Engineering Statistics Handbook Episode 6 Part 5 pptx

Plot of C/C Rational Function Fit We generate a plot of the fitted rational function model with the raw data. The fitted function with the raw data appears to show a reasonable fit. 6-Plot for Model Validation Although the plot of the fitted function with the raw data appears to show a reasonable fit, we need to validate the model assumptions. The 6-plot is an effective tool for this purpose. 4.6.4.5. Cubic/Cubic Rational Function Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd645.htm (3 of 5) [5/1/2006 10:22:58 AM] The 6-plot indicates no significant violation of the model assumptions. That is, the errors appear to have constant location and scale (from the residual plot in row 1, column 2), seem to be random (from the lag plot in row 2, column 1), and approximated well by a normal distribution (from the histogram and normal probability plots in row 2, columns 2 and 3). Residual Plot We generate a full-sized residual plot in order to show more detail. 4.6.4.5. Cubic/Cubic Rational Function Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd645.htm (4 of 5) [5/1/2006 10:22:58 AM] The full-sized residual plot suggests that the assumptions of constant location and scale for the errors are valid. No distinguishing pattern is evident in the residuals. Conclusion We conclude that the cubic/cubic rational function model does in fact provide a satisfactory model for this data set. 4.6.4.5. Cubic/Cubic Rational Function Model http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd645.htm (5 of 5) [5/1/2006 10:22:58 AM] 4. Fit a Q/Q rational function model. 1. Perform the Q/Q fit and plot the predicted values with the raw data. 2. Perform model validation by generating a 6-plot. 3. Generate a full-sized plot of the residuals to show greater detail. 1. The model parameters are estimated. The plot of the predicted values with the raw data seems to indicate a reasonable fit. 2. The 6-plot shows that the residuals follow a distinct pattern and suggests that the randomness assumption for the errors is violated. 3. The full-sized residual plot shows the non-random pattern more clearly. 3. Fit a C/C rational function model. 1. Perform the C/C fit and plot the predicted values with the raw data. 2. Perform model validation by generating a 6-plot. 3. Generate a full-sized plot of the residuals to show greater detail. 1. The model parameters are estimated. The plot of the predicted values with the raw data seems to indicate a reasonable fit. 2. The 6-plot does not indicate any notable violations of the assumptions. 3. The full-sized residual plot shows no notable assumption violations. 4.6.4.6. Work This Example Yourself http://www.itl.nist.gov/div898/handbook/pmd/section6/pmd646.htm (2 of 2) [5/1/2006 10:22:58 AM] Stigler, S.M. (1978) "Mathematical Statistics in the Early States," The Annals of Statistics, Vol. 6, pp. 239-265. Stigler, S.M. (1986) The History of Statistics: The Measurement of Uncertainty Before 1900, The Belknap Press of Harvard University Press, Cambridge, Massachusetts. 4.7. References For Chapter 4: Process Modeling http://www.itl.nist.gov/div898/handbook/pmd/section7/pmd7.htm (2 of 2) [5/1/2006 10:22:58 AM] 4. Process Modeling 4.8. Some Useful Functions for Process Modeling 4.8.1.Univariate Functions Overview of Section 8.1 Univariate functions are listed in this section. They are useful for modeling in their own right and they can serve as the basic building blocks for functions of higher dimension. Section 4.4.2.1 offers some advice on the development of empirical models for higher-dimension processes from univariate functions. Contents of Section 8.1 Polynomials1. Rational Functions2. 4.8.1. Univariate Functions http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd81.htm [5/1/2006 10:22:59 AM] Polynomial Model: Limitations However, polynomial models also have the following limitations. Polynomial models have poor interpolatory properties. High degree polynomials are notorious for oscillations between exact-fit values. 1. Polynomial models have poor extrapolatory properties. Polynomials may provide good fits within the range of data, but they will frequently deteriorate rapidly outside the range of the data. 2. Polynomial models have poor asymptotic properties. By their nature, polynomials have a finite response for finite values and have an infinite response if and only if the value is infinite. Thus polynomials may not model asympototic phenomena very well. 3. Polynomial models have a shape/degree tradeoff. In order to model data with a complicated structure, the degree of the model must be high, indicating and the associated number of parameters to be estimated will also be high. This can result in highly unstable models. 4. Example The load cell calibration case study contains an example of fitting a quadratic polynomial model. Specific Polynomial Functions Straight Line1. Quadratic Polynomial2. Cubic Polynomial3. 4.8.1.1. Polynomial Functions http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd811.htm (2 of 2) [5/1/2006 10:22:59 AM] Statistical Type: Linear Domain: Range: Special Features: None Additional Examples: None 4.8.1.1.1. Straight Line http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8111.htm (2 of 2) [5/1/2006 10:23:00 AM] Statistical Type: Linear Domain: Range: Special Features: None Additional Examples: 4.8.1.1.2. Quadratic Polynomial http://www.itl.nist.gov/div898/handbook/pmd/section8/pmd8112.htm (2 of 5) [5/1/2006 10:23:01 AM] [...]... Quadratic Polynomial http://www.itl.nist.gov/div898 /handbook/ pmd/section8/pmd8112.htm (3 of 5) [5/ 1/20 06 10:23:01 AM] 4.8.1.1.2 Quadratic Polynomial http://www.itl.nist.gov/div898 /handbook/ pmd/section8/pmd8112.htm (4 of 5) [5/ 1/20 06 10:23:01 AM] 4.8.1.1.2 Quadratic Polynomial http://www.itl.nist.gov/div898 /handbook/ pmd/section8/pmd8112.htm (5 of 5) [5/ 1/20 06 10:23:01 AM] . Yourself http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd6 46. htm (2 of 2) [5/ 1/20 06 10:22 :58 AM] Stigler, S.M. (1978) "Mathematical Statistics in the Early States," The Annals of Statistics, Vol. 6, pp. 239- 2 65 . Stigler,. satisfactory model for this data set. 4 .6. 4 .5. Cubic/Cubic Rational Function Model http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd6 45. htm (5 of 5) [5/ 1/20 06 10:22 :58 AM] 4. Fit a Q/Q rational function. tool for this purpose. 4 .6. 4 .5. Cubic/Cubic Rational Function Model http://www.itl.nist.gov/div898 /handbook/ pmd/section6/pmd6 45. htm (3 of 5) [5/ 1/20 06 10:22 :58 AM] The 6- plot indicates no significant