A Global Center for Fire Risk Education
Where do the numbers come from?
Numbers are an easy way to find any point on an L-Curve. The numbers can be
generated by a Computer-aided Fire (CAFire) model, or estimated, as required.
Estimates may come from a tool called a Bubble Diagram.
Concepts for the
Bubble Diagram were developed by Professor Lou Rossman at Worcester Polytechnic
Institute (WPI), adapting Markoff's "Chain Analysis." The use of the initial
Bubble Diagram was awkward. The diagram was improved into its present form by
Mr. Lowell Jackson during an L-Curve course at the University of Wisconsin,
Madison, Wisconsin.
The Bubble Diagram has two parts: 1) its logic; and
2) its use. The L-Curve Method uses 3 principal bubble diagrams, and their 9
subdiagrams (12 diagrams). Since detailed logic of all these 12 diagrams is
beyond this Web Site, one sample will be shown.
Note: The "bar" over a
letter stands for "not". 
Given a full deck of playing cards that are Red and some other color
(A), draw one card. It's either red (B) or not red (not B). Question: What are
the chances it will be red? Answer: 26 in 52 or .5
Given B, you then
enter the decision between "C" and "not C". Obviously "B" plus "not B" are all
of the possibilities. In L-Curve language, "B" plus "not B" must equal 1 just as
"C" plus "not C" must equal 1.
This gives us another advantage. If your
degree of belief is that something will happen (B), then the rest must be (not
B). Thus if B is .37, for example, then not B must be .63 because .37 + .63 = 1.
The Bubble Diagram can be used in both known and unknown situations. For
a known situation, for instance, given a full deck of cards (A), you draw one
card face down. What are the chances that that card will be the Queen of Hearts
when you turn it over? Using the Bubble Diagram, in this shortened 2-layer form,
we can try the system out.
The card will either be red or not red (B or
not B). If it is red (B), then it will either be the Queen of Hearts (C--1 in 26
) or not the Queen of Hearts (not C--25 in 26). The result will be the Queen of
Hearts (D) or not the Queen of Hearts (not D).
Multiply along
solid lines and add up the results of the dotted lines to get the
result. In our example:
The chances of drawing the Queen of Hearts is 1 in 52 or .02. See, it works.
You can try this on any number of different situations. Think of one
yourself. Try it out. Mathematically, the Bubble Diagram will determine the
number. All you have to know is how to add and how to multiply.
Try it; you'll like it.
Computer-Aided Fire (CAFire) models are
presently useful in generating numbers for the fire growth hazard. The fire
growth hazard is one part of the L-Curve process.