5.
Process Improvement
5.5. Advanced topics 5.5.3. How do you optimize a process? 5.5.3.1. Single response case


"Randomness" means that the steepest ascent direction is just an estimate and it is possible to construct a confidence "cone" around this direction estimate 
The direction given by the gradient
g' = (b_{0}, b_{2}, ... ,
b_{k}) constitutes only a single (point) estimate
based on a sample of N runs. If a different set of N
runs were conducted, these would provide different parameter estimates,
which in turn would give a different gradient. To account for this
sampling variability, Box and
Draper gave a formula for constructing a "cone" around the
direction of steepest ascent that with certain probability contains the
true (unknown) system gradient given by
\( (\beta_{1}, \beta_{2}, \ldots , \beta_{k}) \).
The width of the confidence cone is useful to assess how reliable an
estimated search direction is.
Figure 5.4 shows such a cone for the steepest ascent direction in an experiment with two factors. If the cone is so wide that almost every possible direction is inside the cone, an experimenter should be very careful in moving too far from the current operating conditions along the path of steepest ascent or descent. Usually this will happen when the linear fit is quite poor (i.e., when the R^{2} value is low). Thus, plotting the confidence cone is not so important as computing its width. If you are interested in the details on how to compute such a cone (and its width), see Technical Appendix 5B. 

Graph of a confidence cone for the steepest ascent direction 
FIGURE 5.4: A Confidence Cone for the Steepest Ascent Direction in an Experiment with 2 Factors 



Technical Appendix 5B: Computing a Confidence Cone on the Direction of Steepest Ascent  
Details of how to construct a confidence cone for the direction of steepest ascent 
Suppose the response of interest is adequately described by a
firstorder polynomial model. Consider the inequality
with
or inside the 100(1  \( \alpha \))% confidence cone of steepest descent if


Inequality defines a cone  The inequality defines a cone with the apex at the origin and center line located along the gradient of \( \hat{Y} \).  
A measure of goodnes of fit: \( \theta_{\alpha} \) 
A measure of "goodness" of a search direction is given by the fraction
of directions excluded by the
100(1  \( \alpha \))%
confidence cone around the steepest ascent/descent direction (see
Box and Draper, 1987)
which is given by:


Example: Computing \( \theta_{\alpha} \)  
Compute \( s_{b}^{2} \) from ANOVA table and C_{jj} 
From the ANOVA table in the chemical experiment discussed
earlier


Compute \( \theta_{\alpha} \) 


Conclusions for this example  since F_{0.05,1,6} = 5.99. Thus 71% of the possible directions from the current operating point are excluded with 95 % confidence. This is useful information that can be used to select a step length. The smaller \( \theta_{\alpha} \) is, the shorter the step should be, as the steepest ascent direction is less reliable. In this example, with high confidence, the true steepest ascent direction is within this cone of 29% of possible directions. For k=2, 29% of 360^{o} = 104.4^{o}, so we are 95% confident that our estimated steepest ascent path is within plus or minus 52.2^{o} of the true steepest path. In this case, we should not use a large step along the estimated steepest ascent path. 