Direction of steepest ascent

The key to investigating the topology of the electron density p is the gradient vector V p, which is perpendicular to a constant electron density snrface and points in the direction of steepest ascent. Then, a sequence of infinitesimal gradient vectors corresponds to a gradient path. Since gradient vectors are directed, gradient paths also have a direction They can go uphill or downhill. 

Once the transformations have been made, a solution space has been defined. It is only within this framework that the concept of steepest ascents takes meaning. The proper direction in which to proceed can be determined by n + 1 cases if n is the number of controllable variables. In each of the n cases one variable is changed slightly from its value in the base case, while all other variables are held constant. This permits approximating the n partial derivatives of response with respect to each variable. The direction of steepest ascent is given by the vector which is the gradient of the response, R ... 

Before proceeding further it is necessary to decide upon the size of the step to be made in the direction of steepest ascent. This step size, A, can be related to the several partial derivatives by a constant K ... 

Once the constant K has been determined from this equation, the incremental change in each individual variable for steps in the direction of steepest ascent is... 

Now a new case is calculated at the best previous point and a new direction of steepest ascent is determined. The process is repeated as many times as seems advisable. The entire procedure is susceptible to automatic treatment on the computer. The only point of uncertainty is the size of the steps to be taken. It may be necessary to revise this from time to time depending upon the progress of the study. As the optimum is approached, the steps should decrease in size. Even if it is decided that for a particular problem and a particular computer completely automatic calculation is impractical, at least some fairly large combination of operations can be programmed for one computer run. 

No sane cartographer, in drawing a contour map of a hill, would ever choose a scale in the north-BOuth direction different from that in the east west direction. The concept of length, well-defined in the science of geometry, forces him to make the scale independent of the direction. Consequently any cartographer is within his rights to draw small circles and to speak of a direction of steepest ascent. 

Figure 2 shows a yield contour for a particular choice of scales for and x2. The same contour is plotted in Fig. 3 for which the horizontal scale has been doubled. The contour tangent and gradient line at the same point a are given for each choice of horizontal scale. The two gradient lines obviously do not contain the same points. Thus the direction of steepest ascent depends entirely on the relative scales of the... 

Comparing (5.13) and (5.12) we deduce that l(m) really is the direction of steepest ascent,... 

This result comes from the simple fact that if we minimize a functional along some direction, described by a parametric line, the direction of steepest ascent must... 

The first-order design and determining the direction of steepest ascent... 

If the design was for a second-order model and examination of the contour plots or canonical analysis (see below) showed that the optimum probably lay well outside the experimental domain, then the direction for exploration would no longer be a straight line, as for the steepest ascent method. In fact, the "direction of steepest ascent" changes continually and lies on a curve called the optimum path. The calculations for determining it are complex, but with a suitable computer program the principle and graphical interpretation become easy. 

We now return to our first example. Having performed the initial modeling and found the direction of steepest ascent, we go on to perform experiments at the conditions specified in Table 6.3. Doing this we obtain the results in the last column of the table, which are also indicated in Fig. 6.4. 

The coefficients (b s) refer to the values given in Table 12. The gradient, i.e. the direction of steepest ascent, is the vector of partial derivatives of the model with respect to the variables. Differentiating the expression given in Eq. 7 gives in matrix notation... 

The next step in the optimization is to carry out further experiments in the direction indicated by the dotted line in Figure 7.10, at (say) the points numbered 5, 6 and 7. This would indicate point 6 as a rough position for the maximum in this direction. Another factorial experiment is carried out in this region to determine the new direction of steepest ascent. 

Figure 7.10 Contour diagram the initial direction of steepest ascent is shown by the broken line. Further experiments are done at points 5, 6 and 7.

Figure 6 Comparison of the regression between two x-variables and one y-variable by using latent variables from PCA, PCR and MLR. ftpci, direction of first principal component (maximum variance in x-space) Arlsi, direction of first PLS component (maximum covariance with y) Amlr. direction of steepest ascent in y-plane (maximum correlation coefficient with y). Three plots at the bottom show the different correlation coefficients r between the scores of these latent variables and y...

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