1 Introduction
2 Multi-functional nearest-neighbour classifier
2.1 K nearest neighbour
2.2 Fuzzy nearest neighbour
2.3 Multi-functional nearest-neighbour algorithm
2.4 Properties of MFNN
-
Łukasiewicz S-norm: \(S(x,y)=\min (x+y,1)\);
-
Gödel S-norm: \(S(x,y)=\max (x,y)\);
-
Algebraic S-norm: \(S(x,y)=(x+y)-(x*y)\);
-
Einstein S-norm: \(S(x,y) = (x + y) / (1 + x * y)\).
Object |
a
|
b
|
c
|
q
|
---|---|---|---|---|
1 | −0.4 | 0.2 | −0.5 | Yes |
2 | −0.4 | 0.1 | −0.1 | No |
3 | 0.2 | −0.3 | 0 | No |
4 | 0.2 | 0 | 0 | Yes |
2.5 Worked example
Object |
a
|
b
|
c
|
q
|
---|---|---|---|---|
t1 | 0.3 | −0.3 | 0 | No |
t2 | −0.3 | 0.3 | −0.3 | Yes |
3 Relationship between MFNN and FRNN/VQNN
3.1 Fuzzy-rough nearest-neighbour classification
3.2 FRNN and VQNN as special instances of MFNN
4 Popular nearest-neighbour methods as special cases of MFNN
4.1 K nearest neighbour and MFNN
4.2 Fuzzy nearest neighbour and MFNN
5 Experimental evaluation
5.1 Experimental set-up
Dataset | Objects | Attributes |
---|---|---|
Cleveland | 297 | 13 |
Ecoli | 336 | 7 |
Glass | 214 | 9 |
Handwritten | 1593 | 256 |
Heart | 270 | 13 |
Liver | 345 | 6 |
Multifeat | 2000 | 649 |
Olitos | 120 | 25 |
Page-block | 5473 | 10 |
Satellite | 6435 | 36 |
Sonar | 208 | 60 |
Water 2 | 390 | 38 |
Water 3 | 390 | 38 |
Waveform | 5000 | 40 |
Wine | 178 | 14 |
Wisconsin | 683 | 9 |
Dataset | MFNN_G | FRNN | VQNN | FNN | FRNN-O |
kNN |
---|---|---|---|---|---|---|
Cleveland | 53.44 | 53.44 | 58.46 | 49.75 | 46.85* | 55.73 |
Ecoli | 80.57 | 80.57 | 86.85v | 86.55v | 77.95 | 86.20v |
Glass | 73.54 | 73.54 | 68.95 | 68.57 | 71.70 | 63.23* |
Handwritten | 91.13 | 91.13 | 91.37 | 91.40 | 89.94* | 90.18 |
Heart | 76.63 | 76.63 | 82.19v | 66.11* | 66.00* | 81.30 |
Liver | 62.81 | 62.81 | 66.26 | 69.52 | 62.37 | 61.25 |
Multifeat | 97.57 | 97.57 | 97.95 | 94.34* | 96.96 | 97.88 |
Olitos | 78.67 | 78.67 | 80.75 | 63.25* | 67.58* | 81.50 |
Page-block | 96.04 | 96.04 | 95.99 | 95.94 | 96.53 | 95.19* |
Satellite | 90.92 | 90.92 | 90.99 | 90.73 | 90.89 | 90.30 |
Sonar | 85.25 | 85.25 | 79.38* | 73.21* | 85.06 | 75.25 |
Water2 | 84.38 | 84.38 | 85.15 | 77.97* | 79.79* | 84.26 |
Water3 | 79.82 | 79.82 | 81.28 | 74.64* | 73.21* | 80.90 |
Waveform | 73.77 | 73.77 | 81.55v | 83.19v | 79.71v | 80.46v |
Wine | 97.47 | 97.47 | 97.14 | 96.40 | 95.62 | 96.07 |
Wisconsin | 96.38 | 96.38 | 96.69 | 97.20 | 96.00 | 96.92 |
Summary | (v//*) | (0/16/0) | (3/12/1) | (2/8/6) | (1/9/6) | (2/11/3) |
Dataset | MFNN_A | FRNN | VQNN | FNN | FRNN-O |
kNN |
---|---|---|---|---|---|---|
Cleveland | 58.46 | 53.44 | 58.46 | 49.75* | 46.85* | 55.73 |
Ecoli | 86.85 | 80.57* | 86.85 | 86.55 | 77.95 | 86.20 |
Glass | 68.95 | 73.54 | 68.95 | 68.57 | 71.70 | 62.23* |
Handwritten | 91.37 | 91.13 | 91.37 | 91.40 | 89.94* | 90.18* |
Heart | 82.19 | 76.63* | 82.19 | 66.11* | 66.00* | 81.30 |
Liver | 66.26 | 62.81 | 66.26 | 69.52 | 62.37 | 61.25* |
Multifeat | 97.95 | 97.57 | 97.95 | 94.34* | 96.96* | 97.88 |
