scholarly journals Optimal Prediction Variance Capabilities of Inscribed Central Composite Designs

2020 ◽  
pp. 1-8
Author(s):  
Julius C. Nwanya ◽  
Kelechukwu C. N. Dozie
Keyword(s):  
Central Composite Design ◽  
Design Space ◽  
Optimal Prediction ◽  
Prediction Variance ◽  
Central Composite Designs ◽  
Scaled Prediction Variance ◽  
Composite Designs ◽  
Central Composite

This study looks at the effects of replication on prediction variance performances of inscribe central composite design especially those without replication on the factorial and axial portion (ICCD1), inscribe central composite design with replicated axial portion (ICCD2) and inscribe central composite design whose factorial portion is replicated (ICCD3). The G-optimal, I-optimal and FDS plots were used to examine these designs. Inscribe central composite design without replicated factorial and axial portion (ICCD1) has a better maximum scaled prediction variance (SPV) at factors k = 2 to 4 while inscribe central composite design with replicated factorial portion (ICCD3) has a better maximum and average SPV at 5 and 6 factor levels. The fraction of design space (FDS) plots show that the inscribe central composite design is superior to ICCD3 and inscribe central composite design with replicated axial portion (ICCD2) from 0.0 to 0.5 of the design space while inscribe central composite design with replicated factorial portion (ICCD3) is superior to ICCD1 and ICCD2 from 0.6 to 1.0 of the design space for factors k = 2 to 4.

Research Note: Modified Orthogonal Central-Composite Designs

1976 ◽  
Vol 18 (1) ◽  
pp. 95-98 ◽  
Author(s):  
Robert C. Williges
Keyword(s):  
Central Composite Design ◽  
Second Order ◽  
Research Note ◽  
Design Parameters ◽  
Orthogonal Designs ◽  
Central Composite Designs ◽  
Data Points ◽  
Composite Designs ◽  
Central Composite

Simplified formulae for determining the coded value of α are presentedfor rotatable, blocked orthogonal second–order designs in which all data points are replicated an equal number of times. These three central–composite design parameters are compared, and the advantages and limitations of orthogonal designs are presented.

scholarly journals Alternative Axial Distances for Spherical Regions of Central Composite Designs

2020 ◽  
pp. 273-285
Author(s):  
Linus Ifeanyi Onyishi ◽  
F. C. Eze
Keyword(s):  
Central Composite Design ◽  
Optimality Criteria ◽  
Spherical Design ◽  
Central Composite Designs ◽  
Composite Designs ◽  
Central Composite

Alternatives to the existing axial distances of the Central Composite Design (CCD) in spherical design using three axial distances were studied. The aim of this study is to determine a better alternative to already existing axial distances whose prediction properties are more stable in the spherical design regions. Using the concepts of the three Pythagorean means, the arithmetic, harmonic and geometric axial distances for spherical regions were developed. The performances of the alternative axial distances were compared with the existing ones using the D and G optimality criteria. The study shows that the alternative axial distances are better using the D and G optimality criteria.

scholarly journals Alternative Second-Order N-Point Spherical Response Surface Methodology Design and Their Efficiencies

2016 ◽  
Vol 5 (4) ◽  
pp. 22
Author(s):  
Mary Paschal Iwundu
Keyword(s):  
Response Surface Methodology ◽  
Central Composite Design ◽  
Response Surface ◽  
Second Order ◽  
The Face ◽  
Central Composite Designs ◽  
Face Centered ◽  
Composite Designs ◽  
Central Composite ◽  
Exact Designs

