Structural Extension to Logistic Regression
(Auxiliary material)

This webpage presents a number of results related to the paper "Structural Extension to Logistic Regression: Discriminative Parameter Learning of Belief Net Classifiers" (ps; pdf), which appears in the Machine Learning Journal special issue on Probabilistic Graphical Models for Classification -- MLJ 59(3), June 2005, p. 297--322.

The bulk of the material in this website also appears in ps and pdf format.

Report a problem in this page


25 UCI benchmark datasets

We evaluated various algorithms over the standard 25 benchmark datasets used by Friedman et al. [FGG97]: 23 from UCI repository [BM00], plus “mofn-3-7-10” and “corral”, which were developed by [KJ97] to study feature selection.  Also the same 5-fold cross validation and Train/Test learning schemas are followed (see following Table). As part of data preparation, continuous data are always discretized using the supervised entropy-based approach.
(You can download the discretized data here.)

Data set   # Attributes # Classes # Instances
        Train Test
1 australian 14 2 690 CV-5
2 breast 10 2 683 CV-5
3 chess 36 2 2130 1066
4 cleve 13 2 296 CV-5
5 corral 6 2 128 CV-5
6 crx 15 2 653 CV-5
7 diabetes 8 2 768 CV-5
8 flare 10 2 1066 CV-5
9 german 20 2 1000 CV-5
10 glass 9 7 214 CV-5
11 glass2 9 2 163 CV-5
12 heart 13 2 270 CV-5
13 hepatitis 19 2 80 CV-5
14 iris 4 3 150 CV-5
15 letter 16 26 15000 5000
16 lymphography 18 4 148 CV-5
17 mofn-3-7-10 10 2 300 1024
18 pima 8 2 768 CV-5
19 satimage 36 6 4435 2000
20 segment 19 7 1540 770
21 shuttle-small 9 7 3866 1934
22 soybean-large 35 19 562 CV-5
23 vehicle 18 4 846 CV-5
24 vote 16 2 435 CV-5
25 waveform-21 21 3 300 4700

Description of data sets used in the experiments; see also [FGG97: N. Friedman, D. Geiger and M. Goldszmidt. Bayesian Network Classifiers. Machine Learning 29:131-163, 1997 ]

Top


Empirical accuracy of classifiers learned from complete data (25 UCI benchmark datasets)

