1 Introduction
- proposition of four methods for determining the weights of coalitions in a dispersed system using Pawlak analysis model,
- examination of these methods for three approaches of application Pawlak analysis method in a dispersed system,
- comparison and statistical analysis of the obtained results.
2 Application of the Pawlak’s Model for Ensemble of Classifiers
- allied \(R^+(x,y)\), if \(\rho ^*(ag_1,ag_2)<0.5\) (or \(\rho (ag_1,ag_2)<0.5)\),
- in conflict \(R^-(x,y)\), if \(\rho ^*(ag_1,ag_2)>0.5\) (or \(\rho (ag_1,ag_2)>0.5\)),
- neutral \(R^0(x,y)\), if \(\rho ^*(ag_1,ag_2)=0.5\) (or \(\rho (ag_1,ag_2)=0.5\)).
- Approach 1—the first method of defining an information system is used,
- Approach 2—the second method of defining an information system and the function of distance between agents are used,
- Approach 3—the second method of defining an information system and the conflict function are used.
3 Scheme of a Dispersed System’s Operation
4 Weights for Coalitions
5 Example of Applying the Coalitions’ Weights in a Dispersed System
\(a_1\) | \(a_2\) | \(a_3\) | \(a_4\) | \(a_5\) | \(a_6\) | \(a_7\) | \(a_8\) | \(a_9\) | \(a_{10}\) | \(a_{11}\) | \(a_{12}\) | \(a_{13}\) | \(a_{14}\) | \(a_{15}\) | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
x | 1 | 1 | 0 | 2 | 1 | 0 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 1 |
Classifier | \({\bar{\mu }}_{ag}(x)\) | \(\mathbf{r }_{ag}(x)\) | \(sd^x_{ag}\)—standard deviation of normalized \({\bar{\mu }}_{ag}(x)\) |
---|---|---|---|
\(ag_1\) | [0.8, 0.4, 0.6, 0.4, 1] | [2,4,3,4,1] | 0.166 |
\(ag_2\) | [0.6, 0.6, 0.6, 0.6, 0.4] | [1,1,1,1,2] | 0.065 |
\(ag_3\) | [0.2, 0.4, 0.8, 0.6, 1] | [5,4,2,3,1] | 0.215 |
\(ag_4\) | [0.67, 0.67, 0.33, 0.33, 0.33] | [1,1,2,2,2] | 0.159 |
\(ag_5\) | [0.67, 0.67, 0.33, 0.67, 1] | [2,2,3,2,1] | 0.144 |
\(ag_6\) | [1, 0.33, 0.67, 0.67, 0.67] | [1,3,2,2,2] | 0.144 |
\(ag_7\) | [0.8, 0.4, 0.6, 0.8, 0.8] | [1,3,2,1,1] | 0.107 |
5.1 Approach 1
\(ag_1\) | \(ag_2\) | \(ag_3\) | \(ag_4\) | \(ag_5\) | \(ag_6\) | \(ag_7\) | |
---|---|---|---|---|---|---|---|
\(ag_1\) | |||||||
\(ag_2\) | 1 | ||||||
\(ag_3\) | 0 | 1 | |||||
\(ag_4\) | 0.6 | 0.4 | 0.6 | ||||
\(ag_5\) | 0 | 1 | 0 | 0.6 | |||
\(ag_6\) | 0.4 | 0.6 | 0.4 | 0.2 | 0.4 | ||
\(ag_7\) | 0.4 | 0.6 | 0.4 | 0.6 | 0.4 | 0.4 |
Coalition \(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_3\)
|
\(a_4\)
|
\(a_5\)
|
\(a_6\)
|
\(a_7\)
|
\(a_8\)
|
\(a_9\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{13}\)
|
\(a_{14}\)
|
\(a_{15}\)
|
d
|
\(s(x_i,x)\)
|
? | ? | 0 | 0 | 1 | 0 | ? | ? | ? | 2 | 2 | 2 | 2 | 2 | 1 |
\(v_1\)
| 0.800 |
? | ? | 0 | ? | ? | 0 | 1 | 0 | 0 | 2 | 0 | ? | 2 | 2 | 1 |
\(v_1\)
| 0.500 |
1 | 1 | 1 | 2 | 1 | 0 | 1 | 0 | 0 | 2 | 0 | ? | ? | 2 | 1 |
\(v_1\)
| 0.615 |
? | ? | ? | 1 | 1 | 1 | ? | ? | ? | 1 | 1 | 2 | ? | 1 | 1 |
\(v_2\)
| 0.375 |
? | ? | 1 | 1 | 1 | ? | ? | ? | ? | 1 | 1 | 2 | 1 | 2 | ? |
\(v_2\)
| 0.500 |
2 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 2 | 0 | 2 | ? | ? | 1 | 1 |
\(v_2\)
| 0.385 |
? | ? | 1 | ? | ? | ? | 2 | 1 | 2 | 0 | 2 | ? | 1 | 2 | ? |
\(v_2\)
| 0.500 |
? | ? | ? | 2 | 0 | 0 | ? | ? | ? | 0 | 2 | 2 | ? | 2 | 0 |
\(v_3\)
| 0.625 |
? | ? | 2 | 2 | 0 | ? | ? | ? | ? | 0 | 2 | 2 | 1 | 1 | ? |
\(v_3\)
| 0.500 |
? | ? | 2 | ? | ? | ? | 2 | 2 | 1 | 2 | 1 | ? | 1 | 1 | ? |
\(v_3\)
| 0.692 |
1 | 2 | 0 | 2 | 2 | 0 | 2 | 2 | 1 | 2 | 1 | ? | ? | 2 | 0 |
\(v_3\)
| 0.625 |
? | ? | ? | 2 | 1 | 0 | ? | ? | ? | 2 | 2 | 0 | ? | 2 | 2 |
\(v_4\)
| 0.750 |
? | ? | 0 | 2 | 1 | ? | ? | ? | ? | 2 | 2 | 0 | 1 | 0 | ? |
\(v_4\)
| 0.750 |
0 | 0 | 0 | 1 | 1 | ? | 0 | 2 | 1 | 0 | 2 | ? | 1 | 0 | ? |
\(v_4\)
| 0.538 |
0 | 0 | 0 | 1 | 1 | 0 | 0 | 2 | 1 | 0 | 2 | ? | ? | 2 | 2 |
\(v_4\)
| 0.500 |
1 | 1 | 0 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 1 | 2 | 1 |
\(v_5\)
| 0.867 |
Coalition \(\{ag_2, ag_4\}\) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_4\)
|
\(a_5\)
|
\(a_6\)
|
\(a_7\)
|
\(a_8\)
|
\(a_{12}\)
|
d
|
\(s(x_i,x)\)
| |||||||
2 | 1 | 1 | 0 | 0 | 2 | 2 | 2 |
\(v_1\)
| 0.625 | |||||||
1 | 2 | 2 | 0 | 1 | 2 | 2 | 2 |
\(v_2\)
| 0.625 | |||||||
1 | 0 | 2 | 2 | 2 | 2 | 2 | 0 |
\(v_3\)
| 0.500 | |||||||
0 | 0 | 2 | 1 | 0 | 1 | 1 | 2 |
\(v_4\)
| 0.500 | |||||||
0 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |
\(v_5\)
| 0.375 |
Coalition \(\{ag_4, ag_6\}\) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_6\)
|
\(a_{12}\)
|
\(a_{14}\)
|
\(a_{15}\)
|
d
|
\(s(x_i,x)\)
| |||||||||
2 | 1 | 0 | 2 | 2 | 1 |
\(v_1\)
| 0.833 | |||||||||
1 | 2 | 1 | 2 | 1 | 1 |
\(v_2\)
| 0.500 | |||||||||
1 | 0 | 0 | 0 | 2 | 0 |
\(v_3\)
| 0.500 | |||||||||
0 | 0 | 0 | 2 | 2 | 2 |
\(v_4\)
| 0.500 | |||||||||
0 | 1 | 2 | 1 | 2 | 1 |
\(v_5\)
| 0.500 |
Coalition | Vector | Vector transformation | \(SD^x_j\)—standard deviation |
---|---|---|---|
\(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\)
| [0.800,0.500,0.692,0.750,0.867] | [0.222,0.139,0.192,0.208,0.240] | 0.035 |
\(\{ag_2, ag_4\}\)
