Homework 7#

Problem 1: ENSO Phases#

Following Lab 7-1 and Lab 7-2,

A. (1) Use the time series of the phase of the El Niño Southern Oscillation (ENSO) from 1900-2022 to create a lag-1 Markov model of the ENSO phase, where the observed Phases of ENSO are as follows:

  1. warm (El Niño)

  2. neutral (ENSO neutral)

  3. cool (La Niña)

B. (2) Using this Markov model and a random number generator, simulate 5,000 years of ENSO data. Then answer the following questions.

  • According to the model, what is the probability that three warm ENSO years would occur in a row?

  • What is the probability that three cool ENSO years would happen in a row? (Try refreshing the numbers several times to increase the sample size if the condition never happens.)

Check out this blog about why we care about ENSO and the exciting current probability of getting a cool ENSO (La Nina) again in 2023, making it three in a row.

Probelm 2: Rating Curves and Application of Bayes Theorem with MCMC#

Following the class discussion and Lab 7-3, explore how the rating curve and the 95% confidence intervals for the Lyell Fork streamflow site change depending on the method you use to determine the rating curve:

A. (2) Least squares linear regression fitting (with transformed variables) using b = 0.28 m

  • Make 95% confidence intervals around this regression fit

  • Then, assume that we don’t know exactly what b is. Try additional linear regressions using different values of b = 0.10, 0.20, 0.30, 0.40, and 0.50 m (you do not need to calculate 95% confidence intervals for these additional fits)

  • Qualitatively, is the range between these 5 additional lines with different b values larger or smaller than the range between the 95% confidence lines from the original fitted line (the one with b = 0.28 cm)?

B. (1.5) Direct monte carlo parameter estimation

C. (1.5) Bayesian MCMC fitting

Using the code in Lab 7-3, create plots and discuss the differences in the results from these three methods. (2)