  Data set NB+OFE   NB+ELR   TAN+OFE TAN+ELR   GBN+OFE   GBN+ELR  
1 australian 86.81 ± 0.84 84.93 ± 1.06 84.93 ± 1.03 84.93 ± 1.03 86.38 ± 0.98 86.81 ± 1.11
2 breast 97.21 ± 0.75 96.32 ± 0.66 96.32 ± 0.81 96.32 ± 0.70 96.03 ± 0.50 95.74 ± 0.43
3 chess 87.34 ± 1.02 95.40 ± 0.64 92.40 ± 0.81 97.19 ± 0.51 90.06 ± 0.92 90.06 ± 0.92
4 cleve 82.03 ± 2.66 81.36 ± 2.46 80.68 ± 1.75 81.36 ± 1.78 84.07 ± 1.48 82.03 ± 1.83
5 corral 86.40 ± 5.31 86.40 ± 3.25 93.60 ± 3.25 100.00 ± 0.00 100.00 ± 0.00 100.00 ± 0.00
6 crx 86.15 ± 1.29 86.46 ± 1.85 86.15 ± 1.70 86.15 ± 1.70 86.00 ± 1.94 85.69 ± 1.30
7 diabetes 74.77 ± 1.05 75.16 ± 1.39 74.38 ± 1.35 73.33 ± 1.97 75.42 ± 0.61 76.34 ± 1.30
8 flare 80.47 ± 1.03 82.82 ± 1.35 83.00 ± 1.06 83.10 ± 1.29 82.63 ± 1.28 82.63 ± 1.28
9 german 74.70 ± 0.80 74.60 ± 0.58 73.50 ± 0.84 73.50 ± 0.84 73.70 ± 0.68 73.70 ± 0.68
10 glass 47.62 ± 3.61 44.76 ± 4.22 47.62 ± 3.61 44.76 ± 4.22 47.62 ± 3.61 44.76 ± 4.22
11 glass2 81.25 ± 2.21 81.88 ± 3.62 80.63 ± 3.34 80.00 ± 3.90 80.63 ± 3.75 78.75 ± 3.34
12 heart 78.52 ± 3.44 78.89 ± 4.08 78.52 ± 4.29 78.15 ± 3.86 79.63 ± 3.75 78.89 ± 4.17
13 hepatitis 83.75 ± 4.24 86.25 ± 5.38 88.75 ± 4.15 85.00 ± 5.08 90.00 ± 4.24 90.00 ± 4.24
14 iris 92.67 ± 2.45 94.00 ± 2.87 92.67 ± 2.45 92.00 ± 3.09 92.00 ± 3.09 92.00 ± 3.09
15 letter 72.40 ± 0.63 83.02 ± 0.53 83.22 ± 0.53 88.90 ± 0.44 79.78 ± 0.57 81.21 ± 0.55
16 lymphography 82.76 ± 1.89 86.21 ± 2.67 86.90 ± 3.34 84.83 ± 5.18 79.31 ± 2.18 78.62 ± 2.29
17 mofn-3-7-10 86.72 ± 1.06 100.00 ± 0.00 91.60 ± 0.87 100.00 ± 0.00 86.72 ± 1.06 100.00 ± 0.00
18 pima 75.03 ± 2.45 75.16 ± 2.48 74.38 ± 2.81 74.38 ± 2.58 75.03 ± 2.25 74.25 ± 2.53
19 satimage 81.55 ± 0.87 85.40 ± 0.79 88.30 ± 0.72 88.30 ± 0.72 79.25 ± 0.91 79.25 ± 0.91
20 segment 85.32 ± 1.28 89.48 ± 1.11 89.35 ± 1.11 89.22 ± 1.12 77.53 ± 1.50 77.40 ± 1.51
21 shuttle-small 98.24 ± 0.30 99.12 ± 0.21 99.12 ± 0.21 99.22 ± 0.20 97.31 ± 0.37 97.88 ± 0.33
22 soybean-large 90.89 ± 1.31 90.54 ± 0.54 93.39 ± 0.67 92.86 ± 1.26 82.50 ± 1.40 85.54 ± 0.99
23 vehicle 55.98 ± 0.93 64.14 ± 1.28 65.21 ± 1.32 66.39 ± 1.22 48.52 ± 2.13 51.95 ± 1.32
24 vote 90.34 ± 1.44 95.86 ± 0.78 93.79 ± 1.18 95.40 ± 0.63 96.32 ± 0.84 95.86 ± 0.78
25 waveform-21 75.91 ± 0.62 78.55 ± 0.60 76.30 ± 0.62 76.30 ± 0.62 65.79 ± 0.69 65.79 ± 0.69

Top


Empirical accuracy of classifiers learned from incomplete data (25 UCI benchmark datasets with "missing completely at random" at 0.25)

