Appunti Statistica per la sperimentazione e le previsioni in ambito tecnologico (parte 3)

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Summary:

The method of steepest ascent is a procedure used in RSM to quickly move toward the region

where the response is optimized. Often, the initial operating conditions are far from the true

optimum. When this happens a first-order (linear) regression model provides a good local

approximation of the response surface. Instead of exploring the entire experimental space,

the experimenter follows the direction in which the response increases most rapidly.

This direction is called the path of the steepest ascent.

For a first order model, the contours of the response surface are parallel lines, and the

direction of steepest ascent is perpendicular to these contours. Mathematically, this direction

is determined by the regression coeCicients: the step taken along each factor are proportional

to the estimated coeCicients of the model.

Experiments are performed sequentially along this path no further improvement in the

response is observed. At that point, the first-order model is no longer adequate, usually

indicating proximity to the optimum.

If the goal is to minimize the response instead of maximizing it, the same idea is applied in the

opposite direction and is called the method of steepest descent.

9.2 Passage from the 1 to the 2 order

st nd

As well known, the SS of the response variable Y is decomposed in: SSmodel + SSres. The

crucial point in RSM is that SSres is then decomposed in:

Where SSlof is the SS (or deviance) related to the Lack of Fit, e.g. to the good-of-fitness of the

1 order model. In this way, we are dealing with the last 1 order model applied along the

st st

steepest ascent/descent procedure. Instead, the SSpe is the deviance calculated through the

replicates in x0, e.g. n0, where x0 is the design center, in coded values. 61

Please note that the SSlof is strictly related to the model, while the SSpe is strictly related to

the design, and moreover, the SSpe depends on the n0 of replicates.

In order to verify if it is necessary to pass to the 2 order, two hypothesis tests must be carried

nd

out. The first one named LOF test, the second one Curvature test. Obviously, the LOF test can

be also calculated along the steepest ascent/descent procedure, when each single 1 order

st

polynomial model is estimated, however the final LOF test, jointly with the Curvature test, is

mandatory to verify the passage.

Once both test are calculated, then they must be jointly evaluated, in order to verify the

possible passage to the 2 order RSM setting. Four scenarios can occur:

nd

1 scenario both tests are significant, then we must pass to the 2 order.

à

st nd

2 scenario both tests are no significant, then the optimal setting of the factor levels,

à

nd

obtained with the 1 order steepest ascent/descent procedure, is the final optimal setting.

st

3 scenario the Flof test is significant while the Fcurv test is not significant; in this case

à

rd

surely we missed one or several sources of variabilities when start planning the DoE.

4 scenario the Flof test is not significant while the Fcurv test is significant; this situation

à

th

should occur rarely. If it occurs probably the Flof is borderline, e.g. very close to be significant.

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10. Steepest ascent/discent – theory integrated by an

example 63

1 order RS statistical model

st

Having done the DoE we estimate the model 64

System of equations and partial derivates

Not a linear independent system. Formula (4) is Lagrange multiplier.

By setting the partial derivates equal to zero, the following system of k+1 equations is

obtained:

The solution is given by xi according to the following formula:

2

Where bi is the corresponding estimated coeCicient, and is Lagrange multiplier. 65

Therefore, because we do not have a system of linearly independent equations, we must

arbitrarily assign, for a generic i-mo factor, an increment (established or technical

∆,

considerations) so that:

So that: ∆ = 5 ) ∆ .

I must fix a priori (in Montgomery ex. We fixed at least one to estimate

, ∆

Once obtained the estimated value of consequently the values are obtained for the

remaining k-1 factors/variables.

Procedure and Khuri & Cornell example 66

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In RSM this decomposition is important because:

SSPEàDoEàquantify of the variability strictly related to the DoE because we are dealing with

replicates in the center of the design x0.

SSLOFàmodelàif I have applied correctly the 1 order model at the end of the procedure it

st

should contain the variability related to the fact that I have applied a 1 order instead