This coursework comprises 20% of the overall module assessment. It is an individual project and so you must not collaborate when working on this assessment

MTH1004 summative coursework: report

Date set: Friday 4 March 2022

Submission date: Friday 1 April 2022 by noon

Return date: Friday 6 May 2022

Instructions

This coursework comprises 20% of the overall module assessment. It is an individual project and so you must not collaborate when working on this assessment. You should read the College guidelines on collaboration and plagiarism, which are on the following webpage.

http://intranet.exeter.ac.uk/emps/studentinfo/subjects/mathematics/assessment/academicmisconduct

Conduct the investigation outlined below and write a report of your findings, addressing each of the three parts of the investigation in a separate section of your report. Your report should comprise at most three sides of word-processed text (e.g. written with Word or LaTeX) followed by as many pages of figures or tables that you feel is appropriate and then an appendix containing the R code (with explanatory comments but without the R output) that you used to obtain your reported results. No credit will be awarded to additional pages of text. You must use the page size A4 and a font size of at least 11 points, lines must be single-spaced and all margins must be at least one inch wide. You must save your report as a pdf file and upload it to BART at https://bart.exeter.ac.uk by the date and time shown above. Approximately 25 marks are available for each part of the investigation, with another 25 for the presentation (structure, clarity, language etc.) of your report.

Investigation

The Central England Temperature (CET) data set is the longest instrumental temperature record in the world. The data set contains temperatures (in degrees Celsius, ?C) representing the average temperature each month since January 1659 in an area of central England. The Central England Precipitation (CEP) data set is a similar record for precipitation amounts (in millimetres). Precipitation is rainfall, snow, hail etc. The North Atlantic Oscillation (NAO) index is a measure of the difference in atmospheric pressure between the Azores and Iceland. Changes in the NAO index are known to be associated with changes in European weather. The file Climate.RData is available on the module webpage and contains the monthly values of the CET, CEP and NAO data for each December from 1873 to 2020.

1. Consider modelling the precipitation data as realisations of independent and identically distributed random variables. Choose a parametric model for their distribution, estimate the parameters using the method of moments, and assess the realism of your model using appropriate graphs. Include in your report a clear statement of your model and the formulae for your estimates. You do not need to include the derivation of these formulae.

/continued...

1

2. Consider modelling the temperature data as realisations of independent and identically distributed random variables with the cumulative distribution function

y - a

F(y) = 1 - exp -exp

ß

for -8 y 8, where a ? (-8,8) and ß 0 are parameters. The expectation of this distribution is a - 0.5772ß and the variance is p2ß2/6. R functions which compute the pdf, cdf and quantiles of this distribution are given below.

(a) Use the method of moments to estimate the parameters of this model and assess the realism of the model using appropriate graphs. Include in your report a clear statement of the formulae for your estimates. You do not need to include the derivation of these formulae.

(b) Suppose that some households receive financial support if the December mean temperature is below 0?C. Use your fitted model to predict the probability that the mean temperature will be below 0?C in December 2022. State your prediction and your opinion about its reliability.

3. (a) Use a point estimate and confidence interval to assess the difference between the amounts of precipitation in Decembers for which the NAO index is positive and for which it is negative. Include in your report the formulae for your point estimate and interval.

(b) Use a point estimate and confidence interval to assess the difference between the temperatures in Decembers for which the NAO index is positive and for which it is negative. Include in your report the formulae for your point estimate and interval.

(c) Describe qualitatively and very briefly how predictions of the event considered in 2(b) might be affected if we knew in advance whether the NAO index for December 2022 would be positive or negative.

R functions

Here are R functions which compute the cdf, pdf and quantiles of the distribution used in part 2.

pmodel - function(y, alpha, beta) { 1 - exp(-exp((y - alpha) / beta))

} dmodel - function(y, alpha, beta) {

z - exp((y - alpha) / beta) z * exp(-z) / beta

}

qmodel - function(p, alpha, beta) {

alpha + beta * log(-log(1 - p))

}

After you cut and paste these functions in to your R session, you will be able to use them as follows: pmodel(y, alpha, beta) will compute the cdf, F, at y = y for a = alpha and ß = beta; dmodel(y, alpha, beta) will compute the pdf at y = y for a = alpha and ß = beta; qmodel(p, alpha, beta) will compute the p-quantile for a = alpha and ß = beta.

