This thesis uses data-driven techniques to analyse and assess both point and probability forecasts within a prequential framework. Point forecasts are assessed using recursive residuals. Examination of the properties of the recursive residual found them to be unique to this residual. Recursive residuals for the hidden state of HMM are also defined by taking the difference between the one step ahead forecast and the forecast's filtered update. The quality of forecasts generated from different models can be assessed by comparing the information content in their corresponding residuals. When faced with model to correct this misspecification it is shown how this residual can be modelled to correct this misspecification, thereby improving forecasts. It is also shown how the residual content can be used to judge the predictive sufficiency of alternative forecasting methods. Using the theory of probability forecasting, the technique of forecasting assessment by calibration is extended to HMM's to assess how well the one step ahead forecast is explained by its filtered update. A test statistic to test the empirical calibration of the forecasts is also defined and applied to the real world problem of CpG island detection in Human DNA sequences. The distribution of the test statistic is investigated using a prequential frame of reference and is found to be N(0.1). Calibration of HMMs is also examined using smoothed predictions and cross- validation forecasts. A test statistic is defined for this scenario and the its distribution is examined using a cross- validation frame of reference. A prequential estimation algorithm for HMMs is also developed.

Exponential smoothing algorithms are very attractive for the practical world such as in industry. When considering bivariate exponential smoothing methods, in addition to the properties of univariate methods, additional properties give insight to relationships between the two components of a process, and also to the overall structure of the model. It is important to study these properties, but even with the merits the bivariate exponential smoothing algorithms have, exponential smoothing algorithms are nonstatistical/nonstochastic and to study the properties within exponential smoothing may be worthless. As an alternative approach, the (bivariate) ARIMA and the structural models which are classes of statistical models, are shown to generalize the exponential smoothing algorithms. We study these properties within these classes as they will have implications on exponential smoothing algorithms. Forecast properties are studied using the state space model and the Kalman filter. Comparison of ARIMA and structural model completes the study. / Mathematical Sciences / M. Sc. (Statistics)

Exponential smoothing algorithms are very attractive for the practical world such as in industry. When considering bivariate exponential smoothing methods, in addition to the properties of univariate methods, additional properties give insight to relationships between the two components of a process, and also to the overall structure of the model. It is important to study these properties, but even with the merits the bivariate exponential smoothing algorithms have, exponential smoothing algorithms are nonstatistical/nonstochastic and to study the properties within exponential smoothing may be worthless. As an alternative approach, the (bivariate) ARIMA and the structural models which are classes of statistical models, are shown to generalize the exponential smoothing algorithms. We study these properties within these classes as they will have implications on exponential smoothing algorithms. Forecast properties are studied using the state space model and the Kalman filter. Comparison of ARIMA and structural model completes the study. / Mathematical Sciences / M. Sc. (Statistics)