  Data set NB+ELR NB+APN NB+EM   TAN+ELR TAN+APN TAN+EM GBN+ELR GBN+APN GBN+EM
1 australian 78.41 ± 1.01 78.41 ± 0.96 78.55 ± 1.01 77.25 ± 0.59 78.12 ± 0.74 77.25 ± 0.59 74.06 ± 1.06 74.06 ± 1.06 74.78 ± 0.74
2 breast 95.59 ± 1.32 96.03 ± 1.20 96.03 ± 1.20 96.03 ± 1.13 95.88 ± 0.95 96.18 ± 1.02 94.12 ± 1.63 94.85 ± 1.36 94.85 ± 1.36
3 chess 94.56 ± 0.69 89.59 ± 0.94 89.68 ± 0.93 96.15 ± 0.59 93.90 ± 0.73 94.09 ± 0.72 90.34 ± 0.90 90.06 ± 0.92 90.06 ± 0.92
4 cleve 84.07 ± 1.90 82.03 ± 2.05 82.03 ± 2.05 83.73 ± 1.57 83.73 ± 1.57 83.73 ± 1.57 83.05 ± 1.93 81.36 ± 2.34 83.39 ± 1.89
5 corral 81.60 ± 3.25 83.20 ± 3.67 83.20 ± 3.67 88.80 ± 3.67 90.40 ± 1.60 88.80 ± 2.65 92.00 ± 1.79 88.80 ± 2.65 92.00 ± 1.79
6 crx 87.54 ± 1.43 86.00 ± 1.67 86.00 ± 1.67 85.85 ± 1.43 84.62 ± 1.29 85.85 ± 1.43 86.15 ± 1.67 87.23 ± 1.10 86.92 ± 0.97
7 diabetes 75.42 ± 1.84 74.64 ± 1.83 74.64 ± 1.83 74.64 ± 2.06 74.90 ± 2.19 74.90 ± 2.19 73.46 ± 1.99 73.20 ± 1.99 72.81 ± 1.79
8 flare 83.00 ± 1.42 82.35 ± 1.21 82.44 ± 1.24 82.54 ± 0.86 82.35 ± 1.90 82.54 ± 1.52 82.63 ± 1.28 82.63 ± 1.28 82.63 ± 1.28
9 german 74.50 ± 0.89 74.10 ± 1.09 74.00 ± 1.05 72.70 ± 0.54 74.00 ± 0.97 72.90 ± 0.40 73.70 ± 0.68 73.40 ± 0.86 73.70 ± 0.68
10 glass 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33 35.71 ± 4.33
11 glass2 79.38 ± 3.22 77.50 ± 3.03 77.50 ± 3.03 76.25 ± 2.72 76.25 ± 3.37 76.25 ± 2.72 78.13 ± 3.28 77.50 ± 3.75 78.13 ± 3.28
12 heart 75.19 ± 5.13 74.81 ± 4.63 74.81 ± 4.63 72.22 ± 3.26 73.33 ± 4.00 73.33 ± 4.00 73.70 ± 3.95 73.33 ± 4.37 73.33 ± 4.37
13 hepatitis 81.25 ± 7.65 86.25 ± 5.00 86.25 ± 5.00 82.50 ± 5.00 87.50 ± 3.95 86.25 ± 5.00 86.25 ± 3.64 86.25 ± 3.64 86.25 ± 3.64
14 iris 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82 94.67 ± 0.82
15 letter 75.28 ± 0.61 67.24 ± 0.66 67.14 ± 0.66 81.86 ± 0.54 85.25 ± 0.50 84.07 ± 0.52 72.80 ± 0.63 69.81 ± 0.65 68.60 ± 0.66
16 lymphography 84.83 ± 2.80 84.14 ± 1.38 83.45 ± 1.29 82.07 ± 3.84 78.62 ± 2.01 81.38 ± 3.87 78.62 ± 2.29 78.62 ± 2.29 79.31 ± 2.18
17 mofn-3-7-10 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20 82.03 ± 1.20
18 pima 74.90 ± 2.85 74.90 ± 2.85 74.90 ± 2.85 74.25 ± 2.45 73.99 ± 2.28 73.99 ± 2.28 73.99 ± 2.06 74.64 ± 2.25 74.77 ± 2.31
19 satimage 84.90 ± 0.80 81.85 ± 0.86 81.90 ± 0.86 87.70 ± 0.73 87.80 ± 0.73 87.70 ± 0.73 73.95 ± 0.98 76.35 ± 0.95 76.30 ± 0.95
20 segment 89.74 ± 1.09 85.19 ± 1.28 85.19 ± 1.28 89.35 ± 1.11 89.22 ± 1.12 89.09 ± 1.12 77.40 ± 1.51 77.40 ± 1.51 77.40 ± 1.51
21 shuttle-small 99.17 ± 0.21 99.07 ± 0.22 99.07 ± 0.22 99.28 ± 0.19 99.17 ± 0.21 99.17 ± 0.21 99.22 ± 0.20 98.04 ± 0.32 98.04 ± 0.32
22 soybean-large 85.54 ± 1.79 87.68 ± 1.77 86.07 ± 2.37 84.29 ± 1.25 84.64 ± 1.34 86.61 ± 0.80 50.54 ± 1.61 50.18 ± 1.75 48.21 ± 2.43
23 vehicle 62.72 ± 1.69 57.28 ± 1.25 57.51 ± 1.38 64.85 ± 1.29 62.49 ± 1.28 62.60 ± 1.44 49.94 ± 0.91 44.73 ± 1.94 44.73 ± 1.94
24 vote 94.71 ± 0.86 90.80 ± 1.54 91.03 ± 1.52 94.94 ± 0.86 95.40 ± 0.51 95.17 ± 0.67 95.17 ± 0.76 95.63 ± 0.92 95.17 ± 0.76
25 waveform-21 73.34 ± 0.64 73.64 ± 0.64 73.64 ± 0.64 72.26 ± 0.65 72.28 ± 0.65 72.26 ± 0.65 64.38 ± 0.70 55.85 ± 0.72 55.85 ± 0.72

