QBUS1040: Foundations of Business Analytics Homework 3
Semester 2, 2019
This homework consists of eight problems that require you to submit a written response and a coding component. The problems that require a written response are described in this paper. You need to print it and write your answers directly in this paper. You should use scratch paper (which you will not turn in) to do your rough work. You should submit a scanned copy of your written solution as a PDF via Canvas.
Please do not submit photos of your written solution as it is difficult for the marking system to recognise your submission. Also, please do not use a tablet to write your homework as it is very likely that your submission will not be processed correctly through the marking system. Please use a conventional scanner. You should double check your PDF before you submit the file. Your PDF submission must be a 5-page document. The file size of your PDF document should not exceed 128MB, or it will not be accepted by the submission system.
The coding components are described in a separate Jupyter Notebook file. You should also download the Jupyter Notebook file for Homework 3 and enter your code in the space provided. You should submit your code as a Jupyter notebook file via Canvas.
The homework is due by 5pm on Friday, the 27th of September. Late homework will not be accepted. Violation of the above submission instructions may incur a 30% penalty.
For all problems where you are asked for a free-form answer, it must be written in the box below the problem. We won’t read anything outside the boxes.
All problems have equal weight. Some are easy. Others, not so much.
Tutorial time: Tutor’s name:
(For QBUS1040 staff only)
1. (10 points) Centroid interpretations. The n-vectors x1, . . . , xN contain n attributes of N patients admitted to a hospital. The first component, (xi)1, is the age of patient i. The second component, (xi)2, is 1 if the patient is having trouble breathing, and 0 if not. The third component,(xi)3, is the body mass index of the patient. (The other components give other attributes.) A QBUS1040 graduate carries out k-means on this data set, with k = 25. She finds the 18th centroid or group representative is z18 = (41.7,0.36,19.8,…,29.6). Give a simple short interpretation in English of the first three components.
2. (10 points) Orthogonalizing vectors. Suppose that a and b are any n-vectors. Show that we can always find a scalar γ so that (a−γb) ⊥ b, and that γ is unique if b ̸= 0. (Give a formula for the scalar γ.) In other words, we can always subtract a multiple of a vector from another one, so that the result is orthogonal to the original vector. The orthogonalization step in the Gram-Schmidt algorithm is an application of this. You should think about what happens when b = 0 and when b ̸= 0.
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QBUS1040 Homework 3 Semester 2, 2019 3. (10 points) Let α, β, and η be scalars and let a, b, and c be pairwise orthogonal n-vectors. (This
means that a⊥b, a⊥c, and b⊥c.) Express ∥αa + βb + ηc∥ in terms of ∥a∥, ∥b∥, ∥c∥, α, β, and η.
4. (10 points) Running Gram-Schmidt algorithm twice.
You should provide your answer to parts (a)-(c) in the Homework 3 Jupyter Notebook
file and put your answer to part (d) in the space below.
(a) Run the Gram-Schmidt algorithm on the given set of vectors described in the Jupyter Notebook file and return the vectors q1, . . . , q15.
(b) Run the Gram-Schmidt algorithm on vectors q1 , . . . , q15 obtained from the previous part and return the vectors z1, . . . , z15.
(c) Compute the distances between vectors q1 and z1, q2 and z2, …, q15 and z15.
(d) Explain the magnitude of the distances obtained in the previous part. You may want to start by saying what vectors q1,…,q15 and z1,…,z15 are.
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5. (10 points) Matrix sizes. Suppose A, B, and C are matrices that satisfy A + BBT = C. Determine which of the following statements are necessarily true. (There may be more than one true statement.)
(a) A is square.
⃝ True ⃝ False
(b) A and B have the same dimensions. ⃝ True ⃝ False
(c) A, B, and C have the same number of rows. ⃝ True ⃝ False
(d) B is a tall matrix. ⃝ True ⃝ False
6. (10 points) Matrix dimensions. Suppose A is a 5 × 10 matrix, B is a 20 × 10 matrix, and C is a 10 × 10 matrix. Determine whether each of the following expressions make sense. If the expression makes sense, give its dimensions.
?A? (e) B A C
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7. (10 points) Portfolio sector exposures. The n-vector h denotes a portfolio of investments in n assets, with hi the dollar value invested in asset i. We consider a set of m industry sectors, such as pharmaceuticals or consumer electronics. Each asset is assigned to one of these sectors. (More complex models allow for an asset to be assigned to more than one sector.) The exposure of the portfolio to sector i is defined as the sum of investments in the assets in that sector. We denote the sector exposures using the m-vector s, where si is the portfolio exposure to sector i. (When si = 0, the portfolio is said to be neutral to sector i.) An investment advisor specifies a set of desired sector exposures, given as the m-vector sdes. Express the requirement s = sdes as a set of linear equations of the form Ah = b. (You must describe the matrix A and the vector b.) Remark. A typical practical case involves n = 1000 assets and m = 50 sectors. An advisor might specifysdes = 0 if she does not i have an opinion as how companies in that sectori will do in the future; she might specify a positive value for sdes if she thinks the companies i in that sector will do well (i.e., generate positive returns) in the future, and a negative value if she thinks they will do poorly.
8. (10 points) Orthogonality. Please refer to the Jupyter Notebook file for this problem.
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