subset selection in predictive modeling of the cmm digitization uncertainty

Journal ofManr~fucrurin,q Systems

Subset Selection in Predictive Modeling of the CMM Digitization Uncertainty

Chang-Xue (Jack) Feng and Xianfeng Wang, Dept. of Industrial and Manufacturing Engineering, Bradley University, Peoria, Illinois, USA. E-mail: [email protected]

Abstract A number of factors affect digitizing accuracy, such as the

travel speed of the probe, pitch values, probe angles (part orientations), probe sizes, and feature sizes. A proper selection of these parameters in a digitization or automatic inspection process will improve the digitizing accuracy. Factorial design of experiments is used to plan the experiments. Re- gression analysis is applied to developing the empirical models for prediction of digitizing uncertainty. Four criteria, namely the PRESS statistic, the adjusted R2, the C, statistic, and the residual mean square 9, are employed to select the best regression model. Hypothesis testing is conducted to check the goodness of each model in construction and to qualify the validation of the model. It is shown that the prediction model has a satisfactory goodness of fit.The method for model selection and validation as well as the best model selected can be used in both computer-aided reverse engineering and automatic inspection for error prediction and accuracy improvement.

Keywords: Coordinate Measuring Machine, Computer-Aid- ed Reverse Engineering, Quality Design and Control, Regres- sion Analysis, Predictive Process Engineering

Introduction Reliable and accurate digitizing and measuring

results could be obtained by using a coordinate measuring machine (CMM) if proper operation strategies are adopted. With the increasing demand for higher product quality and growing competition in the glo- bal market, coordinate measuring machines have been widely used in industry to improve the efficiency and effectiveness of digitization and inspection (Shaffer 1982; Bosch, Harlow, Thompson 1998).

A number of studies in the field of coordinate measuring techniques have been conducted during the past few years. These studies mainly concen- trated on the accuracy improvement by developing hardware, software, and operating strategies. For example, Balsamo, Marques, and Sartori (1990) studied the correction of temperature-induced length changes and deformation on the coordinate measur-

ing machine or on the workpiece being measured or both. The issues of acquisition and computational correction of the equipment-specific deviation behav- ior of coordinate measuring machines have been examined by Kunzmann, Trapet, and Waldele (1990); Sartori and Zhang (1995); and Zhang and Zhang (1991).

In addition, the error compensation issue of a CMM has been studied in Zhang et al. (1985); Kreuci (1990); Yee and Gavin (1990); Weckenmann, Knauer, and Kunzmann (1998); and Shen and Zhang (1999). Gu and Chan (1996) and Limaiem and ElMaraghy (2000) have recently investigated inspection planning for CMM by means of optimum sequencing and resource allocating.

Two closely related studies are reviewed next. Phillips et al. (1998) provided a study on measurement uncertainty under the guideline of the IS0 standard (IS0 1995). The drawbacks with that research are: (1) its experiments were not based on factorial designs, (2) it did not examine the impact of multiple factors on measurement uncertainty, and (3) it could not examine the interaction issue due to its experimental design and no presence of multiple factors. In addition, Phillips et al. investigated the measurement uncertainty for the center location of a circle, and this paper deals with the radius of a circle. Although each of the previous studies reviewed above has contrib- uted to the study of one or several factors, none of them has systematically examined all the above five factors simultaneously, and nor has an empirical model dealing with digitizing (or inspection) error been developed.

Tang and Sun (2001) dealt with a scanning probe in digitization, and this paper deals with a touch trigger probe; they considered sample size (related to pitch value in our study), speed, stylus deflection (probe length related), internal diameter vs. outside

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diameter, and feature size, but did not include probe size and orientation. It is important to distinguish digitizing with a touch trigger probe in this research and scanning with a true scanning probe. Because a true scanning probe is not available in this research, the digitization process is actually an automatic measurement process once the starting point, ending point, pitch value, and speed are specified. A t-test from our original experimentation has found no statistically significant impact from the probe length when using a touch trigger probe for digitization in this study, as opposed to a scanning probe discussed in Tang and Sun. As a result, the probe length factor was excluded from the experiments and models in this study.

Therefore, when applying a CMM with a touch trigger probe to digitizing a mechanical object, the following five controllable parameters tend to most impact the accuracy of digitizing: probe sizes, part orientations (probe angles), pitch values, travel speeds, and feature sizes. Feng and Pandey (2002) have conducted a comprehensive experimental study of the effect of the above five factors on digitizing (scanning or automatic inspection) uncertainty in reverse engineering. On the basis of their research, this paper presents empirical modeling for predicting digitizing uncertainty. For details of digitization with a CMM for reverse engineering applications, refer to Feng (2003), Feng and Wang (2002b), Wohlers (2001), and Feng and Xiao (2000). Otto and Wood (2001) discussed reverse engineering under the frame- work of the entire product design life cycle.

Improving predictability of the performance of a precision process or product has become a critical concern in industry and academia. Consequently, the National Science Foundation’s two manufacturing programs under the Design and Manufacturing Divi- sion have targeted this area for the past several years. To predict the performance of a manufacturing process, including a digitization and automatic measurement process, a mathematical model is needed. Developing such a model is usually difficult, interesting, and rewarding. A number of techniques have been available in statistics, machine learning, and recently data mining, but they have not been applied to empirical modeling of precision processes for the prediction and simulation of a precision (manufacturing) process or have not caught the attention of domain experts in precision engineering and manufacturing.

Four criteria, namely the PRESS statistic, the ad- justedR2, the C, statistic, and the residual mean square s2, are employed to select the best regression model. Hypothesis testing is conducted to check the goodness of each model in construction and to qualify the validation of the model. It is shown that the prediction model has a satisfactory goodness of fit.

While Feng and Pandey (2002) focused on the design and analysis of factorial experiments in the CMM uncertainty study, this paper concentrates on the follow-up work, that is, developing an empirical model to predict the CMM digitization uncertainty for applications in reverse engineering. Furthermore, this follow-up is not only interested in developing any predictive model, but the best model possible as well as the methodology used to derive such a model. As a result, the model will be useful for future application on the particular CMM, while the method employed for model construction, selection, and cross validation will be of value to similar applications of CMM digitization uncertainty study as well as other precision process studies. For example, the authors have applied the above methodology to predictive modeling of turning surface roughness (Feng and Wang 2003, Feng and Wang 2002a) and honing surface roughness (Feng, Wang, Yu 2002).

Experimental Design and Conditions The fixed bridge-type of Mitutoyo RV507 CMM

under investigation is shown in Figure I. For details about a CMM, refer to Bosch, Harlow, and Thomp- son (1998). The three holes on the two artifacts used in this investigation are from two master gage rings in good condition used to calibrate the respective gage in a production environment. The feature of interest in digitization is the inside diameter of the master rings. The digitization uncertainty is defined by an error percentage calculated in Eq. (l), where the measured radius was obtained by manually measuring the radius 30 times with a digital caliper and by joystick measuring 30 times with the CMM, respectively, and taking the average of them. In joystick measuring of the diameter, the probe started from the center of the hole approximately each time.