Top


Experiments with "Cross Tuning"

Gradient based learners have to determine when to stop climbing. A naive implementation would climb for a fixed pre-set number of iterations, or would continue climbing as long as the empirical accuracy is increasing. Our empirical studies (on both ELR and APN) show that these approaches are problematic, as these systems will typically overfit or underfit. To demonstrate this, we present 5-fold cross validation learning curves from TAN+ELR  training results on the cleve dataset. For each cross validation run, we used a performed 20 iterations over the training data; and we plotted the 'Resubstitution Error' and 'Generalization Error' after each gradient descent iteration (See graphs below.) The 'Generalization Error' is the testing error of the resulting system on the hold-out fold after each training iteration. (I.e., we divided all cleve data into 5 fold: {F1, F2, F3, F4, F5}; in each iteration of the first cross validation run, we used F1+F2+F3+F4 for training, then evaluated the resulting system against the F5 hold-out testing data to produce the 'Generalization Error'). Many of the plots show that ELR's gradient ascent starts overfitting significantly only after a few training iterations. 



Based on the generalization error plots, we see that ELR should stop after   {2, 1, 1, 4, 5}   iterations, for these 5 cross validation runs. Of course, ELR will not know these "optimal iteration numbers" as they are based on the hold-out data, which is NOT available at training time.

Fortunately, ELR estimates these numbers from the available training data, using a standard method we call "cross-tuning",
described on pages 9-10 of the manuscript, to try to identify the number of climbs (iterations) that is appropriate for each specific dataset. Cross-tuning first splits the training set into n parts (folds), then successively trains on n-1 folds and evaluates on the remaining one. In particular, for each instance, it runs the ELR algorithm on n-1 folds for a large number of iterations, and measures the quality of the resulting classifier on the other fold. For each run, it determines which iteration produces the smallest generalization error. Cross-tuning then picks the median value m over these runs. Later, when running on the full dataset (all n folds), it will run for m iterations before stopping.

The paired t-tests of ELR results on the UCI benchmark datasets shows that cross-tuning is essential in ELR learning: (Recall 'x <- y' means 'x is better than y'.)

Here NB+ELR(-xt) is comparable to TAN+ELR(-xt), whose performance was significantly degraded by overfitting. This shows cross-tuning can be effective to prevent overfitting especially when learning parameters of complex BN structures.

The obvious downside of cross-tuning, of course, is computation expense; see timing information.

To demonstrate how cross-tuning works to help avoid overfitting, we revisit the experiments on the cleve dataset. For the first cross validation run, we split the training data from folds {F1,F2, F3, F4} into another 5 folds for cross-tuning; call them 1CT = {1CT1, 1CT2, ..., 1CT5}. (Note: F1 + F2 + F3 + F4 = 1CT1 + 1CT2 + ... + 1CT5.) We then ran 5 fold cross-tuning on 1CT, here by using 4 folds of 1CT for training and the remaining 1CT fold for testing, over 20 iterations. Each cross-tuning run determined an iteration number that produced the smallest testing error on the hold-out 1CT fold. After 5-fold cross-tuning runs, we took the median value of the 5 estimates and used it as the iteration number in the training on the full 1CT set.

For this first cross-validation run, this produced an estimate of 2, which we see (from the "cleve fold 1/5" graph below) is correct. We similarly computed this quantity for the other four cross-validation scenarios, producing {2, 1, 1, 3, 5} respectively for the 5 cross validation runs. Notice cross-tuning identified the correct stopping number in 4 of the 5 cross validation run. The only exception is the fourth one, where it returned 3, not 4.

