On classical and other methods of discriminant analysis and estimation of log-odds

For two multinormal populations with equal covariance matrices the likelihood ratio discriminant function, an alternative allocation rule to the sample linear discriminant function when n1 ≠ n2 ,is studied analytically. With the assumption of a known covariance matrix its distribution is derived and the expectation of its actual and apparent error rates evaluated and compared with those of the sample linear discriminant function. This comparison indicates that the likelihood ratio allocation rule is robust to unequal sample sizes. The quadratic discriminant function is studied, its distribution reviewed and evaluation of its probabilities of misclassification discussed. For known covariance matrices the distribution of the sample quadratic discriminant function is derived. When the known covariance matrices are proportional exact expressions for the expectation of its actual and apparent error rates are obtained and evaluated. The effectiveness of the sample linear discriminant function for this case is also considered. Estimation of true log-odds for two multinormal populations with equal or unequal covariance matrices is studied. The estimative, Bayesian predictive and a kernel method are compared by evaluating their biases and mean square errors. Some algebraic expressions for these quantities are derived. With equal covariance matrices the predictive method is preferable. Where it derives this superiority is investigated by considering its performance for various levels of fixed true log-odds. It is also shown that the predictive method is sensitive to n1 ≠ n2. For unequal but proportional covariance matrices the unbiased estimative method is preferred. Product Normal kernel density estimates are used to give a kernel estimator of true log-odds. The effect of correlation in the variables with product kernels is considered. With equal covariance matrices the kernel and parametric estimators are compared by simulation. For moderately correlated variables and large dimension sizes the product kernel method is a good estimator of true log-odds.

A Delay Vector Variance based marker for an output-only assessment of structural changes in tension leg platforms

Although aspects of power generation of many offshore renewable devices are well understood, their dynamic responses under high wind and wave conditions are still to be investigated to a great detail. Output only statistical markers are important for these offshore devices, since access to the device is limited and information about the exposure conditions and the true behaviour of the devices are generally partial, limited, and vague or even absent. The markers can summarise and characterise the behaviour of these devices from their dynamic response available as time series data. The behaviour may be linear or nonlinear and consequently a marker that can track the changes in structural situations can be quite important. These markers can then be helpful in assessing the current condition of the structure and can indicate possible intervention, monitoring or assessment. This paper considers a Delay Vector Variance based marker for changes in a tension leg platform tested in an ocean wave basin for structural changes brought about by single column dampers. The approach is based on dynamic outputs of the device alone and is based on the estimation of the nonlinearity of the output signal. The advantages of the selected marker and its response with changing structural properties are discussed. The marker is observed to be important for monitoring the as- deployed structural condition and is sensitive to changes in such conditions. Influence of exposure conditions of wave loading is also discussed in this study based only on experimental data.