Error % = IScanned radius - Measured radius1 x 100%

Measured radius (1)

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Fixed bridge type

Figure 1 Configuration and Coordinates of Mitutoyo RV507 CMM

Table 1 Postulation of Error Prediction Model Factors and Levels of Screening Experiments

In regression analysis, postulation of the appropri- ate form of empirical models is of critical importance (Box and Draper 1987; Draper and Smith 1998). In general, an empirical model can be linear or nonlinear. Nonlinear models can be developed with data transformation, for example, logarithmic transformation, or without any data transformation. This research will examine these forms of models. A good tutorial is provided in chapter 9 of Hogg and Ledolter (1992) on how to fit equations to data with statistical modeling methods. For an in-depth discussion of this issue, refer to Daniel and Wood (1980), Draper and Smith (1998), and Montgomery, Peck, and Vining (2001).

Levels

Factors Low High

Probe size (A) 4mm 5mm

Pitch value (B) 2.54 mm 5.08 mm

Speed (C) 5 mm/set. 8 mm/set.

Feature size (D) 31.7134 mm 4 1.3228 mm

Probe angle (E) (90, -9o)* (90, o)**

Notes: * - (90, -90) means YZ plane, and ** - (90,O) means XZ plane.

The factors and levels in the screening experiment are given in Table 1. An early decision was made to divide the 48 experiments into two blocks of 24 experiments on the basis of the part orientation. This is because a significant error from each setup of the part on the CMM was discovered in the preliminary experiments. Thus the “block” is confounded with factor E, that is, the part orientation or the probe angle. A randomized run order was generated with Minitab and applied to each of the two blocks of 24 experiments. The data from the 48 experiments are presented in Table 2.

The experimental study in Feng and Pandey (2002) has eliminated part orientation from further consid-

Jourac~l of Manufacturing Systems Vol. 211No. 6

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eration because their study has shown that the XZ plane in the configuration shown in Figure 1 resulted in the best digitization accuracy. It has also shown that a high pitch value working with a high speed and a low pitch value working with a low speed led to better digitization accuracy. Therefore, these two parameters have been combined to form one composite factor pitch/speed. This has reduced the number of parameters by another one. As a result, only three factors are significant, and we could afford to design a full experiment 3’ to investigate any possible second-order main factor effects, such as n,‘. The factors and levels in the confirmation experiment are shown in Table 3. A replicate number of two is used, leading to 3” x 2 = 54 experiments. Refer to Mont- gomery (2001); DeVor, Chang, and Sutherland ( 1992); and Box, Hunter, and Hunter (1978) for details on design and analysis of experiments. The design of the 33 experiment and the experimental data are presented in TabZe 4.

Simply Linear Models

The functional relationship between digitization uncertainty in terms of the relative error and the digitizing factors under investigation can be postulated as simply as a linear model, shown in Eq. (2).

R<, = cc, + c,P., + czpr + cg,, + c.,jl + CP, (2)

where R,, p,?, pv, ps,, jy, and pa are, respectively, the relative error, probe size, pitch value, probe speed, feature size, and probe angle.

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Table 2 Data for 48 Screening Experiments

Table 3 Factors and Levels of Confirmation Experiment

Factors I Levels

Symbol 1 Name 1 Low 1 Medium 1 High

Equation (2) can be rewritten into the following standard linear mathematical model:

“rl = P, + P,x, + P*x, + P,x, + P&$+ P$x, (3)

where q is the true value of the relative error, x,, x2, x3, x,, and x5 is, respectively, the probe size, pitch value, probe speed, feature size and probe angle, while PO, B,, P2, B3, P4, Bs are the corresponding pawn-

eters to be estimated. Among the five parameters, only

the probe angle is qualitative, and the other four are quantitative.

Due to experimental errors, the true response is y =y-&where y s i th e measured relative error and E is the experimental error.

For simplicity, Eq. (3) is rewritten as:

j= b, + b,x, + b,x, + b,x, + b4x4 + bp 5 (4)

whereiis the predicted relative error value, and b,, b,, b,, b,, b,, and b, is the estimate of parameters B “, P , , P 2, P 3r P 4, and P 5r respectively.

Polynomial Models

Because of the strong nonlinear relationship between the digitizing uncertainty and the main factors and interactions revealed in Feng and Pandey (2002) a nonlinear functional relationship of the second-order form can be postulated without any data transformation by Eq. (5):

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Standard Order

Probe Size

(A)

Table 4 Design of 33 Experiment and Data

Factor % Error

Pitch/Speed Feature Size Replicates Mean Error

(W (E) 1 2 e’)

1 I 3 1 0.137 0.139 0.138

2 3 2 2 0.042 0.042 0.042

3 1 1 1 3 ) 2 ) 0.045 ) 0.045 1 0.045

0.005

.---h-+-y : : :::: :::: 0.142

6 1 3 3 0.003 0.003 0.003

7 3 3 2 0.041 0.044 0.043

I 0.141 ( 0.142 / 0.142

0.045 0.046 0.045

: f 0.003 0.003 0.003

0.042 0.042 0.042

---G--K : : 0.002 0.002 0.002

---k-j-:

0.044 0.044

: : 0.134 ::‘“;z 0.132

I 2 I 7- I 0.045 I 0.045 I ~~~~~~ - 0.045

18 3 3 1 0.127 0.120 0.124

19 I I 1 0.142 0.140 0.141

0.134 0.137 0.135

-+---+- 5 ? : 0.046 0.045 0.045

22 1 2 2 0.044 0.043 0.044

23 2 2 3 0.005 0.005 0.005

24 1 2 3 0.004 0.004 0.004

25 1 I 3 0.004 0.004 0.004

26 3 2 I 0.132 0.132 0.132

27 2 I 3 0.005 0.006 0.005

9 = b. + b,x, + b2x2 + b3x1 + b4x4 + bsxs + b,?x,x2 +

b,,x,x, + b,etx,xb + b,,x,xs + bnxzx3 + bzhxzxd +

b75xzxs + b?s,xixa, + bsx,xg + box4xs + _.

+ 6, ,x; + b?,x; + b,,x; t b,x,2 + b,,x;

(3

Montgomery, Peck, and Vining (2001) referred to the above model as polynomial, a special type of linear regression model, while the authors prefer to call it quadratic (for example, see Miller 2002).

One of the most popularly used data transforma- tions, the logarithmic data transformation, can be used

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to process the input and output data first, and then the regression model is constructed based on the trans- formed data. The beauty of performing such a transformation is that it can obtain a very easy and concise form of model, such as

where j, k, 1, m, and n could be any decimal number, positive or negative. Of course, this is under the as- sumption that the four criteria for model selection outlined earlier will favor such form of the model.

Best Subset Regression Preliminary Analysis

To find the best prediction model, the “best subsets” regression option was used with Minitab (Minitab 2000). The following three issues are of interest for a comparative study: (1) using averaged data vs. individual observations for model construction and validation, (2) using full experiment data or fractional experiment data for model construction and validation, and (3) simply using a linear model vs. polynomial with the second order and interaction terms. The latter will be termed as a polynomial model following the convention of Montgomery, Peck, and Vining (200 1). These three issues will be examined and compared first in this section.

