Effex app documentation – Optimization

Docs: Responses

Mon, 01 Jan 0001 00:00:00 +0000

Specification Limits

Using the options under Specification Limits, the user can specify the optimization specifications for each response variable that is being considered. As an example, consider the following image.

In this example, there are two response variables of interest: Abrasion and Elong. For the response variable Abrasion, the objective is to maximize the value with a lower limit of 120, and for the response variable Elong, the objective is to maintain its value between the range 400 to 600. A reference to the original data set can be found here.

To specify these objectives in the software, first select one of the three choices highlighted in the red box. The description of these choices are as follows:

‘>’ - Maximize the value for the response variable
‘< >’ - Maintain the value for the response varibale between a certain range
‘<’ - Minimize the value for the response variable

After selecting the objective, the user can then set the limits on the slider (highlighted in the purple box in the image above). Setting the limits depends on which one of the three choices was selected.

‘>’ - In this case, the user should specify a lower limit. In the image above, for the response variable Abrasion, we specified a lower limit of roughly 120, as we are interested in finding the settings for the input variables that predict a mean value equal to or above 120 for the response variable ‘Abrasion’.
‘< >’ - In this case, the user should specify a range of values. In the image above, for the response variable Elong, we specify both a lower limit and an upper limit of approximately 400 and 600, as we are interested in finding the settings for the input variables that predict a value between these values for the response variable Elong.
‘<’ - This is the opposite case of ‘>’. In this case, the user should specify an upper limit.

Interval Type

For each prediction value that is calculated for different combinations of settings of the input factors, a statistical interval is also calculated that quantifies the uncertainty around this predicted value. In the software, there are three choices as presented below.

Using the options under Interval Type, the user can specify the type of interval that is of interest. Taking into account the selected models for each response, $\mathbf{X}$ to be the model matrix, $\widehat{\sigma}$ to be the estimate for the root mean squared error, $f(\mathbf{x})$ to the model expansion column vector of $\mathbf{x}$, where $\mathbf{x}$ is a vector with a specific combination of settings for the input factors denoted as $\mathbf{x}$, the different types of intervals are calculated as follows

Confidence interval:

$$\widehat{y} ± t_{1-\frac{\alpha}{2}, n-p} \cdot \widehat{\sigma} \cdot \sqrt{f'(\mathbf{x}){(\mathbf{X’X})}^{-1}f(\mathbf{x})}$$

Prediction interval:

$$\widehat{y} ± t_{1-\frac{\alpha}{2}, n-p} \cdot \widehat{\sigma} \cdot \sqrt{1+f'(\mathbf{x}){(\mathbf{X’X})}^{-1}f(\mathbf{x})}$$

Tolerance interval:

$$\widehat{y} ± z_{\frac{(1+\psi)}{2}} \cdot \widehat{\sigma} \cdot\sqrt{1+f'(\mathbf{x}){(\mathbf{X’X})}^{-1}f(\mathbf{x})} \cdot \sqrt{\frac{n-p}{\chi^{2}_{\alpha,n-p}}}$$

Here, $\psi$ is the proportion of all future predictions that will be covered in the tolerance interval with a confidence of $100(1-\alpha)$%. By default, for all types intervals, $\alpha$ is set to 0.05, so that all intervals have a confidence level of 95%. For the tolerance interval, $\psi$ is set to 0.95 so that the tolerance interval covers 95% of all future observations with a confidence of 95%.

Selecting the type of interval that is of interest, will display such interval in the Profilers tab. The type of interval selected will also influence the calculations for the probability of success (which is presented in the tab Parallel Coordinates Plot).

Docs: Parallel Coordinates Plot

Mon, 01 Jan 0001 00:00:00 +0000

Optimization Graph

The optimization graph is a parallel coordinate plot, similar to the one available in the software for design and model selection. Based on the selected model for each of the response variables, this plot allows the user to visualize the predicted values for the average for each of the response variables for the many combinations of the settings of the input factors. The first few axes allow the user to set limits for the settings of the input factors, followed by axes for the response variables, and finally an axis for the probability of success values. Here is an example of such a plot.

In the above example, the first 8 axes correspond to the input factors, the next 6 axes correspond to the response variables, and the last axis gives the values for the probability of success.

For a given combination of the settings of the input factors (highlighted in the image above using a black line), the probability of success (PoS) quantifies the probability with which the desired optimization goals (specified in the Responses tab) can be achieved for all response variables.

To be precise in our defintion, when

the confidence interval option is selected in the Responses tab, the probability of success (PoS) quantifies the probability with which the desired optimization goals specified (in the Responses tab) can be achieved for the predicted value for the average for all response variables.
the prediction interval option is selected in the Responses tab, the probability of success (PoS) quantifies the probability with which the desired optimization goals specified (in the Responses tab) can be achieved for the predicted value for one single future observation for all response variables.
the tolerance interval option is selected in the Responses tab, the probability of success (PoS) quantifies the probability with which the desired optimization goals specified (in the Responses tab) can be achieved for the predicted value for 95% of all future observations for all response variables.

The PoS values for the average is always going to be greater than the PoS values for one single future observation, which in turn is always going to be greater than the PoS values for 95% of all future observations.

