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4 Dimensionality reduction (PCA & t-SNE)

The document introduces the concept of dimensionality reduction in machine learning, focusing on visualizing multi-dimensional instances in a two-dimensional space. It discusses the importance and techniques of reducing dimensionality to facilitate data analysis and improve model performance. The document includes several visual representations and examples to illustrate these concepts.

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Introduction to Machine Learning (Dimensionality reduction) Dmytro Fishman (dmytro@ut.ee)
So far we have been fearless drawing multi-dimensional instances on two-dimensional planes
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Dimensionality reduction Patients Healthy
The following material was inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0
The following material was inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0 With a bit of fixes, though
Patients Healthy
Patients Healthy Each dot in this plot represents a person
Patients Healthy Every person in its turn could be described by hundreds of protein reactivities Each dot in this plot represents a person
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
Patients Healthy 5, 98, 43, …, 9, 66, Patient
Patients Healthy 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy
Patients Healthy 5, 98, 43, …, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy 3, 45, 13, …, 7, 54, Patient
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?
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?
Introduction to dimensions
1-Dimensional data Person Protein #1 A 24 B 63 C 51 D 34 E 15
1-Dimensional data 300 60 Person Protein #1 A 24 B 63 C 51 D 34 E 15
1-Dimensional data 300 60 300 60 Uniformly distributed data Person Protein #1 A 24 B 63 C 51 D 34 E 15
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
2-Dimensional data 300 60 Protein #1 Person Protein #1 A 24 B 63 C 51 D 34 E 15
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
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
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
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
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
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
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
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
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
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
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
Dimensions 300 60 1-Dimensional
Dimensions 300 60 30 6 3 6 2-Dimensional 1-Dimensional
Dimensions 300 60 30 6 3 6 3-Dimensional 30 6 3 6 3 6 2-Dimensional 1-Dimensional
How about 200-D data?
Are all D are equally important? How about 200-D data?
Are all D are equally important? Let’s go back to 2-D example How about 200-D data?
30 60 Protein#2 300 60 Protein #1 0 Importance of axes
Variation is from left to right 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
Projected data - does not look too different 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
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
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
30 60 Protein#2 300 60 Protein #1 0 Principle components
30 60 Protein#2 300 60 Protein #1 0 Data is mostly spread along this line Principle components
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
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
30 60 Protein #2 30 0 60 Protein #1 0 New X and Y axes Principle components
Principle components Easier to see right/left and above/below variation New X and Y axes
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
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
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
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
Principle components PC1 PC2 If original data would have 3 dimensions, we would have 3rd PC PC3
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
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
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
PCA Now we know what those PC1 and PC2 stand for!
PCA Now we know what those PC1 and PC2 stand for! Wait! But so far we have been talking only about 2 proteins!
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!
Where PCs come from PC1 PC2
PC1 PC2 These are vectors Where PCs come from
PC1 PC2 30 3 Where PCs come from These are vectors All vectors have coordinates
PC1 PC2 30 3 For example, this one has coordinates (3,3) Where PCs come from These are vectors All vectors have coordinates
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
30 60 Protein#2 300 60 Protein #1 0 How did we choose PC1? Where PCs come from
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
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
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
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
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 … … … … …
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)
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 … … … … …
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 … … … … …
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 … … … … …
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
Principle components Principle component 1 Principle component 2 Person B (PC2 score) = 65 Person B (PC1 score) = 60 65 600 0 A B
Principle components Principle component 1 Principle component 2 0 0 A B C D E G
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
PCA is good, but it is a linear algorithm, meaning that it cannot capture complex relationship between features
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…
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
Practice
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
Demo
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)
www.biit.cs.ut.ee www.ut.ee www.quretec.ee

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4 Dimensionality reduction (PCA & t-SNE)

