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4 Dimensionality reduction (PCA & t-SNE)

The fourth lecture from the Machine Learning course series of lectures. This lecture first introduces a problem of visualising multi-dimensional data on fewer dimensions and later discusses one of the most popular methods for reducing dimensionality - principal component analysis (PCA). Later, also t-SNE is mentioned briefly as a non-linear alternative to PCA. A link to my github (https://github.com/skyfallen/MachineLearningPracticals) with practicals that I have designed for this course in both R and Python. I can share keynote files, contact me via e-mail: dmytro.fishman@ut.ee.

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4 Dimensionality reduction (PCA & t-SNE)

  1. 1. Introduction to Machine Learning (Dimensionality reduction) Dmytro Fishman (dmytro@ut.ee)
  2. 2. So far we have been fearless drawing multi-dimensional instances on two-dimensional planes
  3. 3. 28px 28px
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  9. 9. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 126, 213, 34, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19, 227, 254, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 114, 254, 254, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 178, 254, 254, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 60, 236, 254, 254, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78, 254, 254, 254, 84, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78, 254, 254, 231, 56, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 191, 254, 254, 185, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 233, 254, 254, 105, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 233, 254, 254, 30, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 93, 248, 254, 243, 27, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 192, 254, 254, 130, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 32, 254, 254, 248, 95, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 41, 254, 254, 231, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 186, 254, 254, 160, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 21, 203, 254, 188, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 86, 254, 255, 179, 8, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 86, 254, 254, 254, 254, 130, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 68, 240, 254, 254, 250, 103, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 92, 255, 207, 91, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 28px 28px Pixel #X Pixel#Y 41 How about the rest of 782 pixels? 102 But we cannot really visualise all 784 dimensions... or can we?
  10. 10. Dimensionality reduction Patients Healthy
  11. 11. The following material was inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0
  12. 12. The following material was inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0 With a bit of fixes, though
  13. 13. Patients Healthy
  14. 14. Patients Healthy Each dot in this plot represents a person
  15. 15. Patients Healthy Every person in its turn could be described by hundreds of protein reactivities Each dot in this plot represents a person
  16. 16. Patients Healthy Every person in its turn could be described by hundreds of protein reactivities Each dot in this plot represents a person The idea is that people with similar protein reactivity profiles cluster together
  17. 17. Patients Healthy 5, 98, 43, …, 9, 66, Patient
  18. 18. Patients Healthy 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy
  19. 19. Patients Healthy 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy 3, 45, 13, …, 7, 54, Patient
  20. 20. Patients Healthy 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy 3, 45, 13, …, 7, 54, Patient How is that even possible?
  21. 21. Patients Healthy PCA 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy 3, 45, 13, …, 7, 54, Patient How is that even possible?
  22. 22. Introduction to dimensions
  23. 23. 1-Dimensional data Person Protein #1 A 24 B 63 C 51 D 34 E 15
  24. 24. 1-Dimensional data 300 60 Person Protein #1 A 24 B 63 C 51 D 34 E 15
  25. 25. 1-Dimensional data 300 60 300 60 Uniformly distributed data Person Protein #1 A 24 B 63 C 51 D 34 E 15
  26. 26. 1-Dimensional data 300 60 300 60 Uniformly distributed data 30 60 Non-uniform distribution 0 Person Protein #1 A 24 B 63 C 51 D 34 E 15
  27. 27. 2-Dimensional data 300 60 Protein #1 Person Protein #1 A 24 B 63 C 51 D 34 E 15
  28. 28. 2-Dimensional data 30 60 Protein#2 Person Protein #1 Protein #2 A 24 B 63 C 51 D 34 E 15 300 60 Protein #1 0
  29. 29. 2-Dimensional data Person Protein #1 Protein #2 A 24 29 B 63 C 51 D 34 E 15 A30 60 Protein#2 300 60 Protein #1 0
  30. 30. 2-Dimensional data A Person Protein #1 Protein #2 A 24 29 B 63 C 51 D 34 E 15 30 60 Protein#2 300 60 Protein #1 0
  31. 31. 2-Dimensional data A Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 D 34 E 15 B 30 60 Protein#2 300 60 Protein #1 0
  32. 32. 2-Dimensional data A B And so on… Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 D 34 E 15 30 60 Protein#2 300 60 Protein #1 0
  33. 33. 2-Dimensional data Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 32 D 34 56 E 15 4 … … … 30 60 Protein#2 300 60 Protein #1 0
  34. 34. 2-Dimensional data Protein profiles are correlated Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 32 D 34 56 E 15 4 … … … 30 60 Protein#2 300 60 Protein #1 0
  35. 35. 2-Dimensional data Here is no correlation Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 32 D 34 56 E 15 4 … … … 30 60 Protein#2 300 60 Protein #1 0
  36. 36. 3-Dimensional data Person Protein #1 Protein #2 A 24 29 B 63 59 C 51 32 D 34 56 E 15 4 … … … 30 60 Protein#2 300 60 Protein #1 0
  37. 37. 3-Dimensional data Person P1 P2 P3 A 24 29 B 63 59 C 51 32 D 34 56 E 15 4 … … … Protein #330 60 Protein#2 300 60 Protein #1 0 30 60
  38. 38. 3-Dimensional data Person P1 P2 P3 A 24 29 33 B 63 59 C 51 32 D 34 56 E 15 4 … … … A 30 60 Protein #330 60 Protein#2 300 60 Protein #1 0
  39. 39. Dimensions 300 60 1-Dimensional
  40. 40. Dimensions 300 60 30 6 3 6 2-Dimensional 1-Dimensional
  41. 41. Dimensions 300 60 30 6 3 6 3-Dimensional 30 6 3 6 3 6 2-Dimensional 1-Dimensional
  42. 42. How about 200-D data?
