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Analyzing Wind Speed at Arthur's Seat in Edinburgh using a Rayleigh Statistical Model

Project description

In this project we analyse average wind speeds at the top of Arthur’s Seat in Edinburgh. We will model the wind speed by using a Rayleigh statistical model, which we will fit to the data by maximum likelihood estimation and subsequently use to make predictions about future wind speeds.

Project description
You have been asked to analyse the distribution of the wind speed at the top of Arthur’s Seat in Edinburgh. You have access to the values x = (x1, . . . , x1000) of average wind speeds registered on 1000 days in years 2018-2020. The data is provided in the wind.csv file, with values recorded in miles per hour (mph). You are also asked to use this data to make some predictions about the wind speed in the subsequent years.
You are advised to assume a Rayleigh model for the wind speeds, with probability density function given by
f(x, σ) = ( x σ2 exp −x 2 2σ2 , x ≥ 0 0, x < 0,

where σ > 0 is an unknown parameter.
1. Summarise the data by plotting a histogram and calculating numerical summaries, such as the sample mean, standard deviation, median, and other quantiles. Briefly comment on your results. [3 marks]

2. Derive the maximum likelihood estimator for σ (denoted henceforth by σˆ). (Hint: For
any n ≥ 1 we have d dσ 1 σn = − n σn+1 ). [4 marks]

3. Derive the Fisher information for σ and use it to approximate the distribution of σˆ. (Hint: Use the fact that for a random variable X with the Rayleigh distribution with parameter σ > 0, we have E[X2 ] = 2σ 2 ). [4 marks]

4. Using the results of parts 2 and 3, calculate σˆ = ˆσ(x) and report an approximate 95% equal-tailed confidence interval I = [σL(x), σU (x)] for σ. [4 marks]

5. For any i = 1, . . . , 1000, let X′ i ∼ Rayleigh(ˆσ) denote the predicted wind speed on a random day in the future. Let Y ′ = 1 1000 P1000 i=1 X′
i be the predicted mean wind speed in the next 1000 days (assume that predicted individual wind speeds are independent of each other). Use simulation in R to estimate the distribution of Y
Report this distribution by plotting a histogram and calculating numerical summaries. (Hint: Use
the VGAM R package to get access to functions involving the Rayleigh distribution). [4 marks]

Summarizing the Data

6. Climate researchers believe that in the comings years the variance of the wind speed will increase. Using the result of part (5), calculate the probability that this assumption is true based on the data and model available (i.e., calculate P[sd(Z ′ ) > sd(x)], where sd(Z ′ ) is the standard deviation of the predicted sample of future wind speeds Z ′ = (X′ 1 , . . . , X′ 1000) and sd(x) is the standard deviation of the sample x = (x1, . . . , x1000)).

Briefly comment on your results. [3 marks]

7. To assess if your conclusions are robust to errors in the estimation of σ, recalculate the probability from part (6) by using the lower and upper estimates σL(x) and σU (x) derived in part (4). Briefly comment on your results.
[3 marks]

Your findings should be presented in the form of a report, which should:
• have a clear and logical structure;
• include an introduction and clearly stated conclusions that can be understood by any numerate scientist (not necessarily a statistician);
• include details of your mathematical calculations so that your results could be reproduced by another statistician;
• include clearly labelled and correctly referenced tables and diagrams, as appropriate;
• include the R code you used in an appendix (you do not need to explain individual R commands but some comments should be included to indicate the purpose of each section of code);
• include citation and referencing for any material (books, papers, websites etc) used.
• maximum page limit of four (4) pages (11-point font, A4 size, 4 pages = 2 sheets of paper, additional pages are allowed for the R code). Excluding R code, only the first four pages of your submission will be marked. No feedback will be given, or marks awarded, for any work (apart from R code) appearing on subsequent pages.
A total of 5 Marks is available for these aspects of your report. This will be marked according to the rubric given in the Appendix. [Total: 30 Marks]

• This assignment counts for 30% of the course assessment.
• You may discuss this project with your colleagues, but your report must be your own work. Plagiarism is a serious academic offence and carries a range of penalties, some very serious. Copying a friend’s report or code, or copying text into your report from another source (such as a book or website) without citing and referencing that source, is plagiarism. Collusion is also a serious academic offence. You must not share a copy of your report (as a hard copy or in electronic form) or your computer code with anyone else. 

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