The equiradial designs are studied as alternative second-order N-point spherical Response Surface Methodology designs in two variables, for design radius ρ = 1.0. These designs are seen comparable with the standard second-order response surface methodology designs, namely the Central Composite Designs. The D-efficiencies of the equiradial designs are evaluated with respect to the spherical Central Composite Designs. Furthermore, D-efficiencies of the equiradial designs are evaluated with respect to the D-optimal exact designs defined on the design regions of the Circumscribed Central Composite Design, the Inscribed Central Composite Design and the Face-centered Central Composite Design. The D-efficiency values reveal that the alternative second-order N-point spherical equiradial designs are better than the Inscribed Central Composite Design though inferior to the Circumscribed Central Composite Design with efficiency values less than 50% in all cases studied. Also, D-efficiency values reveal that the alternative second-order N-point spherical equiradial designs are better than the N-point D-optimal exact designs defined on the design region supported by the design points of the Inscribed Central Composite Design. However, the N-point spherical equiradial designs are inferior to the N-point D-optimal exact designs defined on the design region supported by the design points of the Circumscribed Central Composite Design and those of the Face-centered Central Composite Design, with worse cases with respect to the design region of the Circumscribed Central Composite Design.

scholarly journals Design Efficiency and Optimal Values of Replicated Central Composite Designs with Full Factorial Portions

2021 ◽  
Vol 4 (3) ◽  
pp. 89-117
Author(s):  
Iwundu M.P. ◽  
Oko E.T.
Keyword(s):  
Central Composite Design ◽  
Optimal Combination ◽  
Efficient Design ◽  
Center Portion ◽  
Design Variables ◽  
Central Composite Designs ◽  
Optimal Values ◽  
Full Factorial ◽  
Composite Designs ◽  
Central Composite

Efficiency and optimal properties of four varieties of Central Composite Design, namely, SCCD, RCCD, OCCD and FCCD and having r_f replicates of the full factorial portion, r_α replicates of the axial portion and r_c replicates of the center portion are studied in four to six design variables. Optimal combination,[r_f: r_α: r_c ] of design points associated with the three portions of each central composite design is presented. For SCCD, the optimal combinations resulting in A- and D- efficient designs generally put emphasis on replicating the center portion of the SCCD. However, replicating the center and axial portions allows for G-optimal and efficient designs. For RCCD, the optimal combinations resulting in A- and D- efficient designs generally put emphasis on replicating the factorial and center portions of the RCCD. However, replicating the center and axial portions allows for G-optimal and efficient designs. For OCCD, the optimal combinations resulting in A- optimal and efficient designs generally put emphasis on replicating the axial and center portion of the OCCD. The optimal combinations resulting in G- optimal and efficient designs generally put emphasis on replicating the factorial and axial portions of the OCCD. To achieve designs that are D-optimal and D-efficient, the optimal combination of design points generally put emphasis on replicating the center portion of the OCCD. For FCCD, the optimal combinations of design points resulting in A-efficient designs put emphasis on replicating the axial portion of the FCCD. The optimal combinations resulting in G- optimal and efficient designs as well as G-optimal and efficient designs generally put emphasis on replicating the factorial and axial portions of the FCCD. It is interesting to note that for FCCD in five design variables, any r^th complete replicate of the distinct design points of the combination [r_f: r_α: r_c ] resulted in a D-efficient design. Many super-efficient designs having efficiency values greater than 1.0 emerged under the D-criterion. Unfortunately, these designs did not perform very well under A- and G-criteria, having some efficiency values much below 0.5 or just about 0.6.

scholarly journals Missing Observations in Split-Plot Central Composite Designs: The Loss in Relative A-, G-, and V- Efficiency

2021 ◽  
Vol 25 (2) ◽  
pp. 239-247
Author(s):  
Y. Yakubu ◽  
A.U. Chukwu
Keyword(s):  
Model Parameters ◽  
Missing Observations ◽  
Prediction Variance ◽  
Efficiency Losses ◽  
Split Plot ◽  
Negative Effect ◽  
Central Composite Designs ◽  
Plot Structure ◽  
Composite Designs ◽  
Central Composite

