127 lines
9.1 KiB
Markdown
127 lines
9.1 KiB
Markdown
## Regression
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### Predictive geographically weighted regression (GWR)
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Predictive GWR builds a spatially-varying regression model to predict unknown values from other known values. Similar to traditional linear regression, GWR takes a linear combination of independent variables and known dependent variables to calculate the best fit of a model. The model coefficients are spatially varying (controlled by the bandwidth parameter), so the model fit varies from geometry to geometry. GWR exposes places where non-stationarity is taking places--that is, where local behavior differs from what would be seen by doing a model without spatial variation.
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#### Arguments
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| Name | Type | Description |
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|------|------|-------------|
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| subquery | TEXT | SQL query that expose the data to be analyzed (e.g., `SELECT * FROM regression_inputs`). This query must have the geometry column name (see the optional `geom_col` for default), the id column name (see `id_col`), and the dependent (`dep_var`) and independent (`ind_vars`) column names. |
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| dep_var | TEXT | Name of the dependent variable in the regression model |
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| ind_vars | TEXT[] | Text array of independent variable column names used in the model to describe the dependent variable. |
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| bw (optional) | NUMERIC | Value of bandwidth. If `NULL` then select optimal (default). |
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| fixed (optional) | BOOLEAN | True for distance based kernel function and False (default) for adaptive (nearest neighbor) kernel function. Defaults to `False`. |
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| kernel (optional)| TEXT | Type of kernel function used to weight observations. One of `gaussian`, `bisquare` (default), or `exponential`. |
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#### Returns
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| Column Name | Type | Description |
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|-------------|------|-------------|
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| coeffs | JSON | JSON object with parameter estimates for each of the dependent variables. The keys of the JSON object are the dependent variables, with values corresponding to the parameter estimate. |
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| stand_errs | JSON | Standard errors for each of the dependent variables. The keys of the JSON object are the dependent variables, with values corresponding to the respective standard errors. |
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| t_vals | JSON | T-values for each of the dependent variables. The keys of the JSON object are the dependent variable names, with values corresponding to the respective t-value. |
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| predicted | NUMERIC | predicted value of y |
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| residuals | NUMERIC | residuals of the response |
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| r_squared | NUMERIC | R-squared for the parameter fit |
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| bandwidth | NUMERIC | bandwidth value consisting of either a distance or N nearest neighbors |
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| rowid | INTEGER | row id of the original row |
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#### Example Usage
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```sql
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SELECT
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g.cartodb_id,
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g.the_geom,
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g.the_geom_webmercator,
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(gwr.coeffs->>'pctblack')::numeric as coeff_pctblack,
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(gwr.coeffs->>'pctrural')::numeric as coeff_pctrural,
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(gwr.coeffs->>'pcteld')::numeric as coeff_pcteld,
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(gwr.coeffs->>'pctpov')::numeric as coeff_pctpov,
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gwr.residuals
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FROM cdb_crankshaft.CDB_GWR_Predict('select * from g_utm'::text,
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'pctbach'::text,
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Array['pctblack', 'pctrural', 'pcteld', 'pctpov']) As gwr
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JOIN g_utm as g
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on g.cartodb_id = gwr.rowid
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```
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Note: See [PostgreSQL syntax for parsing JSON objects](https://www.postgresql.org/docs/9.5/static/functions-json.html).
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### Geographically weighted regression model estimation
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Similar to the prediction-based GWR, this analysis generates the model coefficients for a spatially-varying regression. The model coefficients, along with their respective statistics, allow one to make inferences or describe a dependent variable based on the independent variables that make up the model.
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#### Arguments
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| Name | Type | Description |
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|------|------|-------------|
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| subquery | TEXT | SQL query that expose the data to be analyzed (e.g., `SELECT * FROM regression_inputs`). This query must have the geometry column name (see the optional `geom_col` for default), the id column name (see `id_col`), dependent and independent column names. |
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| dep_var | TEXT | name of the dependent variable in the regression model |
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| ind_vars | TEXT[] | Text array of independent variables used in the model to describe the dependent variable |
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| bw (optional) | NUMERIC | Value of bandwidth. If `NULL` then select optimal (default). |
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| fixed (optional) | BOOLEAN | True for distance based kernel function and False for adaptive (nearest neighbor) kernel function (default). Defaults to false. |
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| kernel | TEXT | Type of kernel function used to weight observations. One of `gaussian`, `bisquare` (default), or `exponential`. |
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#### Returns
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| Column Name | Type | Description |
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|-------------|------|-------------|
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| coeffs | JSON | JSON object with parameter estimates for each of the dependent variables. The keys of the JSON object are the dependent variables, with values corresponding to the parameter estimate. |
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| stand_errs | JSON | Standard errors for each of the dependent variables. The keys of the JSON object are the dependent variables, with values corresponding to the respective standard errors. |
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| t_vals | JSON | T-values for each of the dependent variables. The keys of the JSON object are the dependent variable names, with values corresponding to the respective t-value. |
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| predicted | NUMERIC | predicted value of y |
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| residuals | NUMERIC | residuals of the response |
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| r_squared | NUMERIC | R-squared for the parameter fit |
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| bandwidth | NUMERIC | bandwidth value consisting of either a distance or N nearest neighbors |
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| rowid | INTEGER | row id of the original row |
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#### Example Usage
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```sql
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SELECT
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g.cartodb_id,
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g.the_geom,
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g.the_geom_webmercator,
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(gwr.coeffs->>'pctblack')::numeric as coeff_pctblack,
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(gwr.coeffs->>'pctrural')::numeric as coeff_pctrural,
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(gwr.coeffs->>'pcteld')::numeric as coeff_pcteld,
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(gwr.coeffs->>'pctpov')::numeric as coeff_pctpov,
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gwr.residuals
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FROM cdb_crankshaft.CDB_GWR('select * from g_utm'::text, 'pctbach'::text, Array['pctblack', 'pctrural', 'pcteld', 'pctpov']) As gwr
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JOIN g_utm as g
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on g.cartodb_id = gwr.rowid
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```
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Note: See [PostgreSQL syntax for parsing JSON objects](https://www.postgresql.org/docs/9.5/static/functions-json.html).