  43. 43. Are all D are equally important? How about 200-D data?
  44. 44. Are all D are equally important? Let’s go back to 2-D example How about 200-D data?
  45. 45. 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
  46. 46. Variation is from left to right 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
  47. 47. Projected data - does not look too different 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
  48. 48. We can safely remove 2nd dimension 30 60 Protein#2 300 60 Protein #1 0 300 60 Protein #1 Importance of axes Projected data - does not look too different
  49. 49. 30 60 Protein#2 300 60 Protein #1 0 300 60 Protein #1 The important variation is from left to right We can safely remove 2nd dimension Importance of axes Projected data - does not look too different
  50. 50. 30 60 Protein#2 300 60 Protein #1 0 Principle components
  51. 51. 30 60 Protein#2 300 60 Protein #1 0 Data is mostly spread along this line Principle components
  52. 52. 30 60 Protein#2 300 60 Protein #1 0 Data is mostly spread along this line and a little bit along this line Principle components
  53. 53. 30 60 Protein#2 300 60 Protein #1 0 Data is mostly spread along this line and a little bit along this line What if we make new axes from these lines? Principle components
  54. 54. 30 60 Protein #2 30 0 60 Protein #1 0 New X and Y axes Principle components
  55. 55. Principle components Easier to see right/left and above/below variation New X and Y axes
  56. 56. Principle components PC1 Easier to see right/left and above/below variation PC2 These new axes are called Principle Components or PCs New X and Y axes
  57. 57. Principle components PC1 Easier to see right/left and above/below variation PC1 spans along the direction of the most PC2 These new axes are called Principle Components or PCs New X and Y axes
  58. 58. Principle components PC1 Easier to see right/left and above/below variation PC1 spans along the direction of the most PC2 PC2 spans along the direction of the most These new axes are called Principle Components or PCs New X and Y axes
  59. 59. Principle components PC1 Easier to see right/left and above/below variation PC1 spans along the direction of the most PC2 PC2 spans along the direction of the most These new axes are called Principle Components or PCs Principle Components are always orthogonal one to another New X and Y axes
  60. 60. Principle components PC1 PC2 If original data would have 3 dimensions, we would have 3rd PC PC3
  61. 61. Principle components PC1 PC2 If original data would have 3 dimensions, we would have 3rd PC PC3 There is always as many PCs as there are dimensions
  62. 62. Principle components If original data would have 3 dimensions, we would have 3rd PC There is always as many PCs as there are dimensions Each new PCs is guaranteed to explain less variance PC1 PC2 PC3
  63. 63. Principle components If original data would have 3 dimensions, we would have 3rd PC There is always as many PCs as there are dimensions Usually it is enough to project data points onto first two PCs to see important patterns PC1 PC2 PC3 Each new PCs is guaranteed to explain less variance
  64. 64. PCA Now we know what those PC1 and PC2 stand for!
  65. 65. PCA Now we know what those PC1 and PC2 stand for! Wait! But so far we have been talking only about 2 proteins!
  66. 66. PCA Now we know what those PC1 and PC2 stand for! We are coming to this part Wait! But so far we have been talking only about 2 proteins!
