ME453_HW5_FA23

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University of Illinois, Urbana Champaign *

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Mechanical Engineering

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Dec 6, 2023

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ME 453: Data Science in Manufacturing Quality Control Homework 5 Assigned: October 25, 2023 Due: November 8, 2023 Homework guidelines: 1. The total number of points is 100 and 130 for 3-credit-hour and 4-credit-hour sections, respectively. The assigned points are given beside questions. To get full credit you must SHOW ALL OF YOUR WORK. 2. Problems marked by “[4 credit only]” should be completed by students in the 4-credit-hour section. Students in the 3-credit-hour section are welcome to work on these problems and the earned points will be used to compensate for lost points in homework assignments. 3. Indicates the problem which you need to use Python. 4. A complete submission features the following items: (a) a brief report including all figures and results, and explanations of necessary steps taken to obtain them; and (b) and the source code (Python is recommended). 5. Item (a) can be scanned copies of handwritten or typeset reports. Both items shall be submitted through Canvas. Problem 1 (20 points + 10 points) Suppose the estimated covariance matrix for a data set is: ˆ Σ = 16 5 5 9 Use software to complete the following tasks. (1) Compute the eigenvalues and eigenvectors for the covariance matrix. (10 points) (2) Obtain the projection matrix W , which is no other than eigenvectors stacked horizontally. (5 points) (3) Project an observation x = [5 4] T to the newly mapped space. (5 points) (4) [4 credit only] Calculate the explained variance ratio for each PC (principal component). Which PC is more important? Justify your answer. (10 points) 1

Problem 2 (80 points + 20 points) In the bearing.csv provided on Canvas, the accelerometer signal is collected during a ro- tational test of bearing. A raw signal is plotted with FFT result in the figure below. We are interested in the effect of bearing ball wear, which has four level: slight, light, moderate, and severe. The wearing levels are stored in wearlevel.csv and referred as level 0-3 correspondingly. (1) Perform PCA on the dataset. Note that the mean should be removed at each time step (centered) for best results. (a) Plot scree plot of 30 PCs and determine the number of components to keep. Use α = 0 . 1. (15 points) (b) Visualize the data with the first three PCs in a 3D scatter plot. Distinguish the wear levels by different colors and provide a legend. (10 points) (2) Perform FFT and extract two frequency-domain features and provide a description for each feature. (15 points) (3) Visualize the data using handcrafted features obtained in (2) in a scatter plot. Distinguish the wear levels by different colors and provide a legend. (10 points) (4) Create a feature pool includes two frequency-domain features and first three PCs. Calculate Fisher’s ratio for ”level 0 - level 3” (Fisher’s ratio 1) and ”level 1 - level 2” (Fisher’s ratio 2) for all features. (20 points) (5) Select three features from the feature pool based on the following criterion: max (Fisher’s ratio 1 + Fisher’s ratio 2). Visualize the three selected features in a 3D plot. Use legend to distinguish the data points from different wear levels. (10 points) 2

(6) [4 credit only] Use the intensity from FFT as the data, remove the mean (centered) and perform PCA. Plot scree plot of 30 PCs and determine the number of components to keep. Use α = 0 . 1. (10 points) (7) [4 credit only] Compare and discuss the PCA results from (1) and (6) when using raw signal. (10 points) 3

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