Principal Component Analysis

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Class date(s): 16, 23, 30 September 2011
Video material (part 1)
Download video: Link (plays in Google Chrome) [290 Mb]


Video material(part 2)
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Video material (part 3)
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Video material (part 4)
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Video material (part 5)
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Video material (part 6)
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Video material (part 7)
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Video material (part 8)
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Class 2

Class date: 16 September 2011 [1.65 Mb]
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Background reading

  • Reading for class 2
  • Linear algebra topics you should be familiar with before class 2:
    • matrix multiplication
    • that matrix multiplication of a vector by a matrix is a transformation from one coordinate system to another (we will review this in class)
    • linear combinations (read the first section of that website: we will review this in class)
    • the dot product of 2 vectors, and that they are related by the cosine of the angle between them (see the geometric interpretation section)

This illustration should help better explain what I trying to get across in class 2B

  • \(p_1\) and \(p_2\) are the unit vectors for components 1 and 2.
  • \( \mathbf{x}_i \) is a row of data from matrix \( \mathbf{X}\).
  • \(\hat{\mathbf{x}}_{i,1} = t_{i,1}p_1\) = the best prediction of \( \mathbf{x}_i \) using only the first component.
  • \(\hat{\mathbf{x}}_{i,2} = t_{i,2}p_2\) = the improvement we add after the first component to better predict \( \mathbf{x}_i \).
  • \(\hat{\mathbf{x}}_{i} = \hat{\mathbf{x}}_{i,1} + \hat{\mathbf{x}}_{i,2} \) = is the total prediction of \( \mathbf{x}_i \) using 2 components and is the open blue point lying on the plane defined by \(p_1\) and \(p_2\). Notice that this is just the vector summation of \( \hat{\mathbf{x}}_{i,1}\) and \( \hat{\mathbf{x}}_{i,2}\).
  • \(\mathbf{e}_{i,2} \) = is the prediction error vector because the prediction \(\hat{\mathbf{x}}_{i} \) is not exact: the data point \( \mathbf{x}_i \) lies above the plane defined by \(p_1\) and \(p_2\). This \(e_{i,2} \) is the residual distance after using 2 components.
  • \( \mathbf{x}_i = \hat{\mathbf{x}}_{i} + \mathbf{e}_{i,2} \) is also a vector summation and shows how \( \mathbf{x}_i \) is broken down into two parts: \(\hat{\mathbf{x}}_{i} \) is a vector on the plane, while \( \mathbf{e}_{i,2} \) is the vector perpendicular to the plane.

Geometric-interpretation-of-PCA-xhat-residuals.png


Class 3

Class date: 23, 30 September 2011 [580 Kb]
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Background reading

  • Least squares:
    • what is the objective function of least squares
    • how to calculate the regression coefficient \(b\) for \(y = bx + e\) where \(x\) and \(y\) are centered vectors
    • understand that the residuals in least squares are orthogonal to \(x\)
  • Some optimization theory:
    • How an optimization problem is written with equality constraints
    • The Lagrange multiplier principle for solving simple, equality constrained optimization problems.


Background reading