In Tables 5 to 12, “Vars” stands for the number of variable terms in the model. In Table 13, “Ln( 16 data)” and “Ln(48 data)” means the models with logarithmic transformation for the 16 data and 48 data, respectively, p is the number of parameters in the model including the constant term b,, S is the root mean residual squares, and PRESS is the prediction sum of squares,

where PRESS = cr=,Sf (Walpole et al. 2002).

Given a minimum possible number of p in Table 13, the larger the adjusted R*, the closer the C, statistic to the p value, the smaller the S value and the PRESS value, the better the model is among the four criteria for selecting the best regression model (Draper and Smith 1998; Walpole et al. 2002). Thus, the polynomial model from 48 individually observed data without logarithmic transformation highlighted in Table 13 (model 6) is the best model among all possible options. This model is then obtained using Minitab and provided in Table 14.

The main effect plot of this model is given in Fig- ure 2, and the interaction plot of this model is in Fig- ure 3. Based on the above model presented in Table 14 and without considering any interactions, Figure 2 shows that the larger feature size D = 4 1.3228 mm led to a smaller uncertainty because its p-value in Table 14 is much less than 0.05. Similarly, it has also shown that the X2 plane (level 1 of factor E) led to a smaller uncertainty as itsp-value in Table I4 is much less than 0.05. In addition, because the p-values for the other three factors in Table 14 are also much smaller than 0.05, it can be concluded that the larger probe size A = 5.0000 mm, smaller pitch value B = 2.5400 mm, and higher speed C = 8.0000 mm/s resulted in a smaller uncertainty.

Figure 3a indicates a strong interaction between the probe size and the feature size. For a given larger feature D = 4 I .3228 mm, a larger probe A = 5.0000 mm should be used to reduce the digitizing uncertainty to the minimum level. On the other hand, a smaller probe size D = 4.0000 should be used to reduce the uncertainty of digitization if a smaller feature A = 3 1.7 134 mm is given. Figure 3b also reveals a strong interaction between the pitch value (B) and the speed (C). Either a smaller pitch value B = 2.540 mm working with a slower speed C = 5.000 mm/s, or a larger pitch value B = 5.080 mm working with a faster speed C = 8 mm/s led to a smaller digitizing uncertainty. This observation can guide the design of the next round of experiments to reduce the number of factors because the two parameters go in the same direction to reduce or increase the digitizing uncertainty. Finally, Figure 3c sug- gests that the larger feature working in the YZ plane resulted in the smallest digitization uncertainty, and regardless of the probe angle or part orientation? the larger feature size resulted in smaller digitizing uncertainty within the range of feature sizes investigated in Table I.

Figure 4 graphically compares the 48 observed data with the fitted data from the polynomial model and linear model constructed from these data. Fig- ure 4 also shows that the fitted values from the polynomial model presented in Table 14 are satisfactorily close to the respective observed data. Figure 5 de- picts the relative error from the best polynomial model (model 6) in Table 14 and the best linear model (model 5) in Table 9.

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Table 5 Best Subsets Regression with 16 Averaged Data (Linear model without data transformation)

VXS R-Sq R-Sq(adj) CP S A B C D E

I 72.9 70.9 -1.9 0.0097408 X 2 73.1 69.0 0.0 0.010068 X X 3 73.1 66.4 2.0 0.010473 X X X 4 73.1 63.4 4.0 0.010937 X X X X 5 73.1 59.7 6.0 0.011470 X X X X X

Table 6 Best Subsets Regression with 16 Averaged Data (Polynomial without data transformation)

A A A A B B B C C D VXS R-Sq R-Sq(adj) C,, S A B DEBCDEC DEDEE

I 72.9 70.9 27.9

2 76.8 73.3 24.1

3 81.5 76.8 19.3

4 85.9 80.7 14.8

5 88.4 82.7 13.0

6 94.5 90.8 6.2

7 97.0 94.4 4.4

8 98.3 96.3 4.5

9 98.8 97.0 5.8

10 99.2 97.7 7.1

II 99.3 97.3 9.1

12 99.3 96.5 11.0

13 99.3 94.9 13.0

14 99.3 89.8 15.0

0.0097408

0.0093426

0.0086975

0.0079325

0.0075269 0.0054928

0.0042707

0.0034587

0.0031512

0.0027209

0.0029772

0.0033612

0.0040823

0.0057678

x X X

x x X

x x x X

x x X X X

x x x x X X

x x X X X X X x x x x x x X X x x x x x x X X X

x x x x x x x x X X

x x x x x x x x x X X

x x xxxxxxx x x X

x x xxxxxxx x x x X

x x xxxxxxx xxxxx

Table 7 Best Subsets with 16 Averaged Data (Linear model after data transformation)

VarS R-Sq R-Sq (adj) C” S A B C D E

1 67.7 65.4 -1.1 0.88330 X 2 69.1 64.3 0.4 0.89664 X X 3 70.3 62.8 2.0 0.91543 X X X 4 70.3 59.5 4.0 0.95520 X X X X 5 70.3 55.5 6.0 1.0013 X X X X X

Model Construction

As discussed earlier, the experiment can be sim- plified into three factors, as shown in Table 3, based on the study of Feng and Pandey (2002). Therefore, a full three-level experiment, 33, could be conducted with data shown in Table 4. It is as important to construct models out of these 54 experiments from Table 4 as it is to construct models of those 48 experiments from Table 2. Again, the best subsets regression has been

conducted to compare the goodness of fit of the models from averaged data with those from individual observation, from full experiment with fractional experiment, and from linear with polynomial. The modeling results are reported in TubEes 15 to 18 from the full experiment data.

Table 19 summarizes the statistics of the four best models of one each from Tables 15 to 18. Although the other statistics are comparable, the residual mean

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Table 8 Best Subsets Regression with 16 Averaged Data (Polynomial after data transformation)

AAAABBCD VarS R-Sq R-Sq(adj) Cm s A B CDBCDECDDE

1 67.7 65.4 26.7 0.88330 2 71.7 67.3 23.9 0.85861 3 77.3 71.7 19.1 0.79893 4 83.1 76.9 14.2 0.7208 1 5 84.9 77.4 14.0 0.71341 6 89.9 83.2 10.1 0.61486 7 94.4 89.5 6.7 0.48569 8 96.6 92.7 6.1 0.40425 9 97.2 93.1 7.3 0.39423 10 97.4 92.3 9.1 0.41741 11 97.5 90.5 11.0 0.46229 12 97.5 87.5 13.0 0.53138

X

X x x

x x x x x x x x x x x x x x x x

Table 9 Best Subsets Regression with 48 Individual Observations (Linear model without data transformation)

vars R-Sq R-Sq (adj) CP S A B C D E

1 73.0 72.4 -1.5 0.0093372 X 2 73.2 72.0 0.1 0.0093991 X X 3 73.3 71.4 2.1 0.009498 1 X X X 4 73.3 70.8 4.0 0.0096022 X X X X 5 73.3 70.1 6.0 0.0097156 X X X X X

Table 10 Best Subsets Regression with 48 Individual Observations (Polynomial without data transformation)