| [0.625,0.625,0.500,0.500,0.375] | [0.238,0.238,0.190,0.190,0.143] | 0.036 |
\(\{ag_4, ag_6\}\)
| [0.833,0.500,0.500,0.500,0.500] | [0.294,0.176,0.176,0.176,0.176] | 0.047 |
Coalition | Weight | Vector |
---|---|---|
First method—coefficients \(m^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\) | \(4.5/7=0.643\) | [0.143,0.089,0.123,0.134,0.154] |
\(\{ag_2, ag_4\}\) | \(1.5/7=0.214\) | [0.051,0.051,0.041,0.041,0.031] |
\(\{ag_4, ag_6\}\) | \(1/7=0.143\) | [0.042,0.025,0.025,0.025,0.025] |
Sum | [0.236,0.165,0.189,0.200,0.210] | |
Second method—coefficients \(sd^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\) | 0.595 | [0.132,0.082,0.114,0.124,0.143] |
\(\{ag_2, ag_4\}\) | 0.172 | [0.041,0.041,0.033,0.033,0.025] |
\(\{ag_4, ag_6\}\) | 0.233 | [0.068,0.041,0.041,0.041,0.041] |
Sum | [0.241,0.165,0.188,0.198,0.209] | |
Third method—coefficients \(m^x_{ag}\cdot sd^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\) | 0.704 | [0.156,0.097,0.135,0.146,0.169] |
\(\{ag_2, ag_4\}\) | 0.145 | [0.034,0.034,0.028,0.028,0.021] |
\(\{ag_4, ag_6\}\) | 0.152 | [0.045,0.027,0.027,0.027,0.027] |
Sum | [0.235,0.159,0.189,0.201,0.216] | |
Fourth method—coefficients \(SD^x_j\) | ||
\(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\) | 0.295 | [0.065,0.041,0.057,0.061,0.071] |
\(\{ag_2, ag_4\}\) | 0.401 | [0.072,0.072,0.058,0.058,0.043] |
\(\{ag_4, ag_6\}\) | 0.304 | [0.118,0.071,0.071,0.071,0.071] |
Sum | [0.256,0.184,0.185,0.190,0.185] |
5.2 Approach 2
\(ag_1\) | \(ag_2\) | \(ag_3\) | \(ag_4\) | \(ag_5\) | \(ag_6\) | \(ag_7\) | |
---|---|---|---|---|---|---|---|
\(ag_1\) | |||||||
\(ag_2\) | 0.8 | ||||||
\(ag_3\) | 0.2 | 0.8 | |||||
\(ag_4\) | 0.6 | 0.3 | 0.7 | ||||
\(ag_5\) | 0.3 | 0.6 | 0.4 | 0.5 | |||
\(ag_6\) | 0.4 | 0.5 | 0.5 | 0.5 | 0.5 | ||
\(ag_7\) | 0.4 | 0.4 | 0.5 | 0.5 | 0.4 | 0.3 |
Coalition \(\{ag_1, ag_3, ag_5\}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_3\)
|
\(a_4\)
|
\(a_5\)
|
\(a_7\)
|
\(a_8\)
|
\(a_9\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{13}\)
|
\(a_{14}\)
|
d
|
\(s(x_i,x)\)
|
? | ? | 0 | ? | ? | 1 | 0 | 0 | 2 | 0 | 2 | 2 |
\(v_1\)
| 0.375 |
1 | 1 | 1 | 2 | 1 | 1 | 0 | 0 | 2 | 0 | ? | ? |
\(v_1\)
| 0.500 |
? | ? | 1 | ? | ? | 2 | 1 | 2 | 0 | 2 | 1 | 2 |
\(v_2\)
| 0.500 |
2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 0 | 2 | ? | ? |
\(v_2\)
| 0.400 |
? | ? | 2 | ? | ? | 2 | 2 | 1 | 2 | 1 | 1 | 1 |
\(v_3\)
| 0.625 |
1 | 2 | 0 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | ? | ? |
\(v_3\)
| 0.700 |
0 | 0 | 0 | 1 | 1 | 0 | 2 | 1 | 0 | 2 | 1 | 0 |
\(v_4\)
| 0.500 |
1 | 1 | 0 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | 2 |
\(v_5\)
| 1 |
Coalition \(\{ag_1, ag_5, ag_7\}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_3\)
|
\(a_4\)
|
\(a_5\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{13}\)
|
\(a_{14}\)
|
d
|
\(s(x_i,x)\)
| ||
? | ? | 0 | 0 | 1 | 2 | 2 | 2 | 2 | 2 |
\(v_1\)
| 0.750 | ||
1 | 1 | 1 | 2 | 1 | ? | ? | ? | ? | ? |
\(v_1\)
| 0.800 | ||
? | ? | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 |
\(v_2\)
| 0.500 | ||
2 | 2 | 2 | 2 | 1 | ? | ? | ? | ? | ? |
\(v_2\)
| 0.400 | ||
? | ? | 2 | 2 | 0 | 0 | 2 | 2 | 1 | 1 |
\(v_3\)
| 0.500 | ||
1 | 2 | 0 | 2 | 2 | ? | ? | ? | ? | ? |
\(v_3\)
| 0.600 | ||
? | ? | 0 | 2 | 1 | 2 | 2 | 0 | 1 | 0 |
\(v_4\)
| 0.750 | ||
0 | 0 | 0 | 1 | 1 | ? | ? | ? | 1 | 0 |
\(v_4\)
| 0.429 | ||
1 | 1 | 0 | 2 | 1 | 2 | 2 | 1 | 1 | 2 |
\(v_5\)
| 0.900 |
Coalition \(\{ag_1, ag_6, ag_7\}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_3\)
|
\(a_4\)
|
\(a_5\)
|
\(a_6\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
\(a_{14}\)
|
\(a_{15}\)
|
d
|
\(s(x_i,x)\)
| |
? | ? | ? | 0 | 1 | 0 | 2 | 2 | 2 | 2 | 1 |
\(v_1\)
| 0.875 | |
1 | 1 | 1 | 2 | 1 | 0 | ? | ? | ? | 2 | 1 |
\(v_1\)
| 0.875 | |
? | ? | ? | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 |
\(v_2\)
| 0.375 | |
2 | 2 | 2 | 2 | 1 | 1 | ? | ? | ? | 1 | 1 |
\(v_2\)
| 0.375 | |
? | ? | ? | 2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 |
\(v_3\)
| 0.625 | |
1 | 2 | 0 | 2 | 2 | 0 | ? | ? | ? | 2 | 0 |
\(v_3\)
| 0.625 | |
0 | 0 | 0 | 1 | 1 | 0 | ? | ? | ? | 2 | 2 |
\(v_4\)
| 0.750 | |
? | ? | ? | 2 | 1 | 0 | 2 | 2 | 0 | 2 | 2 |
\(v_4\)
| 0.500 | |
1 | 1 | 0 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 1 |
\(v_5\)
| 0.818 |
Coalition \(\{ag_2, ag_4\}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\)
|
\(a_2\)
|
\(a_4\)
|
\(a_5\)
|
\(a_6\)
|
\(a_7\)
|
\(a_8\)
|
\(a_{12}\)
|
d
|
\(s(x_i,x)\)
| ||||
2 | 1 | 1 | 0 | 0 | 2 | 2 | 2 |
\(v_1\)
| 0.625 | ||||
1 | 2 | 2 | 0 | 1 | 2 | 2 | 2 |
\(v_2\)
| 0.625 | ||||
1 | 0 | 2 | 2 | 2 | 2 | 2 | 0 |
\(v_3\)
| 0.500 | ||||
0 | 0 | 2 | 1 | 0 | 1 | 1 | 2 |
\(v_4\)
| 0.500 | ||||
0 | 1 | 2 | 1 | 1 | 1 | 1 | 1 |
\(v_5\)
| 0.375 |
Coalition \(\{ag_2, ag_7\}\) | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
\(a_4\)
|
\(a_5\)
|
\(a_6\)
|
\(a_7\)
|
\(a_8\)
|
\(a_{10}\)
|
\(a_{11}\)
|
\(a_{12}\)
|
d
|
\(s(x_i,x)\)
| ||||
0 | 1 | ? | ? | ? | 2 | 2 | 2 |