  • 1.
    Introduction to Machine Learning (Dimensionalityreduction) Dmytro Fishman (dmytro@ut.ee)
  • 2.
    So far wehave been fearless drawing multi-dimensional instances on two-dimensional planes
  • 3.
  • 4.
    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
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  • 6.
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  • 8.
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  • 9.
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  • 10.
  • 11.
    The following materialwas inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0
  • 12.
    The following materialwas inspired by: https://www.youtube.com/watch?v=_UVHneBUBW0 With a bit of fixes, though
  • 13.
  • 14.
    Patients Healthy Each dot inthis plot represents a person
  • 15.
    Patients Healthy Every person inits turn could be described by hundreds of protein reactivities Each dot in this plot represents a person
  • 16.
    Patients Healthy Every person inits 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.
    Patients Healthy 5, 98, 43,…, 9, 66, Patient
  • 18.
    Patients Healthy 5, 98, 43,…, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy
  • 19.
    Patients Healthy 5, 98, 43,…, 9, 66, Patient 62, 8, 14, …, 73, 2, Healthy 3, 45, 13, …, 7, 54, Patient
  • 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.
    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.
  • 23.
    1-Dimensional data Person Protein#1 A 24 B 63 C 51 D 34 E 15
  • 24.
    1-Dimensional data 300 60 PersonProtein #1 A 24 B 63 C 51 D 34 E 15
  • 25.
    1-Dimensional data 300 60 30060 Uniformly distributed data Person Protein #1 A 24 B 63 C 51 D 34 E 15
  • 26.
    1-Dimensional data 300 60 30060 Uniformly distributed data 30 60 Non-uniform distribution 0 Person Protein #1 A 24 B 63 C 51 D 34 E 15
  • 27.
    2-Dimensional data 300 60 Protein#1 Person Protein #1 A 24 B 63 C 51 D 34 E 15
  • 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.
    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.
    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.
    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.
    2-Dimensional data A B And soon… 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.
    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.
    2-Dimensional data Protein profilesare 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.
    2-Dimensional data Here isno 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.
    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.
    3-Dimensional data Person P1P2 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.
    3-Dimensional data Person P1P2 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.
  • 40.
  • 41.
    Dimensions 300 60 30 6 3 6 3-Dimensional 306 3 6 3 6 2-Dimensional 1-Dimensional
  • 42.
  • 43.
    Are all Dare equally important? How about 200-D data?
  • 44.
    Are all Dare equally important? Let’s go back to 2-D example How about 200-D data?
  • 45.
  • 46.
    Variation is from leftto right 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
  • 47.
    Projected data - doesnot look too different 30 60 Protein#2 300 60 Protein #1 0 Importance of axes
  • 48.
    We can safely remove2nd 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.
    30 60 Protein#2 300 60 Protein #1 0 30060 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.
  • 51.
    30 60 Protein#2 300 60 Protein #1 0 Datais mostly spread along this line Principle components
  • 52.
    30 60 Protein#2 300 60 Protein #1 0 Datais mostly spread along this line and a little bit along this line Principle components
  • 53.
    30 60 Protein#2 300 60 Protein #1 0 Datais mostly spread along this line and a little bit along this line What if we make new axes from these lines? Principle components
  • 54.
  • 55.
    Principle components Easier tosee right/left and above/below variation New X and Y axes
  • 56.
    Principle components PC1 Easier tosee right/left and above/below variation PC2 These new axes are called Principle Components or PCs New X and Y axes
  • 57.
    Principle components PC1 Easier tosee 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.
    Principle components PC1 Easier tosee 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.
    Principle components PC1 Easier tosee 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.
    Principle components PC1 PC2 If originaldata would have 3 dimensions, we would have 3rd PC PC3
  • 61.
    Principle components PC1 PC2 If originaldata would have 3 dimensions, we would have 3rd PC PC3 There is always as many PCs as there are dimensions
  • 62.
    Principle components If originaldata 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.
    Principle components If originaldata 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.
    PCA Now we knowwhat those PC1 and PC2 stand for!
  • 65.
    PCA Now we knowwhat those PC1 and PC2 stand for! Wait! But so far we have been talking only about 2 proteins!
  • 66.
    PCA Now we knowwhat 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.
    Where PCs comefrom PC1 PC2
  • 68.
  • 69.
    PC1 PC2 30 3 Where PCs comefrom These are vectors All vectors have coordinates
  • 70.
    PC1 PC2 30 3 For example, thisone has coordinates (3,3) Where PCs come from These are vectors All vectors have coordinates
  • 71.
    PC1 PC2 30 3 For example, thisone has coordinates (3,3) In 200-D space vectors have 200 coordinates Where PCs come from These are vectors All vectors have coordinates
  • 72.
    30 60 Protein#2 300 60 Protein #1 0 Howdid we choose PC1? Where PCs come from
  • 73.
    30 60 Protein#2 300 60 Protein #1 0 Howdid we choose PC1? By minimising the distance from points to the vector Where PCs come from
  • 74.
    30 60 Protein#2 300 60 Protein #1 0 Howdid we choose PC1? By minimising the distance from points to the vector Where PCs come from
  • 75.
    30 60 Protein#2 300 60 Protein #1 0 Howdid 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.
    30 60 Protein#2 300 60 Protein #1 0 Howdid 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.
    PCA Example Reactivities PCscoordinates 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.
    PCA Example Reactivities PCscoordinates 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.
    PCA Example Reactivities PCscoordinates 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.
    PCA Example Reactivities PCscoordinates 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.
    PCA Example Reactivities PCscoordinates 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.
    Principle components Principle component1 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.
    Principle components Principle component1 Principle component 2 Person B (PC2 score) = 65 Person B (PC1 score) = 60 65 600 0 A B
  • 84.
    Principle components Principle component1 Principle component 2 0 0 A B C D E G
  • 85.
    Principle components Principle component1 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.
    PCA is good,but it is a linear algorithm, meaning that it cannot capture complex relationship between features
  • 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.
    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.
  • 90.
    https://www.analyticsvidhya.com/blog/2017/01/t-sne-implementation-r-python/ t-SNE visualisation is formhttp://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.
  • 92.
    References • Machine Learningby 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.