Attribute topologies based similarity

Figures & data

Table 1. Information system

U	M	A	E	S
x1	90	97	91	96
x2	88	85	75	80
x3	70	88	79	85
x4	80	88	94	96

Table 2. After similarity with four attributes

M, A, E, S	x1	x2	x3	x4
x1	0	11.5	13	5.5
x2	11.5	0	7.5	11.5
x3	13	7.5	0	9
x4	5.5	11.5	9	0

Table 3. Accuracies with Yao and Pawlak methods when λ ≤ 5

P(X)	P(X^c)	Yao’s method		Accuracy	Pawlak’s method		Accuracy
		X int(x)	Xcl(x)		$\underline{X}$	$\overline{X}$
Ø	X	Ø	Ø	0	Ø	Ø	0
{x1}	{x2, x3, x4}	{x1}	{x1}	1	{x1}	{x1, x4}	1
{x2}	{x1, x3, x4}	{x2}	{x2}	1	{x2}	{x2, x3}	1
{x3}	{x1, x2, x4}	{x3}	{x3}	1	{x3}	{x3}	1
{x4}	{x1, x2, x3}	{x4}	{x4}	1	{x4}	{x4}	1
{x1, x2}	{x3, x4}	{x1, x2}	{x1, x2}	1	{x1, x2}	{x1, x2}	1
{x1, x3}	{x2, x4}	{x1, x3}	{x1, x3}	1	{x1, x3}	{x1, x3}	1
{x1, x4}	{x2, x3}	{x1, x4}	{x1, x4}	1	{x1, x4}	{x1, x4}	1
{x2, x3}	{x1, x4}	{x2, x3}	{x2, x3}	1	{x2, x3}	{x2, x3}	1
{x2, x4}	{x1, x3}	{x2, x4}	{x2, x4}	1	{x2, x4}	{x2, x4}	1
{x3, x4}	{x1, x2}	{x3, x4}	{x3, x4}	1	{x3, x4}	{x3, x4}	1
{x1, x2, x3}	{x4}	{x1, x2, x3}	{x1, x2, x3}	1	{x1, x2, x3}	{x1, x2, x3}	1
{x1, x2, x4}	{x3}	{x1, x2, x4}	{x1, x2, x4}	1	{x1, x2, x4}	{x1, x2, x4}	1
{x2, x3, x4}	{x1}	{x2, x3, x4}	{x2, x3, x4}	1	{x2, x3, x4}	{x2, x3, x4}	1
{x1, x3, x4}	{x2}	{x1, x4}	X	1	{x1, x4}	X	1
X	Ø	X	X	1	X	X	1

Table 4. Accuracies with Yao and Pawlak methods when λ ≤ 10

P(X)	P(X^c)	Yao’s method		Accuracy	Pawlak’s method		Accuracy
		X int(x)	Xcl(x)		$\underline{X}$	$\overline{X}$
Ø	X	Ø	Ø	0	Ø	Ø	0
{x1}	{x2, x3, x4}	Ø	{x1}	0	Ø	{x1, x4}	0
{x2}	{x1, x3, x4}	Ø	{x2}	0	Ø	{x2, x3, x4}	0
{x3}	{x1, x2, x4}	{x3}	{x2, x3}	1/2	{x3}	{x2, x3, x4}	1/2
{x4}	{x1, x2, x3}	{x4}	{x1, x4}	1/2	{x4}	X	1/2
{x1, x2}	{x3, x4}	Ø	{x1, x2}	0	Ø	X	0
{x1, x3}	{x2, x4}	{x3}	{x1, x2, x3}	1/3	{x3}	X	1/4
{x1, x4}	{x1, x4}	{x3}	{x1, x4 }	1	{x1, x4}	X	1/2
{x2, x3}	{x2, x3}	{x2, x3}	{x2, x3}	1	{x2, x3}	{x2, x3, x4}	2/3
{x2, x4}	{x4}	{x4}	{x1, x2, x4}	1/3	{x4}	X	1/4
{x3, x4}	{x3, x4}	{x4}	X	1/2	{x3, x4}	X	1/2
{x1, x2, x3}	{x2, x3}	{x2, x3}	{x1, x2, x3}	2/3	{x2, x3}	X	1/2
{x1, x2, x4}	{x1,x4}	{x1, x4}	{x1, x2, x4}	2/3	{x1, x4}	X	1/2
{x2, x3, x4}	{x2, x3, x4}	{x2, x3, x4}	X	3/4	{x2, x3, x4}	X	3/4
{x1, x3, x4}	{x1, x3, x4}	{x1, x4}	X	3/4	{x1, x4}	X	1/2
X	X	X	X	1	X	X	1

