Difference between revisions of "Principal Component Analysis"
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* Some optimization theory: | * Some optimization theory: | ||
** | ** How an optimization problem is written with equality constraints | ||
** | ** The [http://en.wikipedia.org/wiki/Lagrange_multiplier Lagrange multiplier principle] for solving simple, equality constrained optimization problems. ('''''Understanding the content on this page is very important'''''). | ||
* Reading on [http://literature.connectmv.com/item/12/cross-validatory-estimation-of-the-number-of-components-in-factor-and-principal-components-models cross validation] | * Reading on [http://literature.connectmv.com/item/12/cross-validatory-estimation-of-the-number-of-components-in-factor-and-principal-components-models cross validation] |
Revision as of 19:45, 20 September 2011
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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
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 two regression coefficients
and for - understand that the residuals in least squares are orthogonal to
- 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).
- Reading on cross validation
Update
This illustration should help better explain what I trying to get across in class 2B
and are the unit vectors for components 1 and 2. is a row of data from matrix . = the best prediction of using only the first component. = the improvement we add after the first component to better predict . = is the total prediction of using 2 components and is the open blue point lying on the plane defined by and . Notice that this is just the vector summation of and . = is the prediction error vector because the prediction is not exact: the data point lies above the plane defined by and . This is the residual distance after using 2 components. is also a vector summation and shows how is broken down into two parts: is a vector on the plane, while is the vector perpendicular to the plane.