\[ \newcommand{\Expect}[1]{\mathbb{E}\left[ #1 \right]} \]
This repeats part of the syllabus, for emphasis
Everything you turn in for this assignment must be your own work, or a clearly acknowledged borrowing from an approved source; this includes all mathematical derivations, computer code and output, figures, and text. Any use of permitted sources must be clearly acknowledged in your work, with citations letting the reader verify your source. You are free to consult the textbook and recommended class texts, lecture slides and demos, any resources provided through the class website, solutions provided to this semester’s previous assignments in this course, books and papers in the library, or legitimate online resources, though again, all use of these sources must be acknowledged in your work. (Websites which compile course materials are not legitimate online resources.)
This is a take-home exam, so you are not allowed to discuss the content of the exams with anyone other than the instructors; in particular, you may not discuss the content of the exam with other students in the course.
Carefully study the solutions to the previous homeworks, and the course notes, include the source code for the slides. Everything that is being asked of you has already been covered, in class, in the notes, in homework, or in all three.
Wind power is a renewable, non-polluting energy source, but deciding where to put wind turbines is a complicated issue, with technical, political and social aspects1. On the technical side, an obvious issue is finding locations with high capacity to generate wind power. Here the basic rule of thumb is that the energy which can be extracted from a wind of speed \(v\) is proportional to the cube of the wind speed, \(v^3\). (If you’re curious, read the footnote2.) Because wind speed fluctuates, power companies usually want locations with high average capacity over a typical year3.
The data set on the course website contains measurements of the wind speed over a coastal region, taken at a grid of points, every 6 hours, over a period of several years. The main question is:
What location, within this region, has the highest potential for wind power, averaging over the year?
(Keep in mind that the best location might not be one of the points where we have data.) Accompanying this are two other questions:
What is the expected value of \(v^3\) for this location? What is the standard error in this estimate?
Your write-up should take the form of a report which explains the method by which you reached your conclusion, i.e., how you analyzed the data and arrived at your answer. If you become convinced that this question cannot be sensibly answered on the basis of the data available, you should explain why, and what additional information you would need.
Your need to make choices about the following issues, and your report needs to both explain the choice you made, and why you made that choice:
As usual, this describes the ideal.
Words (15) The text is laid out cleanly, with clear divisions and transitions between sections and sub-sections. The writing itself is well-organized, free of grammatical and other mechanical errors, divided into complete sentences logically grouped into paragraphs and sections, and easy to follow from the presumed level of knowledge.
Numbers (5) All numerical results or summaries are reported to suitable precision, and with appropriate measures of uncertainty attached when applicable.
Pictures (5) All figures and tables shown are relevant to the argument for the ultimate conclusions. Figures and tables are easy to read, with informative captions, axis labels and legends, and are placed near the relevant pieces of text.
Code (10) The code is formatted and organized so that it is easy for others to read and understand. It is indented, commented, and uses meaningful names. It only includes computations which are actually needed to answer the analytical questions, and avoids redundancy. Code borrowed from the notes, from books, or from resources found online is explicitly acknowledged and cited in the comments. Functions or procedures not directly taken from the notes have accompanying tests which check whether the code does what it is supposed to. All code runs, and the Markdown file knits.
Modeling (25) Model specifications are described clearly and in appropriate detail. There are clear explanations of how estimating the model helps to answer the analytical questions, and rationales for all modeling choices. If multiple models are compared, they are all clearly described, along with the rationale for considering multiple models, and the reasons for selecting one model over another, or for using multiple models simultaneously.
Inference (25) The actual estimation of model parameters or estimated functions is technically correct. All calculations based on estimates are clearly explained, and also technically correct. All estimates or derived quantities are accompanied with appropriate measures of uncertainty.
Conclusions (15) The substantive, analytical questions are all answered as precisely as the data and the model allow. The chain of reasoning from estimation results about the model, or derived quantities, to substantive conclusions is both clear and convincing. Contingent answers (``if \(X\), then \(Y\), but if \(Z\), then \(W\)’’) are likewise described as warranted by the model and data. If uncertainties in the data and model mean the answers to some questions must be imprecise, this too is reflected in the conclusions.
The readings on Canvas describe some of the controversies about particular wind power stations, or proposed stations, and some of the implications of wind power in general. One of them is directly relevant to our data.↩
You can skip this note if you’re willing to take the cube-of-wind-speed rule on faith. For those who are still here: The kinetic energy of a body of mass \(m\) and speed \(v\) is \(\frac{1}{2}mv^2\). The kinetic energy per unit volume of a fluid, of density \(\rho\), is thus \(\frac{1}{2}\rho v^2\). The volume of fluid which flows past a turbine of cross-sectional area \(A\) in a time interval \(T\) is \(ATv\). The total kinetic energy of the fluid flowing past the turbine is thus \(\frac{1}{2}AT\rho v^3\). This neglects considerations like whether the turbine is more or less efficient at different speeds, etc., so it’s only start, but good enough for our purposes.↩
Utility companies also need to worry about maximum wind speed (will the turbines break? can the electrical grid handle that much power coming in?), and, to a less extent, minimum speed (how much of the time will the turbine be idle?). We will ignore these complications, and just focus on average capacity.↩
There is a trade-off here between using a very finely-spaced grid, to get a precise location for the maximum, and spending a lot of time on computation. Something that can work well is to use a coarse grid initially. Once you have found the maximum point on the initial grid, create a new grid, with finer spacing but a smaller over-all extent, centered on the initial maximum. You can repeat this any number of times.↩