A A B C D VXS R-Sq R-Sq(adj) C,, S A B C D E B D C D E

1 73.0 72.4 1448.3 0.0093372 X 2 77.0 76.0 1229.0 0.0087125 X X 3 81.5 80.2 981.5 0.0078989 x x X 4 85.5 84.2 762.2 0.0070721 X X X X 5 88.5 87.1 598.4 0.00637 14 x x x x x 6 92.8 91.7 364.8 0.0051130 x x x x X X 7 97.0 96.5 131.6 0.0033157 x x X x x x x 8 99.3 99.2 8.3 0.0016236 xxxxx x x X 9 99.3 99.2 9.6 0.0016313 xxxxx x x x x 10 99.3 99.1 11.0 0.0016394 x x x x x x x x x x

Table 11 Best Subsets Regression with 48 Individual Observations (Linear model after data transformation)

VZiI3 R-Sq R-Sq (adj) CIJ S A B C D E

1 68.2 67.5 1.6 0.82595 X 2 69.6 68.2 1.6 0.81685 X X 3 70.7 68.7 2.0 0.8 1073 X X X 4 70.7 68.0 4.0 0.81976 X X X X 5 70.7 67.3 6.0 0.82944 X X X X X

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Tdle 12 Best Subsets with 48 Individual Data (Polynomial model after data transformation)

AABBCD

vars R-Sq R-Sq(adj) C, S A BC DED ECDDE

I 68.2 67.5 674.8 0.82595 X

2 71.8 70.6 595.0 0.78613 X X

3 77.5 76.0 468.9 0.71061 X X

4 81.4 79.7 381.8 0.65283 X x x 5 84.8 83.0 308.4 0.59834 X X x x 6 89.6 88.1 201.3 0.50052 x x x x x

7 94.3 93.3 97.5 0.37597 X x x x x 8 96.7 96.0 44.3 0.28846 X x x x x x 9 98.4 98.0 9.1 0.20652 x x x x x x x x

10 98.4 98.0 10.1 0.20650 x x x x x x x x x 11 98.4 97.9 12.0 0.20894 x x x x x x x x x

# of Data

Table 13 Comparison of Simply Linear and Polynomial Models

Type of Model Name of Model R2 R’(adj.) P C” S PRESS

16 data

Ln( 16data)

48 data

Ln(48data)

Linear Polynomial

Linear Polynomial

Linear Polynomial

Linear Polynomial

73.1 63.4 5 4 0.01094 0.00278 94.5 90.8 7 6.2 0.005493 0.00095 70.3 59.5 5 4 0.95520 21.2344 94.4 89.5 8 6.7 0.38569 7.72568 73.3 70.8 5 4 0.009602 0.00494 99.3 99.2 9 8.3 0.00162 0.00016 70.7 68.0 5 4 0.81976 36.00700 98.4 98.0 10 9.1 0.20652 2.58590

Table 14 Best Regression Model from 25-’ Experiment (48 individual observations, polynomial, no logarithmic transformation)

Error = -0.250 + 0.0688 A + 0.0202 B + 0.0116 C + 0.00526 D - 0.0393 E - 0.00186 AD - 0.00309 BC + 0.00105 DE

Predictor Coef SE Coef t P

Constant -0.24967 0.01660 A 0.068796 0.003593 B 0.0201662 0.0008206 C 0.0115833 0.000494 1 D 0.0052639 0.0004417 E -0.039332 0.001797

AD -0.00186449 o.ooOO9755

BC -0.0030949 0.0001230 DE 0.00105366 0.00004878

S = 0.001624 R-Sq = 99.3% R-Sq(adj) = 99.2% PRESS = 0.000156 R-Sq(pred) = 98.95%

-15.04 0.000

19.15 0.000

24.57 0.000

23.45 0.000

11.92 0.000

-21.89 0.000

-19.11 0.000

-25.16 0.000

21.60 0.000

Analysis of Variance

Source DF ss MS F P

Regression 8 0.0147377 0.0018422 698.8 1 0.000

Residual Error 39 0.0001028 O.OOOOO26 Total 47 0.0148405

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0.037

0.029

A

w 1

Figure 2 Main Effect Plot of Best Model in Table 14

0.032

31.734 D 41.3228

Probe size (A) and feature size (D) interaction

A ?? 4 1 5

B ?? 2.54 ?? 5.08

5 c 8

Pitch value (B) and speed (0 interaction

::::: \

8 5 0.022-

0.012- __-- -I

___--- ___--- 0.002 - ,__----

D ?? 31.7134 1 41.3228

-1 E I Feature size (D) and probe angle (E) interaction

Figure 3 Interaction Plot of Best Regression Model in Table 14

square S and PRESS favor the polynomial model with 27 averaged data highlighted in Table I9 (model 10). This best model and its key statistics are provided in Table 20. The model’s main effect plot and interaction plot are presented in Figures 6 and 7, respectively. These two kinds of plots confirm earlier observations made of the best model shown in Table 14.

Tables 21 to 24 provide the results from best subsets regression of the 3”.’ experiment data to compare the goodness of fit of the model constructed from the full experiment with that from the fractional 3”’ experiment. The statistics from the four best models of one each from Tables 21 to 24 are summarized in Table 25 for ease of comparison. Table 25 shows that the best polynomial model of the 18 individual observations or model 16 from the 3”-’ experiment is the best among the models 13, 14, 15, and 16 based on all of the four criteria. This model (model 16) and its key statistics are provided in Table 26. The main effect plot and the AC interaction plot are given in Figures 8 and 9, respectively.

For the purpose of a comparative study, computational results of the best regression linear and polynomial model from the 3” full experiment are presented in Table 27, while those from the 33-’ are presented in Table 28. The error statistics of these four best models of one from each kind are summarized in Table 29. Table 29 seems to suggest that the two polynomial models, one from the full experiment and one from the fractional experiment, are better than the two linear models. To rigorously compare the above models, a hypothesis testing has been conducted with Minitab to check the goodness of fit of

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0.05

0.045

0.04

0.035

0.03

j 0.025

Ls 0.02

0.015

0.01

0.005

0

-0.005

No. of Experiments

Figure 4 48 Observed vs. Fitted Data from Model 5 (linear) and Model 6 (polynomial)

Polynomial model

600.00

0.00 1 2 3 4 5 6 7 8 9101112131415161718192021222324252627282930313233343536373839404142434445464748

No. of Experiments

Figure 5 Relative Errors from Model 5 (linear) and Model 6 (polynomial) for 48 Data

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Table 15 Best Subsets Regression with 27 Averaged Data (Linear model)

Vars R-Sq R-Sq(adj) c s A Y B c

1 99.5 99.5 6.3 0.004 1840 X

2 99.6 99.5 3.2 0.0038842 X X

3 99.6 99.5 4.0 0.0038679 X X X

Table 16 Best Subsets Regression with 27 Averaged Data (Polynomial model)

A A B A B C WUS R-Sq R-Sq(adj) C, S ABCB C C A B C

1 99.5 99.5 105.9 0.0041840 X

2 99.6 99.6 75.3 0.0036906 X X

3 99.7 99.7 50.4 0.0032008 X X X

4 99.8 99.8 23.4 0.0024976 X X X X 5 99.9 99.8 17.6 0.0022955 X X X X X

6 99.9 99.9 12.5 0.00208 16 X X x x x x 7 99.9 99.9 6.7 0.0017782 x x x x x x x 8 99.9 99.9 8.0 0.0017925 X x x x x x x x 9 99.9 99.9 10.0 0.0018428 x xxx x x x x x

