MANE 6313

Week 14, Module E

Student Learning Outcome

Select an appropriate experimental design with one or more factors,
Select an appropriate model with one or more factors,
Evaluate statistical analyses of experimental designs,
Assess the model adequacy of any experimental design, and
Interpret model results.

Module Learning Outcome

Apply method of steepest ascent (descent)

Sequential Nature

Method of Steepest Ascent

Search method to find a local maximum
Non-linear optimization techniques usually require:
Search direction, and
Step size
Method of Steepest Descent is used to find a local minimum

First-order Example

Textbook Procedure

Starts with a first-order model with coded data

\[ \widehat{y}=\widehat{\beta_0}+\sum_{i=1}^k\widehat{\beta_i}x_i \]

Assumes that the point \(x_1=x_2=\cdots=x_k=0\) is the base point or origin point

Textbook Procedure, continued

Choose a step size in one of the process variables, say \(\Delta x_j\). Usually, we would select the variable we know the most about, or we would select the variable that has the largest absolute regression coefficient \(|\widehat{\beta_j}|\).
The step size in the other variables is

\[ \Delta_i=\frac{\widehat{\beta_i}}{\widehat{\beta_j}/\Delta_j}\;\;\;i=1,2,\ldots,k\;\;i\neq j \]

Convert the \(\Delta x_i\) from coded variables to the natural variables.

Example 11.1

The fitted, coded regression model is

\[ \widehat{y}=40.44+0.775x_1+0.325x_2 \]

The coded variables are

\[ \begin{aligned} x_1&=\frac{\xi-35}{5}\\ x_2&=\frac{\xi-155}{5} \end{aligned} \]

Example 11.1, Step 1

\(x_1\) is selected as the variable to base the steepest ascent upon

Example 11.1, Step 2

Find \(\Delta_2\)

\[ \Delta_2=\frac{0.325}{0.775/1.0}=0.42 \]

Example 11.1, Step 3

Find \(x_1\) and \(x_2\) for origin plus \(1\Delta\)

\[ \begin{aligned} x_1&=0+1=1\\ x_2&=0+0.42=0.42 \end{aligned} \]

Example 11.1, Step 4

Convert to Natural (uncoded) Variables

\[ \begin{aligned} x_1=\frac{\xi_1-35}{5}\longrightarrow 1.0=\frac{\xi_1-35}{5}\longrightarrow \xi_1=40\\ x_2\frac{\xi_2-155}{5}\longrightarrow 0.42=\frac{\xi_2-155}{5}\longrightarrow \xi_2=157.1 \end{aligned} \]

Example 11.1 Step 5

Run experiment using \(\xi_1\) and \(\xi_2\) to get reponse value

Example 11.1, Step 6

Update table of results (see Table 11.1) and optionally create graph

Example 11.1, Step 7

If local stationary point, stop
Otherwise, increase \(\Delta\) and repeat cycle starting at step 2

Table of Results

Graph of Results

Method of Steepest Descent

You are searching in the opposite direction
Therefore, you must change the sign of \(\Delta_i\)

Non-First Order Models

Note that \(\Delta_i\) is \(\frac{\partial\widehat{y}}{\partial x_i}=\widehat{\beta_i}\)
Therefore, if a non-first order is used then \(\Delta_i=\frac{\partial\widehat{y}}{\partial x_i}\)

steepest() in R

Do not use!
Is entirely model-based
Any mismatch between "real world" and model will result in inaccuracies
Does not require additional experiments to be run so results could be incorrect