Principal Component Analysis

Class notes

<pdfreflow> class_date = 16 September 2011 [1.65 Mb] button_label = Create my projector slides! show_page_layout = 1 show_frame_option = 1 pdf_file = lvm-class-2.pdf </pdfreflow>

• Also download these 3 CSV files and bring them on your computer:
  • Peas dataset: http://datasets.connectmv.com/info/peas
  • Food texture dataset: http://datasets.connectmv.com/info/food-texture
  • Food consumption dataset: http://datasets.connectmv.com/info/food-consumption

I would advise printing the slides out no more than 2 per page (leaving space for extra notes in today's class) <pdfreflow> class_date = 23 September 2011 [580 Kb] button_label = Create my projector slides! show_page_layout = 1 show_frame_option = 1 pdf_file = lvm-class-3.pdf </pdfreflow>

Class preparation

Class 2 (16 September)

• 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)

Class 3 (23 September)

• 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. (Understanding the content on this page is very important).

Class 4 (30 September)

• Reading on cross validation

Update

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.