\(v_1\)
| 0.800 | ||||
1 | 0 | 0 | 2 | 2 | ? | ? | ? |
\(v_1\)
| 0.600 | ||||
1 | 1 | ? | ? | ? | 1 | 1 | 2 |
\(v_2\)
| 0.400 | ||||
2 | 0 | 1 | 2 | 2 | ? | ? | ? |
\(v_2\)
| 0.600 | ||||
2 | 0 | ? | ? | ? | 0 | 2 | 2 |
\(v_3\)
| 0.600 | ||||
2 | 2 | 2 | 2 | 2 | ? | ? | ? |
\(v_3\)
| 0.600 | ||||
2 | 1 | 0 | 1 | 1 | 2 | 2 | 0 |
\(v_4\)
| 0.625 | ||||
2 | 1 | 1 | 1 | 1 | 2 | 2 | 1 |
\(v_5\)
| 0.500 |
Coalition | Vector | Vector transformation | \(SD^x_j\)—standard deviation |
---|---|---|---|
\(\{ag_1, ag_3, ag_5\}\) | [0.500,0.500,0.700,0.500,1] | [0.156,0.156,0.219,0.156,0.313] | 0.062 |
\(\{ag_1, ag_5, ag_7\}\) | [0.800,0.500,0.600,0.750,0.900] | [0.225,0.141,0.169,0.211,0.254] | 0.040 |
\(\{ag_1, ag_6, ag_7\}\) | [0.875,0.375,0.625,0.750,0.818] | [0.254,0.109,0.182,0.218,0.238] | 0.052 |
\(\{ag_2, ag_4\}\) | [0.625,0.625,0.500,0.500,0.375] | [0.238,0.238,0.190,0.190,0.143] | 0.036 |
\(\{ag_2, ag_7\}\) | [0.800,0.600,0.600,0.625,0.500] | [0.256,0.192,0.192,0.200,0.160] | 0.031 |
Coalition | Weight | Vector |
---|---|---|
First method—coefficients \(m^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5\}\) | 1.833/7=0.262 | [0.041,0.041,0.057,0.041,0.082] |
\(\{ag_1, ag_5, ag_7\}\) | 1.167/7=0.167 | [0.038,0.023,0.028,0.035,0.042] |
\(\{ag_1, ag_6, ag_7\}\) | 1.667/7=0.238 | [0.061,0.026,0.043,0.052,0.057] |
\(\{ag_2, ag_4\}\) | 1.5/7=0.214 | [0.051,0.051,0.041,0.041,0.031] |
\(\{ag_2, ag_7\}\) | 0.833/7=0.119 | [0.030,0.023,0.023,0.024,0.019] |
Sum | [0.220,0.164,0.192,0.193,0.230] | |
Second method—coefficients \(sd^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5\}\) | 0.299 | [0.047,0.047,0.065,0.047,0.093] |
\(\{ag_1, ag_5, ag_7\}\) | 0.238 | [0.054,0.033,0.040,0.050,0.060] |
\(\{ag_1, ag_6, ag_7\}\) | 0.238 | [0.060,0.026,0.043,0.052,0.056] |
\(\{ag_2, ag_4\}\) | 0.128 | [0.030,0.030,0.024,0.024,0.018] |
\(\{ag_2, ag_7\}\) | 0.098 | [0.025,0.019,0.019,0.020,0.016] |
Sum | [0.216,0.155,0.192,0.193,0.244] | |
Third method—coefficients \(m^x_{ag}\cdot sd^x_{ag}\) | ||
\(\{ag_1, ag_3, ag_5\}\) | 0.342 | [0.053,0.053,0.075,0.053,0.107] |
\(\{ag_1, ag_5, ag_7\}\) | 0.163 | [0.037,0.023,0.028,0.034,0.041] |
\(\{ag_1, ag_6, ag_7\}\) | 0.235 | [0.060.0.026,0.043,0.051,0.056] |
\(\{ag_2, ag_4\}\) | 0.192 | [0.046,0.046,0.037,0.037,0.027] |
\(\{ag_2, ag_7\}\) | 0.068 | [0.017,0.013,0.013,0.014,0.011] |
Sum | [0.213,0.161,0.195,0.189,0.242] | |
Fourth method—coefficients \(SD^x_j\) | ||
\(\{ag_1, ag_3, ag_5\}\) | 0.279 | [0.044,0.044,0.061,0.044,0.087] |
\(\{ag_1, ag_5, ag_7\}\) | 0.183 | [0.041,0.026,0.031,0.039,0.046] |
\(\{ag_1, ag_6, ag_7\}\) | 0.235 | [0.060,0.026,0.043,0.051,0.056] |
\(\{ag_2, ag_4\}\) | 0.162 | [0.039,0.039,0.031,0.031,0.023] |
\(\{ag_2, ag_7\}\) | 0.142 | [0.036,0.027,0.027,0.028,0.023] |
Sum | [0.219,0.161,0.193,0.193,0.235] |
5.3 Approach 3
\(ag_1\) | \(ag_2\) | \(ag_3\) | \(ag_4\) | \(ag_5\) | \(ag_6\) | \(ag_7\) | |
---|---|---|---|---|---|---|---|
\(ag_1\) | |||||||
\(ag_2\) | 1 | ||||||
\(ag_3\) | 0.4 | 1 | |||||
\(ag_4\) | 1 | 0.4 | 0.8 | ||||
\(ag_5\) | 0.4 | 1 | 0.8 | 0.8 | |||
\(ag_6\) | 0.8 | 0.6 | 0.6 | 0.2 | 0.8 | ||
\(ag_7\) | 0.6 | 0.6 | 0.4 | 0.6 | 0.8 | 0.4 |
Coalition \(\{ag_1, ag_3\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\) | \(a_2\) | \(a_3\) | \(a_4\) | \(a_5\) | \(a_7\) | \(a_8\) | \(a_9\) | \(a_{10}\) | \(a_{11}\) | d | \(s(x_i,x)\) |
1 | 1 | 1 | 2 | 1 | 1 | 0 | 0 | 2 | 0 | \(v_1\) | 0.500 |
2 | 2 | 2 | 2 | 1 | 2 | 1 | 2 | 0 | 2 | \(v_2\) | 0.400 |
1 | 2 | 0 | 2 | 2 | 2 | 2 | 1 | 2 | 1 | \(v_3\) | 0.700 |
0 | 0 | 0 | 1 | 1 | 0 | 2 | 1 | 0 | 2 | \(v_4\) | 0.500 |
1 | 1 | 0 | 2 | 1 | 2 | 2 | 1 | 2 | 2 | \(v_5\) | 1 |
Coalition \(\{ag_1, ag_5\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\) | \(a_2\) | \(a_3\) | \(a_4\) | \(a_5\) | \(a_{13}\) | \(a_{14}\) | d | \(s(x_i,x)\) | |||
? | ? | 0 | ? | ? | 2 | 2 | \(v_1\) | 0.667 | |||
1 | 1 | 1 | 2 | 1 | ? | ? | \(v_1\) | 0.800 | |||
? | ? | 1 | ? | ? | 1 | 2 | \(v_2\) | 0.667 | |||
2 | 2 | 2 | 2 | 1 | ? | ? | \(v_2\) | 0.400 | |||
? | ? | 2 | ? | ? | 1 | 1 | \(v_3\) | 0.333 | |||
1 | 2 | 0 | 2 | 2 | ? | ? | \(v_3\) | 0.600 | |||
0 | 0 | 0 | 1 | 1 | 1 | 0 | \(v_4\) | 0.429 | |||
1 | 1 | 0 | 2 | 1 | 1 | 2 | \(v_5\) | 1 |
Coalition \(\{ag_2, ag_4\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\) | \(a_2\) | \(a_4\) | \(a_5\) | \(a_6\) | \(a_7\) | \(a_8\) | \(a_{12}\) | d | \(s(x_i,x)\) | ||
2 | 1 | 1 | 0 | 0 | 2 | 2 | 2 | \(v_1\) | 0.625 | ||
1 | 2 | 2 | 0 | 1 | 2 | 2 | 2 | \(v_2\) | 0.625 | ||
1 | 0 | 2 | 2 | 2 | 2 | 2 | 0 | \(v_3\) | 0.500 | ||
0 | 0 | 2 | 1 | 0 | 1 | 1 | 2 | \(v_4\) | 0.500 | ||
0 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | \(v_5\) | 0.375 |
Coalition \(\{ag_3, ag_7\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_4\) | \(a_5\) | \(a_7\) | \(a_8\) | \(a_9\) | \(a_{10}\) | \(a_{11}\) | \(a_{12}\) | d | \(s(x_i,x)\) | ||
0 | 1 | ? | ? | ? | 2 | 2 | 2 | \(v_1\) | 0.800 | ||
? | ? | 1 | 0 | 0 | 2 | 0 | ? | \(v_1\) | 0.200 | ||
