School of Computing and Information Systems,
Singapore Management University
24 Nov 2023
Introducing Regression Modelling
What is Spatial Non-stationary
Introducing Geographically Weighted Regression
Interpreting and Visualising
Why condominium units located at the central part of Singapore were transacted at relatively higher prices than others?
A set of statistical processes for explaining the relationships among variables.
The focus is on the relationship between a dependent variable (y) and one or more independent variables (x)
Estimate outcomes based on the relationships modelled.
The formula:
Linearity assumption. The relationship between the dependent variable and independent variables is (approximately) linear.
Normality assumption. The residual errors are assumed to be normally distributed.
Homogeneity of residuals variance. The residuals are assumed to have a constant variance (homoscedasticity).
The residuals are uncorrelated with each other.
(Optional) The errors (residuals) are normally distributed and have a 0 population mean.]
Residuals vs Fitted plot - Used to check the linear relationship assumptions. A horizontal line, without distinct patterns is an indication for a linear relationship, what is good.
The myth: - We should transform the values of the y variable when they are large.
| y | |||
| Predictors | Estimates | CI | p |
| (Intercept) | 1.98 | 1.66 – 2.30 | <0.001 |
| x | 2.35 | 2.13 – 2.57 | <0.001 |
| Observations | 100 | ||
| R2 / R2 adjusted | 0.819 / 0.817 | ||
Despite the values of the dependent variable is rather similar to the values of the independent variable, the diagnostic plot shows that the linearity assumption has been violated.
| y | |||
| Predictors | Estimates | CI | p |
| (Intercept) | 0.51 | 0.36 – 0.65 | <0.001 |
| x [exp] | 1.01 | 0.98 – 1.04 | <0.001 |
| Observations | 100 | ||
| R2 / R2 adjusted | 0.977 / 0.977 | ||
The diagnostic plot on the right shows that the linearity assumption has been conformed.
Warning
This is the test on the residual and not on the dependent variable.
The purpose of this test is to ensure the residuals of a multiple regression are time independent.
Spatial non-stationarity: the same stimulus provokes a different response in different parts of the study region.
Global models: statements about processes which are assumed to be stationary and as such are location independent.
Local models: spatial decompositions of global models, the results of local models are location dependent – a characteristic we usually anticipate from geographic (spatial) data.
The residuals are assumed to be distributed at random over geographical space.
To test if the relationships in the model are non-stationary.
Local statistical technique to analyze spatial variations in relationships.
Spatial non-stationarity is assumed and will be tested.
Based on the “First Law of Geography”: everything is related with everything else, but closer things are more related.
Package Model
Results of Global Regression
Results of Geographically Weighted Regression
SDF: A SpatialPointDataFrame
Brunsdon, C., Fotheringham, A.S., and Charlton, M. (2002) “Geographically weighted regression: A method for exploring spatial nonstationarity”. Geographical Analysis, 28: 281-289.
Brunsdon, C., Fotheringham, A.S. and Charlton, M., (1999) [“Some Notes on Parametric Significance Tests for Geographically Weighted Regression”](https://onlinelibrary-wiley-com.libproxy.smu.edu.sg/doi/abs/10.1111/0022-4146.00146. Journal of Regional Science, 39(3), 497-524.
Mennis, Jeremy (2006) “Mapping the Results of Geographically Weighted Regression”, The Cartographic Journal, Vol.43 (2), p.171-179.
Stephen A. Matthews ; Tse-Chuan Yang (2012) “Mapping the results of local statistics: Using geographically weighted regression”, Demographic Research, Vol.26, p.151-166.
