Assignment 1

<rst> <rst-options: 'toc' = False/> This quick assignment considers the `food texture <http://datasets.connectmv.com/info/food-texture>`_ data (introduced in class 2). There are 5 variables in the data table:

1. . ``Oil``: percentage oil in the pastry
2. . ``Density``: the product’s density (the higher the number, the more dense the product)
3. . ``Crispy``: a crispiness measurement, on a scale from 7 to 15, with 15 being more crispy.
4. . ``Fracture``: the angle, in degrees, through which the pasty can be slowly bent before it fractures.
5. . ``Hardness``: a sharp point is used to measure the amount of force required before breakage occurs.

Please provide answers to these questions:

1. . Calculate the mean centering vector (a :math:`1 \times 5` vector).
2. . Calculate the scaling vector (a :math:`1 \times 5` vector) and indicate whether you multiply or divide columns in \(\mathbf{X}\) by the corresponding entries in your vector.
3. . Draw a scatter plot for ``Oil`` vs ``Density`` using all 50 data points from the raw data table.
4. . Draw a scatter plot for ``Oil`` vs ``Density`` after you have centered and scaled the data. Any observations when you compare it to the previous scatter plot?
5. . Use the software to calculate PCA model and report the :math:`R^2` value for the first and second component. What is the total :math:`R^2` using 2 components?
6. . Report the cumulative :math:`R^2` value for each of the 5 variables after adding (a) one component and (b) two components.
7. . Write down the values of the :math:`p_1` loading vector.
8. . What are the characteristics of pastries with large negative :math:`t_1` values?
9. . What is the second component in the model describing?
10. . Replicate the calculation for the :math:`t_1` value for pastry B758. Show each of the 5 terms that make up this linear combination.

• Hand in your answers at the next class; we will go through the assignment interactively during the next class*.

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