1 | 1 | ? | ? | ? | 1 | 1 | 2 | \(v_2\) | 0.400 | ||
? | ? | 2 | 1 | 2 | 0 | 2 | ? | \(v_2\) | 0.400 | ||
2 | 0 | ? | ? | ? | 0 | 2 | 2 | \(v_3\) | 0.600 | ||
2 | ? | ? | 2 | 1 | 2 | 1 | ? | \(v_3\) | 0.800 | ||
2 | 1 | ? | ? | ? | 2 | 2 | 0 | \(v_4\) | 0.800 | ||
0 | ? | ? | 2 | 1 | 0 | 2 | ? | \(v_4\) | 0.600 | ||
2 | 1 | 2 | 2 | 1 | 2 | 2 | 1 | \(v_5\) | 0.875 |
Coalition \(\{ag_6, ag_7\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_4\) | \(a_5\) | \(a_6\) | \(a_{10}\) | \(a_{11}\) | \(a_{12}\) | \(a_{14}\) | \(a_{15}\) | d | \(s(x_i,x)\) | ||
0 | 1 | 0 | 2 | 2 | 2 | 2 | 1 | \(v_1\) | 0.875 | ||
1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | \(v_2\) | 0.375 | ||
2 | 0 | 0 | 0 | 2 | 2 | 2 | 0 | \(v_3\) | 0.625 | ||
2 | 1 | 0 | 2 | 2 | 0 | 2 | 2 | \(v_4\) | 0.750 | ||
2 | 1 | 2 | 2 | 2 | 1 | 2 | 1 | \(v_5\) | 0.750 |
Coalition \(\{ag_4, ag_6\}\) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
\(a_1\) | \(a_2\) | \(a_6\) | \(a_{12}\) | \(a_{14}\) | \(a_{15}\) | d | \(s(x_i,x)\) | ||||
2 | 1 | 0 | 2 | 2 | 1 | \(v_1\) | 0.833 | ||||
1 | 2 | 1 | 2 | 1 | 1 | \(v_2\) | 0.500 | ||||
1 | 0 | 0 | 0 | 2 | 0 | \(v_3\) | 0.500 | ||||
0 | 0 | 0 | 2 | 2 | 2 | \(v_4\) | 0.500 | ||||
0 | 1 | 2 | 1 | 2 | 1 | \(v_5\) | 0.500 |
Coalition | Vector | Vector transformation | \(SD^x_j\)—standard deviation |
---|---|---|---|
\(\{ag_1, ag_3\}\) | [0.500,0.400,0.700,0.500,1] | [0.161,0.129,0.226,0.161,0.323] | 0.069 |
\(\{ag_1, ag_5\}\) | [0.800,0.667,0.600,0.429,1] | [0.229,0.191,0.172,0.123,0.286] | 0.055 |
\(\{ag_2, ag_4\}\) | [0.625,0.625,0.500,0.500,0.375] | [0.238,0.238,0.190,0.190,0.143] | 0.036 |
\(\{ag_3, ag_7\}\) | [0.800,0.400,0.800,0.800,0.875] | [0.218,0.109,0.218,0.218,0.238] | 0.046 |
\(\{ag_6, ag_7\}\) | [0.875,0.375,0.625,0.750,0.750] | [0.259,0.111,0.185,0.222,0.222] | 0.050 |
\(\{ag_4, ag_6\}\) | [0.833,0.500,0.500,0.500,0.500] | [0.294,0.176,0.176,0.176,0.176] | 0.047 |
Coalition | Weight | Vector |
---|---|---|
First method—coefficients \(m^x_{ag}\) | ||
\(\{ag_1, ag_3\}\) | 1/7=0.143 | [0.023,0.018,0.032,0.023,0.046] |
\(\{ag_1, ag_5\}\) | 1.5/7=0.214 | [0.049,0.041,0.037,0.026,0.061] |
\(\{ag_2, ag_4\}\) | 1.5/7=0.214 | [0.051,0.051,0.041,0.041,0.031] |
\(\{ag_3, ag_7\}\) | 1/7=0.143 | [0.031,0.016,0.031,0.031,0.034] |
\(\{ag_6, ag_7\}\) | 1/7=0.143 | [0.037,0.016,0.026,0.032,0.032] |
\(\{ag_4, ag_6\}\) | 1/7=0.143 | [0.042,0.025,0.025,0.025,0.025] |
Sum | [0.233,0.167,0.193,0.178,0.229] | |
Second method—coefficients \(sd^x_{ag}\) | ||
\(\{ag_1, ag_3\}\) | 0.212 | [0.034,0.027,0.048,0.034,0.069] |
\(\{ag_1, ag_5\}\) | 0.173 | [0.040,0.033,0.030,0.021,0.050] |
\(\{ag_2, ag_4\}\) | 0.125 | [0.030,0.030,0.024,0.024,0.018] |
\(\{ag_3, ag_7\}\) | 0.180 | [0.039,0.020,0.039,0.039,0.043] |
\(\{ag_6, ag_7\}\) | 0.140 | [0.036,0.016,0.026,0.031,0.031] |
\(\{ag_4, ag_6\}\) | 0.169 | [0.050,0.030,0.030,0.030,0.030] |
Sum | [0.229,0.155,0.197,0.180,0.240] | |
Third method—coefficients \(m^x_{ag}\cdot sd^x_{ag}\) | ||
\(\{ag_1, ag_3\}\) | 0.190 | [0.031,0.025,0.043,0.031,0.061] |
\(\{ag_1, ag_5\}\) | 0.227 | [0.052,0.043,0.039,0.028,0.065] |
\(\{ag_2, ag_4\}\) | 0.145 | [0.034,0.034,0.028,0.028,0.021] |
\(\{ag_3, ag_7\}\) | 0.161 | [0.035,0.018,0.035,0.035,0.038] |
\(\{ag_6, ag_7\}\) | 0.126 | [0.033,0.014,0.023,0.028,0.028] |
\(\{ag_4, ag_6\}\) | 0.152 | [0.045,0.027,0.027,0.027,0.027] |
Sum | [0.229,0.161,0.195,0.176,0.240] | |
Fourth method—coefficients \(SD^x_j\) | ||
\(\{ag_1, ag_3\}\) | 0.227 | [0.037,0.029,0.051,0.037,0.073] |
\(\{ag_1, ag_5\}\) | 0.181 | [0.042,0.035,0.031,0.022,0.052] |
\(\{ag_2, ag_4\}\) | 0.118 | [0.028,0.028,0.022,0.022,0.017] |
\(\{ag_3, ag_7\}\) | 0.153 | [0.033,0.017,0.033,0.033,0.036] |
\(\{ag_6, ag_7\}\) | 0.166 | [0.043,0.018,0.031,0.037,0.037] |
\(\{ag_4, ag_6\}\) | 0.155 | [0.046,0.027,0.027,0.027,0.027] |
Sum | [0.228,0.154,0.196,0.179,0.243] |
Approach | Coalitions | Method of | Result | |
---|---|---|---|---|
calculating weights | \(\varepsilon = 0.019\) | \(\varepsilon = 0.01\) | ||
Approach 1 | \(\{ag_1, ag_3, ag_5, ag_6, ag_7\}\), \(\{ag_2, ag_4\}\), \(\{ag_4, ag_6\}\) | 1 2 3 4 | \(\{v_1\}\) \(\{v_1\}\) \(\{v_1,v_3,v_4,v_5\}\) \(\{v_1\}\) | \(\{v_1\}\) \(\{v_1\}\) \(\{v_1\}\) \(\{v_1\}\) |
Approach 2 | \(\{ag_1, ag_3, ag_5\}\), \(\{ag_1, ag_5, ag_7\}\), \(\{ag_1, ag_6, ag_7\}\), \(\{ag_2, ag_4\}\), \(\{ag_2, ag_7\}\) | 1 2 3 4 | \(\{v_1,v_5\}\) \(\{v_5\}\) \(\{v_5\}\) \(\{v_1,v_5\}\) | \(\{v_1,v_5\}\) \(\{v_5\}\) \(\{v_5\}\) \(\{v_5\}\) |
Approach 3 | \(\{ag_1, ag_3\}\), \(\{ag_1, ag_5\}\), \(\{ag_6,ag_7\}\), \(\{ag_4,ag_6\}\) | 1 2 3 4 | \(\{v_1,v_5\}\) \(\{v_1,v_5\}\) \(\{v_1,v_5\}\) \(\{v_1,v_5\}\) | \(\{v_1,v_5\}\) \(\{v_5\}\) \(\{v_5\}\) \(\{v_5\}\) |
6 Experimental Protocol
- \(DS_{Ag 3}\)—dispersed system with 3 tables,
- \(DS_{Ag 5}\)—dispersed system with 5 tables,
- \(DS_{Ag 7}\)—dispersed system with 7 tables,
- \(DS_{Ag 9}\)—dispersed system with 9 tables,
- \(DS_{Ag 11}\)—dispersed system with 11 tables.