Table 17 Best Subsets Regression with 54 Individual Observations (Linear model)

vars R-Sq R-Sq(adj) C” s A B C

1 99.4 99.4 12.8 0.0042656 X

2 99.5 99.5 4.4 0.0039348 X X

3 99.5 99.5 4.0 0.0038824 X X X

Table 18 Best Subsets Regression with 54 Individual Observations (Polynomial model)

A AB ABC VXS R-Sq R-Sq(adj) C,, S A BCB CCABC

1 99.4 99.4 193.9 0.0042656 X

2 99.6 99.5 140.3 0.0037841 X X

3 99.7 99.7 93.2 0.0032863 X X X

4 99.8 99.8 40.0 0.0025792 X X X X

5 99.8 99.8 26.5 0.0023530 X X X X X

6 99.9 99.8 17.9 0.0021863 X x x x X X

7 99.9 99.9 7.3 0.0019551 X X X x X x x

8 99.9 99.9 8.1 0.0019506 X xx xxxxx

9 99.9 99.9 10.0 0.0019694 xxxxxxxxx

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2002

0.09

j 0.06

0.03

Figure 6 Main Effect Plot of Best Model from Full 3’ Experiment

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0.00

12.734 31.713 41.323 C

A

?? 2 ?? 4

5

0.10 !z o 0.08 5:

0.06

12.734 31.;13 41.;23 C

B ?? 1 + 2

3

1

Figure 7 Interaction Plot of Best Model from Full 3’ Experiment

models 9 and 10 from Table 19 and models 15 and 16 from Table 25.

Refer to Montgomery and Runger (2003) for a detailed discussion of hypothesis testing. The hypothesis testing results are given in Table 30. These hypothesis tests compared the observed data with the

fitted data from each of the four models. Because all the p-values in Table 30 are greater than 0.05, much greater as a matter of fact, each of the four models has a statistically satisfactory goodness of fit. The next section will validate these four models, respectively.

Model Validation To follow the train-test-validate strategy outlined

in some recent texts on data mining (Groth 1998; Witten and Frank 2000; Hastie, Tibshirani, Friedman 2001), the data in Table 2 will be used to validate the different kinds of best models. Table 31 presents the computational results for validation of the two linear models (models 9 and 15), while Tuble 32 presents the two polynomial models (models 10 and 16). Table 33 summarizes the error statistics of the computational results from Tables 31 and 32. A hypothesis testing has been conducted using Minitab to check the goodness of prediction of these four models, and the results are given in Table 34. The hypothesis tests compared the predicted value with the respective observed value for the four models, respectively. Be- cause all of the p-values in Table 34 are greater than or equal to 0.05, they all can provide a statistically satisfactory goodness of prediction. The t-tests comparing the means for the two linear models (models 9 and 15 or RA 1 and RA3) have a p-value close to 0.05, meaning that they provide a marginally satisfactory goodness of prediction. Between the two polynomial models 10 and 16, model 16 would be recommended because it is simpler and the other statistics from the four criteria in Tables 19 and 2.5 and from the hypothesis tests in Tables 30 and 34 are comparable.

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Journal of Manufacturing Sysrems Vol. 21/No. 6 2002

Table 19 Comparison of Best Subsets Models (Full 35 exaeriment)

# of Data Type of Model Name of Model R? R’(adi.) 1’ C S PRESS

” 27 Linear 9 99.6 99.5 4 4 0.003868 0.000483

Polynomial 10 99.9 99.9 8 6.7 0.001778 0.000134

54 Linear 11 99.5 99.5 4 4 0.003882 0.000889

Polynomial 12 99.9 99.9 8 7.3 0.0019s5 0.000257

Table 20 Best Model from Full 33 Experiment (27 averaged data, polynomial)

Error = 0.211 + 0.00764A - 0.00654 C + 0.000108 AC + 0.000103 BC - 0.00176 A’ - 0.000985 B* + 0.000024 C’

Predictor Coef

Constant 0.211030

A 0.007636

C -0.0065387

AC 0.00010794

BC 0.00010318

AA -0.0017593

BB -0.0009854

cc 0.00002398

s = 0.001778 R-Sq = 99.9%

PRESS = 0.000134 R-Sq(pred) = 99.84%

SE Coef

0.005300

0.002634

0.0002409

0.000023 10

0.00003337

0.0003696

0.0002556

0.00000405

R-Sq(adj) = 99.9%


t P

39.82 0.000

2.90 0.009

-27.14 0.000

4.67 0.000

3.09 0.006

~I.76 0.000

-3.86 0.001

5.92 0.000

Source DF ss MS F P

Regression 7 0.083560 0.011937 3775.14 0.000

Residual Error 19 0.000060 0.000003

Total 26 0.083620

B C

0.00 -I_ 2 b 5

--\

___-___ _____.

Figure 8 Main Effect Plot of Best Model from 33-’ Experiment

432

0.06 -i \

12.713 31.713 41.323

C

A ?? 2 * 4

5

Figure 9 Interaction Plot of Best Model from 3’.’ Experiment

Table 21 Best Subsets Regression of

9 Averaged Data from 3’-’ Experiment (Linear model)

Van R-Sq R-Sq(aW C,, S A B C

1 99. I 99.0 3.3 0.0057737 X

2 99.3 99.1 3.3 0.0054570 x X

3 99.5 99.2 4.0 0.0053137 X X X

Finally, it appears to be interesting in comparing the predicted values of these models in pairs directly to draw a conclusion on if a polynomial model is better than a simply linear one, or if a full experiment led to a better model than a fractional experiment. Table 35 is constructed based on hypothesis tests using Minitab to answer the above questions in the processes of model construction and validation based on the data in Table 4. The simply linear and polynomial model test was for model 9 (linear) and model 10 (polynomial) in Table 19, while the full and fractional test was to compare model 10 from 27 averaged data

.loumal ofMan@7rturing Systems Vol. 21/No. 6

2002

Table 23 Best Subsets Regression of 18 Individual Observations

from 3 ’ Experiment (Linear model)

Van R-Sq R-Sq(adj) C,, S A B C

I 99.1 99.0 9.6 0.005575 1 X

2 99.3 99.2 6.1 0.0050327 X X

3 99.5 99.3 4.0 0.0045871 X X X

and model 16 in Table 25 from 18 individual observations.

Because all the p-values in Table 35 are much greater than 0.05, the equality null hypothesis cannot be rejected in each case. Therefore, there is no statistical evidence that one kind of model is better than the other, or that the full experiment would be better than the fraction experiment in model construction and validation. One implication from these hypothesis tests is that in future digitization uncertainty study, we might recommend the use of the fractional experiment because it is more cost and time effective and it did not lead to a worse model than a full experiment.

