the EM algorithm is used since it has reasonable fast computational time when compared to other algorithms. The EM algorithm finds the optimum model parameters by iteratively refining GMM parameters to increase the likelihood of the estimated model for the given fault feature modal vector. For the EM equations for training a GMM, the reader is referred to [19]. Fault detection or diagnosis using this classifier is then achieved by computing the likelihood of the unknown modal data of the different fault models. This likelihood is given by [18] 

 $\hat{\mathbf{s}}=\arg\max_{\|\mathbf{f}\|\leq\mathbf{F}}\sum_{\mathbf{k}=\mathbf{l}}^{\mathbf{k}}\log\mathbf{p}(\mathbf{x}_{\mathbf{k}}\,|\,\lambda_{\mathbf{f}}\,)$  where $\mathbf{F}$ represent the number of faults. 
K
 (14) 
where, F, represent the number of faults to be diagonalized, X x{ x, **,...,** x } = 1 2 K
 is the unknown D-dimension fault modal data and p(xk|λf**) is the mixture density function** 
given by [18] 

$\mathrm{p}(\mathrm{x}\,|\,\lambda)=\sum_{i=1}^{M}\mathrm{w}_{i}\mathrm{p}(\mathrm{x})$  with,  $\mathrm{p}_{i}(\mathrm{x}_{i})=\frac{1}{\left(2\pi\right)^{D/2}}\sqrt{\sum_{i}}\exp\{-\frac{1}{2}\left(\mathrm{x}_{k}-\mu_{i}\right)^{T}\left(\sum_{i}\right)^{-1}\left(\mathrm{x}_{k}-\mu_{i}\right)\}$
It should be noted that the mixture weights, wi**, satisfy the constrains,** w 1

$\sum_{\mathbf{v}}^{\mathbf{M}}\mathbf{w}_{\mathbf{v}}$
$$(14)$$
$$(15)$$
$$(16)$$
i 1
∑ i =
=
$_{\text{in}}\;=\;1$ . 

## 5. Input Data

This section describes the inputs that are used to test the SVM, MLP and GMM. When modal analysis is used for fault classification it is often found that there are more parameters extracted from the vibration data than can be possibly used for MLP, SVM and GMM training. These data must therefore be reduced because of a phenomenon called the curse of dimensionality [18], which refers to the difficulties associated with the feasibility of density estimation in many dimensions. However, this reduction process must be conducted such that the loss of essential information is minimized. The techniques implemented in this paper to reduce the dimension of the input data remove parts of the data that do not contribute significantly to the dynamics of the system being analysed or those that are too sensitive to irrelevant parameters. To achieve this, we implement the principal component analysis, which is discussed in the next section. 

## 5.1 Principal Component Analysis

In this paper we use the principal component analysis (PCA) [20;21] to reduce the input data into independent input data. The PCA orthogonalizes the components of the input vector so that they are uncorrelated with each other. In the PCA, correlations and interactions among variables in the data are summarised in terms of a small number of underlying factors. 

## 6. Foundations Of Dynamics

As indicated earlier, in this paper modal properties i.e. natural frequencies and mode shapes are extracted from the measured vibration data and used for fault classification. For this reason the foundation of these parameters are described in this section. All elastic structures may be described the time domain as [22] 