Engineering Statistics Handbook Episode 6 Part 9 docx

Statistical 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 cub

Trang 2

4.8.1.2.9 Quadratic / Cubic Rational Function

http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (3 of 4) [5/1/2006 10:23:12 AM]

Trang 3

4.8.1.2.9 Quadratic / Cubic Rational Function

http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8129.htm (4 of 4) [5/1/2006 10:23:12 AM]

Trang 4

Statistical

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

http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (2 of 4) [5/1/2006 10:23:13 AM]

Trang 5

Trang 6

Trang 7

f( ) = 0, this implies n < m

●

f( ) = constant, 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'( ) = an/bm

●

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, Pn, 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 k3,

then n is greater than or equal to k3. 4.8.1.2.11 Determining m and n for Rational Function Models

http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (2 of 13) [5/1/2006 10:23:15 AM]

Trang 8

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), k4, n+m-1.

●

if n = m, the number of real roots of R'(x), k4, 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 k4, then n+m-1 is k4 Similarly, if the fitted function f(x) is such that, for finite x and n =

m, the number of times f'(x) = 0 is k4, then n+m-2 k4.

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.

0 constant

n < m

n = m

n > m

0 constant

n < m + 1

n = m +1

n > m + 1

3 For finite x, desired number, k3,

of times f(x) = 0

Relation of n to k3

4 For finite x, desired number, k4,

of times f'(x) = 0

Relation of n to k4 and m

k4 (n m)

k4 (n = m)

n (1 + k4) - m

n (2 + k4) - m

4.8.1.2.11 Determining m and n for Rational Function Models

Trang 9

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

Trang 10

Shape 3

Trang 11

Shape 4

Trang 12

Shape 5

Trang 13

Shape 6

Trang 14

Shape 7

Trang 15

Shape 8

Trang 16

Shape 9

Trang 17

Shape 10

Trang 18

Định dạng
Số trang	18
Dung lượng	132,29 KB