Statistics for Environmental Engineers

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The method was first demonstrated by Box and Wilson (1951) in a paper that Margolin (1985) describes as follows:

The paper… is one of those rare, truly pioneering papers that completely shatters the existing paradigm of how to solve a particular problem.. The design strategy… to attain an optimum operating condition is brilliant in both its logic and its simplicity. Rather than exploring the entire continuous experimental region in one fell swoop, one explores a sequence of subregions. Two distinct phases of such a study are discernible. First, in each subregion a classical two-level fractional factorial design is employed… This first-phase process is iterative and is terminated when a region of near stationarity is reached. At this point a new phase is begun, one necessitating radically new designs for the successful culmination of the research effort.

Response Surface Methodology

The strategy is to explore a small part of the experimental space, analyze what has been learned, and then move to a promising new location where the learning cycle is repeated. Each exploration points to a new location where conditions are expected to be better. Eventually a set of optimal operating conditions can be determined. We visualize these as the peak of a hill (or bottom of a valley) that has been reached after stopping periodically to explore and locate the most locally promising path. At the start we imagine the shape of the hillside is relatively smooth and we worry mainly about its steepness. Figure 43.1 sketches the progress of an iterative search for the optimum conditions in a process that has two active independent variables. The early explorations use two-level factorial experimental designs, perhaps augmented with a center point. The main effects estimated from these designs define the path of steepest ascent (descent) toward the optimum. A two-level factorial design may fail near the optimum because it is located astride the optimum and the main effects appear to be zero. A quadratic model is needed to describe the optimal region. The experimental design to fit a quadratic model is a two-level factorial augmented with stars points, as in the optimization stage design shown in Figure 43.1.

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