1. (5 points cach) The study of how musical pieces and artists influence each other is...

Transcribed Image Text:

1. (5 points cach) The study of how musical pieces and artists influence each other is called music genomics. In this project, we apply singular value decomposition to our study of music genomics. This technique requires a binary matrix M (a matrix of zeros and ones). Use the raw data from Figure 1 to create a song-trait matrix. In order to do so, use the following values to determine whether to assign a zero or a one to each raw data value. In the crescendo category, assign a 0 for raw data less than 8 and assign a 1 otherwise, and well call 8 the cut off for this category. Use cutoffs of 2 for decrescendos, 22 for staccato, 10 for portamento, 109 for tempo, and 4 for intervals. Decrescendo SONG Beat It How Will I Know Together Again No Scrubs Poker Face Crazy in Love Only Girl Say My Name Dreamlover Like a Virgin Billie Jean I Wanna Dance All for You Waterfalls Born This Way Single Ladies Umbrella Bootylicious Fantasy Like a Prayer Crescendo 4 17 4 10 4 7 7 18 10 6 4 9 10 0 3 6 9 8 5 14 7 3 3 4 2 4 2 4 2 2 4 4 0 4 2 4 2 1 0 al 6 2 1 1 Staccato 18 34 16 14 16 19 56 21 21 19 29 14 14 38 10 Ar 15 14 36 34 20 16 10 Portamento FIGURE 1: Raw data for 20 the number 1 Billboard Hit choruses 0 0 0 24 8 24 11 26 8 D 0 9 12 15 8 9 28 4 16 7 0 Tempo 132 120 120 90 120 100 126 70 104 118 117 120 114 84 124 96 86 104 112 120 Intervals 2.417 5.281 5.875 5.563 4.75 5.563 3.24 3.25 5.25 5,375 2.938 3.15 1.563 4.389 1.5 2.656 5.292 4.375 7.75 5.375 4.375 (a) S = MMT is called the similarity matrix related to the binary song-trait matrix. Find the similarity matrix S and describe what S₁, and S, where i ‡ j represent. (b) Compute the singular value decomposition of S keeping only the 2 largest singular values. Type: [U, W,V] suds(S.2). (c) With 20 songs in the data set, the matrix U will be a 20 x 2 matrix. The rows of U are the new coordinates for each of the songs, s, based on this decomposition. Plot the new coordinates of the songs to see which songs are closest together based on the SVD algorithm. Note that you might want to create a labeling of the songs before plotting them. To do this type: labels = cellstr(num2str ([1: length(U)]')); plot (U(:,1),U(: ,2), 'rx'); text (U( 1),U: 2), labels) (d) Let us now assume that there is one song sy that we wish to compare with all of the others. We will use the cosine distance from the new coordinates of s₁, Michael Jacksons Beat It, to rank the original songs. The distance from si to s, is d. = Find d; for each song, other than si, and rank the songs based on how close they are to 8₁. (e) It is possible that an artist has been influenced by Michael Jackson but not his song Beat It. Define q as the average of the new coordinates for 81 and 811, both Michael Jackson songs. Find the distance from each song to gi, d=and rank the songs based on how close they are to ql. 1. (5 points cach) The study of how musical pieces and artists influence each other is called music genomics. In this project, we apply singular value decomposition to our study of music genomics. This technique requires a binary matrix M (a matrix of zeros and ones). Use the raw data from Figure 1 to create a song-trait matrix. In order to do so, use the following values to determine whether to assign a zero or a one to each raw data value. In the crescendo category, assign a 0 for raw data less than 8 and assign a 1 otherwise, and well call 8 the cut off for this category. Use cutoffs of 2 for decrescendos, 22 for staccato, 10 for portamento, 109 for tempo, and 4 for intervals. Decrescendo SONG Beat It How Will I Know Together Again No Scrubs Poker Face Crazy in Love Only Girl Say My Name Dreamlover Like a Virgin Billie Jean I Wanna Dance All for You Waterfalls Born This Way Single Ladies Umbrella Bootylicious Fantasy Like a Prayer Crescendo 4 17 4 10 4 7 7 18 10 6 4 9 10 0 3 6 9 8 5 14 7 3 3 4 2 4 2 4 2 2 4 4 0 4 2 4 2 1 0 al 6 2 1 1 Staccato 18 34 16 14 16 19 56 21 21 19 29 14 14 38 10 Ar 15 14 36 34 20 16 10 Portamento FIGURE 1: Raw data for 20 the number 1 Billboard Hit choruses 0 0 0 24 8 24 11 26 8 D 0 9 12 15 8 9 28 4 16 7 0 Tempo 132 120 120 90 120 100 126 70 104 118 117 120 114 84 124 96 86 104 112 120 Intervals 2.417 5.281 5.875 5.563 4.75 5.563 3.24 3.25 5.25 5,375 2.938 3.15 1.563 4.389 1.5 2.656 5.292 4.375 7.75 5.375 4.375 (a) S = MMT is called the similarity matrix related to the binary song-trait matrix. Find the similarity matrix S and describe what S₁, and S, where i ‡ j represent. (b) Compute the singular value decomposition of S keeping only the 2 largest singular values. Type: [U, W,V] suds(S.2). (c) With 20 songs in the data set, the matrix U will be a 20 x 2 matrix. The rows of U are the new coordinates for each of the songs, s, based on this decomposition. Plot the new coordinates of the songs to see which songs are closest together based on the SVD algorithm. Note that you might want to create a labeling of the songs before plotting them. To do this type: labels = cellstr(num2str ([1: length(U)]')); plot (U(:,1),U(: ,2), 'rx'); text (U( 1),U: 2), labels) (d) Let us now assume that there is one song sy that we wish to compare with all of the others. We will use the cosine distance from the new coordinates of s₁, Michael Jacksons Beat It, to rank the original songs. The distance from si to s, is d. = Find d; for each song, other than si, and rank the songs based on how close they are to 8₁. (e) It is possible that an artist has been influenced by Michael Jackson but not his song Beat It. Define q as the average of the new coordinates for 81 and 811, both Michael Jackson songs. Find the distance from each song to gi, d=and rank the songs based on how close they are to ql.