Engineering Statistics Handbook Episode 6 Part 9 docx

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] 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] 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 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (2 of 4) [5/1/2006 10:23:13 AM] 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (3 of 4) [5/1/2006 10:23:13 AM] 4.8.1.2.10. Cubic / Cubic Rational Function http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812a.htm (4 of 4) [5/1/2006 10:23:13 AM] 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 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (2 of 13) [5/1/2006 10:23:15 AM] 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 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (3 of 13) [5/1/2006 10:23:15 AM] 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 http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (4 of 13) [5/1/2006 10:23:15 AM] Shape 3 4.8.1.2.11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd812b.htm (5 of 13) [5/1/2006 10:23:15 AM] [...]... http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (6 of 13) [5/1/20 06 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 5 http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (7 of 13) [5/1/20 06 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 6 http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (8 of 13) [5/1/20 06 10:23:15... http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (9 of 13) [5/1/20 06 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 8 http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (10 of 13) [5/1/20 06 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models Shape 9 http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (11 of 13) [5/1/20 06 10:23:15... 4.8.1.2.11 Determining m and n for Rational Function Models Shape 10 http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (12 of 13) [5/1/20 06 10:23:15 AM] 4.8.1.2.11 Determining m and n for Rational Function Models http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (13 of 13) [5/1/20 06 10:23:15 AM] . 4.8.1.2 .9. Quadratic / Cubic Rational Function http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd81 29. htm (3 of 4) [5/1/20 06 10:23:12 AM] 4.8.1.2 .9. Quadratic / Cubic Rational. Models http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (7 of 13) [5/1/20 06 10:23:15 AM] Shape 6 4.8.1.2.11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm. Models http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm (10 of 13) [5/1/20 06 10:23:15 AM] Shape 9 4.8.1.2.11. Determining m and n for Rational Function Models http://www.itl.nist.gov/div 898 /handbook/ pmd/section8/pmd812b.htm