Conclusions Empirical models for prediction of digitization

uncertainty with a CMM in terms of percent relative error have been developed based on factorial design of digitization experiments. Four criteria from statistics are used to select the best model. and hypothesis tests have been conducted to check the goodness of fit and goodness of prediction. Validation computa- tion shows that the predictive models are statistically satisfactory. The models developed by means of regression analysis can be used for estimating the values of relative error of digitization (scanning and/or

Table 22 Best Subsets Regression of 9 Averaged Data from 3’-’ Experiment (Polynomial model)

A A B A VaEY R-Sq R-Sq(adj) C,J S A B C B C C A

I 99.1 99.0 -1.5 0.0057737 X

2 99.5 99.3 -0.8 0.0049707 X X

3 99.7 99.5 0.3 0.0042463 X X X

3 99.1 99.5 2.1 0.0042989 X X X X

5 99.7 99.3 4.0 0.0048373 x x x X X

6 99.7 99.0 6.0 0.0058517 X X X X X X

I 99.8 98.0 8.0 0.0082201 X x x X x x x

433

Journal of Manufacturing Systems Vol. 21/No. 6 2002

Table 24 Best Subsets Regression of 18 Individual Observations from 33-’ Experiment (Polynomial model)

A B A c ValY? R-Sq R-Sq(adj) C” S A B C C C A C

1 99.1 99.0 35.7 0.005575 1 X

2 99.3 99.3 23.8 0.0048888 X X

3 99.6 99.6 9.8 0.0037577 X X X

4 99.7 99.7 5.9 0.0032723 X X X X

5 99.8 99.7 5.3 0.0030643 X X X X X

6 99.8 99.7 6.2 0.0030530 X X X X X X

7 99.8 99.7 8.0 0.003 1629 X X X X X X X

Table 25 Comparison of Best Models from 3”” Experiment

# of Data Type of Model Name of Model R* R*(adj.) p C” S PRESS

9 data Linear 13 99.5 99.2 4 4 0.005314 0.000462

Polynomial 14 99.7 99.0 7 6 0.005852 0.000456

18 data Linear 15 99.5 99.3 4 4 0.004582 0.000485

Polvnomial 16 99.8 99.7 7 6.2 0.003053 0.000259

Table 26 Best Regression Model from 3j-l Experiment (18 data, polynomial)

Error = 0.204 + 0.0105 A - 0.00116 B - 0.00623 C + 0.000136 AC - 0.00239 A2 + 0.000021 C’

Predictor Coef SE Coef t P

Constant 0.20429 0.01098 18.60 0.000

A 0.010459 0.005640 1.85 0.091

B 4.001162 0.001114 -1.04 0.319

C -0.0062334 0.0005052 -12.34 0.000

AC 0.00013554 0.00006133 2.21 0.049

AA -0.0023889 0.0007773 -3.07 0.011

cc 0.00002131 0.0000085 1 2.50 0.029

s = 0.003053 R-Sq = 99.8% R-Sq(adj) = 99.7%

PRESS = 0.000259 R-Sq(pred) = 99.53%

Source DF


ss MS F P

Regression 6 0.0545344 0.0090891 975.11 0.000 Residual Error 11 0.0001025 0.0000093

Total 17 0.0546369

automatic inspection) given certain digitizing parameters in reverse engineering or for assisting the selection of digitizing parameters given a required level of minimum accuracy. This model has been implemented in Java and posted on Dr. Jack Feng’s home page at

http://hilltop.bradleyedu/-cfeng/cfeng/research.html. The model has been used in related courses as a labo- ratory tool for reverse engineering and CMM digitization uncertainty study. The methodology used in this research for model selection and model valida-

434


2002

Table 27 Computational Results of Best Regression Models from 3’ Experiment

Run No. A Observed Fitted Error Absolute Error Relative Error

B C Error RAl RA2 RAl RA2 RAI RA2

3

5

6

I

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

1

3

1

2

I I

3

3

2

2

2

3

3

3

I

3

2

3

I

2

2

I

2

I

I

3

2

3

2

3

3

2

3

3

2

2

1

3

1

1 3

1

1

1

3

1

3

2

2

2

2

1

2

1

1

2

2

3

I

3

2

3

1

1 2.

3

2

3

2

I

2

1

1 I

2

2

3

3

3

1

3

0.138

0.042

0.045

0.005

0.142

0.003

0.043

0.002

0.141

0.142

0.045

0.003

0.042

0.002

0.044

0.132

0.045

0.124

0.141

0.135

0.045

0.044

0.005

0.004

0.004

0.132

0.005

0.137

0.045

0.048

0.000

0.138

0.003

0.044

0.000

0.135

0.136

0.045

0.001

0.046

-0.001

0.050

0.135

0.047

0.133

0.139

0.134

0.046

0.049

0.001

0.004

0.005

0.134

0.002

Notes: RAI-Linear model (model 9); RA2-Polynomial model (model 10)

0.138 0.001 0.000 1 .oo 0.042 0.003 0.000 6.94

0.044 0.003 0.001 6.53

0.006 0.005 0.001 92.98

0.141 0.004 0.001 3.08

0.003 0.000 0.000 I.13

0.040 0.001 0.003 2.13

0.003 0.002 0.00 1 99.50

0.138 0.006 0.003 4.30

0.140 0.006 0.002 4.27

0.045 0.000 0.000 0.57

0.002 0.002 0.00 I 66.33

0.041 0.004 0.001 9.32

0.002 0.003 0.000 149.50

0.045 0.006 0.001 13.50

0.133 0.003 0.001 I .97

0.046 0.002 0.001 5.01

0.128 0.009 0.004 6.93

0.143 0.002 0.002 1.69

0.135 0.00 1 0.000 0.79

0.046 0.001 0.001 2.79

0.045 0.005 0.001 11.23

0.007 0.004 0.002 72.98

0.004 0.000 0.000 0.85

0.002 0.001 0.002 25.85

0.131 0.002 0.001 1.21

0.005 0.003 0.000 52.98

0.21

0.72

2.64

18.66

0.48

4.97

6.87

43.00

1.96

1.50

0.40

48.23

1.47

9.80

2.61

0.80

2.57

3.06

I .40

0.28

3.27

3.33

31.94

12.15

44.83

0.45

5.78

tion is applicable to developing empirical models for similar manufacturing processes and systems, for example, Feng and Wang (2003) and Feng and Wang (2002a).

Acknowledgments

This research has been partially funded by Cater- pillar Fellowship #25- 1 1 - 154 and Bradley University Research Excellence Award #I 3-26233 granted to Dr. Jack Feng.

References Balsamo, A.; Marques, D.; and &tori, S. (1990). “A method for thermal-

deformation corrections of CMMs.” Annals qf’ the CIRP (~39, nl), ~~557-560.

Bosch, T.; Harlow, R.; and Thompson, R.L. (1998). Fundamentals of Di- mensional Metrology, 3rd ed. Albany, NY: Delmar Publishers.

Box, G.E.P. and Draper, N.R. (1987). Empirical Model-BuiEding and Re- sponse Surfaces. New York: John Wiley & Sons.

Box, G.E.P.; Hunter, W.G.; and Hunter, J.S. (1978). Statisticsfor Experi- menters: An Introduction to Design, Data Analysis, and Model Build- ing. New York: John Wiley & Sons.

Daniel, C. and Wood, ES. (1980). Fitting Equations to Data. 2nd ed. New York: John Wiley & Sons.