Top


Experiments on UCIrvine databases with missing data

classification errors for 20 UCI incomplete datasets

classification errors NB+EM   NB+APN   NB+ELR   TAN+EM   TAN+APN   TAN+ELR  
agaricus-lepiota 4.41 0.3 4.35 0.32 0 0 0.01 0.01 0 0 0 0
allbp 4.22 0 4.22 0 3.09 0 4.12 0 4.12 0 3.5 0
allhyper 2.78 0 2.78 0 1.85 0 2.37 0 1.85 0 1.75 0
allrep 3.5 0 3.6 0 3.29 0 2.47 0 2.67 0 2.78 0
anneal 5.79 1.66 4.65 1.84 1.76 0.67 6.54 1.64 5.16 1.86 1.89 0.4
bands 30 1.96 29.81 1.79 25.56 1.39 25.37 2.06 24.63 2.24 26.48 2.24
breast-cancer 2.59 0.84 2.59 0.84 3.74 1.14 5.18 0.89 5.76 1.27 5.04 0.85
cleve 15.67 3.23 15.67 3.23 16 2.72 18 1.62 17.33 1.63 18 1.62
crx 14.06 1.11 14.06 1.04 13.33 0.93 15.22 0.51 15.07 0.77 15.22 0.51
dermatology 2.19 0.82 2.19 1.11 1.92 0.7 4.66 1.11 3.29 1.11 3.29 1.27
dis 2.11 0.56 2.11 0.6 1.39 0.26 1.71 0.27 1.57 0.3 1.43 0.22
horse-colic 19.73 1.66 19.73 1.66 17.81 1.15 18.08 1.01 18.36 0.82 19.73 0.93
hypothyroid 2.25 0.6 1.99 0.63 1.96 0.54 2.31 0.6 2.24 0.6 2.15 0.55
imports-85 37.56 4.27 37.56 3.33 40 2.51 34.63 1.79 34.15 3.45 33.17 3.05
monk1-corrupt 36.11 0 36.11 0 34.72 0 22.92 0 22.22 0 16.67 0
primary-tumor 51.64 2.69 50.15 3.01 50.45 2.89 51.94 4.51 54.93 3.38 51.94 4.51
sick 4.71 1.21 4.89 1.36 4.11 0.77 4.46 0.85 4.46 0.88 4.18 0.71
sick-euthyroid 7.03 0.93 6.96 0.89 6.36 0.99 7.25 0.89 7.15 0.91 6.46 1.1
soybean-large 11.97 0 7.71 0 8.51 0 8.78 0 10.11 0 10.37 0
water-treatment 47.31 1.91 47.31 1.91 47.31 1.91 47.31 1.91 47.31 1.91 47.31 1.91
average 15.2815   14.922   14.158   14.1665   14.119   13.568  

Paired T-tests ('x <- y' means 'x is better than y'):

NB+ELR <- NB+EM (p < 0.00559)

NB+ELR <- NB+APN (p < 0.026125)

TAN+ELR <- TAN+EM (p < 0.083164)

TAN+ELR <- TAN+APN (p < 0.077631)
 

dataset information * instance # attribute# class#   missing ratio missing total/attris
agaricus-lepiota 8124 22 2 CV5 1.39% 2480/1
allbp 2800/972 29 3 train/test 5.54% 4556+1508
allhyper 2800/972 29 5 train/test 5.54% 4556+1508
allrep 2800/972 29 4 train/test 5.54% 4556+1508
anneal 798 38 6 CV5 64.94% 19692/28
bands 540 29 2 CV5 1.93% 302
breast-cancer 699 10 2 CV5 0.23% 16
cleve 303 13 2 CV5 0.18% 7
crx 690 15 2 CV5 0.65% 67/7
dermatology 366 34 6 CV5 0.06% 8/1
dis 2800 29 2 CV5 5.61% 4556
horse-colic 368 22 2 CV5 23.80% 1927
hypothyroid 3163 25 2 CV5 6.74% 5329
imports-85 205 25 7 CV5 1.15% 59/7
monk1-corrupt 288/144 6 2 train/test 30.17% 521+261
primary-tumor 339 17 22 CV5 3.90% 225/5
sick 2800 29 2 CV5 5.61% 4556
sick-euthyroid 3163 25 2 CV5 6.74% 5329
soybean-large 307/376 25 2 train/test 4.32% 705/33
water-treatment 523 38 13 CV5 2.97% 591/31

* Note all datasets have >200 instances

Top


ELR vs other Learning Algorithms

The page summarizes all the results on complete data experiments from various papers. In short, we found that x+ELR performed comparably to C4.5 and SNB.
The following table summarizes our results when comparing ELR vs SVM-light. (Note that we only ran over the datasets with BINARY class labels.) The page presents further details on these SVM experiments.
 

