MA3202-MA4202-MA7202 Introduction to Functional Data Analysis 2024-25

Coursework 2

Deadline:

Friday, 28 March 2025, at 12:00.

Student Details: Full Name: Student Number:

Submission Instructions:

Please submit your work electronically as a single PDF file (including R code, graphs, explana- tions, and descriptions) via Blackboard. No more than 15 pages.

Please do Not submit R scripts, you can copy the code into the PDF file.

It is your responsibility to double-check whether you submit the correct file.

Email submissions will not be accepted.

Marking Criteria:

Coursework is marked out of 100 points. The number of marks of each main question is indicated at the beginning of each.

The work should be your own. Copying from other students or resources will result in a ZERO

mark.

Late Submission: In cases where there are no accepted mitigating circumstances, late submission of coursework will lead automatically to the imposition of a penalty. Your mark will be reduced in line with the Universitys penalties scheme for late submission of coursework. The Universitys regulations about the late submission of coursework are found in Senate Regulation 7: governing the Assessment of Taught Programmes. Refer specifically to 7.94 for details on penalties for late submission.

Question [100 marks]

Australian age-specific fertility data

This project examines historical fertility patterns and demographic changes in Australia from 1921 to 2015, using data from the Australian Bureau of Statistics (Cat.No.3105.0.65.001, Table 38). The data consist of annual fertility rates by the age of the mother in single years from ages 15 to 49, defined as the number of live births per 1,000 female residents of the same age group as of June 30 each year. By analysing variations in fertility rates across different age groups and time periods, this study aims to identify key demographic shifts, the impact of social and economic factors on fertility trends, and potential policy implications.

Therefore, we will investigate the fertility data for Australian women aged 15 49 from 1921 to 2015. The data file Australiafertility is a built-in dataset in the fda package, which is a non-periodic dataset.

Implement the following R code to observe the raw dataset.

Perform the following tasks based on the Australiafertility functional dataset above.

Task 1 [20 marks] Implement Functional Smoothing Techniques.

[10 marks] Identify the relative information from the Australiafertility functional dataset, and use the order 4 B-splines to smooth this dataset. Determine the smoothing parameter ? by Generalized Cross-Validation, and explain your choice.

[10 marks] Using the selected value of ?, smooth the dataset and generate a plot of the result- ing functional data. Then, adjust the smoothing parameter ? as needed to prevent overfitting or oversmoothing.

Task 2 [35 marks] Implement Descriptive Statistical Analysis.

[10 marks] Evaluate the mean and standard deviation of this functional dataset and plot them. Briefly describe your results.

[10 marks] Evaluate the bivariate covariance of this functional dataset and plot it. Briefly describe your results.

[5 marks] Generate the contour plot of the variance surface. Briefly describe your results.

[10 marks] Generate the Phase-Plane plots. Briefly describe your results.

Task 3 [45 marks] Implement Functional Principal Component Analysis (FPCA).

[15 marks] Use the scree-plot method to determine the number of principal components (or har- monics). And explain your results.

[15 marks] After determining the number of principal components (or harmonics), calculate the eigenvalues of the principal components (or harmonics) and their respective proportions of the total variation. Provide a description of these principal components (or harmonics).

[15 marks] Implement the VARIMAX rotation technique and compute the eigenvalues of the princi- pal components (or harmonics) along with their respective proportions of the total variation. Provide a description of these rotated principal components (or harmonics).