Apr 22, 2002
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- Introduction to least squares approximations

- Given three points so far we have attempted to find
	- a line (assumes points are collinear)
	- piecewise lines

- What if we need a line but the three points are not collinear?
	- settle for least-squares approximations

- Objective here is not to pass through the points,
	- but to find a line with minimal error

- Formulating error functions for least squares
	- partial derivatives and finding coefficients

- Examples of least squares fitting
	- lines
	- quadratics
	- non-polynomials
	- trigonometric functions
	- practically, anything!

- Worked out problems from book

Apr 24, 2002
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- More on least-squares

- Solving linear systems in least-squares sense
	- Normal equations: A^T A x = A^T b

- Fitting power-laws
	- e.g., movie rating datasets

- Two options
	- traditional least squares formulation leads to
	  non-linear equation => solve a non-linear equation
		OR
	- use log(x) and log(y) instead of x and y, to get
	  a linear equation

- BUT (a big but)
	- the constants estimated by these different ways 
          may *not* be the same
          	- because you are minimizing different error
          	  functions in each case

Apr 26, 2002
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- Applications of numerical methods in search engines
	- CLEVER
	- Google

- Two types of matrices used
	- hyperlink matrix (which web page links to which?)
	- term-document matrix (which web page contains what words?)

- Power iteration to find "most authoritative" web pages
	- e.g., "Japanese automobile manufacturers"
	- e.g., "computer companies"