hare question

In the Introduction to the lecture notes, we referred to the data from the Hudson Bay Company. This graph suggests that the population of snowshoe hares follows a periodic function, although it is not a true sine or cosine wave, due to their interaction with their primary predator the lynx. We noted in the beginning of the lecture section on differentiation of trigonometric functions that the work of Fourier showed that any function can be modeled with the summation of a series of sine waves.

In this question you will use the work of Fourier, and Excel's solver, to model the quantity of the hare pelts turned into the Hudson Bay Trading Company for years after 1900. The data for hare pelts is shown below, and also on the downloadable Excel spreadsheet.

Year	Hares (x1000)	Year	Hares (x1000)
1900	30
1901	47.2	1911	40.3
1902	70.2	1912	57
1903	77.4	1913	76.6
1904	36.3	1914	52.3
1905	20.6	1915	19.5
1906	18.1	1916	11.2
1907	21.4	1917	7.6
1908	22	1918	14.6
1909	25.4	1919	16.2
1910	27.1	1920	24.7

We would like to approximate these data with a model composed of a sum of sine functions of the form

where we need to find the appropriate constants a_i, w, and f_i.

a. Begin by trying to fit the data with a simple sine function and a constant, which satisfies the equation

P(t) = a₀ + a₁sin(wt - f₁).

Use Excel's solver to find the constants a₀, a₁, w, and f₁, which give a least squares best fit to the data, and write these values in your lab report. Also, give the value of the sum of least squares error that is computed. Graph both the the model and the data from 1900 to 1920. What is the period of this approximation? Find the percent error between the model and the data for the years 1903, 1909, 1914, and 1919. Find the year and value of the absolute minimum and the absolute maximum for this approximation. (Give only the first occurrence as it is periodic.)

b. The next step in this problem is to see how much better the data are fit using another sine function in the Fourier series. Thus, we want to fit the function

P(t) = a₀ + a₁sin(wt - f₁) + a₂sin(2wt - f₂).

Use Excel's solver to find the constants a₀, a₁, a₂, w, f₁, and f₂, which give a least squares best fit to the data, and write these values in your lab report. Again, give the value of the sum of least squares error that is computed. Graph both the the model and the data from 1900 to 1920. What is the period of this approximation? Find the percent error between the model and the data for the years 1903, 1909, 1914, and 1919. Find the year and value of the absolute minimum and the absolute maximum for this approximation. (Give only the first occurrence as it is periodic.)

c. Repeat the process in Part b. adding a third term, a₃sin(3wt - f₃), to the Fourier series. Then repeat this process adding a fourth term, a₄sin(4wt - f₄), to the Fourier series.

d. Write a brief discussion of how much the different coefficients change as you add more terms to the Fourier series. What is the trend for the sum of least squares error and for the calculated percent error at the dates requested? Do you expect a much better fit by adding another term to the Fourier series?