
$$E_{t}=\sum_{t:H_{i}(x_{i})\neq y_{i}}D_{t}(i)$$
= **(11)** 
$$H_{t}=\operatorname*{arg\,max}_{y\in Y}\sum_{k=1}^{K}\sum_{t:H_{t}(x)=y}\log(1/\beta_{t})\tag{14}$$
1 ξ ξ γ (16)  ∑= = C c c c c
$$1.5$$

$$\langle\mathbf{l}\mid\mathbf{l}$$

the accuracy of the results but the confidence of the system on its own decision. 

If the error is greater than 0.5, the current hypothesis is discarded and the new training and testing data are selected according to the distribution DT. **Otherwise,** 
if the error is less than 0.5, the normalized error of the composite hypothesis is computed as: 

$\eqref{eq:walpha}$ . 

## 10 Experimental Results For On-Line Learning

The first experiment evaluates the incremental capability of the algorithm using first level fault diagnosis. The data used were collected from bushing over a period of 2.5 years from bushings in services. The algorithm is implemented with 1500 training examples and 4000 validation examples. The training data were divided into five databases each with 300 training instances. In each training session, Learn++ 
is provided with each database and 20 hypotheses are generated. The weakLearner uses an MLP with 10 input layer neurons, 5 hidden layer neurons and one output layer neuron. To ensure that the method retains previously learned data, the previous database is tested at each training session. The first row of Table 3 shows the performance of the Learn++ on the training data for different databases. On average, the weakLearner gives 60% **classification rate on its** 
training dataset, which improves to 98% when the hypotheses are combined. This demonstrates the performance improvement of Learn++ as inherited from AdaBoost on a single database. Fig. 5 shows the performance of Learn++ on training dataset against the number of classifiers for a single database. 

Each column shows the performance of current and previous databases and this is to show that Learn ++ 
does not forget previously learned information, when new data are introduced. The last row of Table 3 shows the classifiers performances on the testing dataset, which gradually improves from 65.7% to 95.8% as new databases become available, demonstrating incremental capability of Learn++. Fig. 

6 shows the performance of Learn++ on one dataset against the number of datasets. Table 4 shows that the confidence of the framework increases as new data are introduced. 

 The second experiment was performed to evaluate whether the frameworks can accommodate new classes. The faulty data were divided into 1000 training examples and 2000 validation data, which contained all the three classes. The training data were divided into five databases, each with 200 training instances. The first and second database contained, training examples of PD and thermal faults. 

$$B_{t}=\Big\langle\Big\rangle_{1-E_{t}}$$

=1 **(12)** 
The error is used in the distribution update rule, where the weights of the correctly classified instances are reduced, consequently increasing the weights of the misclassified instances. This ensures that instances that were misclassified by the current hypothesis have a higher probability of being selected for the subsequent training set. The distribution update rule is given by 1 [| ( ) |]
1)(t i i H x y t t Bt w w i
− ≠
+ = × **(13)** 
Once the T **hypothesis is created for each database,** 
the final hypothesis is computed by combining the hypothesis using weighted majority voting given by:

4.2.Confidence measurement 
A simple procedure is used to determine the 
confidence of the algorithm on its own decision. A vast majority of hypothesis agreeing on a given 
instances can be interpreted as an algorithm having
confidence on the decision. Let us assume that a total 
of T hypothesis are generated in k **training sessions** 
for a C-class problem. For any given example, the final classification class, if the total vote class c 
receives is given by [21][22]: 
$$\xi_{c}=\sum_{t:h_{t}(x)=c}\psi_{t}$$
ξ ψ **(15)** 
where ψt denotes the voting weights of the tth, hypothesis ht. Normalizing **the votes received by each** 
class gives: 
γc **can be interpreted as a measure of confidence on a** 
scale of 0 to 1. A high value of γc **shows high** 
confidence in the decision and consequently, a low value of γc **shows low confidence in the decision. It** 
should be noted that the γc **value does not represent** 