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Robust parameter designs, introduced by Taguchi, are experimental designs used to exploit the interaction between control and uncontrollable noise variables by robustification -- finding the settings of the control factors that minimize response variation from uncontrollable factors.〔Brewer, K., Carraway, L., and Ingram, D. (2010) "Forward Selection as a Candidate for Constructing Nonregular Robust Parameter Designs." Arkansas State University.〕 Control variables are variables of which the experimenter has full control. Noise variables lie on the other side of the spectrum, and while these variables may be easily controlled in an experimental setting, outside of the experimental world they are very hard, if not impossible, to control. Robust parameter designs use a naming convention similar to that of FFDs. A 2''(m1+m2)-(p1-p2)'' is a 2-level design where ''m1'' is the number of control factors, ''m2'' is the number of noise factors, ''p1'' is the level of fractionation for control factors, and ''p2'' is the level of fractionation for noise factors. Consider an RPD cake baking example from Montgomery (2005), where an experimenter wants to improve the quality of cake.〔Montgomery, D. (2005), Design and Analysis of Experiments. 6th ed. Wiley.〕 While the cake manufacturer can control the amount of flour, amount of sugar, amount of baking powder, and coloring content of the cake; other factors are uncontrollable, such as oven temperature and bake time. The manufacturer can print instructions for a bake time of 20 minutes but in the real world has no control over consumer baking habits. Variations in the quality of the cake can arise from baking at 325o instead of 350o or from leaving the cake in the oven for a slightly too short or too long period of time. Robust parameter designs seek to minimize the effects of noise factors on quality. For this example, the manufacturer hopes to minimize the effects in fluctuation of bake time on cake quality, and in doing this the optimal settings for the control factors is required. RPDs are primarily used in a simulation setting where uncontrollable noise variables are easily controlled. Whereas in the real world noise factors are hard to control, in an experimental setting control over these factors is easily maintained. For the cake-baking example, the experimenter can fluctuate bake time and oven temperature to understand the effects of such fluctuation that may occur when control is no longer in his hands. Robust parameter designs very similar to fractional factorial designs (FFDs) in that the optimal design can be found using Hadamard matrices, principles of effect hierarchy and factor sparsity are maintained, and aliasing is present when full RPDs are fractionated. Much like FFDs, RPDs are screening designs and can provide a linear model of the system at hand. What is meant by effect hierarchy for FFDs is that higher-order interactions tend to have a negligible effect on the response.〔Wu, C.F.J. and Hamada, M. (2000), Experiments: Planning, Analysis, and Parameter Design Optimization. Wiley.〕 As stated in Carraway, main effects are most likely to have an effect on the response, then two-factor interactions, then three-factor interactions, and so on.〔Carraway, L. (2008). “Investigating the Use of Computational Algorithms for Constructing Non-Regular Robust Parameter Designs,” Masters Thesis, Arkansas State University.〕 The concept of effect sparsity is that not all factors will have an effect on the response. These principles are the foundation for fractionating Hadamard matrices. By fractionating, experimenters can form conclusions in fewer runs and with fewer resources. Oftentimes, RPDs are used at the early stages of an experiment. Because two-level RPDs assume linearity among factor effects, other methods may be used to model curvature after the number of factors has been reduced. ==Construction of RPDs== Hadamard matrices are square matrices consisting of only + and -. If a Hadamard matrix is normalized and fractionated, a design pattern is obtained. However, not all designs are equal. This means that some designs are better than others, and specific design criteria is used to determine which design is best. After obtaining a design pattern, experimenters know to which setting each factor should be set. Each row in the pattern indicates a run, and each column indicates a factor. For the partial design pattern shown left, the experimenter has identified 7 factors that may have an effect on the response and hopes to gain insight as to which factors have an effect in 8 runs. In the first run, factors 1, 4, 5, and 6 are set to high levels while factors 2, 3, and 7 are set to low levels. Low levels and high levels are settings typically defined by the subject matter expert. These values are extremes but not so extreme that the response is pushed into non-smooth regions. After each run, results are obtained; and by fluctuating multiple factors in single runs instead of using the OFAT method, interactions between variables may be estimated as well as the individual factor effects. If two factors interact, then the effect one factor has on the response is different depending on the settings of another factor. Fractionating Hadamard matrices appropriately is very time-consuming. Consider a 24-run design accommodating 6 factors. The number of Hadamard designs from each Hadamard matrix is 23 choose 6; that's 100,947 designs from each 24x24 Hadamard matrix. Since there are 60 Hadamard matrices of that size, the total number of designs to compare is 6,056,820. Fortunately, Leoppky, Bingham, and Sitter (2006) used complete search methodology and have listed the best RPDs for 12, 16, and 20 runs. Because complete search work is so exhaustive, the best designs for larger run sizes are often not readily available. In that case, other statistical methods may be used to fractionate a Hadamard matrix in such a way that allows only a tolerable amount of aliasing. Efficient algorithms such as forward selection and backward elimination have been produced for FFDs, but due to the complexity of aliasing introduced by distinguishing control and noise variables, these methods have not yet been proven effective for RPDs.〔Ingram, D. (2000), The construction of generalized minimum aberration designs by efficient algorithm. Dissertation, University of Memphis.〕〔Ingram, D. and Tang, B. (2001), Efficient Computational Algorithms for Searching for Good Designs According to the Generalized Minimum Aberration Criterion, American Journal of Mathematical and Management Sciences, 21 325-344.〕〔Ingram, D. And Tang, B. (2005), Construction of minimum G-aberration Designs via Efficient Computational Algorithms, Journal of Quality Technology, 37 101-114.〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Robust parameter design」の詳細全文を読む スポンサード リンク
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