Decades ago in the U.S., clear lines delineated which neighborhoods were acceptable for certain people and which were not. Techniques such as steering and biased mortgage practices continue to perpetuate a segregated outcome for many residents. In contrast, ethnic enclaves and age restricted communities are viewed as voluntary segregation based on cultural and social amenities. This diversity surrounding the causes of segregation are not just region-wide characteristics, but can vary within a region. Local segregation analysis aims to uncover this local variation, and hence open the door to policy solutions not visible at the global scale. The centralization index, originally introduced as a global measure of segregation focused on spatial concentration of two population groups relative a region's urban center, has lost relevancy in recent decades as regions have become polycentric, and the index's magnitude is sensitive to the particular point chosen as the center. These attributes, which make it a poor global measure, are leveraged here to repurpose the index as a local measure. The index's ability to differentiate minority from majority segregation, and its focus on a particular location within a region make it an ideal local segregation index. Based on the local centralization index for two groups, a local multigroup variation is defined, and a local space-time redistribution index is presented capturing change in concentration of a single population group over two time periods. Permutation based inference approaches are used to test the statistical significance of measured index values. Applications to the Phoenix, Arizona metropolitan area show persistent cores of black and white segregation over the years 1990, 2000 and 2010, and a trend of white segregated neighborhoods increasing at a faster rate than black. An analysis of the Phoenix area's recently opened light rail system shows that its 28 stations are located in areas of significant white, black and Hispanic segregation, and there is a clear concentration of renters over owners around most stations. There is little indication of statistically significant change in segregation or population concentration around the stations, indicating a lack of near term impact of light rail on the region's overall demographics.