Multivariate decision trees for machine learning

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6.2.Results for Classification and Regression Trees

For the rest of these results, the definition given in Table 6.2.1 applies.

TABLE 6.2.1 Definition of tree-based methods

Name of the Method	Uni/Multi	Impurity Measure	Pruning	Feature Selection
ID3	Uni	Information Gain	Pre-pruning	No
CART	Multi	Information Gain	Pre-pruning	No
FSCART	Multi	Information Gain	Pre-pruning	Yes

In this section, the multivariate method CART and the univariate method ID3 are compared. Also feature selection is applied to CART to see if there will be a difference in accuracy or node size. The results of this comparison are shown in Table 6.2.2 and in Figure 6.2.1. Node results are shown in Table 6.2.3 and Figure 6.2.2. Learning time results are shown in Table 6.2.3, Figure 6.2.3 and Figure 6.2.4.

ID3 is statistically significantly better than CART in six out 20 data sets. ID3 is better then FSCART in two cases with 95% and FSCART is better then ID3 in one case with 99% confidence. So we can say that no one of the three methods is clearly the best.

Concerning the tree size, generally multivariate techniques FSCART and CART perform better than ID3 and this is significant in six data sets. Note that a CART node internally is more complex then an ID3 node.

The learning times of univariate technique ID3 is significantly better than multivariate techniques CART and FSCART.

Feature selection improves accuracy and node size generally but increases learning time significantly.

TABLE 6.2.2 Accuracy results for ID3 and CART

Data set name	ID3	CART	FSCART	Significance
Breast	94.111.24	94.851.44	94.651.23
Bupa	62.265.33	61.743.38	61.502.48
Car	80.971.26	83.842.03	78.347.40	1>3,2>>3
Cylinder	68.502.22	59.524.05	N/A	1>2
Dermatology	92.842.37	80.874.56	83.727.66	1>>2
Ecoli	78.103.57	74.743.80	76.903.89	1>>3
Flare	85.262.03	81.553.60	85.753.89
Glass	60.655.97	53.934.20	58.136.58
Hepatitis	78.443.71	78.964.04	78.583.72
Horse	87.551.98	76.963.02	N/A	1>>2
Iris	93.872.75	89.334.44	90.404.48
Ironosphere	87.633.15	86.844.03	84.782.78	1>3
Monks	92.2710.15	91.206.89	82.878.09
Mushroom	99.700.06	93.451.75	N/A	1>>2
Ocrdigits	78.401.47	81.352.08	N/A
Pendigits	85.731.01	87.102.91	91.470.86	3>>>1
Segment	91.081.16	88.071.69	92.461.71	3>2
Vote	94.941.06	90.303.17	90.443.88	1>>2
Wine	88.653.72	87.304.40	93.033.62
Zoo	92.064.80	69.929.69	69.338.93	1>>2,1>>3

TABLE 6.2.3 Node results for ID3 and CART

Data set name	ID3	CART	FSCART	Significance
Breast	17.002.11	11.602.67	10.802.39	1>2,1>3
Bupa	53.405.48	43.203.82	40.604.20	1>2,1>>3
Car	25.400.70	29.003.40	30.004.14
Cylinder	54.105.90	45.004.90	N/A
Dermatology	20.402.67	28.004.74	20.803.46	3>>1
Ecoli	33.802.70	34.005.01	31.403.24
Flare	37.904.51	33.806.20	25.809.48	1>>2
Glass	38.205.90	42.404.12	38.204.34
Hepatitis	19.603.78	14.003.43	11.601.90	1>>3
Horse	55.805.92	28.005.19	N/A	1>>2
Iris	8.401.35	10.202.35	8.201.40
Ironosphere	19.203.05	16.403.78	16.003.68
Monks	25.4013.53	17.8010.16	11.402.27	1>2,1>>>3
Mushroom	23.000.00	43.006.53	N/A	2>>>1
Ocrdigits	74.404.01	70.803.98	N/A	1>>2
Pendigits	81.805.51	77.8010.08	71.005.16
Segment	41.803.79	45.208.97	36.804.57
Vote	18.203.16	17.205.29	18.205.75
Wine	10.401.35	9.402.27	9.001.33
Zoo	15.001.89	25.204.94	16.402.32	2>>1,2>>>3

TABLE 6.2.4 Learning time results for ID3 and CART

Data set name	ID3	CART	FSCART	Significance
Breast	20	10717	482122	3>>>2>>>1
Bupa	31	25223	829119	3>>>2>>>1
Car	50	1178148	130561661	3>>>2>>>1
Cylinder	102	4589343	N/A	2>>>1
Dermatology	30	858170	105531748	3>>>2>>>1
Ecoli	30	22125	85976	3>>>2>>>1
Flare	20	1032203	88923203	3>>2>>>1
Glass	30	32025	1481154	3>>>2>>>1
Hepatitis	10	20947	1709265	3>>>2>>>1
Horse	40	34811101	N/A	2>>1
Iris	00	3111	6917	3>2>>1
Ironosphere	397	54494	86641662	3>>>2>>>1
Monks	21	12661	27355	3>>>2>>1
Mushroom	11333	336132942	N/A	2>>>1
Ocrdigits	2079	9148713	N/A	2>>>1
Pendigits	47622	3311350	245441374	3>>>2>>>1
Segment	34510	1212170	9173905	3>>>2>>>1
Vote	10	805167	130583920	3>>>2>>>1
Wine	10	8426	566147	3>>>2>>>1
Zoo	10	45361	2088375	3>>>2>>>1

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