
## 6.1.Applying Ids And Cog

![0_Image_1.Png](0_Image_1.Png)

Assume that the data points in the plane X-Y have a circular structure. They can also have any other structure which does not satisfy the structure of functions, ݔ∃)ݔ)݂ :ଵݕ = (ݔ),݂ଶݕ =/ ଵݕ ܽ݊݀ ଶݕ = ().

By applying IDS and COG on this data plane, the 

![0_image_3.png](0_image_3.png)

structure of data points are ruined. It is because in each column of data plane, Center of Gravity tries to select the average as a delegate of the column.

9 In addition, a more complicated example is stated in 5.3. which demonstrates the strength of Extended Thinning.

From the viewpoint of the geometry, Center of Gravity reduces a 2-dimensional space to some 1-dimensional spaces (which we previously called columns) which are located side by side. It is because COG selects only one point, called delegate point for each column in the data plane. Calculating delegates is independent of data in the neighbor columns. Since this space reduction causes destroying some pieces of information, the structure of original data points is not necessarily preserved.

## 6.2.Applying Extended Thickening And Extended Thinning

Thinning is capable of preserving the data structure. In the next stage of original ALM, data domain is divided to find narrow paths. Previously, ALM had to break narrow paths based on variance due to lack of information. On the contrary, the modified ALM divides these narrow paths by the width of thickened data in any vertical lines. Thus ALM arrives to an admissible answer faster. Using these new operators, the

![0_image_0.png](0_image_0.png) fuzzy rules made by ALM are more accurate and less in numbers with respect to original ALM. 

## 6.3.Comparing The Couple Operators

![0_Image_2.Png](0_Image_2.Png)

By giving a complicated sample, the drawback of COG
will be highlighted. In this sample, data points have a structure like four chained circles and a separated half circle.

-Figure 6.4: original data points (structure consisting of 

![0_image_4.png](0_image_4.png)

four attached circles and a half circle)
-Figure 6.5: Ink Drop Spread (IDS) and Center of Gravity 

![0_image_5.png](0_image_5.png)

applied on original data points shown in Fig. 6.4 In original ALM, Center of Gravity destroyed the structure of data due to its dimension reduction property. With this diminished data, original ALM had to break data points to inaccurate data sets. Now, the modified ALM divides data points based on the width of data in 