435

Journal of Manufacturing System Vol. 21/No. 6 2002

Table 28 Computational Results of Best Regression Models from 3”” Experiment

Observed Predicted Error Absolute Error Relative Error RunNo. A B C Error RA3 RA4 RA3 RA4 RA3 RA4

I 1 1

2 1 2

3 2 I

4 3 1

5 I 3

6 2 2

7 2 3

8 3 2

9 3 3

10 1 1

11 I 2

12 2 1

13 3 1

14 1 3

I.5 2 2

16 2 3

17 3 2

18 3 3

0.142 0.140

0.004 0.006

0.045 0.048

0.003

0.045

0.140

0.005 -0.00 I

0.042 0.044

0.127 0.129

0.140 0.140

0.004 0.006

0.045 0.048

0.003

0.045

0.141

0.004 -0.001

0.042 0.044

0.120 0.129

0.002

0.047

0.133

0.002

0.047

0.133

Notes: RA3-Linear model (model 15); RA4-Polynomial model (model 16)

0.142 0.002

0.003 0.002

0.048 0.003

0.003 0.001

0.045 0.002

0.137 0.007

0.006 0.006

0.040 0.002

0.126 0.002

0.142 0.000

0.003 0.002

0.048 0.003

0.003 0.001

0.045 0.002

0.137 0.008

0.006 0.005

0.040 0.002

0.126 0.009

0.000 I .24 0.10

0.00 I 39.58 16.40

0.003 7.75 5.98

0.000 29.77 16.00

0.000 5.00 I .07

0.003 4.66 2.38

0.001 123.58 12.72

0.002 4.22 5.27

0.00 1 I .39 0.64

0.002 0.17 1.53

0.001 39.58 16.40

0.003 7.75 5.98

0.000 29.77 16.00

0.000 5.00 1.07

0.004 5.34 3.07

0.002 129.48 40.90

0.002 4.22 5.27

0.006 7.30 5.16

Table 29 Error Statistics of Best Models (Models 9,10,15, and 16)

Model Name RMS Error

RAl (model 9) 0.003570

RA2 (model IO) 0.001492

RA3 (model 15) 0.004046

RA4 (model 16) 0.002387

Relative Error

Mean SD Max Error Min Error

23.90 38.88 0.009 0.000

9.38 14.66 0.003 0.000

24.77 39.27 0.009 0.000

8.66 10.02 0.006 0.000

Table 30 Hypothesis Testing to Check Goodness of Fit of Each Model

p-value

95% C.I. RAI RA2 RA3 RA4

Paired r-test for means 0.959 1.000 0.956 0.772 F-test for variances 0.996 0.997 0.984 0.992

PRESS 0.000483 0.000134 0.000485 0.000259

436

Journc~l qf’ Manufacturing Systems Vol. 21/No. 6

2002

Run No. A B C

Table 31 Computational Results for Linear Model Validation

Observed Predicted Error Absolute Error Error RAl RA3 RAl RA3

Relative Error RAI RA3

1 -1

2 -1

3 I

4 I 5 -1

6 -1

I 1

8 1

9 -1

10 -1

II 1

12 1

-1 -1 0.019 0.028 0.029 0.009 0.010 41.37 52.63

1 -1 0.017 0.026 0.024 0.009 0.007 52.94 41.18

-1 1 0.001 0.001 0.003 0.000 0.002 32.16 158.10

1 I 0.002 -0.00 1 -0.003 0.003 0.005 133.92 237.95

-1 -1 0.019 0.028 0.029 0.009 0.010 47.37 52.63

1 -1 0.018 0.026 0.024 0.008 0.006 44.44 33.33

-1 1 0.002 0.00 I 0.003 0.001 0.001 33.92 29.05

1 1 0.002 -0.001 -0.003 0.003 0.005 133.92 237.95

-1 -1 0.019 0.028 0.029 0.009 0.010 47.37 52.63

1 -I 0.018 0.026 0.024 0.008 0.006 44.44 33.33

-1 1 0.001 0.001 0.003 0.000 0.002 32.16 158.10

1 I 0.002 -0.001 -0.003 0.003 0.005 133.92 237.95

Notes: RA i-27 data linear model; RA3-18 data linear model

Table 32 Computational Results for Polynomial Model Validation

Run No. A B C Observed Predicted Error

Error RA2 RA4 Absolute Error RA2 RA4

Relative Error RA2 RA4

1 -1 -1 -1 0.019 0.026 0.027 0.007 0.008 36.84 42.11

2 -1 1 -1 0.017 0.025 0.025 0.008 0.008 47.06 41.06

3 1 -I 1 0.001 0.002 0.002 0.001 0.001 5 1.54 I IO.67

4 I I 1 0.002 0.002 0.000 0.000 0.002 7.40 110.66

5 -1 -I -1 0.019 0.026 0.027 0.007 0.008 36.84 42.11 6 -1 1 -1 0.018 0.025 0.025 0.007 0.007 38.89 38.89

7 I -1 I 0.002 0.002 0.002 0.000 0.000 24.23 5.34

8 I I I 0.002 0.002 0.000 0.000 0.002 7.40 110.66

9 -1 -1 -1 0.019 0.026 0.027 0.007 0.008 36.84 42.11

10 -1 I -1 0.018 0.025

II I -1 1 0.001 0.002

12 1 1 1 0.002 0.002

Notes: RA2-27 data polynomial model; RAL18 data polynomial model

0.025 0.007 0.007 38.89 38.89

0.002 0.001 0.001 51.54 110.67

0.000 0.000 0.002 7.40 1 10.66

Dek’or. R.E.; Chang, T-H.; and Sutherland, J.W. ( 1992). Statistical Quality Design and Control: Contemporary Concept.7 and Methods. New York: John Wiley & Sons.

Draper, N.R. and Smith, H. (1998). Applied Regre.\.vion Analvsis, 3rd ed. New York: John Wiley & Sons.

Feng, C-X. (2003). “Internet-based reverse engineering.” lnt’1 Journal of Advanced Mfg. Technology (~21, n2). ~~138-144.

Feng. C-X. and Wang, X. (2003). “Surface roughness predictive model- mg: Regression vs. neural networks.” IIE Trans. (~35, nl), ppl l-27.

Feng, C-X. and Pandey, V. (2002). “Experimental study of the effect of digitizing parameters on digitizing uncertainty with a CMM.” In?‘1 Jour- ruczl of Production Research (~40. n3). pp683-697.

Table 33 Error Statistics of Four Best Models frum Data in Table 4

RAl

RA2

RA3

RA4

RMS Error

0.006286

0.00508 1

0.006418

0.005561

Relative Error

Mean SD

65.33 41.88

32.07 16.60

110.40 88.76

64.49 39.51

Max Mitt

Error Error

0.009 0.001

0.008 0.000

0.010 0.00 I

0.008 0.000

437

Journal of Manufacturing Systems Vol. 2l/No. 6 2002

Table 34 Hypothesis Testing to Check Goodness of Prediction

95% C.I. RAl RA2

p-value

RA3 RA4

Paired t-test for means 0.050 0.399 0.081 0.45 1

F-test for variances 0.125 0.273 0.125 0.194

T&le 35 Summary of Hypothesis Testing for Model Comparison

Model Construction

Linear vs. Polynomial Full vs. Fractional

Model Validation

Linear vs. Polynomial Full vs. Fractional

95% C.I.

f-test for means

F-test for variances

0.956 0.991

0.992 0.982

p-value

0.674 0.962

0.648 0.834

Feng, C-X. and Wang, X-F. (2002a). “Development of empirical models for surface roughness prediction in finish turning.” Znt’l Journal of Advanced Mfg. Technology (~20, n5), ~~348-356.

