Convolution of Frequency-Impact Distributions
The convolution dialog allows the user to compute the
convolution of frequency-impact distributions where the
convolution is computed as a weighted sum of independent
variables distributed as impact with the number of terms
in the sum defined by frequency. The frequency node
represents the probability distribution over the number of
events.
The assumption of the convolution functionality is that the frequency
node predicts the expected number of events in one time period while
the impact node predicts the cost of one given event. The frequency
and impact nodes may dependent of the value of other nodes. The
convolution of the frequency and impact distributions is the total
impact.
The convolution of a frequency distribution and an impact distribution
is computed as the sum of independent variables as illustrated in this
formula:
where F is the frequency node and the I's are independent
random variables distributed as the impact node.
The convolution is a sum of distributions where the kth term is
the probability of k events multiplied with the distribution of
a sum of k cost variables where each cost variable is a random
variable with a distribution equal to that of impact and all
cost variables are independent.
Convolution of frequency-impact distributions is supported for a pair
nodes consisting of a numbered node and an interval node. The numbered
and interval nodes must be well-formed. The interval node is
well-formed if its intervals cover the range from zero to
infinity (and it otherwise meets the requirements for an interval
node). The numbered node is well-formed if its state values are
zero, one, two, three, e.t.c. (and it otherwise meets the requirements
for a numbered node). The numbered node is interpreted as the
frequency node and the interval node is interpreted as the impact
node.
The convolution of the frequency and impact distributions is
computed with respect to the states of an interval node. This is the
total impact node. The intervals of the total impact node are equal to
the intervals of the impact node.
The convolution algorithm is based on the Table Generator
implementation, i.e., for each interval a predefined number of values
are used in the calculations (as opposed to only using the middle
value of an interval).
Convolution is supported for a domain in sum-normal equilibrium.
Figure 1 shows the convolution pane appearing after activating
the convoltion analysis. In this pane the result of convoluting the
frequency and impact distributions is shown (Figure 2). Initially,
Figure 1 shows the (posterior) probability distributions of the
frequency node and the impact node.
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Figure 1: The dialog shows the distributions of the frequency node and the impact node.
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The total impact distribution (i.e., the convolution of frequency
and impact) is computed once the Convolute button is depressed.
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Figure 2: The total impact distribution.
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Figure 2 shows the total impact distribution. The total impact
distribution is computed as the convolution of the frequency and
impact distributions. The convolution is computed over the intervals
of the impact node.
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