Olitos | 80.75 | 78.67 | 80.75 | 63.25* | 67.58* | 81.50 |
Page-block | 95.99 | 96.04 | 95.99 | 95.94 | 96.53 | 95.19* |
Satellite | 90.99 | 90.92 | 90.99 | 90.73 | 90.89 | 90.30* |
Sonar | 79.38 | 85.25v | 79.38 | 73.21* | 85.06v | 75.25 |
Water2 | 85.15 | 84.38 | 85.15 | 77.97* | 79.79* | 84.26 |
Water3 | 81.28 | 79.82 | 81.28 | 74.64* | 73.21* | 80.90 |
Waveform | 81.55 | 73.77* | 81.55 | 83.19v | 79.71* | 80.46 |
Wine | 97.14 | 97.47 | 97.14 | 96.40 | 95.62 | 96.07 |
Wisconsin | 96.69 | 96.38 | 96.69 | 97.20 | 96.00 | 96.92 |
Summary | (v//*) | (1/12/3) | (0/16/0) | (1/8/7) | (1/7/8) | (0/11/5) |
5.2 Influence of number of neighbours
5.3 Comparison with other nearest neighbour methods
Dataset | MFNN_E | FRNN | VQNN | FNN | FRNN-O |
kNN |
---|---|---|---|---|---|---|
Cleveland | 53.64 | 53.44 | 58.46 | 49.75 | 46.85* | 55.73 |
Ecoli | 81.93 | 80.57* | 86.85v | 86.55v | 77.95 | 86.20v |
Glass | 74.29 | 73.54 | 68.95 | 68.57 | 71.70 | 63.23* |
Handwritten | 91.20 | 91.13 | 91.37 | 91.40 | 89.94* | 90.18 |
Heart | 76.70 | 76.63 | 82.19v | 66.11* | 66.00* | 81.30 |
Liver | 63.07 | 62.81 | 66.26 | 69.52 | 62.37 | 61.25 |
Multifeat | 97.59 | 97.57 | 97.95 | 94.34* | 96.96 | 97.88 |
Olitos | 78.83 | 78.67 | 80.75 | 63.25* | 67.58* | 81.50 |
Page-block | 96.18 | 96.04 | 95.99 | 95.94 | 96.53 | 95.19* |
Satellite | 91.52 | 90.92* | 90.99 | 90.73* | 90.89* | 90.30* |
Sonar | 85.35 | 85.25 | 79.38* | 73.21* | 85.06 | 75.25* |
Water2 | 85.21 | 84.38 | 85.15 | 77.97* | 79.79* | 84.26 |
Water3 | 80.77 | 79.82 | 81.28 | 74.64* | 73.21* | 80.90 |
Waveform | 74.97 | 73.77* | 81.55v | 83.19v | 79.71v | 80.46v |
Wine | 97.47 | 97.47 | 97.14 | 96.40 | 95.62 | 96.07 |
Wisconsin | 96.38 | 96.38 | 96.69 | 97.20 | 96.00 | 96.92 |
Summary | (v//*) | (0/13/3) | (3/12/1) | (2/7/7) | (1/8/7) | (2/10/4) |
5.4 Comparison with the state of the art: use of different aggregators
-
PART (Witten and Frank 1998, 2000) generates rules by means of repeatedly creating partial decision trees from the data. The algorithm adopts a divide-and-conquer strategy such that it removes instances already covered by the current ruleset during the learning processing. Essentially, a rule is created by building a pruned tree for the current set of instances; the branch leading to a leaf with the highest coverage is promoted to a classification rule. In this paper, this method is empirically learned with a confident factor of 0.25.
-
J48 is based on ID3 (Quinlan 1993) and creates decision trees by choosing the most informative features and recursively partitioning a training data table into subtables based on the values of such features. Each node in the tree represents a feature, with the subsequent nodes branching from the possible values of this node according to the current subtable. Partitioning stops when all data items in the subtable have the same classification. A leaf node is then created to represent this classification. In this paper, J48 is set with the pruning confidence threshold \(C=0.25\).
