• Title/Summary/Keyword: rotatability

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On Slope Rotatability of Central Composite Designs of the Second Type

  • Kim, Hyuk-Joo;Ko, Yun-Mi
    • Communications for Statistical Applications and Methods
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    • v.11 no.1
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    • pp.121-137
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    • 2004
  • Kim(2002) proposed a second type of central composite design (CCD2), in which the positions of the axial points are indicated by two numbers. In this paper, we study properties of CCD2 when we are interested in estimating the slope of a response surface. Conditions are obtained for CCD2 to be slope-rotatable over axial directions, and some CCD2's are presented that have slope rotatability over axial directions. Also values of a measure of slope rotatability over axial directions are tabulated for various CCD2's. Finally, it is shown that CCD2 is always slope-rotatable over all directions.

Slope-Rotatability over All Directions in Third Order Response Surface Models

  • Park, Sung-Hyun;Lee, Min-Soo
    • Journal of the Korean Statistical Society
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    • v.24 no.2
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    • pp.519-536
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    • 1995
  • In the design of experiments for response surface analysis, sometimes it is of interest to estimate the difference of responses at two points. If differences at points close together are involved, the design that reliably estimates the slope of response surface is important. This idea was conceptualized by slope rotatability by Hader & Park (1978) and Park (1987). Until now, second order polynomial models were only studied for slope ratatability. In this paper, we will propose the necessary and sufficient conditions for slope rotatability over all directions for the thired order polynomial models in two, three and four independent variables. Also practical slope rotatable designs over all directions for two independent variables are suggested.

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Statistical Properties of Second Type Central Composite Designs (제2종의 중심합성계획의 통계적 성질)

  • Kim Hyuk-Joo;Park Sung-Hyun
    • The Korean Journal of Applied Statistics
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    • v.19 no.2
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    • pp.257-270
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    • 2006
  • Kim(2002) proposed a second type of central composite design in which the positionsof the axial points are indicated by two numbers, and called it CCD2. In the present paper, we have studied CCD2 further and obtained several new facts. We have obtained CCD2's that have both orthogonality and rotatability, both orthogonality and slope rotatability, and both rotatability and uniform precision. We also have applied Park and Kim's (1992) measure of slope rotatability to such CCD2's and observed some useful results.

MEASURES FOR STABILITY OF SLOPE ESTIMATION ON THE SECOND ORDER RESPONSE SURFACE AND EQUALLY-STABLE SLOPE ROTATABILITY

  • Park, Sung H.;Kang, Ho-Seog;Kang, Kee-Hoon
    • Journal of the Korean Statistical Society
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    • v.32 no.4
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    • pp.337-357
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    • 2003
  • This paper introduces new measures for the stability of slope estimation on the second order response surface at a point and on a sphere. As a measure of point stability of slope estimation, we suggest a point dispersion measure of slope variances over all directions at a point. A spherical dispersion measure is also proposed as a measure of spherical stability of slope estimation on each sphere. Some designs are studied to explore the usefulness of the proposed measures. Using the point dispersion measure, another concept of slope rotatability called equally-stable slope rotatability is proposed as a useful property of response surface designs. We provide a set of conditions for a design to have equally-stable slope rotatability.

A Class of Multi-Factor Designs for Estimating the Slope of Response Surfaces

  • Park, Sung H.
    • Journal of Korean Society for Quality Management
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    • v.14 no.1
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    • pp.26-32
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    • 1986
  • A class of multi-factor designs for estimating the slope of second order response surfaces is presented. For multi-factor designs the variance of the estimated slope at a point is a function of the direction of measurement of the slope and the design. If we average the variance over all possible directions, the averaged variance is only a function of the point and the design. By choice of design, it is possible to make this variance constant for all points equidistant from the design origin. This property is called "slope-rotatability over all directions", and the necessary and sufficient conditions for a design to have this property are given and proved. The class of design with this property is mainly discussed.

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SLOPE ROTATABILITY OF ICOSAHEDRON AND DODECAHEDRON DESIGNS

  • Kim Hyuk-Joo;Park Sung-Hyun
    • Journal of the Korean Statistical Society
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    • v.35 no.1
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    • pp.25-36
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    • 2006
  • Icosahedron and dodecahedron designs are experimental designs which can be used for response surface analysis for the case when three independent variables are involved. When we are interested in estimating the slope of a response surface, slope rot at ability is a desirable property. In this paper, we derive conditions for icosahedron and dodecahedron designs to have slope rotatability, and actually obtain some slope-rotatable icosahedron and dodecahedron designs. We also apply Park and Kim (1992)'s measure of slope rotatability to icosahedron and dodecahedron designs, and observe resultant facts.

Construction of Second Order Slope Rotatable Designs Using Symmetrical Unequal Block Arrangements with Two Unequal Block Sizes

  • Babu, B.Re.Victor
    • Journal of the Korean Statistical Society
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    • v.31 no.2
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    • pp.153-161
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    • 2002
  • A new method of construction of second order slope rotatable designs (SOSRD) using symmetrical unequal block arrangements (SUBA) with two unequal block sizes is suggested. The proposed method may sometimes lead to designs with less number of design points than those available in the literature. Further, bounds for the parameters of SOSRD are also obtained.

A Study on the Influence of a Missing Cell in a Class of Central Composite Designs

  • Park, Sung-Hyun;Noh, Hyun-Gon
    • Journal of the Korean Statistical Society
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    • v.27 no.1
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    • pp.133-152
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    • 1998
  • The central composite design is widely used in the response surface analysis, because it can fit the second order model with small experimental points. In practice, the experimental data are not always obtained on all the points. When there are missing observations, many problems due to the missing cells can occur. In this paper, the influence of a missing cell on the central composite design is discussed. First, the influences of a missing cell on the variances of estimated regression coefficents are compared as $\alpha$ varies. Second, how the average predition variance is affected by a missing sell is discussed. And the influence on rotatability is investigated. Third, the influence of a missing cell on optimality, especially on D-optimality and A-optimality, is examined.

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A Measure of Slope Rotatability for Mixture Experiments

  • Jung-Il Kim
    • Communications for Statistical Applications and Methods
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    • v.3 no.1
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    • pp.51-59
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    • 1996
  • A measure that quantifies the amount of slope retatability for the second degree Scheffe polynomial model for mixture experiments is proposed and used to compare the several mixture designs which met the symmetric momemts conditions in this article.

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