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## Advanced reading
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* Fotheringham, A. Stewart, Chris Brunsdon, and Martin Charlton. 2002. Geographically Weighted Regression: The Analysis of Spatially Varying Relationships. John Wiley & Sons. <http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471496162.html>
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* Brunsdon, Chris, A. Stewart Fotheringham, and Martin E. Charlton. 1996. "Geographically Weighted Regression: A Method for Exploring Spatial Nonstationarity." Geographical Analysis 28 (4): 281–98. <http://onlinelibrary.wiley.com/doi/10.1111/j.1538-4632.1996.tb00936.x/abstract>
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* Brunsdon, Chris, Stewart Fotheringham, and Martin Charlton. 1998. "Geographically Weighted Regression." Journal of the Royal Statistical Society: Series D (The Statistician) 47 (3): 431–43. <http://onlinelibrary.wiley.com/doi/10.1111/1467-9884.00145/abstract>
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* Fotheringham, A. S., M. E. Charlton, and C. Brunsdon. 1998. "Geographically Weighted Regression: A Natural Evolution of the Expansion Method for Spatial Data Analysis." Environment and Planning A 30 (11): 1905–27. doi:10.1068/a301905. <https://www.researchgate.net/publication/23538637_Geographically_Weighted_Regression_A_Natural_Evolution_Of_The_Expansion_Method_for_Spatial_Data_Analysis>
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### GWR for prediction
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* Harris, P., A. S. Fotheringham, R. Crespo, and M. Charlton. 2010. "The Use of Geographically Weighted Regression for Spatial Prediction: An Evaluation of Models Using Simulated Data Sets." Mathematical Geosciences 42 (6): 657–80. doi:10.1007/s11004-010-9284-7. <https://www.researchgate.net/publication/225757830_The_Use_of_Geographically_Weighted_Regression_for_Spatial_Prediction_An_Evaluation_of_Models_Using_Simulated_Data_Sets>
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### GWR in application
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* Cahill, Meagan, and Gordon Mulligan. 2007. "Using Geographically Weighted Regression to Explore Local Crime Patterns." Social Science Computer Review 25 (2): 174–93. doi:10.1177/0894439307298925. <http://isites.harvard.edu/fs/docs/icb.topic923297.files/174.pdf>
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* Gilbert, Angela, and Jayajit Chakraborty. 2011. "Using Geographically Weighted Regression for Environmental Justice Analysis: Cumulative Cancer Risks from Air Toxics in Florida." Social Science Research 40 (1): 273–86. doi:10.1016/j.ssresearch.2010.08.006. <http://scholarcommons.usf.edu/cgi/viewcontent.cgi?article=2985&context=etd>
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* Ali, Kamar, Mark D. Partridge, and M. Rose Olfert. 2007. "Can Geographically Weighted Regressions Improve Regional Analysis and Policy Making?" International Regional Science Review 30 (3): 300–329. doi:10.1177/0160017607301609. <https://www.researchgate.net/publication/249682503_Can_Geographically_Weighted_Regressions_Improve_Regional_Analysis_and_Policy_Making>
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* Lu, Binbin, Martin Charlton, and A. Stewart Fotheringhama. 2011. "Geographically Weighted Regression Using a Non-Euclidean Distance Metric with a Study on London House Price Data." Procedia Environmental Sciences, Spatial Statistics 2011: Mapping Global Change, 7: 92–97. doi:10.1016/j.proenv.2011.07.017. <https://www.researchgate.net/publication/261960122_Geographically_weighted_regression_with_a_non-Euclidean_distance_metric_A_case_study_using_hedonic_house_price_data>
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