In this paper, we address fundamental issues in BPM-based observations and present methods to analyze beam dynamics in an accelerator. The analysis methods do not rely on any particular machine model, and therefore are referred to as Model Independent Analysis (MIA). There are two major parts in MIA. One is noise reduction and degree-of-freedom analysis using a singular value decomposition of BPM-reading matrix. The other is a physical base decomposition of the BPM-reading matrix based on the time structure of pulse-by-pulse beam and/or machine parameters, The combination of these two methods allow one to break the resolution limit set by individual BPMs and observe beam dynamics at more accurate levels.