CS 525, Simulation
Quiz 4, 23 March 2005
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Of the two approaches we discussed for determining when a simulation enters the
steady state, which is better able to identify regenerative behavior as
non-steady state? Explain your answer.
The two main properties of a regenerative state is that the underlying distribution is always varying and the variations are cyclic. The variance monitoring technique in general will fail with non-stationary distributions, so just having variations should be enough to reject the steady state.
The moving-average technique, however, smooths out variations. If the moving average covers several cycles, then the moving average will smooth them out, potentially indicating a steady state when none exists. (You could make a similar augment for variance monitoring, assuming the sample rates fell below the Nyquist sampling rate. However, any stability testing procedure will fail under those conditions, so it's not a good argument.)
None of the answers hit the key point, which is that the moving-average technique smooths out variations and may run into trouble with regenerative behavior.
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Explain the steps you would take to fit a probability distribution to a
sequence of trials x1, x2, ..., xn where each xi
represents either a success or a failure. The choice of which of the two
parameter-estimation techniques to use, as well as the initial distribution
guess, is up to you. You needn't go into excruciating mathematical detail, but
you should present enough detail to indicate you understand what's being asked
of you and that you know how to deal with it.
The three steps used in matching a probability distribution to a sample set are:
- Gather candidate distributions. In this case, the samples were identified as success-failure trials, which pretty much fixes the candidate distributions to the Bernoulli distribution.
- Estimate Parameters. The Bernoulli distribution has a single parameter p giving the success probability. You have a choice of using the maximum likelyhood estimator or the method of moments to determine the parameter.
- Determine goodness of fit. The Bernoulli distribution is discrete, which calls for the chi-square test. One obvious choice is a two-bucket histogram test with one degree of freedom. If there were enough samples, a second test could be based on a binomial distribution with parameters p and n; n consecutive samples would be grouped into a binomial trial and then dumped into one of n buckets based on the number of successes in the grouped sample.
There was much confusion in the answers to this question, with various techniques wandering away from their categories (the chi-square test being used to estimate parameters, for example).
| This page last modified on 26 March 2005. |
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