What is the relationship between mathematics and science?
Week 4: The Rayleigh Distribution Activity
By the start of class, please read the Unreasonable Effectiveness of Mathematics in Science by Eugene Wigner. Try to think of (or find) one example of a surprising connection between mathematics and science. In a paragraph or so, state and briefly explain this example; make sure to give appropriate references.
During this class, we will learn about one application of mathematics to science: the Rayleigh distribution. This has applications to oceanography (where it can represent wave heights [1]), nutrition (where is can represent nutrient responses of human and non-human animals [2]), and medicine (where it can represent noise in MRI images [3]) , among others.
To learn about the Rayleigh distribution, we will complete an activity and then study a real-life application.
Part I: Activity
This activity will take the form of a dart game. To the left we depict a dart board. Below, we provide a random number generator and a calculator. Now imagine that we are throwing darts at the board, but we are not good at darts (this is realistic, in my case). Although all darts hit the board, they hit randomly. In particular, we will describe the location of each dart by the coordinate pair (x,y), where x is the horizontal distance from the center of the board, and y is the vertical distance from the center of the board. These x and y values will take random values from 0 to 10.
The straight line distance of each point to the board's center is
If we were to conduct this experiment many times, we would find that the probability of a dart hitting the board at a distance of r from its center is
Here, ⍺ is the mode of the distribution. This is the value which appears most often in the set. To complete the activity, please follow these steps.
Imagine now that you throw 20 darts at the board. Generate 20 pairs of random values for the x and y values of the darts, using the random number generator below. Create values with three significant figures (in this case, two decimal points; for instance: 8.39.) You may also use Excel or Numbers to generate random numbers; for instance, the Excel function RAND() produces a random number between 0 and 1.
Input these values into Excel or Numbers. Excel is suggested, since I give directions below only in Excel.
Compute r for each dart.
We will assign the following point values to the throws:
0< r < 2: 1.95 points
2< r < 4: 1.66 points
4< r < 6: 1.21 points
6< r < 8: 0.76 points
8< r < 10: 0.40points
10< r < 12: 0.18 points
12< r < 14: 0.07 points
r>14: 0 points
Now create a separate column entitled "Point Values"; to obtain these, multiply each r value by its point value. Produce a histogram of your data which separates the point values into the following ranges (these are often called "bins"): 0 to 1, 1 to 2, 2 to 3, .... To accomplish this in Excel, first create the bins by creating a column consisting of integers from 1 to the maximum point value. Then, select the "Data" tab and then click "Data Analysis". Select "Histogram". The "Input Range" should consist of the point values. The "Bin Range" should be the aforementioned column of integers. Click "Output Range" and then select a blank cell. A table will be created from this cell. Finally, check "Chart Output". This should produce a table of the frequencies of r values, and an associated histogram.
Now answer the following questions regarding your data set. Please type your answers in a Word (or equivalent) document, which you will submit below. This is an assignment, and not a presentation.
What is ⍺? If there is a not a single mode, average the modes to produce a single value.
Given that we are studying a distribution of radii, does this value of ⍺ make intuitive sense?
Perform a literature search to find a plot of the Rayleigh distribution, and cite your sources. (Alternatively, you could make a plot of the probability function given above.) Does your histogram look like an exact Rayleigh distribution? If not, how could this experiment be tweaked to produce a better facsimile? Does this say anything about the usefulness of statistical distributions with different sample sizes?
Part II: Application to Nutrient Response
Please open and read this paper and answer the following questions based upon it.
In your own words, what do you think is the main hypothesis of the paper?
Describe how the authors attempted to falsify the hypothesis.
In Figure 1, there are six figures. The data is plotted with error bars, which the caption indicates are standard errors. Look up the definitions of standard error and standard deviation. In a random, unbiased data set, what the probability that a randomly selected point will fall with one standard deviation of the mean? What is the probability that such a point would fall outside of one standard deviation from the mean?
Do you think the study convincingly demonstrates that the hypothesis is true?
References
1. Thornton, Edward B., and R. T. Guza. "Transformation of wave height distribution." Journal of Geophysical Research: Oceans 88.C10 (1983): 5925-5938.
2. Ahmadi, Hamed. "A mathematical function for the description of nutrient-response curve." PloS one 12.11 (2017): e0187292.
3. Sijbers, Jan, et al. "Parameter estimation from magnitude MR images." International Journal of imaging systems and technology 10.2 (1999): 109-114.
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