2

MTH1004 summative coursework: report

Date set: Friday 4 March 2022

Submission date: Friday 1 April 2022 by noon

Return date: Friday 6 May 2022

Instructions

This coursework comprises 20% of the overall module assessment. It is an individual project and so you must not collaborate when working on this assessment. You should read the College guidelines on collaboration and plagiarism, which are on the following webpage.

http://intranet.exeter.ac.uk/emps/studentinfo/subjects/mathematics/assessment/academicmisconduct

Conduct the investigation outlined below and write a report of your findings, addressing each of the three parts of the investigation in a separate section of your report. Your report should comprise at most three sides of word-processed text (e.g. written with Word or LaTeX) followed by as many pages of figures or tables that you feel is appropriate and then an appendix containing the R code (with explanatory comments but without the R output) that you used to obtain your reported results. No credit will be awarded to additional pages of text. You must use the page size A4 and a font size of at least 11 points, lines must be single-spaced and all margins must be at least one inch wide. You must save your report as a pdf file and upload it to BART at https://bart.exeter.ac.uk by the date and time shown above. Approximately 25 marks are available for each part of the investigation, with another 25 for the presentation (structure, clarity, language etc.) of your report.

Investigation

The Central England Temperature (CET) data set is the longest instrumental temperature record in the world. The data set contains temperatures (in degrees Celsius, ?C) representing the average temperature each month since January 1659 in an area of central England. The Central England Precipitation (CEP) data set is a similar record for precipitation amounts (in millimetres). Precipitation is rainfall, snow, hail etc. The North Atlantic Oscillation (NAO) index is a measure of the difference in atmospheric pressure between the Azores and Iceland. Changes in the NAO index are known to be associated with changes in European weather. The file Climate.RData is available on the module webpage and contains the monthly values of the CET, CEP and NAO data for each December from 1873 to 2020.

1. Consider modelling the precipitation data as realisations of independent and identically distributed random variables. Choose a parametric model for their distribution, estimate the parameters using the method of moments, and assess the realism of your model using appropriate graphs. Include in your report a clear statement of your model and the formulae for your estimates. You do not need to include the derivation of these formulae.

/continued...

1

2. Consider modelling the temperature data as realisations of independent and identically distributed random variables with the cumulative distribution function

y - a

F(y) = 1 - exp -exp

ß

for -8 y 8, where a ? (-8,8) and ß 0 are parameters. The expectation of this distribution is a - 0.5772ß and the variance is p2ß2/6. R functions which compute the pdf, cdf and quantiles of this distribution are given below.

(a) Use the method of moments to estimate the parameters of this model and assess the realism of the model using appropriate graphs. Include in your report a clear statement of the formulae for your estimates. You do not need to include the derivation of these formulae.

(b) Suppose that some households receive financial support if the December mean temperature is below 0?C. Use your fitted model to predict the probability that the mean temperature will be below 0?C in December 2022. State your prediction and your opinion about its reliability.

3. (a) Use a point estimate and confidence interval to assess the difference between the amounts of precipitation in Decembers for which the NAO index is positive and for which it is negative. Include in your report the formulae for your point estimate and interval.

(b) Use a point estimate and confidence interval to assess the difference between the temperatures in Decembers for which the NAO index is positive and for which it is negative. Include in your report the formulae for your point estimate and interval.

(c) Describe qualitatively and very briefly how predictions of the event considered in 2(b) might be affected if we knew in advance whether the NAO index for December 2022 would be positive or negative.

R functions

Here are R functions which compute the cdf, pdf and quantiles of the distribution used in part 2.

pmodel - function(y, alpha, beta) { 1 - exp(-exp((y - alpha) / beta))

} dmodel - function(y, alpha, beta) {

z - exp((y - alpha) / beta) z * exp(-z) / beta

}

qmodel - function(p, alpha, beta) {

alpha + beta * log(-log(1 - p))

}

After you cut and paste these functions in to your R session, you will be able to use them as follows: pmodel(y, alpha, beta) will compute the cdf, F, at y = y for a = alpha and ß = beta; dmodel(y, alpha, beta) will compute the pdf at y = y for a = alpha and ß = beta; qmodel(p, alpha, beta) will compute the p-quantile for a = alpha and ß = beta.

2

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