Feb 29 2008

MAIN POINTS
The reading first defines "inconsistent" systems of equations, which have no solution. These are probably extremely common in scientific computation because real world data (and data represented inside computers) as coefficients rarely would give an intersection with more than two lines, because a miniscule error in the coefficient could throw it all off. The reading writes a system in vector form, then reminds us that we can calculate the shortest distance from a point to a plane by using the concept of orthogonality, which would give a dot product of 0. This is used to create an equation for finding the least squares for any system.
4.1.2 Generalizes this equation for the specific purpose of finding data to fit a curve.
The solution follows the same steps. Fortunately, the text gives a MATLAB command to calculate least squares.

The second half of the reading takes an example we've touched upon in class — periodic temperature. This leads to a large number of data points being punched into the formula. Then it covers exponential growth, going on to Moore's law and concentration of checmicals in the bloodstream.... a hearty dose of examples.

CHALLENGES
I see that there is equation 4.6. I don't understand what the operation A^T is doing— it's turning A on its side, but what are the implications of this? Also, I don't get how they solve for the final least-squares solution with Gaussian elimination (on the top of page 197). Because of this, I still don't really get the process of turning this large system into a least square number, especially when dealing with curves that aren't lines. I would like to see an example calculation before we try this in MATLAB!

REFLECTIONS
The fact that models such as sines and exponential functions can work inside this matrix model for best fit is interesting. I've used simple equations for regression lines in high school biology and chemistry, and generated them automatically in excel for college chemistry, and never understood the reasoning behind them. This could be very useful in robotics— estimating things like movement and vision in which a limited number of discrete data points is available. The text also points out how it is useful for compression, and I'm curious about what kinds of compression use these ideas.