- estimator of classification errorwhere \(I(d(x) \notin {\hat{d}}_{DS}(x))=1\), when \(d(x_i) \notin {\hat{d}}_{DS}(x)\) and \(I(d(x) \notin {\hat{d}}_{DS}(x))=0\), when \(d(x) \in {\hat{d}}_{DS}(x)\); \({\hat{d}}_{DS}(x)\) is a set of global decisions generated by the system for the test object x$$\begin{aligned} e=\frac{1}{card\{U_{test}\}}\ \sum _{x\in U_{test}} I(d(x) \notin {\hat{d}}_{DS}(x)), \end{aligned}$$
- estimator of classification ambiguity errorwhere \(I(d(x) \not = {\hat{d}}_{DS}(x))=1\), when \(\{d(x)\} \not = {\hat{d}}_{DS}(x)\) and \(I(d(x) \not = {\hat{d}}_{DS}(x))=0\), when \(\{d(x)\} = {\hat{d}}_{DS}(x)\)$$\begin{aligned} e_{ONE}=\frac{1}{card\{U_{test}\}} \sum _{x\in U_{test}} I(d(x) \not = {\hat{d}}_{DS}(x)), \end{aligned}$$
- the average size of the global decisions sets$$\begin{aligned} {\overline{d}}_{DS}=\frac{1}{card\{U_{test}\}} \sum _{x\in U_{test}} card\{{\hat{d}}_{DS}(x)\}. \end{aligned}$$
7 Experiments and Discussion
7.1 Results with the Soybean Data Set
- the version of dispersion (System)
- the methods of determining the strength of the coalition (Weights): 1—the size of the coalition, 2—the unambiguous of the decisions made by the classifiers, 3—the size of the coalition and the unambiguous of the taken decisions, 4—the unambiguous of the decisions made by the coalition
- the optimal parameters values \(m_1\), \(m_2\) and \(\varepsilon\)
- the measures to determine the quality of inference: e, \(e_{ONE}\) and \({\overline{d}}_{DS}\)
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/3/0.0031 | 0.021 | 0.271 | 1.886 | ||||
2 | 1/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
3 | 1/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
4 | 1/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
\(DS_{Ag 5}\) | 1 | 8/2/0.0028 | 0.032 | 0.295 | 2.093 | 8/2/0.0021 | 0.053 | 0.255 | 1.529 |
2 | 8/2/0.0026 | 0.035 | 0.295 | 2.037 | 8/2/0.0021 | 0.053 | 0.255 | 1.516 | |
3 | 8/2/0.0027 | 0.035 | 0.295 | 2.027 | 8/2/0.0021 | 0.053 | 0.255 | 1.524 | |
4 | 8/2/0.0023 | 0.035 | 0.290 | 2.043 | 8/2/0.0016 | 0.059 | 0.234 | 1.476 | |
\(DS_{Ag 7}\) | 1 | 10/5/0.003 | 0.053 | 0.383 | 1.965 | 10/5/0.0018 | 0.069 | 0.316 | 1.529 |
2 | 10/5/0.0026 | 0.051 | 0.375 | 2.112 | 10/5/0.0016 | 0.074 | 0.309 | 1.521 | |
3 | 10/5/0.0029 | 0.053 | 0.386 | 2.085 | 10/5/0.0017 | 0.072 | 0.314 | 1.529 | |
4 | 10/5/0.0017 | 0.069 | 0.338 | 2.005 | 10/5/0.0011 | 0.077 | 0.311 | 1.564 | |
\(DS_{Ag 9}\) | 1 | 6/2/0.0027 | 0.043 | 0.351 | 2.000 | 6/2/0.0017 | 0.061 | 0.301 | 1.548 |
2 | 6/2/0.0026 | 0.045 | 0.343 | 1.963 | 6/2/0.0015 | 0.064 | 0.285 | 1.492 | |
3 | 6/2/0.0028 | 0.037 | 0.354 | 2.112 | 6/2/0.0016 | 0.064 | 0.285 | 1.495 | |
4 | 1/3/0.0024 | 0.032 | 0.391 | 1.984 | 1/3/0.0015 | 0.045 | 0.319 | 1.521 | |
\(DS_{Ag 11}\) | 1 | 7/2/0.0024 | 0.109 | 0.426 | 1.968 | 7/2/0.0016 | 0.144 | 0.388 | 1.540 |
2 | 7/2/0.0024 | 0.104 | 0.420 | 1.880 | 7/2/0.0015 | 0.138 | 0.383 | 1.559 | |
3 | 7/2/0.0024 | 0.104 | 0.420 | 1.894 | 7/2/0.0016 | 0.138 | 0.386 | 1.551 | |
4 | 4/3/0.0015 | 0.136 | 0.410 | 1.904 | 4/3/0.0009 | 0.144 | 0.391 | 1.561 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | ||||
2 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
3 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
4 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
\(DS_{Ag 5}\) | 1 | 9/2/0.00273 | 0.037 | 0.290 | 2.013 | 9/2/0.0021 | 0.053 | 0.255 | 1.508 |
2 | 9/2/0.00259 | 0.035 | 0.295 | 2.043 | 9/2/0.0021 | 0.053 | 0.253 | 1.468 | |
3 | 9/2/0.00266 | 0.037 | 0.290 | 2.024 | 9/2/0.0021 | 0.053 | 0.253 | 1.473 | |
4 | 9/2/0.00238 | 0.035 | 0.290 | 2.040 | 9/2/0.00168 | 0.056 | 0.245 | 1.535 | |
\(DS_{Ag 7}\) | 1 | 4/6/0.0029 | 0.040 | 0.356 | 2.080 | 4/6/0.00196 | 0.064 | 0.314 | 1.513 |
2 | 10/10/0.00266 | 0.045 | 0.346 | 2.093 | 10/10/0.00196 | 0.053 | 0.311 | 1.556 | |
3 | 4/6/0.0028 | 0.043 | 0.348 | 2.069 | 4/6/0.00175 | 0.064 | 0.311 | 1.527 | |
4 | 6/6/0.00224 | 0.051 | 0.367 | 1.989 | 6/6/0.00147 | 0.064 | 0.314 | 1.543 | |
\(DS_{Ag 9}\) | 1 | 8/2/0.00238 | 0.035 | 0.324 | 1.902 | 8/2/0.00168 | 0.051 | 0.279 | 1.545 |
2 | 8/2/0.00266 | 0.032 | 0.327 | 2.069 | 8/2/0.00154 | 0.043 | 0.276 | 1.556 | |
3 | 8/2/0.00273 | 0.029 | 0.335 | 2.082 | 8/2/0.00161 | 0.048 | 0.274 | 1.532 | |
4 | 10/2/0.00168 | 0.032 | 0.319 | 1.955 | 10/2/0.00091 | 0.051 | 0.274 | 1.492 | |
\(DS_{Ag 11}\) | 1 | 4/2/0.00245 | 0.066 | 0.370 | 1.785 | 4/2/0.00168 | 0.085 | 0.330 | 1.551 |
2 | 4/1/0.0028 | 0.056 | 0.420 | 2.104 | 4/1/0.00168 | 0.077 | 0.338 | 1.527 | |
3 | 4/1/0.00259 | 0.059 | 0.386 | 1.856 | 4/1/0.00168 | 0.080 | 0.332 | 1.511 | |
4 | 4/3/0.00182 | 0.072 | 0.394 | 1.785 | 4/3/0.00119 | 0.077 | 0.338 | 1.537 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | ||||
2 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
3 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
4 | 2/3/0.0031 | 0.021 | 0.271 | 1.886 | |||||
\(DS_{Ag 5}\) | 1 | 1/2/0.0027 | 0.016 | 0.303 | 2.106 | 1/2/0.0017 | 0.032 | 0.242 | 1.439 |
2 | 1/2/0.0027 | 0.021 | 0.306 | 2.088 | 1/2/0.0016 | 0.040 | 0.245 | 1.439 | |
3 | 1/2/0.0027 | 0.019 | 0.311 | 2.011 | 1/2/0.0014 | 0.040 | 0.234 | 1.386 | |
4 | 1/2/0.00259 | 0.024 | 0.324 | 2.077 | 1/2/0.0014 | 0.051 | 0.239 | 1.410 | |
\(DS_{Ag 7}\) | 1 | 1/2/0.0022 | 0.013 | 0.322 | 1.793 | 1/2/0.0017 | 0.027 | 0.293 | 1.574 |
