# GUI for evaluation of a custom three variables multilevel DoE

Ivan Brezani (ivan.brezani@tuke.sk)

This Graphical User Interface (GUI) was designed as a tool for evaluation of results based on custom Design of Experiment (DoE). This three factor, multilevel design is based on Central Composite Design (CCD), but with more levels for each factor. 41 experiments (combinations) are used.

## Contents

## Starting the GUI

GUI can be started from matlab command line from the directory with extracted files by simply running the function:

>> DoE_GUI

Window will appear that will look like:

Main window consist of the input tables (located on the left side) and graph (located on the right side of the window). There are two input/output tables. First one (top) can be used to change minimum and maximum values of variables to be used in the design. It also allows user to change the name of three factors (A, B and C) that will be used as axis labels.

Second table is generated based on values in the first table and it shows locations of 41 points (combinations of factors) of the design. Each of the variables (A, B and C) is represented by numerical value.

DoE consisting of 41 rows and three columns (A, B and C) can be copied to clipboard and pasted into spreadsheet editor, such as MS Excel. Column Y will be used for input of the response variable.

To the right of the firs table there are three buttons. First one (top) can be used to paste values of response variable (Y) into the second table. Second button (middle) can be used to open current graph in a new figure window for further customization and for saving. Third button can be used to save four output variables (A, B, C and Y). Size of each is 11x11x11. Matrices A, B and C represent a three dimensional grid consisting of 11*11*11 = 1331 points between minimum and maximum value of each of the three variables. Before the approximation is run, Y matrix will consist of only NaN values (if no Y values were input) or 41 typed Y values and NaNs. After the approximation is run it will represent approximated (and extrapolated) values at individual grid points.

To the right side of the second table there is a area where panels for some basic customization of graphs will be displayed. When the GUI is started showing scatter of the basic grid the only option available is turning the numbering of individual data points on/off.

## Design of Experiment

This design is custom and only applicable on three variables, which can be expressed numerically - minimum and maximum value of each variable. Basic grid is then calculated based on these values. Following figure represents basic grid where all three variables are in the range [-5 5].

When experiments are finished then the output should consist of 41 values of response variable (Y). These values can be either typed manually into each cell (41 rows of the Y column) or copied from a spreadsheet (values must be numerical and the size of the copied area must be 41 rows by 1 column). After entering the 41 response values, new panel appears (red rectangle)

## Calculate statistics - ANOVA

ANOVA panel can be used to calculate N-way ANOVA using Matlab ANOVAN function. Before approximation (using the newly appeared Approximate button) it can only be calculated based on 41 entered values. Pressing the button opens a dialog box for choosing interaction model (Linear, Interaction or Full) of ANOVAN function. After % running the approximation, options of calculating ANOVA based on 87 or 495 points become available (for more informations on how the 87 and 495 points are calculated read the "Approximation" section. Please keep in mind that statistics calculated based on more than 41 original values will be (more or less) influenced by approximation. I recommend to use 41 points for calculation of effect of individual variables and 87 points for calculation of effects of interactions between variables.

## Approximation / extrapolation

Approximation uses Matlab function `griddata` for interpolation in 2D space on 3 basic planes (XY, XZ and YZ) and 6 diagonal planes. There are 13 data points (experimental values) on each basic plane as shown on picture below.

On each diagonal plane, there are 15 data points (experimental values) as shown on picture below.

These 3 basic planes and 6 diagonal planes intersect each other at basic axes (X, Y, Z). There are 4 approximations for each axis from such intersections. Approximation also takes into account two values approximated using planes of inner blue cube (i.e. value between points 1 and 2 is also calculated using approximation of values inside of square consisting of points 18 to 21). Number of approximated points rises from 41 (13 on basic axes and 28 outside these axes) to 59 (31 on basic axes and 28 outside these axes).

Then approximation of basic planes and diagonals is run again and values between points on inner blue cube are calculated. Values calculated from red/blue rhombuses are also taken into account (i.e. value between points 14 and 18 is calculated as a mean of value from approximation of one basic plane - 8,9,12,13, one diagonal plane - 22,25,26,29 and a rhombus - 14,18,11,7). This adds 12 new points and rises the number of values to 71.

In a third step approximation of diagonals is run again based on newest data and values on each of 6 diagonal axes is calculated as an intersection of two approximated diagonal planes. This raises the number of calculated values by 16 to 87.

Approximation is then used to estimate 735 values. `inpaint_nans` function is then used for extrapolation of the remaining values. Final matrix consist of 1331 values.

Please note that `inpaint_nans` function must be downloaded from File Exchange server.

## Display results graphically

Popupmenu that appears above the graph after approximation can be used to select one of the three methods for displaying the results.

Scatter plot uses filled circles with color (hsv colormap) and size varying with value of estimated response variable. Color map and size can be inverted as an option. (Example figure is displayed on the right side of the below image).

Slice plot is an intersection of three planes. Position of the planes can be changed using the settings button. (Example figure is displayed on the left side of the below image).

Isosurface can be used to find "shapes" of hyper surfaces with the same value of the output variable.