crankshaft/pg/doc/02_moran.md

Moran's I

What is Moran's I and why is it significant for CartoDB?

Moran's I is a geostatistical calculation which gives a measure of the global clustering and presence of outliers within the geographies in a map. Here global means over all of the geographies in a dataset. Imagine mapping the incidence rates of cancer in neighborhoods of a city. If there were areas covering several neighborhoods with abnormally low rates of cancer, those areas are positively spatially correlated with one another and would be considered a cluster. If there was a single neighborhood with a high rate but with all neighbors on average having a low rate, it would be considered a spatial outlier.

While Moran's I gives a global snapshot, there are local indicators for clustering called Local Indicators of Spatial Autocorrelation. Clustering is a process related to autocorrelation -- i.e., a process that compares a geography's attribute to the attribute in neighbor geographies.

For the example of cancer rates in neighborhoods, since these neighborhoods have a high value for rate of cancer, and all of their neighbors do as well, they are designated as "High High" or simply HH. For areas with multiple neighborhoods with low rates of cancer, they are designated as "Low Low" or LL. HH and LL naturally fit into the concept of clustering and are in the correlated variables.

"Anticorrelated" geogs are in LH and HL regions -- that is, regions where a geog has a high value and it's neighbors, on average, have a low value (or vice versa). An example of this is a "gated community" or placement of a city housing project in a rich region. These deliberate developments have opposite median income as compared to the neighbors around them. They have a high (or low) value while their neighbors have a low (or high) value. They exist typically as islands, and in rare circumstances can extend as chains dividing LL or HH.

Strong policies such as rent stabilization (probably) tend to prevent the clustering of high rent areas as they integrate middle class incomes. Luxury apartment buildings, which are a kind of gated community, probably tend to skew an area's median income upwards while housing projects have the opposite effect. What are the nuggets in the analysis?

Two functions are available to compute Moran I statistics:

• cdb_moran_local computes Moran I measures, quad classification and significance values from numerial values associated to geometry entities in an input table. The geometries should be contiguous polygons When then queen w_type is used.
• cdb_moran_local_rate computes the same statistics using a ratio between numerator and denominator columns of a table.

The parameters for cdb_moran_local are:

• table name of the table that contains the data values
• attr name of the column
• signficance significance threshold for the quads values
• num_ngbrs number of neighbors to consider (default: 5)
• permutations number of random permutations for calculation of pseudo-p values (default: 99)
• geom_column number of the geometry column (default: "the_geom")
• id_col PK column of the table (default: "cartodb_id")
• w_type Weight types: can be "knn" for k-nearest neighbor weights or "queen" for contiguity based weights.

The function returns a table with the following columns:

• moran Moran's value
• quads quad classification ('HH', 'LL', 'HL', 'LH' or 'Not significant')
• significance significance value
• ids id of the corresponding record in the input table

Function cdb_moran_local_rate only differs in that the attr input parameter is substituted by numerator and denominator.