2 | 1/2/0.0021 | 0.016 | 0.338 | 1.832 | 1/2/0.0014 | 0.029 | 0.295 | 1.519 | |
3 | 1/2/0.0019 | 0.013 | 0.324 | 1.771 | 1/2/0.0014 | 0.029 | 0.285 | 1.516 | |
4 | 5/6/0.00147 | 0.037 | 0.324 | 1.936 | 5/6/0.00105 | 0.048 | 0.285 | 1.564 | |
\(DS_{Ag 9}\) | 1 | 2/3/0.0027 | 0.021 | 0.338 | 2.066 | 2/3/0.0017 | 0.053 | 0.247 | 1.319 |
2 | 2/3/0.0027 | 0.035 | 0.332 | 2.029 | 2/3/0.0016 | 0.061 | 0.245 | 1.311 | |
3 | 2/3/0.0026 | 0.035 | 0.332 | 2.008 | 2/3/0.0017 | 0.053 | 0.253 | 1.351 | |
4 | 1/3/0.00245 | 0.011 | 0.335 | 1.973 | 1/3/0.00119 | 0.045 | 0.255 | 1.346 | |
\(DS_{Ag 11}\) | 1 | 2/4/0.003 | 0.029 | 0.311 | 1.883 | 2/4/0.0022 | 0.037 | 0.277 | 1.511 |
2 | 2/2/0.0027 | 0.029 | 0.332 | 1.777 | 2/2/0.0022 | 0.037 | 0.293 | 1.505 | |
3 | 2/2/0.0024 | 0.027 | 0.314 | 1.593 | 2/2/0.0021 | 0.035 | 0.279 | 1.505 | |
4 | 2/5/0.00203 | 0.040 | 0.407 | 1.894 | 2/5/0.00133 | 0.064 | 0.314 | 1.471 |
System | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | t |
---|---|---|---|---|---|
Approach 1 | |||||
\(DS_{Ag 3}\) | 1/6/0.0043 | 0.021 | 0.295 | 1.939 | 0.17 |
\(DS_{Ag 5}\) | 6/2/0.0043 | 0.051 | 0.290 | 1.973 | 0.10 |
6/2/0.0024 | 0.061 | 0.253 | 1.484 | 0.10 | |
\(DS_{Ag 7}\) | 10/6/0.005 | 0.059 | 0.378 | 2.027 | 2.58 |
10/6/0.0032 | 0.069 | 0.327 | 1.561 | 2.58 | |
\(DS_{Ag 9}\) | 6/3/0.0044 | 0.048 | 0.340 | 1.997 | 2.27 |
6/3/0.0024 | 0.064 | 0.309 | 1.559 | 2.27 | |
\(DS_{Ag 11}\) | 8/3/0.00425 | 0.106 | 0.399 | 2.037 | 5.25 |
8/3/0.0023 | 0.141 | 0.378 | 1.481 | 5.25 | |
Approach 2 | |||||
\(DS_{Ag 3}\) | 1/6/0.0043 | 0.021 | 0.295 | 1.861 | 0.16 |
\(DS_{Ag 5}\) | 9/2/0.003 | 0.053 | 0.261 | 1.564 | 0.10 |
\(DS_{Ag 7}\) | 4/10/0.0044 | 0.059 | 0.319 | 1.676 | 5.36 |
4/10/0.0041 | 0.061 | 0.311 | 1.535 | 5.36 | |
\(DS_{Ag 9}\) | 5/3/0.0026 | 0.037 | 0.316 | 1.718 | 2.19 |
5/3/0.0023 | 0.048 | 0.279 | 1.527 | 2.19 | |
\(DS_{Ag 11}\) | 4/2/0.00575 | 0.056 | 0.407 | 1.992 | 1.55 |
4/2/0.00425 | 0.074 | 0.319 | 1.516 | 1.55 | |
Approach 3 | |||||
\(DS_{Ag 3}\) | 1/3/0.0041 | 0.021 | 0.277 | 1.883 | 0.08 |
\(DS_{Ag 5}\) | 1/2/0.00605 | 0.019 | 0.290 | 2.045 | 0.07 |
1/2/0.00365 | 0.048 | 0.245 | 1.481 | 0.07 | |
\(DS_{Ag 7}\) | 1/10/0.0082 | 0.019 | 0.362 | 2.003 | 0.46 |
1/10/0.006 | 0.029 | 0.306 | 1.585 | 0.46 | |
\(DS_{Ag 9}\) | 6/3/0.0066 | 0.045 | 0.364 | 2.011 | 2.05 |
6/3/0.004 | 0.066 | 0.279 | 1.383 | 2.05 | |
\(DS_{Ag 11}\) | 2/5/0.0101 | 0.045 | 0.348 | 1.819 | 7.11 |
2/5/0.0077 | 0.056 | 0.293 | 1.505 | 7.11 |
7.2 Results with the Vehicle Silhouettes Data Set
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/9/0.0021 | 0.118 | 0.476 | 1.520 | 1/9/0.0009 | 0.185 | 0.366 | 1.217 |
2 | 1/9/0.0022 | 0.122 | 0.508 | 1.543 | 1/9/0.001 | 0.201 | 0.374 | 1.213 | |
3 | 1/9/0.0022 | 0.122 | 0.508 | 1.543 | 1/9/0.001 | 0.201 | 0.374 | 1.213 | |
4 | 2/9/0.0022 | 0.134 | 0.531 | 1.539 | 2/9/0.001 | 0.213 | 0.417 | 1.236 | |
\(DS_{Ag 5}\) | 1 | 3/8/0.0016 | 0.169 | 0.524 | 1.516 | 3/8/0.0009 | 0.205 | 0.413 | 1.256 |
2 | 3/8/0.0018 | 0.157 | 0.539 | 1.535 | 3/8/0.0009 | 0.217 | 0.406 | 1.228 | |
3 | 3/8/0.0018 | 0.157 | 0.539 | 1.535 | 3/8/0.0009 | 0.217 | 0.406 | 1.228 | |
4 | 3/10/0.0016 | 0.240 | 0.610 | 1.524 | 3/10/0.0007 | 0.327 | 0.508 | 1.244 | |
\(DS_{Ag 7}\) | 1 | 4/8/0.0019 | 0.094 | 0.465 | 1.531 | 4/8/0.0007 | 0.197 | 0.378 | 1.232 |
2 | 5/8/0.0019 | 0.098 | 0.480 | 1.551 | 5/8/0.0008 | 0.177 | 0.370 | 1.240 | |
3 | 5/8/0.0019 | 0.098 | 0.480 | 1.551 | 5/8/0.0008 | 0.177 | 0.370 | 1.240 | |
4 | 1/5/0.0014 | 0.189 | 0.594 | 1.555 | 1/5/0.0007 | 0.272 | 0.480 | 1.256 | |
\(DS_{Ag 9}\) | 1 | 5/4/0.001 | 0.126 | 0.528 | 1.555 | 5/4/0.0005 | 0.224 | 0.421 | 1.248 |
2 | 5/4/0.0011 | 0.142 | 0.512 | 1.528 | 5/4/0.0005 | 0.240 | 0.425 | 1.224 | |
3 | 5/5/0.0012 | 0.130 | 0.520 | 1.551 | 5/5/0.0005 | 0.244 | 0.429 | 1.232 | |
4 | 4/5/0.001 | 0.193 | 0.591 | 1.579 | 4/5/0.0005 | 0.276 | 0.461 | 1.280 | |
\(DS_{Ag 11}\) | 1 | 2/2/0.0009 | 0.122 | 0.480 | 1.524 | 2/2/0.0004 | 0.185 | 0.354 | 1.224 |
2 | 3/1/0.0009 | 0.134 | 0.512 | 1.512 | 3/1/0.0005 | 0.193 | 0.398 | 1.244 | |
3 | 3/1/0.001 | 0.146 | 0.528 | 1.539 | 3/1/0.0005 | 0.193 | 0.382 | 1.228 | |
4 | 2/3/0.0008 | 0.181 | 0.567 | 1.535 | 2/3/0.0003 | 0.248 | 0.437 | 1.217 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/9/0.0022 | 0.110 | 0.469 | 1.539 | 1/9/0.001 | 0.189 | 0.370 | 1.232 |
2 | 3/2/0.0024 | 0.110 | 0.484 | 1.520 | 3/2/0.0011 | 0.197 | 0.382 | 1.217 | |
3 | 3/2/0.0024 | 0.110 | 0.488 | 1.524 | 3/2/0.0011 | 0.197 | 0.382 | 1.217 | |
4 | 3/2/0.0022 | 0.130 | 0.496 | 1.520 | 3/2/0.0011 | 0.213 | 0.398 | 1.224 | |
\(DS_{Ag 5}\) | 1 | 2/1/0.0016 | 0.154 | 0.543 | 1.555 | 2/1/0.0009 | 0.224 | 0.433 | 1.260 |
2 | 7/1/0.0019 | 0.157 | 0.547 | 1.528 | 7/1/0.0009 | 0.236 | 0.449 | 1.260 | |
3 | 6/1/0.0017 | 0.167 | 0.547 | 1.520 | 6/1/0.0008 | 0.232 | 0.429 | 1.236 | |
4 | 6/1/0.0016 | 0.236 | 0.591 | 1.535 | 6/1/0.0008 | 0.291 | 0.476 | 1.260 | |
\(DS_{Ag 7}\) | 1 | 2/8/0.0021 | 0.110 | 0.472 | 1.524 | 2/8/0.0008 | 0.177 | 0.362 | 1.232 |
2 | 2/5/0.0022 | 0.091 | 0.480 | 1.535 | 2/5/0.0009 | 0.173 | 0.354 | 1.228 | |
3 | 6/6/0.0019 | 0.114 | 0.472 | 1.508 | 6/6/0.0008 | 0.177 | 0.346 | 1.220 | |
4 | 6/6/0.0018 | 0.146 | 0.516 | 1.528 | 6/6/0.0009 | 0.248 | 0.409 | 1.228 | |
