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The
simulation method uses the Monte Carlo technique to generate
samples and uses descriptive statistics to estimate the
moments. Although the method is quite different than the
enumeration method, the forms used in the two cases are
similar. To create a model for the simulation choose Add
Function from the menu. Information required for
the model is the same, except the Max,Min and
Interval fields are neglected and the Simulation
button is checked. |
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The form constructed by the add-in
is shown below. The first row of the form shows the name and
analysis method. The random variables are defined next in rows
2 through 5. The Binomial distribution describes a
discrete random variable and has two parameters, the number
of trials, n, and the probability of a success, p.
The random variable is the number of successes in 5 trials and
can assume the values of 0 through 5. |
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Rows 6 through 9 hold
information used in the simulation process. . The analyst should
enter the names of the variables in row 6. Row 7 holds random
numbers generated by the add-in using the Excel built-in random
number generator. The initial form has 0.5 for all entries in
this row. Row 8 holds the inverse probability function provided
by the add-in. The RV_Inverse function returns the smallest
value of the random variable whose cumulative distribution value
is greater than or equal to the random number immediately above
the function. This operation is equivalent to Monte Carlo simulation.
Rows 9 through 16 hold information regarding the
functions and the results of the analysis. The names of the
output variables should be placed in row 9. The maroon cells
in row 10 are for the formulas that compute the functions to
be analyzed. Rows 12 through 16 will hold the results of the
analysis with rows 12 and 13 holding the mean and variance.
Row 14 holds the number of simulated values used in the statistical
estimates of the moments. Row 16 holds the two-sided confidence
interval for the sample mean. The confidence level in B15 can
be changed to find different confidence intervals.
The figure below shows the functions used by the
example. Excel equations implementing these functions are in
row 10 of the simulation form.
The figure below shows the form with random numbers in row
7. The values in row 8 are simulated from these numbers using
the Monte Carlo method. |
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To begin the simulation we choose
Moments from the menu. The dialog below accepts the name
of the form to be analyzed, the number of simulation iterations
and the calculation option. |
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After 100 simulated observations,
the results are placed on the form. |
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The results can be compared to the
accurate moments found with the enumeration method. |
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A simulation using 1000 observations
yields more accurate results as indicated by the smaller confidence
interval. |
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Other Distributions |
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The Monte Carlo technique is the
same for both discrete and continuous random variables. No additional
approximation is required. The results below are an example
using a variety of distributions. We present the enumeration
results below the simulation results for comparison. |
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Form for Both Simulation and Enumeration |
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When constructing the function form, the add-in provides an
option to construct a form that handles both enumeration and
simulation. This might be handy for comparing the results for
the two methods on the same functions. This form for the example
is shown below. |
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The analysis method is chosen on the Moments
dialog. Both methods are available for this form. |
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The results of a simulation analysis
are below. The Lower Limit, Upper Limit, Interval and the three
Probability rows are irrelevant for the simulation analysis. |
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The results of the enumeration analysis
are shown below. The Rand., Sample Size and Confidence Interval
rows are irrelevant for this analysis. |
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Simulation Statistics |
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The moment estimates from the simulation
are obtained with descriptive statistics.
The confidence interval formula is based on the Central
Limit Theorem. We use the standard Normal distribution
rather than the t-distribution because the sample size
is usually very large for simulation analyses. |
Comparing Simulation and Enumeration |
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Although both the simulation and
enumeration methods have the goal of finding the moments of
functions of random variables, the process used for the two
methods are quite different. For discrete distributions enumeration
will yield accurate answers. For continuous distributions the
results are approximate, but no measure of accuracy is available.
For problems with few random variables with a small integer
range, enumeration is no doubt the best choice. For larger problems
the curse of dimensionality will probably make this analysis
impractical.
Simulation always provides results subject to statistical variability.
Every simulation analysis draws samples from the random variable
distributions and computes statistical moments of the function
evaluation. Two simulations of a system will have different
results. Partially moderating this disadvantage is the availability
of a confidence interval on the sample mean. This provides a
reliability measure. The confidence interval decreases as the
sample size grows. Unfortunately the interval is proportional
to the inverse of the square root of the sample size. Increasing
the sample size from 100 to 1000 decreases the width of the
interval (assuming the same sample variance) by a factor of
the square root of 10. The curse of dimensionality really does
not hold for simulation, because the analyst can simply select
the number of simulation iterations. But the accuracy of the
results is severally impacted for small sample sizes.
Both methods will have a place in the analyst's tool box and
we illustrate several applications on these pages and throughout
this web site. |
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