To learn how to interact and filter with this graph, see [1] and [2].

The discretization of the levels for each input factor can be controlled using the Discretization level option. For the finest grid of combinations of the settings of the inputs factors, choose the maximum discretization level value which is 11. In the example above, the discretization level is set to 5.

Note that the discretization level option takes into account any filters placed in the input factor axes in the parallel coordinate plot. On specifying a filter on one or more axes corresponding to the input factors, click on the Zoom button to discretize the levels of the input factor over the filtered range.

Technical note on the calculations for the probability of success (PoS) (click to unfold)

For a given response variable, the probability of success (PoS) calculation for its corresponding combination of the settings of the input factors is calculated as follows:

If the value for the said response variable is to be maximized beyond a minimum value as entered in the Responses tab, the PoS value is the area under the probability curve for the distribution concerning the specific type of interval (specified in the Responses tab) that lies above the minimum value specified for the response.
If the value for the said response variable is to be minimized beyond a maximum value as entered in the Responses tab, the PoS value is the area under the probability curve for the distribution concerning the specific type of interval (specified in the Responses tab) that lies below the maximum value specified for the response.
If the value for the said response variable is to be maintained between a range with a specified minimum and maximum value as entered in the Responses tab, the PoS value is the area under the probability curve for the distribution concerning the specific type of interval (specified in the Responses tab) that lies between the minimum and maximum value for the response.

In the Responses tab,

when the confidence interval option is selected, for the PoS, the probability curve of the distribution for the mean of the predictions for the given combination of settings of the input factors is considered.
when the prediction interval option is selected, for the PoS, the probability curve of the distribution for a single future observation for the given combination of settings of the input factors is considered.
when the tolerance interval option is selected, for the PoS, the probability curve of the distribution for 95% of all future observations for the given combination of settings of the input factors is considered.

When there are multiple response variables, a PoS value is calculated for each response, and a combined product is reported as the final PoS value.

Note: For combinations of the settings of the input factors for which the predicted value for the average for any one of the response variables is outside the specified ranges provided in the Responses tab, the calculated PoS value is reported as 0.

Selected Points

The Selected Points table lists 5, 10 or 15 best combinations of the input factors that have the largest value for the PoS. You can specify the number of points to be displayed using the following option

If any filters are placed by the user in the parallel coordinate plot, then this table will only show points that match the specified filtering criteria. Here is an example.

In the above example, a filter is specified for the axis with PoS values so that we get only the points or combinations of the the settings of the input factors that achieve a PoS value of 0.797 or higher. In this case, there are only 4 points that satisfy this constraint, and these are displayed in the Selected Points table, with all points being ranked by its PoS value.

Docs: Profilers

Mon, 01 Jan 0001 00:00:00 +0000

The profiler can be used to study the change in (i) predicted value for the average or (ii) PoS values, for each response variable, based on changing the values set for a single input factor, while keeping the values for the remaining input factors fixed.

Controls

First specify, the Criterion that you would like to study using the following option.

Here there are two choices:

y-predicted: This concerns the predicted values for the average for each response.
PoS: This concerns the calculated probability of success values for each response.

A profiler plot is available for each combination of the input factor and response variable. The settings for all input factors that appear in the selected models can be changed using the sliders that are available at the bottom of the screen. An example is provided below.

Selecting y-predicted

In a single profiler plot for a given combination of the input factor and response variable, the black line represents the relationship between the input factor and the predicted value for the average of the specific response variable. Here is an example of such a plot.

If the black line is straight, this shows that based on the selected model for the said response variable, the effect of changing the setting of the input factor (while keeping the settings of the remaining input factors fixed) results in a linear or proportional change in the predicted value for the average of the specific response variable.
If the black line is a curve (not straight), this shows that based on the selected model for the said response variable, the effect of changing the setting of the input factor (while keeping the settings of the remaining input factors fixed) results in a non-linear or non-proportional change in the predicted value for the average of the specific response variable. This indicates that a quadratic effect corresponding to this input factor is present in the selected model corresponding to the given response variable. In the example plot above, the plot shows that the factors Silica and Silane have their corresponding quadratic effects present in the selected model.

It is possible that changing the setting of a specific input factor results in a tilting effect on the black lines present in other profiler plots concerning different input factors but for the same response variable. For example, for a given response variable, if change in the levels for factor A, results in the tilting of the black line present in the profiler concerning factor B, then this is an indication that there is an interaction effect A x B present in the selected model. The greater the degree of tilting of the black line in the profiler for factor B, the greater is the effect of this interaction effect. It is possible that a single factor may have interactions with multiple other factors.

In each profiler plot,

The green and blue lines represent the upper and lower bounds corresponding to the interval type selected in the Responses tab.
The dashed black lines make it easier to get the exact values for the predicted mean for a given value of the input factor.
The shaded purple area indicates the range of response values that have been prespecified in the Responses tab as desirable.

Selecting PoS

Here the black line now represents the relationship between the input factor and the PoS value concerning the specific response variable. Here is an example of such a plot.

In each profiler plot, the dashed black lines make it easier to get the exact values for the PoS values for a given value of the input factor.