\(DS_{Ag 9}\) | 1 | 5/4/0.001 | 0.122 | 0.512 | 1.543 | 5/4/0.0005 | 0.213 | 0.378 | 1.220 |
2 | 2/4/0.0012 | 0.126 | 0.524 | 1.535 | 2/4/0.0006 | 0.228 | 0.421 | 1.248 | |
3 | 5/4/0.0011 | 0.134 | 0.524 | 1.539 | 5/4/0.0005 | 0.217 | 0.413 | 1.256 | |
4 | 5/4/0.0011 | 0.165 | 0.535 | 1.539 | 5/4/0.0005 | 0.264 | 0.417 | 1.228 | |
\(DS_{Ag 11}\) | 1 | 1/1/0.0009 | 0.122 | 0.484 | 1.520 | 1/1/0.0005 | 0.177 | 0.378 | 1.256 |
2 | 1/1/0.0011 | 0.118 | 0.504 | 1.528 | 1/1/0.0005 | 0.197 | 0.374 | 1.236 | |
3 | 1/1/0.0011 | 0.130 | 0.496 | 1.531 | 1/1/0.0006 | 0.185 | 0.378 | 1.252 | |
4 | 1/1/0.001 | 0.154 | 0.520 | 1.539 | 1/1/0.0004 | 0.244 | 0.402 | 1.236 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/7/0.0022 | 0.114 | 0.457 | 1.504 | 1/7/0.001 | 0.193 | 0.378 | 1.228 |
2 | 5/10/0.0026 | 0.110 | 0.496 | 1.531 | 5/10/0.0012 | 0.189 | 0.366 | 1.217 | |
3 | 5/10/0.0026 | 0.110 | 0.496 | 1.531 | 5/10/0.0012 | 0.189 | 0.366 | 1.217 | |
4 | 2/4/0.0026 | 0.110 | 0.496 | 1.543 | 2/4/0.0011 | 0.209 | 0.378 | 1.213 | |
\(DS_{Ag 5}\) | 1 | 1/8/0.00168 | 0.150 | 0.531 | 1.559 | 1/8/0.00102 | 0.224 | 0.433 | 1.264 |
2 | 9/4/0.002 | 0.189 | 0.555 | 1.500 | 9/4/0.0011 | 0.248 | 0.433 | 1.236 | |
3 | 9/4/0.002 | 0.189 | 0.555 | 1.500 | 9/4/0.0011 | 0.248 | 0.433 | 1.236 | |
4 | 1/1/0.0018 | 0.213 | 0.571 | 1.504 | 1/1/0.0008 | 0.287 | 0.484 | 1.232 | |
\(DS_{Ag 7}\) | 1 | 1/7/0.0019 | 0.102 | 0.472 | 1.528 | 1/7/0.0009 | 0.189 | 0.374 | 1.236 |
2 | 5/7/0.00202 | 0.087 | 0.469 | 1.535 | 5/7/0.0008 | 0.181 | 0.335 | 1.217 | |
3 | 5/7/0.00202 | 0.087 | 0.469 | 1.535 | 5/7/0.0008 | 0.181 | 0.335 | 1.217 | |
4 | 1/8/0.0018 | 0.161 | 0.551 | 1.539 | 1/8/0.0007 | 0.236 | 0.417 | 1.228 | |
\(DS_{Ag 9}\) | 1 | 5/4/0.00105 | 0.126 | 0.528 | 1.543 | 5/4/0.0005 | 0.232 | 0.421 | 1.248 |
2 | 5/1/0.0011 | 0.146 | 0.516 | 1.535 | 5/1/0.0005 | 0.256 | 0.429 | 1.228 | |
3 | 4/2/0.0012 | 0.154 | 0.539 | 1.547 | 4/2/0.0006 | 0.248 | 0.425 | 1.232 | |
4 | 4/4/0.001 | 0.201 | 0.551 | 1.492 | 4/4/0.0005 | 0.299 | 0.465 | 1.224 | |
\(DS_{Ag 11}\) | 1 | 1/1/0.001 | 0.114 | 0.500 | 1.535 | 1/1/0.0005 | 0.181 | 0.370 | 1.260 |
2 | 5/3/0.001 | 0.130 | 0.480 | 1.496 | 5/3/0.0005 | 0.177 | 0.346 | 1.224 | |
3 | 5/3/0.0011 | 0.118 | 0.496 | 1.547 | 5/3/0.0005 | 0.181 | 0.350 | 1.224 | |
4 | 2/3/0.001 | 0.173 | 0.539 | 1.535 | 2/3/0.0005 | 0.217 | 0.394 | 1.236 |
System | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | t |
---|---|---|---|---|---|
Approach 1 | |||||
\(DS_{Ag 3}\) | 1/2/0.0036 | 0.134 | 0.500 | 1.535 | 0.05 |
1/2/0.0018 | 0.197 | 0.425 | 1.272 | 0.05 | |
\(DS_{Ag 5}\) | 1/2/0.0033 | 0.228 | 0.626 | 1.535 | 0.06 |
1/2/0.00195 | 0.295 | 0.480 | 1.224 | 0.06 | |
\(DS_{Ag 7}\) | 1/7/0.00405 | 0.201 | 0.614 | 1.547 | 0.47 |
1/7/0.0018 | 0.268 | 0.492 | 1.260 | 0.47 | |
\(DS_{Ag 9}\) | 10/2/0.00225 | 0.232 | 0.614 | 1.508 | 0.15 |
10/2/0.0015 | 0.303 | 0.528 | 1.276 | 0.15 | |
\(DS_{Ag 11}\) | 10/4/0.0027 | 0.197 | 0.602 | 1.532 | 5.10 |
10/4/0.00135 | 0.252 | 0.449 | 1.236 | 5.10 | |
Approach 2 | |||||
\(DS_{Ag 3}\) | 9/2/0.003 | 0.118 | 0.500 | 1.539 | 0.07 |
9/2/0.0018 | 0.181 | 0.417 | 1.268 | 0.07 | |
\(DS_{Ag 5}\) | 2/10/0.00315 | 0.217 | 0.602 | 1.531 | 0.52 |
2/10/0.00135 | 0.283 | 0.465 | 1.220 | 0.52 | |
\(DS_{Ag 7}\) | 10/6/0.0039 | 0.146 | 0.535 | 1.535 | 1.02 |
10/6/0.0021 | 0.205 | 0.425 | 1.276 | 1.02 | |
\(DS_{Ag 9}\) | 10/5/0.003 | 0.205 | 0.567 | 1.528 | 9.43 |
10/5/0.0015 | 0.264 | 0.457 | 1.252 | 9.43 | |
\(DS_{Ag 11}\) | 9/3/0.00315 | 0.154 | 0.531 | 1.531 | 2.46 |
9/3/0.00135 | 0.205 | 0.421 | 1.252 | 2.46 | |
Approach 3 | |||||
\(DS_{Ag 3}\) | 1/2/0.00495 | 0.102 | 0.472 | 1.535 | 0.04 |
1/2/0.0024 | 0.177 | 0.390 | 1.256 | 0.04 | |
\(DS_{Ag 5}\) | 1/9/0.0048 | 0.177 | 0.520 | 1.524 | 0.11 |
1/9/0.0021 | 0.264 | 0.449 | 1.232 | 0.11 | |
\(DS_{Ag 7}\) | 10/6/0.0063 | 0.154 | 0.516 | 1.500 | 0.21 |
10/6/0.0033 | 0.205 | 0.413 | 1.244 | 0.21 | |
\(DS_{Ag 9}\) | 1/1/0.0054 | 0.189 | 0.543 | 1.512 | 0.08 |
1/1/0.00255 | 0.268 | 0.457 | 1.260 | 0.08 | |
\(DS_{Ag 11}\) | 3/1/0.00465 | 0.154 | 0.520 | 1.520 | 0.40 |
3/1/0.0024 | 0.213 | 0.409 | 1.272 | 0.40 |
7.3 Results with the Landsat Satellites Data Set
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/4/0.0013 | 0.044 | 0.409 | 1.745 | ||||
2 | 1/4/0.0013 | 0.044 | 0.409 | 1.745 | |||||
3 | 1/4/0.0013 | 0.044 | 0.409 | 1.745 | |||||
4 | 1/4/0.0013 | 0.044 | 0.409 | 1.745 | |||||
\(DS_{Ag 5}\) | 1 | 1/4/0.0015 | 0.015 | 0.381 | 1.656 | 1/4/0.0007 | 0.045 | 0.222 | 1.240 |
2 | 1/4/0.0015 | 0.015 | 0.381 | 1.656 | 1/4/0.0007 | 0.045 | 0.222 | 1.240 | |
3 | 1/4/0.0015 | 0.015 | 0.381 | 1.656 | 1/4/0.0007 | 0.045 | 0.222 | 1.240 | |
4 | 1/4/0.0015 | 0.015 | 0.381 | 1.656 | 1/4/0.0007 | 0.045 | 0.223 | 1.241 | |
\(DS_{Ag 7}\) | 1 | 2/4/0.0013 | 0.015 | 0.371 | 1.612 | 2/4/0.0006 | 0.054 | 0.226 | 1.237 |
2 | 2/4/0.00135 | 0.016 | 0.384 | 1.647 | 2/4/0.0006 | 0.054 | 0.226 | 1.237 | |
3 | 2/4/0.00135 | 0.016 | 0.384 | 1.647 | 2/4/0.0006 | 0.054 | 0.226 | 1.237 | |
4 | 2/4/0.00135 | 0.015 | 0.384 | 1.650 | 2/4/0.0006 | 0.053 | 0.226 | 1.238 | |
\(DS_{Ag 9}\) | 1 | 1/2/0.00105 | 0,035 | 0,404 | 1,612 | 1/2/0.0005 | 0.078 | 0.264 | 1.236 |
2 | 1/2/0.00105 | 0.034 | 0.404 | 1.613 | 1/2/0.0005 | 0.082 | 0.265 | 1.234 | |
3 | 1/2/0.00105 | 0.035 | 0.405 | 1.612 | 1/2/0.0005 | 0.079 | 0.263 | 1.237 | |
