SONFIS is Root Mean Square Error (RMSE), given as below: 

$RMSE=\sqrt{\frac{\sum_{i=1}^{m}\left(t_{i}-t_{i}^{\ast}\right)^{2}}{m}}$; Where $t_{i}$ is output of SONFIS and $t_{i}^{\ast}$ is real answer; $m$ is the number of 
test data (test objects). In the rest of paper, let m=93 and *number of training data set =600*. 

Figures 9, 10 indicate the results of the aforesaid system (so, performance of selected SONFIS-R on the test data). Two indicated positions in figure 10 state minimum RMSE over same rules, are near to each other. In such case, we have two opportunities: selection of SONFIS with min number of rules (n.r) or involved min objects. Figure 10 show second occasion. With augmenting of close-open iterations SONFIS-R or range of n.r, our system emerges more near min RMSE, not defiantly lowest value (figure 10-a, c). By employing of (5) in SONFIS-AR, and α =1.01; β =.001 and γ =.5; the general pattern of RMSE vs. 

neuron growth (in first layer of algorithm) can be observed (figure11). With neglecting of some details, the same trend of error fluctuation is distinguishable (highlighted by colored windows). It is worth noting that by α =.8 and *n.r=2;* SONFISAR reveals a general chaos form (figure 12). The main reason of this can be followed in the first layer property: regulation of neurons in SOM may get in to the "dead station" and random selection of weights in such layer. Other reason is about the range of error vacillation. In fact in this case our system has a high senility to the error, and then to the neuron growth. In this case, we can determine two new balance measures: durability of neurons and distribution of points in a neuron-error space. First measure gets lesser than 20 neurons while in second measure system after 100 iterations falls in the "balance hole" with nearly 50 neurons size. 

Figure 13 shows our mean about structure detection. 

Under SONFIS-R (figure10-c) figure 13-a, declares three major clusters in *lugeon-T.W.R*. With more neurons in SOM, we can acquire like patterns but may lose the supposed balancing criteria. This proves the balance measure even if get min RMSE 
but losses the data distributions or major structures. 

In employing of part of second algorithm (figure6), we use- -for in this case- only exact rules i.e., one decision class in right hand of an *if-then* rule. Figure 14 and 15 depict the scaling process by 1-D SOM (3 neurons) and the performance of SORST-R over 7 random selection of SOM structure, respectively. 

The applied error measure is: 

![0_image_0.png](0_image_0.png)

| Type of weathering   | Ascribed code   |
|----------------------|-----------------|
| Fresh-MW             | 1.5             |
| SW-MW                | 2               |
| Fresh-SW             | .5              |
| Fresh                | 0               |
| MW                   | 3               |
| CW                   | 2.5             |
| SW                   | 1               |
| HW-MW                | 3.5             |
| HW                   | 4               |

It must be noticed that for unrecognizable objects in test data (elicited by rules) a fix value such 4 is ascribed. So for measure part when any object is not identified, 1 is attributed. This is main reason of such swing of MSE in reduced data set 6 (figure 15b). Clearly, in data set 7 SORST gains a lowest error (26 neurons in SOM). The extruded rules in the optimum case can be purchased in table 2. We have explained application of SORST in back analysis in other study [14]. 