[EM] Finding the constants for the approximating function
Michael Ossipoff
mikeo2106 at msn.com
Tue Jan 30 09:59:17 PST 2007
Some days ago I posted a suggestion for finding the values of the constants
A & B in the approximating function B/(q+A), by least-squares. The same
method could be used for B*exp(-A*q).
That's probably the most accurate way to evaluate the constants.
I'd noticed that Warren finds the constants based on the fact that the U.S.
population and the number of states are know. Yes, those two known
quantities make it possible to solve two equations to find the constants A &
B.
The integral, with respect to q, of the frequency distribution
approximating function, from 0 to Q
represents the number of states from population zero out to a population of
Q Hare quotas.
For the least-squares solution, I'd suggested fitting that integral function
to the data points consisting of the states's Hare quotas and their
cumulative numbers (The smallest state's cumulative number is 1. The 2nd
smallest state's cumulative number is 2...etc), to get the values of the two
constants.
But, for the two equations solution, you could set that integral, from 0 out
to 52 Hare quotas, equal to the number of states. That's one of the two
equations.
That integral function gives the cumulative number of states as a function
of q. Its inverse gives q as a function of the cumulative number of states.
Write the integral of that inverse from 0 to 50, and set it equal to 435
(the U.S. population in Hare quotas). That's the 2nd equation. Solve those
two equations for A and B.
Least squares is almost surely more accurate. If the distribution function
really is the exponential function, then both methods are completely
accurate. But if it isn't, then the least squares approximation gives the
exponential function that gives the best approximation.
Another way to make two equations would be set the integral of the
approximatng function equal to 34 when q has the value of the number of Hare
quotas possessed by the 34th state. And to set the integral of the
approximating function equal to 17 when q has the value of the number of
Hare quotas possessed by the 17th state.
I'd considered that, and preferred least-squares, because, as I said, it's
the closest B*exp(-A*q) approximation to the distribution when the
distribution isn't really that function.
Mike Ossipoff
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