-
SMO (Smola and Schölkopf 1998) is an algorithm for efficiently solving optimisation problems which arise during the training of a support vector machine (Cortes and Vapnik 1995). It breaks optimisation problems into a series of smallest possible subproblems, which are then resolved analytically. In this paper, SMO is set with \(C=1\), tolerance \(L=0.001\), round-off error=\(10^{-12}\), data running on normalised and polynomial kernel.
-
NB (Naive Bayes) (John and Langley 1995) is a simple probabilistic classifier, directly applying Bayes’ theorem (Papoulis 1984) with strong (naive) independence assumptions. Depending on the precise nature of the probability model used, naive Bayesian classifiers can be trained very efficiently in a supervised learning setting. The learning only requires a small amount of training data to estimate the parameters (means and variances of the variables) necessary for classification.
Dataset | MFNN_G | PART | J48 | SMO | NB |
---|---|---|---|---|---|
Cleveland | 53.44 | 52.44 | 53.39 | 58.31 | 56.06 |
Ecoli | 80.57 | 81.79 | 82.83 | 83.48 | 85.50v |
Glass | 73.54 | 69.12 | 68.08 | 57.77* | 47.70* |
Handwritten | 91.13 | 79.34* | 76.13* | 93.58v | 86.19* |
Heart | 76.63 | 77.33 | 78.15 | 83.89v | 83.59v |
Liver | 62.81 | 65.25 | 65.84 | 57.98 | 54.89 |
Multifeat | 97.57 | 94.68* | 94.62* | 98.39v | 95.27* |
Olitos | 78.67 | 67.00* | 65.75* | 87.92v | 78.50 |
Page-block | 96.04 | 96.93v | 96.99v | 92.84* | 90.01* |
Satellite | 90.92 | 86.63* | 86.41* | 86.78* | 79.59* |
Sonar | 85.25 | 77.40* | 73.61* | 76.60* | 67.71* |
Water2 | 84.38 | 83.85 | 83.18 | 83.64 | 69.72* |
Water3 | 79.82 | 82.72 | 81.59 | 87.21v | 85.49v |
Waveform | 73.77 | 77.62v | 75.25 | 86.48v | 80.01v |
Wine | 97.47 | 92.24* | 93.37 | 98.70 | 97.46 |
Wisconsin | 96.38 | 95.68 | 95.44 | 97.01 | 96.34 |
Summary | (v//*) | (2/8/6) | (1/10/5) | (6/6/4) | (4/5/7) |
Dataset | MFNN_E | PART | J48 | SMO | NB |
---|---|---|---|---|---|
Cleveland | 53.64 | 52.44 | 53.39 | 58.31 | 56.06 |
Ecoli | 81.93 | 81.79 | 82.83 | 83.48 | 85.50 |
Glass | 74.29 | 69.12 | 68.08 | 57.77* | 47.70* |
Handwritten | 91.20 | 79.34* | 76.13* | 93.58v | 86.19* |
Heart | 76.70 | 77.33 | 78.15 | 83.89v | 83.59v |
Liver | 63.07 | 65.25 | 65.84 | 57.98 | 54.89 |
Multifeat | 97.59 | 94.68* | 94.62* | 98.39v | 95.27* |
Olitos | 78.83 | 67.00* | 65.75* | 87.92v | 78.50 |
Page-block | 96.18 | 96.93v | 96.99v | 92.84* | 90.01* |
Satellite | 91.52 | 86.63* | 86.41* | 86.78* | 79.59* |
Sonar | 85.35 | 77.40* | 73.61* | 76.60* | 67.71* |
Water2 | 85.21 | 83.85 | 83.18 | 83.64 | 69.72* |
Water3 | 80.77 | 82.72 | 81.59 | 87.21v | 85.49v |
Waveform | 74.97 | 77.62v | 75.25 | 86.48v | 80.01v |
Wine | 97.47 | 92.24* | 93.37 | 98.70 | 97.46 |
Wisconsin | 96.38 | 95.68 | 95.44 | 97.01 | 96.34 |
Summary | (v//*) | (2/8/6) | (1/10/5) | (6/6/4) | (3/6/7) |
Dataset | MFNN_A | PART | J48 | SMO | NB |