Feng, C-X. and Wang, X-F. (2002b). “Digitizing uncertainty modeling for reverse engineering applications: Regression versus neural networks.” Journal of Intelligent Mfg. (~13, n3), ppl89-200.

Feng, C-X. and Xiao, S. (2000). “Computer-aided reverse engineering with CMM for digitization and LOM for duplication.” Proc. of 4th Int’l Conf. on Frontiers of Design and Mfg., June 17-19,2000, Hangzhou, China. Beijing: Int’l Academic Publishers, ~~256-262.

Feng, C-X.; Wang, X.; and Yu, Z. (2002). “Neural networks modeling of honing surface roughness parameters defined by ISO13565.” Journal of Manufacturing Systems (~21, n5), ~~395-408.

Groth, R. (1998). Data Mining: A Hands-on Approach for Business Pro- fessionals. Upper Saddle River, NJ: Prentice-Hall.

Gu, P. and Chan, K. (1996). “Generative inspection process and probe path planning for coordinate measuring machines.” Journal of Manufactur- ing Systems (~15, n4), ~~240-255.

Hastie, T.; Tibshirani, R.; and Friedman, J. (2001). The Elements of Statistical Learning: Data Mining, Inference, and Prediction. New York: Springer.

Hogg, R.V. and Ledolter, J. (1992). Applied Statistics for Engineers and Physical Scientists, 2nd ed. New York: Macmillan.

International Organization for Standardization (ISO) (1995). Guide to the Expression of Uncertainty in Measurement. Geneva, Switzerland.

Kreuci, J.V. (1990). “CMM measurement enhancement using probe compensation algorithms.” SME Technical Paper MS90-09. Dearborn, MI: Society of Manufacturing Engineers.

Kunzmann, H.; Trapet, E.; and Wlldele, F. (1990). “A uniform concept for calibration, acceptance test, and periodic inspection of coordinate measuring machines using reference objects.” Annals of the CIRP (~39, nl), ~~561-564.

Limaiem, A. and ElMaraghy, H.A. (2000). “Integrated accessibility analysis and measurement operations sequencing for CMMs.” Journal (if Manufacturing Systems (~19, n2), ~~83-93.

Miller, A.J. (2002). Subset Selection in Regression, 2nd ed. London: Chapman & Hall, and personal communications.

Minitab (2000). Meet Minitab. Release 13. State College, PA: Minitab Inc. Montgomery, DC. (2001). Design and Analysis of Experiments, 5th ed.

New York: John Wiley & Sons. Montgomery, D.C. and Runger, G.C. (2003). Applied Statistics and Prob-

ability for Engineers, 3rd ed. New York: John Wiley & Sons.

Montgomery, DC.; Peck, E.A.; and Vining, G.G. (2001). Introduction to Linear Regression Analysis, 3rd ed. New York: John Wiley & Sons.

Otto, K. and Wood, K. (2001). Product Design: Techniques in Reverse Engineering and New Product Development. Upper Saddle River, NJ: Prentice-Hall.

Phillips, S.D.; Borchardt, B.; Estler, W.T.; and Buttreee, J. (1998). “The estimation of measurement uncertainty of small circular features measured by coordinate measuring machines.” Precision Engg. (v22), ~~87-97.

Sartori, S. and Zhang, G.X. (1995). “Geometric error measurement and compensation of machines.” Annals of the CZRP (~44, n2), ~~599-609.

Shaffer, G. (1982). “Taking the measure of CMMs.” American Machinisr (Oct. 1982), ~~145-160.

She”, Y.L. and Zhan, X.P. (1999). “Pretravel compensation for horizon- tally oriented touch trigger probes with straight styli.” Journal ofManu- facturing Systems (~18, n3), ~~175-186.

Tang, S-Y. and Sun, A. (2001). “An analysis of CMM scanning measurement uncertainty using design of experiments.” Proc. of 2001 Indus- trial Engg. Research Conf. Paper No. 2045. Norcross. GA: Institute of Industrial Engineers.

Walpole, R.E.; Myers, R.H.; Myers, S.L.; and Ye, K. (2002). Pmbability and Statisticsfor Engineers and Scientists, 7th ed. Upper Saddle River. NJ: Prentice-Hall.

Weckenmann, A.; Knauer, M.; and Kunzmann, H. (1998). “The influence of measurement strategy on the uncertainty of CMM-measurements.” Annals of the CIRP (~47, nl), pp451-454.

Witten, I.H. and Frank, E. (2000). Data Mining: Practical Machining Learn- ing Tools and Techniques with Java Implementations. San Francisco: Morgan Kaufmann Publishers.

Wohlers, Terry (2001). “Rapid prototyping & tooling - State of the industry 2001 world wide progress report.” Fort Collins, CO: Wohlers Asso- ciates, and persona1 communications.

Yee, K.W. and Gavin, R.J. (1990). “Implementing fast part probing and error compensation on machine tools.” NIST Report 4447. Gaithersburg, MD: Nat’1 Institute of Standards and Technology.

Zhang, G.; Veale, R.; Charlton, T.; Borchardt, B.; and Hocken, R. (1985). “Error compensation of coordinate measuring machines.” Annals of the CIRP (~34, nl), pp445-448.

Zhang, G.X. and Zhang, Y.F. (1991). “A method for machine geometric calibration using 1-D ball array.” Annals of the CIRP (~40, nl), ~~519-522.

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Authors’ Biographies Chang-Xue (Jack) Feng has been an associate professor of industrial

and manufacturing engineering at Bradley University since 1998. He received his PhD and MS degrees in industrial engineering and MS and BS degrees in mechanical engineering. He has applied computational tools including statistics, optimization. computational neural networks, and fuzzy logic in integrated product and process development, agile/lean manufacturing, and quality and precision engineering. His recent research focuses on web-driven data mining with Java technology for design, modeling, prediction, and optimization of precision manufacturing processes and systems, He has coauthored three books and published more than 40 technical

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papers in a number of journals and conference proceedings. His past research has been sponsored by Bradley University, Caterpillar Inc., China NSF, Cincinnati Machine, John Deere, Pennsylvania State University, Rockwell International, Rubbermaid, and the Society of Manufacturing Engineers. He has been a member of ASME, ASQ. BE, and SME.

Xianfeng (David) Wang received his BS, MS, and PhD in mechanical engineering from Wuhan University of Technology (Wuhan, China) and MS in endustrial engineering from Bradley University. He has completed more than 10 research projects and published about 30 technical papers in journals and professional conferences. More than 10 of his technical papers have won various prizes.

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subset selection in predictive modeling of the cmm digitization uncertainty

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