Table 5. Business statement

U	c1	c2	c3	c4	c5	c6	c7	c8	d
x1	2,497.23	1.36	6.31	9,565.70	70.93	172.95	2,708.80	1.36	1
x2	2,830.22	60.01	5.01	1,731.24	144.99	256.40	2,830.22	60.01	2
x3	2,807.22	1.88	6.04	8,753.61	19.54	24.73	2,807.22	1.88	0
x4	1,578.80	0.19	5.25	8,575.66	56.80	78.09	1,578.80	0.19	0
x5	1,525.40	2.63	0.89	1,483.68	54.93	56.26	1,525.40	2.63	1
x6	1,107.52	1.52	0.34	1,089.49	32.23	63.99	1,107.52	1.52	0
x7	1,100.50	1.21	0.99	1,089.54	72.23	670.57	1,100.50	1.21	0
x8	870.05	−1.43	0.69	884.13	−7.16	96.73	870.05	−1.43	0
x9	795.94	7.32	1.28	734.84	40.09	50.26	795.94	7.32	1
x10	789.92	0.23	3.32	707.31	21.78	64.33	789.92	1.23	1

Table 6. Discretion of Table 5

U	c1	c2	c3	c4	c5	c6	c7	c8	d
x1	2	1	3	3	2	3	2	1	1
x2	2	3	3	2	3	3	2	3	2
x3	2	1	3	3	1	1	2	1	0
x4	1	1	3	3	1	1	1	1	0
x5	1	2	2	1	1	1	1	2	1
x6	1	1	1	1	1	1	1	1	0
x7	1	1	2	1	2	3	1	1	0
x8	1	1	1	1	1	1	1	1	0
x9	1	2	2	1	1	1	1	2	1
x10	1	1	3	1	1	1	1	1	1

Table 7. Discretion of Table 6

U	C1	C2	C3	C4	C5	C6	d
X1	2	1	3	3	2	3	1
X2	2	3	3	2	3	3	2
X3	2	1	3	3	1	1	0
X4	1	1	3	3	1	1	0
X5	1	2	2	1	1	1	1
X6	1	1	1	1	1	1	0
X7	1	1	2	1	2	3	0
X8	1	1	3	1	1	1	1

Table 8. Removing attributes

	Removing attributes
Number of elementary sets	None	C1	C2	C3	C4	C5	C6
	8	7	8	7	7	8	8

Table 9. Discuss attributes C₂, C₅, and C₆

U	C2	C5	C6	D
X1	1	2	3	1
X2	3	3	3	2
X3	1	1	1	0
X4	1	1	1	0
X5	2	1	1	1
X6	1	1	1	0
X7	1	2	3	0
X8	1	1	1	1

Table 10. Classification attributes C₂, C₅, and C₆

T	U/C	C2	C5	C6
t1	{X1, X7 }	1	2	3
t2	{X2 }	3	3	3
t3	{X3, X4, X6, X8}	1	1	1
t4	{X5}	2	1	1

Table 11. Similarity about attribute C₂

C2	t1	t2	t3	t4
t1	0	2	0	1
t2	2	0	2	1
t3	0	2	0	1
t4	1	1	1	0

Table 12. Accuracies with Yao and Pawlak methods when λ ≥ 1

P(T)	P(T^c)	Yao’s method		Accuracy-1.1	Pawlak’s method		Accuracy-2.1
		T int(x)	Tcl(x)		$\underline{T}$	$\overline{T}$
Ø	T	Ø	Ø	0	Ø	Ø	0
{t1}	{t2, t3, t4}	Ø	{t1, t3}	0	Ø	T	0
{t2}	{t1, t3, t4}	{t2}	{t2}	1	Ø	T	0
{t3}	{t1, t2, t4}	Ø	{t1, t3}	0	Ø	T	0
{t4}	{t1, t2, t3}	{t4}	{t4}	1	Ø	T	0
{t1, t2}	{t3, t4}	{t2}	{t1, t2, t3}	1/3	Ø	T	0
{t1, t3}	{t2, t4}	{t1, t3}	{t1, t3}	1	Ø	T	0
{t1, t4}	{t2, t3}	{t4}	{t1, t3, t4}	1/3	Ø	T	0
{t2, t3}	{t1, t4}	{t2}	{t1, t2, t3}	1/3	Ø	T	0
{t2, t4}	{t1, t3}	{t2, t4}	{t2, t4}	1	{t2, t4}	T	1/2
{t3, t4}	{t1, t2}	{t4}	{t1, t3, t4}	1/3	Ø	T	0
{t1, t2, t3}	{t4}	{t1, t2, t3}	{t1, t2, t3}	1	{t1, t2, t3}	T	3/4
{t1, t2, t4}	{t3}	{t2, t4}	T	1/2	Ø	T	0
{t2, t3, t4}	{t1}	{t2, t4}	T	1/2	{t2, t4}	T	1/2
{t1, t3, t4}	{t2}	{t1, t3, t4}	{t1, t3, t4}	1	{t1, t3, t4}	T	3/4
T	Ø	T	T	1	T	T	1

