Applications Of Data Science And Statistical Modelling - Principal Component Analysis - Data Analytics Assignment Help

Applications Of Data Science And Statistical Modelling - Data Analytics Assignment Help

Assignment Task:

1. Use R to solve the following system of linear equations: x + y + z + w = 13, 2x + 3y − w = −1, −3x + 4y + z + 2w = 10, x + 2y − z + w = 1.

2. Download the food.csv dataset from the ELE page. The data is from the DEFRA website showing the consumption in grams (per person, per week) of 17 different types of foodstuff measured and averaged in the four countries of the United Kingdom in 1997.

a. As a first step read in the data and transform it into a form that can be used for Principal Component Analysis (PCA). Hint: When transforming the data keep in mind that PCA can only be applied on numeric datasets and that the food dataset has 17 features and 4 observations.

b. Perform PCA (without reducing the dimensions) similarly to Exercise 4 of the first Problem Sheet. Find the eigenvalues of the covariance matrix, and determine the minimum number of eigenvalues we would have to choose in order to explain at least 95% of the variation in the data.

c. Project the data onto its first principal component and plot the result. Next, project the data onto its first two principal components and plot the result. Using these plots, can you identify any patterns in the data? In particular, is there a country that appears to be different from the rest? If so which one is it?

d. Using your conclusions from part c., can you identify food types in the original dataset that might cause the observed pattern?

3. The aim of this exercise is to compress a colour RBG-image (that is, one that’s not greyscaled) of your choice using Principal Component Analysis (PCA) similarly to Exercise 5 of the first Problem Sheet. Note that when you import a colour RBG-image into R that image will be stored in a 3-dimensional array. This array contains three matrices, where the first matrix gives the Red colour channel, the second matrix gives the Green colour channel, and the third matrix gives the Blue colour channel.

a. Separate the Red, Green and Blue colour channels, and perform PCA on the resulting matrices separately.

b. Decide how many principal components you want to use, then construct the compressed version of each colour channel. Recall that a colour RBG-image is stored in a 3- dimensional array where each matrix corresponds to a colour channel. Thus in order to get the compressed version of your original image you need to recreate this 3-dimensional array from the compressed colour components.

c. Export at least 4 images with different resolutions. At least one of the four images should be unrecognisable (or barely recognisable), and at least one should be easily recognisable. Find the compression ratio for each of the four images

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