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Details of the OPTEX Procedure |
The search procedures available in the OPTEX procedure offer various compromises between speed and reliability in finding the optimum. In general, the longer an algorithm takes to arrive at an answer, the more efficient is the resulting design, although this is not invariably true. The right search procedure for any specific case depends on the size of the problem, the relative importance of using the best possible design as opposed to a very good one, and the computing resources available.
It follows, for example, that adding x multiplies the determinant of the information matrix by 1+x'Vx, and likewise deleting y multiplies the determinant by 1 - y'Vy. For any point z, the quantity z'Vz is proportional to the prediction variance at the point z. Thus, the point x whose addition to the design maximizes the determinant of the information is the point whose prediction variance calculated from the present design is largest. The point whose deletion from the design costs the least in terms of the determinant is the point with the smallest prediction variance.
Similar rank-one update formulas can be derived for A-optimality, which is based on the trace of the inverse of the information matrix instead of its determinant. However, in this case there is no single quantity that can be examined for both adding and deleting a point. Instead, the trace that results from adding a point x depends on
There are no useful rank-one update formulas for the distance-based design criteria.
Since the sequential search method involves no randomness, it requires only one try to find a design. The sequential procedure is by far the fastest of any of the search methods, but in difficult design situations it is also the least reliable in finding a globally optimal design. Also, the fact that it always finds the same design (due to the lack of randomness mentioned previously) makes it inappropriate when you want to find a design that represents a compromise between several optimality criteria.
Johnson and Nachtsheim (1983) introduce a generalization of both the simple exchange algorithm and the modified Fedorov search algorithm of Cook and Nachtsheim (1980), which is described later in this list. In the modified Fedorov algorithm, each of the points in the current design is considered for exchange with a candidate point; in the generalized version, only the k design points with smallest variance in the current design are considered for exchange. You can specify k-exchange as the search procedure for OPTEX by giving a value for k in parentheses after METHOD=EXCHANGE. When k=N_{D}, the size of the design, k-exchange is equivalent to the modified Fedorov algorithm; when k=1, it is equivalent to the simple exchange algorithm. Cook and Nachtsheim (1980) indicate that k < N_{D} / 4 is typically sufficient.
At each step, the Fedorov algorithm (Fedorov 1972) seeks the pair (x,y) of one candidate point and one design point that optimizes the change in the optimality criterion, and then switches x for y in the design. Thus, after computing for all possible pairs of candidate and design points, the Fedorov algorithm performs only one switch.
The modified Fedorov algorithm of Cook and Nachtsheim (1980) computes the same number of 's on each step but switches each point y in the design with the candidate point x that maximizes (x,y). This procedure is generally as reliable as the simple Fedorov algorithm in finding the optimal design, but it can be up to twice as fast.
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