Table 13. Similarity about attribute C₅

C5	t1	t2	t3	t4
t1	0	1	1	1
t2	1	0	2	2
t3	1	2	0	0
t4	1	2	0	0

Table 14. Accuracies with Yao and Pawlak methods with C₅ and λ ≥ 1

P(T)	P(T^c)	Yao’s method		Accuracy-1.2	Pawlak’s method		Accuracy-2.2
		Tint(x)	Tcl(x)		$\underline{T}$	$\overline{T}$
Ø	T	Ø	Ø	0	Ø	Ø	0
{t1}	{t2, t3, t4}	{t1}	{t1}	1	{t1}	T	1/3
{t2}	{t1, t3, t4}	{t2}	{t2}	1	{t2}	T	1/3
{t3}	{t1, t2, t4}	Ø	{t3, t4}	0	Ø	T	0
{t4}	{t1, t2, t3}	Ø	{t3, t4}	0	Ø	T	0
{t1, t2}	{t3, t4}	{t1, t2}	{t1, t2}	1	{t1, t2}	T	1/2
{t1, t3}	{t2, t4}	{t1}	{t1, t3, t4}	1/3	{t1}	T	1/3
{t1, t4}	{t2, t3}	{t1}	{t1, t3, t4}	1/3	{t1}	T	1/3
{t2, t3}	{t1, t4}	{t2}	{t2, t3, t4}	1/3	{t2}	T	1/3
{t2, t4}	{t1, t3}	{t2}	{t2, t3, t4}	1/3	{t2}	T	1/3
{t3, t4}	{t1, t2}	{t3, t4}	{t3, t4}	1	{t3, t4}	T	1/2
{t1, t2, t3}	{t4}	{t1, t2}	T	1/2	{t1, t2}	T	1/2
{t1, t2, t4}	{t3}	{t1, t2}	T	1/2	{t1, t2}	T	1/2
{t2, t3, t4}	{t1}	{t2, t3, t4}	{t2, t3, t4}	1	{t2, t3, t4}	T	3/4
{t1, t3, t4}	{t2}	{t1, t3, t4}	{t1, t3, t4}	1	{t1, t3, t4}	T	3/4
T	Ø	T	T	1	T	T	1

Table 15. Similarity about attribute C₂ and C₅

(C2 + C5)/2	t1	t2	t3	t4
t1	0	3/2	1/2	1
t2	3/2	0	2	3/2
t3	1/2	2	0	1/2
t4	1	2/3	1/2	0

Table 16. Accuracies with Yao and Pawlak methods with C₂, C₅ and λ ≥ 1

P(T)	P(T^c)	Yao’s method		Accuracy-1.3	Pawlak’s method		Accuracy-2.3
		T int(x)	Tcl(x)		$\underline{T}$	$\overline{T}$
Ø	T	Ø	Ø	0	Ø	Ø	0
{t1}	{t2, t3, t4}	{t1}	{t1, t3}	1/2	{t1}	T	1/4
{t2}	{t1, t3, t4}	{t2}	{t2}	1	{t2}	{t1, t2, t4}	1/3
{t3}	{t1, t2, t4}	Ø	{t3}	0	Ø	{t1, t3, t4}	0
{t4}	{t1, t2, t3}	{t4}	{t3, t4}	1/2	{t4}	T	1/4
{t1, t2}	{t3, t4}	{t1, t2}	{t1, t2, t3}	2/3	{t1, t2}	T	1/2
{t1, t3}	{t2, t4}	{t1}	{t1, t3}	1/2	{t1}	T	1/4
{t1, t4}	{t2, t3}	{t1, t4}	{t1, t3, t4}	2/3	{t1, t4}	T	1/2
{t2, t3}	{t1, t4}	{t2}	{t2, t3}	1/2	{t2}	T	1/4
{t2, t4}	{t1, t3}	{t2, t4}	{t2, t3, t4}	2/3	{t2, t4}	T	1/2
{t3, t4}	{t1, t2}	{t4}	{t3, t4}	1/2	{t4}	T	1/4
{t1, t2, t3}	{t4}	{t1, t2}	{t1, t3, t4}	2/3	{t1, t2}	T	1/2
{t1, t2, t4}	{t3}	{t1, t2, t4}	T	3/4	{t1, t2, t4}	T	3/4
{t2, t3, t4}	{t1}	{t2, t4}	{t2, t3, t4}	2/3	{t2, t4}	T	1/2
{t1, t3, t4}	{t2}	{t1, t3, t4}	{t1, t3, t4}	1	{t1, t3, t4}	T	3/4
T	Ø	T	T	1	T	T	1