Spatial Pattern: Smoothing and Trend Surface Analysis

This module combines some previous ideas for a new application
Search for spatial pattern (module 2): aids in helping us to understand the geographic structure of our datasets
Analysis by multiple regression (module 4): aids in generating models to generalize our models

A couple of key questions that we
can address from this combined perspective
1. Can we �cut through the noise� to focus on the major trends in a map?
Perhaps not important to represent every local variation in our model: major features only
2. Can we build a model that represents the spatial patterns that we observe?

Smoothing can be thought of as simply �getting rid of the noise�
A Simple Example: Smoothing a Time Series
Although short-term variations can exist in a time series, they may be much less important
than the longer-term trends reflected in the dataset

Basic Methods
1. Interpolation: �filling in the gaps�
Could simply do a visual estimate in a very simple situation (subjective)
Better solution: mathematical interpolation, such as using an �inverse
squared model� (objective result)
Idea of mathematical interpolation is to calculate estimates based on data values
at other nearby points

Spatial Smoothing
Problems with two dimensional filter mapping
1. how do you move the window?
2. what window size and shape do you use?
3. what do you do at the edges of your study area?

Spatial smoothing is a very useful analytical method
But for the results to be meaningful for your study, you must think through your options


Trend Surface Analysis
Basic Concepts
TSA is a technique for combining modeling with smoothing
TSA is really just multiple regression with location coordinates (longitude, latitude)
as the independent variables
The variable of interest for your research (elevation, temperature, income, average age, etc.) is the dependent variable

Typically start with a linear, or first order, model (a plane)
Elevation a b x b y 1 2 = + +
Could next consider a second order model (a quadratic surface) if needed
Could continue to higher order models (cubic models and on, only if necessary)

Straightforward technique and application, but several potential problems, particularly
with higher order models (i.e. cubic and higher equations)
Surfaces can become �unstable� at map edges: the edge effect

Other complications
Watch out for numerical problems, particularly with some coordinate systems (e.g. UTM)
Can be helpful to standardize coordinates
Important point: make sure you don�t mix coordinate systems within a single model (one variable measured in one coordinate system, a second variable in a different coordinate system)


Residuals are likely to show spatial autocorrelation
Remember what we said earlier in the course: this is not always a �problem� for geographers

What�s �exciting� about studying spatial autocorrelation with residuals in TSA?
When we uncover spatial autocorrelation we are really finding some clues that can help us
build even better spatial models
Essentially, spatial autocorrelation is often an input to new theory generation (a good thing)

Let�s use an example to see how this residual analysis can really help
Case study: trend surface analysis of settlement development in Pennsylvania
Goal: generate a spatial model of when towns and cities were established across the state
This entire analysis (basic TSA model + follow-up examination of residuals) can
provide real insight into the processes that guided city establishment in PA

Smoothing and Trend Surface
Quantitative methods and applications in GIS
Spatial analysis