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We can use truth
tables to convert an IF-THEN
statement to an INCLUSIVE OR. Read about Boolean
algebras for more on this
important topic. You've seen this at various points in your math career;
sometimes it's easier to attack the contrapositive rather
than the original statement.
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A major theme of today's
lecture was building complicated functions out of simpler ones. In the end,
we can get trunctations, maximum and minimums, and absolute values from
IF-THEN and other operations. This shows that we can build complicated
functions out of simpler pieces. Some of these we haven't done yet but will
either do on Monday, or I'll just leave to the books.
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In calculus the absolute
value function wreaks havoc, as it is not differentiable. In linear
programming it's effect is far less, as we can essentially linearize it (at
the cost of introducing binary indicator variables). This can lead to
conversations on how to measure errors. The
Method of Least Squares is one of my favorites in statistics (click
here for the Wikipedia page,
and click
here for my notes).
The Method of Least Squares is a great way to find best fit parameters.
Given a hypothetical relationship y = a x + b, we observe values of y for
different choices of x, say (x1, y1), (x2, y2), (x3, y3) and so on. We then
need to find a way to quantify the error. It's natural to look at the
observed value of y minus the predicted value of y; thus it is natural that
the error should be Sum_{i=1 to n} h(yi - (a xi +
b)) for some function h. What is a good choice? We could try h(u) = u, but
this leads to sums of signed errors (positive and negative), and thus we
could have many errors that are large in magnitude canceling out. The next
choice is h(u) = |u|; while this is a good choice, it is not analytically
tractable as the absolute value function is not differentiable. We thus use
h(u) = u2;
though this assigns more weight to large errors, it does lead to a
differentiable function, and thus the techniques of calculus are applicable.
We end up with a very nice, closed form expression for the best fit values
of the parameters.
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It is possible to get so caught up in reductions and compactifications that
the resulting equation hides all meaning. A
terrific example is the great physicist Richard Feynman's reduction of all of
physics to one equation, U = 0, where U represents the unworldliness of the
universe.
Suffice it to say, reducing all of physics to this one equation does not
make it easier to solve physics problems / understand physics (though, of
course, sometimes good notation does assist us in looking at things the
right way). I'm mentioning this here as what he does involves measuring
errors by squaring, the topic discussed a moment ago. For each physics
equation look at the square of the left hand side minus the right hand side,
then sum everything and call that U. Thus one term is say (F - ma)^2, and
thus we see that the only way U = 0 is if each summand is zero, and thus
each physics equation must hold.
Suffice it to say, reducing all of physics to this one equation does not
make it easier to solve physics problems / understand physics (though, of
course, sometimes good notation does assist us in looking at things the
right way).
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Video online from 2014:
http://youtu.be/TPDlEMpXBjg