Data set NB+ELR   TAN+ELR   GBN+ELR   svm-light c0.05 t1 d2 *   svm-light best value  
australian 84.93 1.06 84.93 1.03 86.81 1.11 70.29 9.11 77.10 2.88
breast 96.32 0.66 96.32 0.70 95.74 0.43 93.97 1.21 96.62 1.23
chess 95.40 0.64 97.19 0.51 90.06 0.92 97.65 0.00 98.97 0.00
cleve 81.36 2.46 81.36 1.78 82.03 1.83 72.54 4.39 80.34 3.08
corral 86.40 3.25 100.00 0.00 100.00 0.00 96.80 5.22 100.00 0.00
crx 86.46 1.85 86.15 1.70 85.69 1.30 70.15 8.34 70.31 6.43
diabetes 75.16 1.39 73.33 1.97 76.34 1.30 69.28 5.77 76.34 3.50
flare 82.82 1.35 83.10 1.29 82.63 1.28 82.06 3.81 82.91 3.13
german 74.60 0.58 73.50 0.84 73.70 0.68 66.20 1.75 68.70 5.75
glass2 81.88 3.62 80.00 3.90 78.75 3.34 79.37 8.45 79.37 8.45
heart 78.89 4.08 78.15 3.86 78.89 4.17 76.67 2.81 83.33 3.21
hepatitis 86.25 5.38 85.00 5.08 90.00 4.24 86.25 5.23 86.25 5.23
mofn-3-7-10 100.00 0.00 100.00 0.00 100.00 0.00 100.00 0.00 100.00 0.00
pima 75.16 2.48 74.38 2.58 74.25 2.53 70.59 4.03 75.95 2.03
vote 95.86 0.78 95.40 0.63 95.86 0.78 93.10 1.15 95.17 1.50
average 85.43   85.92   86.05   81.66   84.76  
                   
* We tried many settings, and found this specific setting, [c=0.05, poly 2 (t=1, d=2)] produced the best average for SVM. (As this is based on ALL data, this does give svm-light a slight advantage.)          
         
       

Paried T-tests ('x <-- y' means 'x is better than y'):

NB+ELR <-- SVM-light[best_ave]   (p< 0.023)
TAN+ELR <-- SVM-light[best_ave]   (p< 0.036)
GBN+ELR  <-- SVM-light[best_ave]   (p< 0.0078)

See here for the 'training time' data for various algorithms.

Top


Compare to the work in [GGS97]

Finally, our companion paper [GGS97] also considers learning the parameters of a given structure towards optimizing performance on a distribution of queries. Our results here differ, as we are considering a different learning model: [GGS97] tries to minimize the squared-error score, a variant of  Equation 9 that is based on two different types of samples --- one over tuples, to estimate P(C | E), and the other over queries, to estimate the probability of seeing each ``What is P(C | E = e)?'' query. By contrast, the current paper tries to minimize classification error (Equation 3) by seeking the optimal ``conditional likelihood'' score (Equation 4), wrt a single sample of labeled instances. Moreover, our current paper includes new theoretical results, a different algorithm, and completely new empirical data.

Recall from the paper,

Top


Earlier publications

The current paper significantly extends the earlier short papers
  1. Structural Extension to Logistic Regression: Discriminant Parameter Learning of Belief Net Classifiers
    Russell Greiner and Wei Zhou
    Proceedings of the Eighteenth Annual National Conference on Artificial Intelligence (AAAI-02),  p. 167-173, Edmonton, Aug 2002.
  2. Discriminative Parameter Learning of General Bayesian Network Classifiers
    Bin Shen, Xiaoyuan Su, Russell Greiner, Petr Musilek, Corrine Cheng
    Proceedings of the Fifteenth IEEE International Conference on Tools with Artificial Intelligence (ICTAI-03),  Sacramento, Nov 2003.
as it provides It also assembles the various pieces into a single coherent story.

Top


For problems or questions regarding this web page, contact [Bin Shen]
Last updated: 2004/10/11