The purpose of this paper is to study nth order integral equations. The integrals studied in this paper are of the Riemann type.

<p> The very sophisticated control systems of today are built around computers. It is felt that an improved form of cost function in vector or matrix form is needed to fully and most easily utilize the computer's advantages. After defining a vector cost function G , the problem of adapting and learning simplifies to the solution of a partial difference equation. Total system properties are easily defined as matrix arrays which are "learned" in an adapting and "learning" control loop.</p> <p> The relative merits of open and closed loop adaptive systems were investigated. The Nth order adaptive control system was finally chosen to be closed loop after developing two criterion equations in two unknowns which, if satisfied guaranteed improved system sensitivity with the closed loop configuration.</p> <p> Finally, several simple examples are given in experiment form to demonstrate the applicability of the proposed control system techniques.</p> / Thesis / Master of Engineering (MEngr)

Time series data mining is one branch of data mining. Time series analysis and prediction have always played an important role in human activities and natural sciences. A Pseudo-Periodical time series has a complex structure, with fluctuations and frequencies of the times series changing over time. Currently, Pseudo-Periodicity of time series brings new properties and challenges to time series analysis and prediction. This thesis proposes two original computational approaches for time series analysis and prediction: Moving Average of nth-order Difference (MANoD) and Series Features Extraction (SFE). Based on data-driven methods, the two original approaches open new insights in time series analysis and prediction contributing with new feature detection techniques. The proposed algorithms can reveal hidden patterns based on the characteristics of time series, and they can be applied for predicting forthcoming events. This thesis also presents the evaluation results of proposed algorithms on various pseudo-periodical time series, and compares the predicting results with classical time series prediction methods. The results of the original approaches applied to real world and synthetic time series are very good and show that the contributions open promising research directions.

Time series data mining is one branch of data mining. Time series analysis and prediction have always played an important role in human activities and natural sciences. A Pseudo-Periodical time series has a complex structure, with fluctuations and frequencies of the times series changing over time. Currently, Pseudo-Periodicity of time series brings new properties and challenges to time series analysis and prediction. This thesis proposes two original computational approaches for time series analysis and prediction: Moving Average of nth-order Difference (MANoD) and Series Features Extraction (SFE). Based on data-driven methods, the two original approaches open new insights in time series analysis and prediction contributing with new feature detection techniques. The proposed algorithms can reveal hidden patterns based on the characteristics of time series, and they can be applied for predicting forthcoming events. This thesis also presents the evaluation results of proposed algorithms on various pseudo-periodical time series, and compares the predicting results with classical time series prediction methods. The results of the original approaches applied to real world and synthetic time series are very good and show that the contributions open promising research directions.