Motivation of dimensionality reduction, Principal Component Analysis (PCA), and applying PCA.

1. Motivation

1a. Motivation I: Data Compression

  • You are able to reduce the dimension of the data from 2D to 1D
    • For example, pilot skill and pilot happiness can be reduced to pilot’s aptitude
    • Generally, you can reduce x1 and x2 to z1
  • Your are able to reduce the dimension of the data from 3D to 2D
    • Project the data such that they lie on a plane
    • Specify two axes
      • z1
      • z2
    • You would then be able to reduce the data’s dimension from 3D to 2D

1b. Motivation II: Visualization

  • Given a set of data, how are able to examine the data such as this?
  • We can use reduce the data’s dimensionality from 50D to 2D
    • Typically we do not know what the 2 dimensions’ meanings are
    • But we can make sense of out of the 2 dimensions

2. Principal Component Analysis (PCA)

2a. PCA Problem Formation

  • Let’s say we have the following 2D data
    1. We can project with a diagonal line (red line)
      • PCA reduces the blue lines (the projection error)
        • Before performing PCA, perform mean normalization (mean = 0) and feature scaling
    2. We can also project with another diagonal line (magenta)
      • But the projection errors are much larger
      • Hence PCA would choose the red line instead of this magenta line
  • Goal of PCA
    • It’s trying to find a lower dimensional surface onto which to project the data, so as to minimize this squared projection error
    • To minimize the square distance between each point and the location of where it gets projected.
  • PCA is not linear regression
    • PCA is a minimization of the orthogonal distance

2b. Principal Component Analysis Algorithm

  • Data pre-processing step
    • You must always do this before doing PCA
  • PCA intuition
    • You need to compute the vector or vectors
      • Left graph: compute vector z(1)
      • Right graph: compute vector z(1) and z(2)
  • Procedure
    • You can use eig (eigen) or svd (singular value decomposition) but the later is more stable
      • You can use any library in other languages that does singular value decomposition
      • You will get 3 matrices: U, S and V
      • But we only need matrix U where we manipulate to get z that is a k x 1 vector
  • Summary of PCA algorithm in octave

3. Applying PCA

3a. Reconstruction from Compressed Representation

  • We can go from lower dimensionality to higher dimensionality

3b. Choosing the Number of Principal Components

  • k is the number of principal components
    • But how do we choose k?
  • There is a more efficient method on the right compared to the left
    • We then use the S matrix for calculations
  • You would realise that PCA can retain a high percentage of the variance even after compressing the number of dimensions of the data

3c. Advice for Applying PCA

  • Supervised learning
    • For many data sets, we can reduce by 5-10x easily to ensure our learning algorithm runs much faster
  • Application of PCA
    1. Compression
      • Reduce memory or disk needed to store data
      • Speed up learning algorithm
        • We choose k by percentage of variance retained
    2. Visualization
      • We choose only k = 2 or k = 3
  • Bad uses of PCA
    1. To prevent over-fitting
      • Regularization is better because it is less likely to throw away valuable information as it knows the labels
    2. Running PCA without consideration