这个作业是用Matlab预测城市人口变化

AMATH 301 – Spring 2020
Homework #6
Due on Friday, May 15, 2020
Instructions for submitting:
• Problems 1, 3, 5, and 7 should be submitted to MATLAB Grader. You have 3 attempts
for each problem.
• Problems 2, 4, 6, 8, and 9 should be submitted to Gradescope. The solutions and the
code used to get those solutions should be submitted as a single pdf. All code should be
at the end of the file. You must select which page each problem is on when you submit
to Gradescope.
Part I: Population Data
The file SeaPopData.mat, which is included with the homework, contains the following population data for the city of Seattle.
Year Population
1860 188
1870 1151
1880 3533
1890 42837
1900 80671
1910 237194
1920 315312
1930 365583
1940 368302
1950 467591
1960 557087
1970 530831
1980 493846
1990 516259
2000 563374
2010 608660
Data from:
1. https://en.wikipedia.org/wiki/Demographics_of_Seattle
2. https://www.seattle.gov/opcd/population-and-demographics
You should load this data into MATLAB using the load command. Be sure that the file
SeaPopData.mat has been downloaded into the same directory as your script file. MATLAB
Grader has its own copy of this file. If the load command is successful, you will have two
new vectors in your workspace, t and Seattle Pop. The values of the vector t are the number
of years since 1860. Therefore, t = 0 is 1860 and t = 150 is 2010. The vector Seattle Pop has
the corresponding populations from the table above.
(20 points) Problem 1: MATLAB Grader
(a) Find the line of best fit for the data. That is, find a line P = mt + b where t is the number
of years since 1860 and P is the population of Seattle. Store the slope of the line in the
variable ans1. The most recent population estimate for Seattle is that the 2019 population
was 747,300. Use the equation of the best fit line to predict the population in 2019. Store
this value in the variable ans2.
(b) Find the best fit quadratic function for the data. Use this curve to predict the population
in 2019, and store the prediction in the variable ans3. Repeat this process for the best
fit polynomials of degree 5 and degree 9. Create a 1 × 2 row vector named ans4 with
the predictions for each. The prediction from the degree 5 polynomial should be the first
component of the vector.
(15 points) Problem 2: Gradescope
(a) Create a plot the contains the Seattle population data and all four of the best fit polynomials
that you computed in Problem 1. Your plot should have the following features:
i. The data should be plotted as black circles.
ii. Your plot should show from t = 0 to t = 160 and the y-axis should show from P = 0
to P = 800, 000.
iii. The line of best fit should be blue, the quadratic fit should be red, the degree 5 polynomial should be magenta, and the degree 9 polynomial should be green.
iv. All circles and lines should be large/thick enough to be easily visible when you upload
the plot on the writeup portion of your homework.
v. Label the x-axis with “Years since 1860”.
vi. Label the y-axis with “Seattle Population”.
vii. Include a legend. Use legend labels “data”, “deg 1”, “deg 2”, “deg 5”, and “deg 9”.
(b) What is the real-world interpretation of the slope of the line of best fit for the Seattle
population data? What does it tell us about how the population is changing?
(c) Of the different polynomial fits you tried in Problem 1, which gave the most accurate
prediction of the 2019 population and which gave the least accurate? If you had to predict
the population in 2050, which polynomial fit would you trust the most? Justify your answer.
Part II: Atmospheric CO2 Data
The amount of CO2 in the atmosphere is regularly measured at the Mauna Loa observatory in
Hawaii. The file CO2 data.mat, which is included with the homework, contains the monthly
averages since 1958. Shown below is a plot of the data. The data has an overall upward trend
as well as seasonal oscillations.
You should load this data into MATLAB using the load command. Be sure that the file
CO2 data.mat has been downloaded into the same directory as your script file. MATLAB
Grader has its own copy of this file. If the load command is successful, you will have two
new vectors in your workspace, t and CO2. The values of the vector t are the number of years
since 1958 corresponding to each month from March 1958 to April 2020. So the first value is
t(1)= 3/12 because March 1958 is the third month (out of 12) since the beginning of 1958.
