The Microguide to Process Modeling in Bpmn 2.0 by MR Tom Debevoise and Rick Geneva_15 doc

4.8.1.2.8. Linear / Cubic Rational Function
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4. Process Modeling
4.8. Some Useful Functions for Process Modeling
4.8.1. Univariate Functions
4.8.1.2. Rational Functions
4.8.1.2.9.Quadratic / Cubic Rational
Function
Function:
4.8.1.2.9. Quadratic / Cubic Rational Function
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Function
Family: Rational
Statistical
Type: Nonlinear
Domain:
with undefined points at the roots of
There will be 1, 2, or 3 roots, depending on the particular values of the parameters.
Explicit solutions for the roots of a cubic polynomial are complicated and are not
given here. Many mathematical and statistical software programs can determine the
roots of a polynomial equation numerically, and it is recommended that you use one
of these programs if you need to know where these roots occur.
Range:
with the possible exception that zero may be excluded.
Special
Features:
Horizontal asymptote at:
and vertical asymptotes at the roots of
There will be 1, 2, or 3 roots, depending on the particular values of the parameters.
Explicit solutions for the roots of a cubic polynomial are complicated and are not
given here. Many mathematical and statistical software programs can determine the

roots of a polynomial equation numerically, and it is recommended that you use one
of these programs if you need to know where these roots occur.
Additional
Examples:
4.8.1.2.9. Quadratic / Cubic Rational Function
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4.8.1.2.9. Quadratic / Cubic Rational Function
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4.8.1.2.9. Quadratic / Cubic Rational Function
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4. Process Modeling
4.8. Some Useful Functions for Process Modeling
4.8.1. Univariate Functions
4.8.1.2. Rational Functions
4.8.1.2.10.Cubic / Cubic Rational Function
Function:
4.8.1.2.10. Cubic / Cubic Rational Function
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Function
Family: Rational
Statistical
Type: Nonlinear
Domain:
with undefined points at the roots of
There will be 1, 2, or 3 roots, depending on the particular values of the parameters.
Explicit solutions for the roots of a cubic polynomial are complicated and are not
given here. Many mathematical and statistical software programs can determine the
roots of a polynomial equation numerically, and it is recommended that you use one
of these programs if you need to know where these roots occur.
Range:

with the exception that y = may be excluded.
Special
Features:
Horizontal asymptote at:
and vertical asymptotes at the roots of
There will be 1, 2, or 3 roots, depending on the particular values of the parameters.
Explicit solutions for the roots of a cubic polynomial are complicated and are not
given here. Many mathematical and statistical software programs can determine the
roots of a polynomial equation numerically, and it is recommended that you use one
of these programs if you need to know where these roots occur.
Additional
Examples:
4.8.1.2.10. Cubic / Cubic Rational Function
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4.8.1.2.10. Cubic / Cubic Rational Function
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4.8.1.2.10. Cubic / Cubic Rational Function
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4. Process Modeling
4.8. Some Useful Functions for Process Modeling
4.8.1. Univariate Functions
4.8.1.2. Rational Functions
4.8.1.2.11.Determining m and n for Rational
Function Models
General
Question
A general question for rational function models is:
I have data to which I wish to fit a rational function to. What degrees n and m should I use
for the numerator and denominator, respectively?
Four

Questions
To answer the above broad question, the following four specific questions need to be answered.
What value should the function have at x =
? Specifically, is the value zero, a constant,
or plus or minus infinity?
1.
What slope should the function have at x =
? Specifically, is the derivative of the
function zero, a constant, or plus or minus infinity?
2.
How many times should the function equal zero (i.e., f (x) = 0) for finite x?3.
How many times should the slope equal zero (i.e., f '(x) = 0) for finite x?4.
These questions are answered by the analyst by inspection of the data and by theoretical
considerations of the phenomenon under study.
Each of these questions is addressed separately below.
Question 1:
What Value
Should the
Function
Have at x =
?
Given the rational function
or
then asymptotically
From this it follows that
if n < m, R(
) = 0●
if n = m, R( ) = a
n
/b

