explain why the colored points are where they are. (Hint: Remember, the observed storm came after an interval of 30 years.)
Figure RQ9- 2: Likelihood function for the observation of a 30-year interval before the just-observed “100-year” amplitude storm.
  1. The likelihood ratio between the C and S hypotheses, as evidenced by the observed 30-year interval, is 2.015. Explain what features in Fig RQ9- 2 indicates this value for the likelihood ratio.
  1. Translate the likelihood ratio of 2.015 into a verbal description of the “strength” of the C hypothesis compared to the S hypothesis.
  1. The likelihood ratio and verbal description in (5) and (6) are based on only one observation: a 30-year interval between so-called “100-year” storms. Refer to Fig RQ9- 1, is there any possible observation that would give even “moderate” evidence for a preference for one of the three hypotheses.

Note: It’s hard to draw conclusions from just a single observation. Instead, the likelihoods from each of many observations would be combined to form a conclusion. Importantly, those observations need to take the form of not just observed intervals between storms, but also observations for places that have not yet had a storm. For such no-storm-yet locales, the relative probability functions under the three hypotheses would have a different shape than in Fig RQ9- 1. In order to explain that new shape, we will need techniques from Chapter 13. Stay tuned!