Monday, April 24, 2017
A Fit of the Equinox Deviations Using Expected Values
I got a little bogged down with some technical details associated with copying formulas from one Excel worksheet to another. It's a little annoying when Excel crashes, restarts and you have to redo the stuff you haven't saved. It may have been how an error crept into one of my previous pages. You literally lose track of what you are doing. One needs to constantly check one's formulas and trace dependencies when transferring material from one page to another.
I used the expected value formulas to do the fit for the deviations of the Equinox times. The results were similar. The frequencies that I've been using were relative frequencies defined in as the ratio of counts for an interval to the total count. One can also define the function f as a probability density or probability per unit interval. I had to use this definition to get the fit to work properly for the Equinox times. The value of f here is the previous value divided by the width of the interval dx.
The error bounds are nominal in the sense that they are typical of the observed variations for a trapazoidal distribution. The fit values for the trapazoidal distribution are a=4.470 min and b=15.089.
Supplemental (Apr 24): The trapazoidal distribution has an interesting series for formulas for its expected values. The pattern holds for higher powers of x. Technically this might be called a folded trapazoidal distribution since the probabilities for the positive and negative values of x are combined.
Supplemental (Apr 25): The variations in the relative frequencies are scaled down versions of the expected variation in the counts for an interval as this derivation shows.
In evaluating f and δf in the table above I used the observed values for the interval's density, obs_f (=ni/n/dx), as an approximation. It was intended as a check of the trapazoidal density formula whose maximum value is 2/(a+b)=0.1023. Using the same letter for the relative frequencies of the counts and the probability density formula may have be a little too confusing. So the error bounds in the plot are a little too large. Using f* for the density the correct formula for the expected rms error in the density would be as follows with Δx=dx.