Intro Matlab Coding

AMATH 301 - Fall 2018 Due Friday, 16 November, 5:00

Homework 6

Problem 1: Salmon Population

The file salmon.mat (included with the homework) contains the annual Chinook salmon counts taken at Bonneville on the Columbia river from the years 1938 to 2017. (You can find more information at www.cbr.washington.edu. The data is from the DART adult passage annual summary.) You can load this file into matlab with the command load(’salmon.mat’). Do not upload this file to scorelator. Scorelator has its own copy of salmon.mat, with counts from a different species. If the load command is successful, you will have two new vectors in your workspace: t and salmon. The vector t represents years since 1938, so t = 0 is the year 1938 and t = 79 is the year 2017. Each entry in salmon represents the number of Chinook salmon that passed through Bonneville in the corresponding year. In this homework, we will try many methods to predict how many Chinook salmon will pass through Bonneville next year (t = 81). (At the time of this writing, 334,651 salmon made it through in 2018, but since the year is not over I have not included that value in the dataset.)

You should plot the data alongside every curve that we fit to it. This will not only give you some insight into salmon population dynamics, but also provides a good sanity check as to whether or not your code is working correctly. However, before you submit your assignment to scorelator you should comment out all plot commands.

(a) Find the best fit line for this data. That is, find a line y = mt + b, where t is the time in years and y is the salmon population. Your line should be “best” in the sense of minimizing root-mean-square error. Save the slope of this line in A1.dat. Now use this formula to predict the population of salmon in 2019 and save the predicted population in A2.dat. Find the root-mean-square error of this best fit line and save it in A3.dat.

Things to think about: Is the slope positive or negative? What does this tell you about the salmon population dynamics? Has the salmon population increased or decreased over the last three years? Is that trend reflected in the best fit line?

(b) Now find the best fit quadratic for this data (i.e., the best fit polynomial of order 2, where “best” is in the root-mean-square sense). Use this curve to predict the

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AMATH 301 - Fall 2018 Due Friday, 16 November, 5:00

salmon population in 2019 and save this prediction in A4.dat. Repeat this process for the best fit polynomials of orders 5 and 20. Save your predictions in A5.dat and A6.dat, respectively.

Things to think about: What are the root-mean-square errors for each of these fits? Which would you most trust to predict this years population? To see what Matlab means by the warning “polynomial is badly conditioned”, try this problem with a 25th order polynomial (or even higher). Some of what you are seeing is the polynomial wiggle phenomenon that we discussed in class, but most of it is rounding error. Try typing out the polynomial explicitly (for example, for a second order polynomial try y = a*x.^2 + b*x + c versus using polyval. For low orders, you should not see a difference, but for high order polynomials these do not produce the same results. polyval uses an algorithm called Horner’s method. Look up this method - can you see why it is faster / more accurate? (Hint: Think about how rounding error might accumulate.)

(c) It is often reasonable to model population growth with an exponential equation of the form y = N0e

rt. Here, N0 is the population at time 0 and r is the population growth rate. Rewrite this equation by hand as z = mt + b, where z = ln y. (Note that m and b are related to N0 and r, but not necessarily the same. This transformation is described in the notes and the videos.) Find the best fit line (in the root mean square sense) between t and z. Use your answer to calculate the best values of N0 and r. Store these two values as a 1 × 2 row vector [N0, r] and save this row vector in A7.dat. Use the exponential model to predict the salmon population in 2019 and save your prediction in A8.dat.

Things to think about: Do you get the same parameters if you use fminsearch to find N0 and r directly? Why do you think this is/is not? What did you get for a root-mean-square error? What does this tell you?

(d) Suppose that a biologist proposes the following function as a model for salmon population growth:

y = eat 2+bt+c,

where a, b and c are parameters. Write a function to compute the root-mean- square error between this curve and the population data salmon for any given values of a, b and c. Use fminsearch to find the parameters that minimize this error (using an initial guess of a = 0.001, b = −0.01 and c = 10), then use this optimal set of parameters to predict the chinook population in 2019. Save your prediction in A9.dat.

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AMATH 301 - Fall 2018 Due Friday, 16 November, 5:00

Things to think about: What was the root-mean-square error? What does that tell you about the fit? How does this error compare to that from part (c)? How are these two models related? Do you think this is a biologically plausible model?

(e) It is not unreasonable to think that the salmon populations over the last few years are much more relevant to our prediction than the population from 1938. One way we could focus on more recent data is to use a cubic spline. Use a spline of the data to predict the salmon population in 2019. Save your prediction in A10.dat.

Things to think about: Do you think it is wise to use a spline for extrapola- tion? Can you think of a better extrapolation method that primarily uses the more recent data?

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