The trace (A), maximum average prediction variance (G), and integrated average prediction variance (V) criteria are experimental design evaluation criteria, which are based on precision of estimates of parameters and responses. Central Composite Designs(CCD) conducted within a split-plot structure (split-plot CCDs) consists of factorial (𝑓), whole-plot axial (𝛼), subplot axial (𝛽), and center (𝑐) points, each of which play different role in model estimation. This work studies relative A-, G- and V-efficiency losses due to missing pairs of observations in split-plot CCDs under different ratios (d) of whole-plot and sub-plot error variances. Three candidate designs of different sizes were considered and for each of the criteria, relative efficiency functions were formulated and used to investigate the efficiency of each of the designs when some observations were missing relative to the full one. Maximum A-efficiency losses of 19.1, 10.6, and 15.7% were observed at 𝑑 = 0.5, due to missing pairs 𝑓𝑓, 𝛽𝛽, and 𝑓𝛽, respectively, indicating a negative effect on the precision of estimates of model parameters of these designs. However, missing observations of the pairs- 𝑐𝑐, 𝛼𝛼, 𝛼𝑐, 𝑓𝑐, and 𝑓𝛼 did not exhibit any negative effect on these designs' relative A-efficiency. Maximum G- and Vefficiency losses of 10.1,16.1,0.1% and 0.1, 1.1, 0.2%, were observed, respectively, at 𝑑 = 0.5, when the pairs- 𝑓𝑓, 𝛽𝛽, 𝑐𝑐, were missing, indicating a significant increase in the designs' maximum and average variances of prediction. In all, the efficiency losses become insignificant as d increases. Thus, the study has identified the positive impact of correlated observations on efficiency of experimental designs. Keywords: Missing Observations, Efficiency Loss, Prediction variance

scholarly journals Evaluation and Comparison of Three Classes of Central Composite Designs

2021 ◽  
pp. 31-47
Author(s):  
Fidelia Chinenye Kiwu-Lawrence ◽  
Lawrence Chizoba Kiwu ◽  
Desmond Chekwube Bartholomew ◽  
Chukwudi Paul Obite ◽  
Akanno Felix Chikereuba
Keyword(s):  
Response Surface Methodology ◽  
Central Composite Design ◽  
Response Surface ◽  
Central Composite Designs ◽  
Face Centered ◽  
Composite Designs ◽  
Composite Face ◽  
Central Composite

Three classes of Central Composite Design: Central Composite Circumscribed Design (CCCD), Central Composite Inscribed Design (CCID) and Central Composite Face-Centered Design (CCFD) in Response Surface Methodology (RSM) were evaluated and compared using the A-, D-, and G-efficiencies for factors, k, ranging from 3 to 10, with 0-5 centre points, in other to determine the performances of the designs under consideration. The results show that the CCDs (CCCD, CCFD and CCID) are at their best when the G-efficiency is employed for all the factors considered while the CCID especially behaves poorly when using the A- and D-efficiencies.

Response Surface Methodology Central-Composite Design Modifications for Human Performance Research

1973 ◽  
Vol 15 (4) ◽  
pp. 295-310 ◽  
Author(s):  
Christine Clark ◽  
Robert C. Williges
Keyword(s):  
Response Surface Methodology ◽  
Central Composite Design ◽  
Response Surface ◽  
Human Performance ◽  
Experimental Point ◽  
Multiple Observations ◽  
Central Composite Designs ◽  
Composite Designs ◽  
Performance Research ◽  
Central Composite

Selected response surface methodology (RSM) designs that are viable alternatives in human performance research are discussed. Two major RSM designs that are variations of the basic, blocked, central-composite design have been selected for consideration: (1) central-composite designs with multiple observations at only the center point, and (2) central-composite designs with multiple observations at each experimental point. Designs of the latter type are further categorized as: (a) designs which collapse data across all observations at the same experimental point; (b) between-subjects designs in which no subject is observed more than once, and observations at each experimental point may be multiple and unequal or multiple and equal; and (c) within-subject designs in which each subject is observed only once at each experimental point. The ramifications of these designs are discussed in terms of various criteria such as rotatability, orthogonal blocking, and estimates of error.

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