  67. 67. Where PCs come from PC1 PC2
  68. 68. PC1 PC2 These are vectors Where PCs come from
  69. 69. PC1 PC2 30 3 Where PCs come from These are vectors All vectors have coordinates
  70. 70. PC1 PC2 30 3 For example, this one has coordinates (3,3) Where PCs come from These are vectors All vectors have coordinates
  71. 71. PC1 PC2 30 3 For example, this one has coordinates (3,3) In 200-D space vectors have 200 coordinates Where PCs come from These are vectors All vectors have coordinates
  72. 72. 30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? Where PCs come from
  73. 73. 30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? By minimising the distance from points to the vector Where PCs come from
  74. 74. 30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? By minimising the distance from points to the vector Where PCs come from
  75. 75. 30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? Coordinates of this vector are coordinates of PC1 By minimising the distance from points to the vector Where PCs come from PC1
  76. 76. 30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? Coordinates of this vector are coordinates of PC1 By minimising the distance from points to the vector Where PCs come from Coordinates of PC2 are chosen in similar fashion but PC2 should be orthogonal to PC1 PC1 PC2
  77. 77. PCA Example Reactivities PCs coordinates PC1 PC2 2 3 4 12 … … -5 0 Let’s find coordinates of Person A in terms of PCs Person Pr1 Pr2 … PrN A 24 29 … 11 B 63 59 … 1 C 51 32 … 23 D 34 56 … 2 E 15 4 … 8 … … … … …
  78. 78. PCA Example Reactivities PCs coordinates PC1 PC2 2 3 4 12 … … -5 0 Let’s find coordinates of Person A in terms of PCs Person Pr1 Pr2 … PrN A 24 29 … 11 B 63 59 … 1 C 51 32 … 23 D 34 56 … 2 E 15 4 … 8 … … … … … asmanyasthere areproteins(N)
  79. 79. PCA Example Reactivities PCs coordinates Let’s find coordinates of Person A in terms of PCs Person A (PC1 score) = PC1 PC2 2 3 4 12 … … -5 0 Person Pr1 Pr2 … PrN A 24 29 … 11 B 63 59 … 1 C 51 32 … 23 D 34 56 … 2 E 15 4 … 8 … … … … …
  80. 80. PCA Example Reactivities PCs coordinates Let’s find coordinates of Person A in terms of PCs Person A (PC1 score) = 24*2 + 29*4 +… + 11*(-5) = 11 PC1 PC2 2 3 4 12 … … -5 0 Person Pr1 Pr2 … PrN A 24 29 … 11 B 63 59 … 1 C 51 32 … 23 D 34 56 … 2 E 15 4 … 8 … … … … …
  81. 81. PCA Example Reactivities PCs coordinates Person A (PC2 score) = 24*3+ 29*12 + … +11*0 = 21 Reactivities PCs coordinates Let’s find coordinates of Person A in terms of PCs Person A (PC1 score) = 24*2 + 29*4 +… + 11*(-5) = 11 PC1 PC2 2 3 4 12 … … -5 0 Person Pr1 Pr2 … PrN A 24 29 … 11 B 63 59 … 1 C 51 32 … 23 D 34 56 … 2 E 15 4 … 8 … … … … …
  82. 82. Principle components Principle component 1 Principle component 2 21 110 0 A Person A (PC2 score) = 24*3+ 29*12 + … +11*0 = 21 Person A (PC1 score) = 24*2 + 29*4 +… + 11*(-5) = 11
  83. 83. Principle components Principle component 1 Principle component 2 Person B (PC2 score) = 65 Person B (PC1 score) = 60 65 600 0 A B
  84. 84. Principle components Principle component 1 Principle component 2 0 0 A B C D E G
  85. 85. Principle components Principle component 1 Principle component 2 0 0 A B C D E G The idea is that similar points in multi-dimensional space get located close in fewer dimensions Healthy Patients
  86. 86. PCA is good, but it is a linear algorithm, meaning that it cannot capture complex relationship between features
  87. 87. PCA is good, but it is a linear algorithm, meaning that it cannot capture complex relationship between features There are always alternative options to consider…
  88. 88. https://www.analyticsvidhya.com/blog/2017/01/t-sne-implementation-r-python/ t-Distributed Stochastic Neighbour Embedding (t-SNE) visualisation is form http://distill.pub/2016/misread-tsne/ t-SNE is non-linear dimensionality reduction algorithm t-SNE uses conditional probabilities to represent similarity between points Has been shown to better represent underlying structure of multidimensional data
  89. 89. Practice
  90. 90. https://www.analyticsvidhya.com/blog/2017/01/t-sne-implementation-r-python/ t-SNE visualisation is form http://distill.pub/2016/misread-tsne/ PCA O(N2) makes it very slow Meaning of features is lost Has been shown to capture the structure of multi-dimensional data better (than PCA) Works relatively fast even on big datasets Transformed features could be traced back Usually is not so good in figuring out hidden structure
  91. 91. Demo
  92. 92. References • Machine Learning by Andrew Ng (https://www.coursera.org/learn/machine- learning) • Introduction to Machine Learning by Pascal Vincent given at Deep Learning Summer School, Montreal 2015 (http://videolectures.net/ deeplearning2015_vincent_machine_learning/) • Welcome to Machine Learning by Konstantin Tretyakov delivered at AACIMP Summer School 2015 (http://kt.era.ee/lectures/aacimp2015/1-intro.pdf) • Stanford CS class: Convolutional Neural Networks for Visual Recognition by Andrej Karpathy (http://cs231n.github.io/) • Data Mining Course by Jaak Vilo at University of Tartu (https://courses.cs.ut.ee/ MTAT.03.183/2017_spring/uploads/Main/DM_05_Clustering.pdf) • Machine Learning Essential Conepts by Ilya Kuzovkin (https:// www.slideshare.net/iljakuzovkin) • From the brain to deep learning and back by Raul Vicente Zafra and Ilya Kuzovkin (http://www.uttv.ee/naita?id=23585&keel=eng)
  93. 93. www.biit.cs.ut.ee www.ut.ee www.quretec.ee

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