4 | 1/2/0.00105 | 0.037 | 0.408 | 1.610 | 1/2/0.0005 | 0.086 | 0.270 | 1.233 | |
\(DS_{Ag 11}\) | 1 | 1/1/0.00105 | 0.022 | 0.386 | 1.616 | 1/1/0.0005 | 0.063 | 0.231 | 1.223 |
2 | 1/1/0.00105 | 0.024 | 0.383 | 1.607 | 1/1/0.0005 | 0.064 | 0.231 | 1.220 | |
3 | 1/1/0.00105 | 0.022 | 0.384 | 1.611 | 1/1/0.0005 | 0.062 | 0.233 | 1.226 | |
4 | 1/1/0.0011 | 0.025 | 0.398 | 1.646 | 1/1/0.0005 | 0.064 | 0.236 | 1.229 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 2/4/0.0013 | 0.041 | 0.405 | 1.738 | ||||
2 | 2/4/0.0013 | 0.041 | 0.404 | 1.734 | |||||
3 | 2/4/0.0013 | 0.041 | 0.404 | 1.734 | |||||
4 | 4/4/0.0013 | 0.041 | 0.404 | 1.733 | |||||
\(DS_{Ag 5}\) | 1 | 1/2/0.0015 | 0.015 | 0.381 | 1.651 | 1/2/0.0007 | 0.044 | 0.229 | 1.240 |
2 | 1/4/0.0015 | 0.018 | 0.380 | 1.640 | 1/4/0.0007 | 0.045 | 0.216 | 1.224 | |
3 | 2/4/0.0015 | 0.018 | 0.378 | 1.636 | 2/4/0.0007 | 0.044 | 0.222 | 1.233 | |
4 | 2/4/0.0015 | 0.017 | 0.378 | 1.638 | 2/4/0.0007 | 0.047 | 0.223 | 1.235 | |
\(DS_{Ag 7}\) | 1 | 1/1/0.0013 | 0.014 | 0.371 | 1.619 | 1/1/0.00062 | 0.043 | 0.231 | 1.244 |
2 | 1/1/0.0013 | 0.013 | 0.372 | 1.628 | 1/1/0.0006 | 0.045 | 0.229 | 1.236 | |
3 | 2/4/0.00135 | 0.014 | 0.377 | 1.628 | 2/4/0.0006 | 0.050 | 0.218 | 1.222 | |
4 | 1/2/0.00135 | 0.011 | 0.376 | 1.637 | 1/2/0.0006 | 0.048 | 0.226 | 1.242 | |
\(DS_{Ag 9}\) | 1 | 1/2/0.00115 | 0.024 | 0.413 | 1.648 | 1/2/0.0005 | 0.065 | 0.252 | 1.230 |
2 | 1/2/0.00115 | 0.028 | 0.412 | 1.641 | 1/2/0.0005 | 0.063 | 0.254 | 1.234 | |
3 | 1/2/0.00115 | 0.027 | 0.411 | 1.642 | 1/2/0.0005 | 0.067 | 0.252 | 1.226 | |
4 | 1/4/0.0012 | 0.026 | 0.395 | 1.650 | 1/4/0.0005 | 0.063 | 0.250 | 1.240 | |
\(DS_{Ag 11}\) | 1 | 1/1/0.0011 | 0.016 | 0.395 | 1.646 | 1/1/0.0005 | 0.056 | 0.232 | 1.236 |
2 | 1/2/0.00115 | 0.017 | 0.403 | 1.656 | 1/2/0.0005 | 0.052 | 0.235 | 1.240 | |
3 | 1/3/0.00115 | 0.018 | 0.395 | 1.644 | 1/3/0.0005 | 0.050 | 0.228 | 1.236 | |
4 | 1/2/0.0011 | 0.017 | 0.399 | 1.653 | 1/2/0.0005 | 0.055 | 0.245 | 1.250 |
System | Weights | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) |
---|---|---|---|---|---|---|---|---|---|
\(DS_{Ag 3}\) | 1 | 1/4/0.0014 | 0.023 | 0.406 | 1.768 | ||||
2 | 1/4/0.0014 | 0.022 | 0.402 | 1.751 | |||||
3 | 1/4/0.0014 | 0.021 | 0.402 | 1.754 | |||||
4 | 1/4/0.00135 | 0.022 | 0.396 | 1.721 | |||||
\(DS_{Ag 5}\) | 1 | 5/2/0.00147 | 0.013 | 0.372 | 1.643 | 5/2/0.0007 | 0.042 | 0.230 | 1.241 |
2 | 2/2/0.0015 | 0.014 | 0.378 | 1.632 | 2/2/0.0007 | 0.045 | 0.224 | 1.232 | |
3 | 3/2/0.00155 | 0.013 | 0.384 | 1.653 | 3/2/0.0007 | 0.044 | 0.224 | 1.230 | |
4 | 4/5/0.0015 | 0.017 | 0.376 | 1.643 | 4/5/0.0007 | 0.045 | 0.220 | 1.232 | |
\(DS_{Ag 7}\) | 1 | 1/1/0.0014 | 0.012 | 0.383 | 1.655 | 1/1/0.0007 | 0.045 | 0.227 | 1.244 |
2 | 1/1/0.0014 | 0.014 | 0.378 | 1.653 | 1/1/0.0007 | 0.042 | 0.225 | 1.242 | |
3 | 1/1/0.0014 | 0.013 | 0.377 | 1.636 | 1/1/0.0007 | 0.043 | 0.225 | 1.242 | |
4 | 1/2/0.00135 | 0.011 | 0.385 | 1.640 | 1/2/0.0006 | 0.051 | 0.230 | 1.236 | |
\(DS_{Ag 9}\) | 1 | 1/2/0.00125 | 0.017 | 0.392 | 1.630 | 1/2/0.0006 | 0.050 | 0.240 | 1.244 |
2 | 1/5/0.0014 | 0.017 | 0.387 | 1.639 | 1/5/0.0007 | 0.051 | 0.234 | 1.242 | |
3 | 3/5/0.0013 | 0.023 | 0.407 | 1.667 | 3/5/0.0006 | 0.050 | 0.247 | 1.258 | |
4 | 1/2/0.0012 | 0.021 | 0.393 | 1.633 | 1/2/0.0006 | 0.063 | 0.257 | 1.253 | |
\(DS_{Ag 11}\) | 1 | 1/1/0.00125 | 0.014 | 0.387 | 1.632 | 1/1/0.0006 | 0.044 | 0.224 | 1.245 |
2 | 1/1/0.0013 | 0.013 | 0.387 | 1.627 | 1/1/0.0006 | 0.046 | 0.218 | 1.229 | |
3 | 1/1/0.0013 | 0.014 | 0.389 | 1.648 | 1/1/0.0006 | 0.044 | 0.222 | 1.239 | |
4 | 1/1/0.0012 | 0.018 | 0.384 | 1.622 | 1/1/0.0005 | 0.053 | 0.218 | 1.220 |
System | \(m_1\)/\(m_2\)/\(\varepsilon\) | e | \(e_{ONE}\) | \({\overline{d}}_{DS}\) | t |
---|---|---|---|---|---|
Approach 1 | |||||
\(DS_{Ag 3}\) | 1/3/0.0013 | 0.041 | 0.409 | 1.758 | 4.47 |
\(DS_{Ag 5}\) | 1/1/0.0014 | 0.018 | 0.360 | 1.607 | 5.19 |
1/1/0.0007 | 0.045 | 0.235 | 1.245 | 5.19 | |
\(DS_{Ag 7}\) | 3/5/0.0013 | 0.014 | 0.366 | 1.596 | 20.34 |
3/5/0.0006 | 0.051 | 0.224 | 1.241 | 20.34 | |
\(DS_{Ag 9}\) | 2/50.0012 | 0.042 | 0.400 | 1.618 | 42.58 |
2/5/0.0006 | 0.077 | 0.263 | 1.253 | 42.58 | |
\(DS_{Ag 11}\) | 1/3/0.00153 | 0.041 | 0.376 | 1.632 | 38.55 |
1/3/0.0009 | 0.071 | 0.264 | 1.259 | 38.55 | |
Approach 2 | |||||
\(DS_{Ag 3}\) | 1/3/0.0013 | 0.038 | 0.401 | 1.720 | 4.49 |
\(DS_{Ag 5}\) | 4/3/0.00154 | 0.019 | 0.379 | 1.627 | 6.41 |
4/3/0.0008 | 0.048 | 0.238 | 1.258 | 6.41 | |
\(DS_{Ag 7}\) | 1/5/0.0015 | 0.027 | 0.375 | 1.611 | 18.06 |
1/5/0.0008 | 0.043 | 0.231 | 1.243 | 18.06 | |
\(DS_{Ag 9}\) | 1/4/0.0015 | 0.043 | 0.378 | 1.606 | 27.49 |
1/4/0.0008 | 0.066 | 0.256 | 1.239 | 27.49 | |
\(DS_{Ag 11}\) | 1/3/0.00157 | 0.039 | 0.387 | 1.622 | 34.09 |
1/3/0.0008 | 0.062 | 0.245 | 1.233 | 34.09 | |
Approach 3 | |||||
\(DS_{Ag 3}\) | 1/3/0.0015 | 0.035 | 0.388 | 1.711 | 5.02 |
\(DS_{Ag 5}\) | 2/2/0.0018 | 0.028 | 0.382 | 1.607 | 5.59 |
2/2/0.001 | 0.047 | 0.240 | 1.245 | 5.59 | |
\(DS_{Ag 7}\) | 1/3/0.0021 | 0.031 | 0.371 | 1.612 | 7.39 |
1/3/0.0013 | 0.049 | 0.246 | 1.240 | 7.39 | |
\(DS_{Ag 9}\) | 1/1/0.0027 | 0.043 | 0.389 | 1.613 | 6.56 |
1/1/0.0015 | 0.071 | 0.253 | 1.237 | 6.56 | |
\(DS_{Ag 11}\) | 1/1/0.0032 | 0.041 | 0.377 | 1.635 | 7.45 |
1/1/0.0019 | 0.063 | 0.244 | 1.245 | 7.45 |