Friday, April 28, 2017

Interpreting Error Bounds for the Trapazoidal Distribution Fit

  I've got a better handle on the error bounds for the fit of a trapazoidal distribution to the deviations of the time of the Equinox. Here's a review and a corrected plot with error bounds showing the expected deviation from expected value for δt.

The error bounds used the values of the estimate of the deviation, δf*, for the probability densities, obs_f*, for the intervals in the table above. One can get a better understanding of what the error bounds mean by looking at the expected relative frequency, fi=ni/n, for the intervals chosen. The x values indicate the center of the interval. Using the values of a and b for the fitted trapazoidal distribution we can compare the observed counts with the expected counts, k=nf and their expected rms deviation of the counts, δk=√[nf(1-f)]. The expected variation in the relative frequency will then be δf=√[f(1-f)/n]. But what does all this tell us about the observations themselves? One can look at the terms of the binomial distribution with p=f and determine the probability of observing exactly k counts in each interval. Then we can add up the probabilities for those values of k which are within a distance of δk from the expected value for k. The last column on the right shows the probability of this occurring for each interval. A calculation shows the odds aren't uniform for the intervals but equal to 0.6252 ± 0.0375. The probabilities associated with the error bounds are less than those for a normal distribution and fluctuate a little because we are dealing with a discrete probability distribution and taking sum of those values of k between <k>-δk and <k>+δk. One can show that probabilities associated with bounds for a given number of standard deviations in a normal distribution is equal to P(k)=erf(k/√2). So we would expect slightly more observations to be outside the error bounds for the binomial distribution than would be the case for a normal distribution

The binomial distributions for each of the intervals can be plotted together for comparison.

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