Alzheimer's disease~(AD) is the most frequent neurodegenerative dementia and a growing health problem. Early and accurate diagnosis and prediction of AD is crucial because treatment may be most efficacious if introduced as early as possible. Neuropsychological testing, which is clinically used, sometimes fails to recognize probable dementia, especially to recognize the disease at an early time point such as the mild cognitive impairment~(MCI), which is the prodromal stage of AD. Recently, there has been a realization that magnetic resonance imaging~(MRI) may help diagnoses of AD and MCI. In this dissertation, we introduce an MRI-analysis based algorithm to help diagnose the illness before irreversible neuronal loss has set in, and to help detect brain changes between MCI patients who may convert and may not convert to AD. Given a set of brain MR images, the algorithm first calculates the distance between each pair of images via a registration process. Then images are projected from a high dimensional Euclidean space to a low dimensional Euclidean subspace based on the calculated distances, with a dimension reduction method. Finally classical supervised classification approaches are employed to assign images to appropriate groups in the low dimensional space. The classification accuracy rates we obtained in our experiments are higher than, or at least comparable to, those reported in recently published papers. Moreover, this algorithm can be extended to explore the pathology distribution of AD. Exploring the distribution of AD pathology is of great importance to reveal AD related regional atrophy at specific stages of the disease and provide insight into longitudinal sequence of disease progression. Calculating distances between different brain structures produces different classification accuracy. Those structures yielding higher classification accuracy are considered as pathological regions. Our experimental results on pathology localization are also compared with the reproduced results using other existing popular algorithms; the observations are consistent. / Ph. D.