m
●
if n > m, R( ) = ●
Conversely, if the fitted function f(x) is such that
4.8.1.2.11. Determining m and n for Rational Function Models
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f( ) = 0, this implies n < m●
f( ) = constant, this implies n = m●
f( ) = , this implies n > m●
Question 2:
What Slope
Should the
Function
Have at x =
?
The slope is determined by the derivative of a function. The derivative of a rational function is
with
Asymptotically
From this it follows that
if n < m, R'(
) = 0●
if n = m, R'( ) = 0●
if n = m +1, R'( ) = a
n
/b
m
●
if n > m + 1, R'( ) = ●
Conversely, if the fitted function f(x) is such that
f'(

) = 0, this implies n m●
f'( ) = constant, this implies n = m + 1●
f'( ) = , this implies n > m + 1●
Question 3:
How Many
Times Should
the Function
Equal Zero
for Finite
?
For fintite x, R(x) = 0 only when the numerator polynomial, P
n
, equals zero.
The numerator polynomial, and thus R(x) as well, can have between zero and n real roots. Thus,
for a given n, the number of real roots of R(x) is less than or equal to n.
Conversely, if the fitted function f(x) is such that, for finite x, the number of times f(x) = 0 is k
3
,
then n is greater than or equal to k
3
.
4.8.1.2.11. Determining m and n for Rational Function Models
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Question 4:
How Many
Times Should
the Slope
Equal Zero
for Finite
?

The derivative function, R'(x), of the rational function will equal zero when the numerator
polynomial equals zero. The number of real roots of a polynomial is between zero and the degree
of the polynomial.
For n not equal to m, the numerator polynomial of R'(x) has order n+m-1. For n equal to m, the
numerator polynomial of R'(x) has order n+m-2.
From this it follows that
if n
m, the number of real roots of R'(x), k
4
, n+m-1.●
if n = m, the number of real roots of R'(x), k
4
, is n+m-2.●
Conversely, if the fitted function f(x) is such that, for finite x and n m, the number of times f'(x)
= 0 is k
4
, then n+m-1 is k
4
. Similarly, if the fitted function f(x) is such that, for finite x and n =
m, the number of times f'(x) = 0 is k
4
, then n+m-2 k
4
.
Tables for
Determining
Admissible
Combinations
of m and n
In summary, we can determine the admissible combinations of n and m by using the following

four tables to generate an n versus m graph. Choose the simplest (n,m) combination for the
degrees of the intial rational function model.
1. Desired value of f( ) Relation of n to m
0
constant
n < m
n = m
n > m
2. Desired value of f'( ) Relation of n to m
0
constant
n < m + 1
n = m +1
n > m + 1
3. For finite x, desired number, k
3
,
of times f(x) = 0
Relation of n to k
3
k
3
n k
3
4. For finite x, desired number, k
4
,
of times f'(x) = 0
Relation of n to k
4

and m
k
4
(n m)
k
4
(n = m)
n (1 + k
4
) - m
n
(2 + k
4
) - m
4.8.1.2.11. Determining m and n for Rational Function Models
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Examples for
Determing m
and n
The goal is to go from a sample data set to a specific rational function. The graphs below
summarize some common shapes that rational functions can have and shows the admissible
values and the simplest case for n and m. We typically start with the simplest case. If the model
validation indicates an inadequate model, we then try other rational functions in the admissible
region.
Shape 1
Shape 2
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 3
4.8.1.2.11. Determining m and n for Rational Function Models

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Shape 4
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 5
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 6
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 7
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 8
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 9
4.8.1.2.11. Determining m and n for Rational Function Models
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Shape 10
4.8.1.2.11. Determining m and n for Rational Function Models
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4.8.1.2.11. Determining m and n for Rational Function Models
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