The vector CO2 has the corresponding CO2 levels.
Data from:
1. https://www.esrl.noaa.gov/gmd/ccgg/trends/data.html
(10 points) Problem 3: MATLAB Grader
It looks like the overall trend of the data might be captured well by using an exponential
fit. Recall from the video “Least-Squares Fitting Methods” that this can be done by data
linearization. Here is a recap of that method:
Let’s say that the data is stored in vectors T and Y . We wish to fit an exponential function
y = aert to the data. You can take a natural logarithm of Y and perform a linear fit on the data
vectors T and ln(Y ). This gives a linear function y = mt + b. You can get the values of a and
r for the exponential fit of the original data by using that m = r and b = ln(a).
Use data linearization to get an exponential fit y = aert for the CO2 data. Store the value of r
in the variable ans5 and the value of a in the variable ans6.
(10 points) Problem 4: Gradescope
Create a plot the contains the atmospheric CO2 data and your exponential fit from Problem 3.
Your plot should have the following features:
i. The data should be plotted as black dots connected by black lines by using the line specification ‘-k.’.
ii. Your plot should show from t = 0 to t = 65.
iii. The exponential fit should be plotted as a red curve and it should be thick enough to be
easily visible when you upload the plot on the writeup portion of your homework.
iv. Label the x-axis with “Years since 1958”.
v. Label the y-axis with “Atmospheric CO2”.
vi. Include a legend. Use legend labels “data”, “fit curve”.
(10 points) Problem 5: MATLAB Grader The exponential fit from Problems 3 and 4 is
not great at capturing the overall trend of the data. It can be improved by fitting an exponential
function that is shifted up by a constant:
y = aert + b.
For this function, the data linearization trick will not work. Instead, you must create a MATLAB
function that takes the values of a, r, and b as inputs and calculates the sum of squared errors
as output. Then you can use fminsearch to find the values of a, r, and b that minimize the
sum of squared errors (and therefore minimize the root-mean squared error). Use this method
to find the best fit curve of the form y = aert + b. For fminsearch, use initial guesses of
a = 30, r = 0.03, and b = 300. Make a 3 × 1 column vector named ans7 with the optimal
parameters [a; r; b]. Calculate the minimum value of the sum of squared errors and store
it in the variable ans8.
Hint: If you use fminsearch properly, the answers to ans7 and ans8 can both be output by
fminsearch.
(10 points) Problem 6: Gradescope
Create a plot the contains the atmospheric CO2 data and your best fit curve from Problem 5.
Use the same specifications as given in Problem 4.
(10 points) Problem 7: MATLAB Grader The best fit curve from Problems 5 and 6 does
a much better job of capturing the overall trend of the data, but it still does not capture the
seasonal oscillations. In order to capture the oscillations, we can find a best fit curve of the
form
y = aert + b + c sin(d(t − e))
Create an error function and use fminsearch to find the values of the parameters that minimize
the sum of squared errors. For your initial guess, use the values of a, r, and b that you stored in
ans7 and use c = 3, d = 6 and e = 0. Make a 6×1 column vector named ans9 with the optimal
parameters [a; r; b; c; d; e]. Calculate the minimum value of the sum of squared errors
and store it in ans10.
(10 points) Problem 8: Gradescope
Create a plot the contains the atmospheric CO2 data and your best fit curve from Problem 7.
When you plot the best fit curve, make sure you use enough points to capture the oscillations.
Use the same specifications as given in Problem 4.
(5 points) Problem 9: Gradescope Which of the following types of error is most resistant
to outliers (i.e. is least affected by the presence of outliers)?
(a) average error
(b) maximum error
(c) root-mean square error
Gradescope Deliverables Your Gradescope writeup should contain the following:
• Problem 2: The plot with the population data a polynomial fit curves, and answers to
the questions in (b) and (c).
• Problem 4: The plot
• Problem 6: The plot
• Problem 8: The plot
• Problem 9: A letter corresponding to your answer choice
• Code: Code for problems 2, 4, 6, and 8