---|---|---|---|---|---|
Cleveland | 58.46 | 52.44* | 53.39 | 58.31 | 56.06 |
Ecoli | 86.85 | 81.79* | 82.83* | 83.48 | 85.50 |
Glass | 68.95 | 69.12 | 68.08 | 57.77* | 47.70* |
Handwritten | 91.37 | 79.34* | 76.13* | 93.58v | 86.19* |
Heart | 82.19 | 77.33 | 78.15 | 83.89 | 83.59 |
Liver | 66.26 | 65.25 | 65.84 | 57.98* | 54.89* |
Multifeat | 97.95 | 94.68* | 94.62* | 98.39 | 95.27* |
Olitos | 80.75 | 67.00* | 65.75* | 87.92v | 78.50 |
Page-block | 95.99 | 96.93v | 96.99v | 92.84* | 90.01* |
Satellite | 90.99 | 86.63* | 86.41* | 86.78* | 79.59* |
Sonar | 79.38 | 77.40 | 73.61 | 76.60 | 67.71* |
Water2 | 85.15 | 83.85 | 83.18 | 83.64 | 69.72* |
Water3 | 81.28 | 82.72 | 81.59 | 87.21v | 85.49v |
Waveform | 81.55 | 77.62* | 75.25* | 86.48v | 80.01 |
Wine | 97.14 | 92.24* | 93.37 | 98.70 | 97.46 |
Wisconsin | 96.69 | 95.68 | 95.44 | 97.01 | 96.34 |
Summary | (v//*) | (1/7/8) | (1/9/6) | (4/8/4) | (1/7/8) |
Dataset | MFNN_AW | PART | J48 | SMO | NB |
---|---|---|---|---|---|
Cleveland | 58.46 | 52.44* | 53.39 | 58.31 | 56.06 |
Ecoli | 87.12 | 81.79* | 82.83* | 83.48 | 85.50 |
Glass | 66.76 | 69.12 | 68.08 | 57.77* | 47.70* |
Handwritten | 91.32 | 79.34* | 76.13* | 93.58v | 86.19* |
Heart | 82.78 | 77.33 | 78.15 | 83.89 | 83.59 |
Liver | 64.65 | 65.25 | 65.84 | 57.98* | 54.89* |
Multifeat | 98.05 | 94.68* | 94.62* | 98.39 | 95.27* |
Olitos | 82.42 | 67.00* | 65.75* | 87.92 | 78.50 |
Page-block | 96.40 | 96.93v | 96.99v | 92.84* | 90.01* |
Satellite | 90.76 | 86.63* | 86.41* | 86.78* | 79.59* |
Sonar | 81.31 | 77.40 | 73.61* | 76.60 | 67.71* |
Water2 | 85.90 | 83.85 | 83.18 | 83.64 | 69.72* |
Water3 | 81.15 | 82.72 | 81.59 | 87.21v | 85.49v |
Waveform | 81.09 | 77.62* | 75.25* | 86.48v | 80.01 |
Wine | 96.53 | 92.24 | 93.37 | 98.70 | 97.46 |
Wisconsin | 96.66 | 95.68 | 95.44 | 97.01 | 96.34 |
Summary | (v//*) | (1/8/7) | (1/8/7) | (3/9/4) | (1/7/8) |
5.5 Comparison with the state of the art: use of different similarity metrics
Dataset | MFNN_AR | PART | J48 | SMO | NB |
---|---|---|---|---|---|
Cleveland | 57.21 | 52.44 | 53.39 | 58.31 | 56.06 |
Ecoli | 87.68 | 81.79* | 82.83* | 83.48* | 85.50 |
Glass | 74.98 | 69.12 | 68.08 | 57.77* | 47.70* |
Handwritten | 91.29 | 79.34* | 76.13* | 93.58v | 86.19* |
Heart | 81.93 | 77.33 | 78.15 | 83.89 | 83.59 |
Liver | 64.88 | 65.25 | 65.84 | 57.98* | 54.89* |
Multifeat | 97.75 | 94.68* | 94.62* | 98.39v | 95.27* |
Olitos | 80.67 | 67.00* | 65.75* | 87.92v | 78.50 |
Page-block | 96.09 | 96.93v | 96.99v | 92.84* | 90.01* |
Satellite | 90.01 | 86.63* | 86.41* | 86.78* | 79.59* |
Sonar | 77.02 | 77.40 | 73.61 | 76.60 | 67.71* |
Water2 | 85.03 | 83.85 | 83.18 | 83.64 | 69.72* |
Water3 | 80.82 | 82.72 | 81.59 | 87.21v | 85.49v |
Waveform | 80.14 | 77.62* | 75.25* | 86.48v | 80.01 |
Wine | 95.97 | 92.24 | 93.37 | 98.70 | 97.46 |
Wisconsin | 96.35 | 95.68 | 95.44 | 97.01 | 96.34 |
Summary | (v//*) | (1/9/6) | (1/9/6) | (5/6/5) | (1/7/8) |