EECS 16B Spring 2019 - UC Berkeley Final Exam

Aug 30, 2026

Uploaded by Rhys

UC Berkeley EECS 16B Spring 2019 final exam materials cover pre-exam procedures and a set of open questions followed by detailed problem-solving on principal component analysis. The PCA task models microphone recordings as column vectors, requiring identification of the first principal component using geometry and a singular value decomposition approach. It emphasizes rank-1 structure, normalization, zero-mean assumptions, and the correct selection of singular vectors consistent with column-wise data projection.

LanguageEnglish

CategoryExam

FormatPDF

Pages33

Reading Time1 h 23 min

AccessFree to download

Section 0: Pre-exam questions

Question 1

Question 2

PCA problem

Part (a)

Part (b)

| EECS 16B Spring 2019 | Designing Information Devices and Systems II
UC Berkeley Final Exam | | --- | --- | Exam location: GPB 100 PRINT your student ID: PRINT AND SIGN your name: , (ﬁrst and last) (signature) PRINT your discussion section and GSI(s) (the one(s) you attend): Row Number (front row is 1): Seat Number (left most is 1): Name and SID of the person to your left: Name and SID of the person to your right: Name and SID of the person in front of you: Name and SID of the person behind you: Section 0: Pre-exam questions (3 points) 1. What has been your favorite course at UC Berkeley so far? (1 pt) 2. What would you do on your perfect vacation? Describe how you would feel. (2 pts) Do not turn this page until the proctor tells you to do so. You can work on Section 0 above before time starts. Final Exam ©UCBEECS 16B, Spring 2019 . All Rights Reserved. This may not be publicly shared without explicit permission. 1 PRINT your name and student ID: [Extra page. If you want the work on this page to be graded, make sure you tell us on the problem's main page.] Final Exam ©UCBEECS 16B, Spring 2019 . All Rights Reserved. This may not be publicly shared without explicit permission. 2 PRINT your name and student ID: 3. PCA (14 pts) In this problem, we are going to think of our data points as being given in columns. You can imagine that the data points are recordings from a microphone. We take many such recordings. Our goal is to identify the principal components so that we could, in the future, project fresh recordings from the microphone onto those principal components to help us better understand what was being said. (a) (2 pts) Suppose for this part, that you have four observed data vectors (say corresponding to the same spoken word, being repeated four times) and all of them just happened to be multiples ofthe following 2 3 3 6 􀀀4 7 6-dimensional vector = 6 5 7 : (For your convenience, note that kk = 10.) 6 4 7 4􀀀3 5 You arrange the data vectors as the columns of a matrix A given by: 2 ~ ~ ~ ~3 A = 6 􀀀v 􀀀2v v 2v 7 (1) 4 5 You want to perform PCA to better understand your data. Find the ﬁrst principal component vector of A to explain the nature of your data points. (HINT: You don't need to compute any covariance matrices or compute any eigenvalue/eigenvectorsin this simple case. Also, be sure to think about what size vector you want as the answer. Don't forget to normalize!) Solution: Principal component analysis is in general about understanding how best to approximate our (potentially) high-dimensional data (like recordings from a microphone) with its lower-dimensional essence. The ﬁrst principal component is about seeing which one-dimensional line best approximates the data points—i.e. which is the line for which projecting the data points onto it results in “estimates” that are as close as possible to the data points. In the case of this problem, every point is explicitly given as a multiple of a single vector and so the data already lies on a straight line going through the origin. So, the ﬁrst principal component is just along the direction of . Because a principal component represents a direction, it is conventional to normalize the vector to have unit length. In this case, we are told that the vector has length 10, and so the answer is ~~1~~ . (Because the line is all that matters, you could also have used the negative of this 􀀀 ~~1~~.) A more methodical way to do PCA is to invoke the SVD. First, however, you need to make sure that your data is zero-mean because the SVD will only give you directions relative to the origin. In this problem, all the data is zero-mean by construction. The singular value decomposition of a matrix A is a way of decomposing A into a sum of rank 1 matrices. In this sum the ith rank 1 matrix is formed from taking the outer product of normalized column vectors i and normalized row vectors

FAQs

What is the purpose of Section 0 in this exam?

Section 0 includes pre-exam questions that students answer before starting the main timed work. The document specifies points for each question and allows work on Section 0 before time starts.

How does the PCA problem define the data points?

Data points are given as column vectors, interpreted as microphone recordings repeated for a spoken word. The goal is to find principal components so future recordings can be projected onto those directions.

Why must the data be zero-mean before applying SVD for PCA?

The solution explains that SVD yields directions relative to the origin, so centering is needed. In this problem, the data is constructed to be zero-mean by design.

Aug 30, 2026

Uploaded by Rhys