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Glen Johnson, PhD Lehman College / CUNY School of Public Health [email protected]

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Introduction to Smoothing and Spatial Regression

Glen Johnson, PhD

Lehman College / CUNY School of Public Health

Consider points distributed in space

“Pure” Point process:

Only coordinates locating some “events”.

Set of points,

S ={s1,s2, … ,sn}

Points represent locations of something that is measured.

Values of a random variable, Z, are observed for a set S of locations, such that the set of measurements are

Z(s) ={Z(s1), Z(s2), … , Z(sn)}

- _____________________
- Examples include
- location of burglaries
- location of disease cases
- location of trees, etc.

- ___________________________
- Examples include
- cases and controls (binary outcome) identified by location of residence
- Population-based count (integer outcome) tied to geographic centroids
- PCBs measured in mg/kg (continuous outcome) in soil cores taken at specific point locations

Example of a Pure Point Process: Baltimore Crime Events

Question: How to interpolate a smoothed surface that shows varying “intensity” of the points?

(source: http://www.people.fas.harvard.edu/~zhukov/spatial.html)

Kernel Density Estimation

From: Cromely and McLafferty. 2002. GIS and Public Health.

Kernel Density Estimation

Estimate “intensity” of events at regular grid points as a function of nearby observed events. General formula for any pointx is:

where xiare “observed” pointsfor i= 1, …, nlocations in the study area, k(.) is a kernel function that assigns decreasing weight to observed points as they approach the bandwidth h. Points that lie beyond the bandwidth, h, are given zero weighting.

Results from Kernel Density Smoothing in R

Kernel Density Surface of Bike Share Locations in NYC

Source: http://spatialityblog.com/2011/09/29/spatial-analysis-of-nyc-bikeshare-maps/

Examples of Values Observed at Point Locations, Z(s):

Question: How to interpolate a smoothed surface that captures variation in Z(s)?

- First, consider “deterministic” approaches to spatial interpolation:
- Deterministic models do not acknowledge uncertainty.
- Only real advantage is simplicity; good for exploratory analysis
- Several options, all with limitations. We will consider Inverse Distance Weighted (IDW) because of its common usage.

Inverse Distance Weighted Surface Interpolation

Define search parameters

Define power of distance-decay function

Illustration: Tampa Bay sediment total organic carbon

True “geostatistical” models assume the data,Z(S) = {Z(s1), Z(s2), … , Z(sn)}, are a partial realization of a random field.

Note that the set of locations S are a subset of some 2-dimensional spatial domain D, that is a subset of the real plane.

General Protocol:

Characterize properties of spatial autocorrelation through variogram modeling;

Predict values for spatial locations where no data exist, throughKriging.

A semivariogram is defined as

for distanceh between the two locations, and is estimated as for nh pairs separated by distance hj (called a “lag”).

After repeating for different lags, say j =1, … 10, the semivariance can be plotted as a function of distance.

Given any location si, all other locations are treated as within distance h if they fall within a search window defined by the direction, lag h, angular tolerance and bandwidth.

bandwidth

Adapted from Waller and Gotway. Applied Spatial Statistics for Public Health. Wiley, 2004.

Example semivariogram cloud for pairwise differences (red dots) , with the average semivariance for each lag (blue +), and a fitted semivariogram model (solid blue line)

Characteristics of a semivariogram

Range = the distance within which positive spatial autocorrelation exists

Nugget = spatial discontinuity + observation error

Sill = maximum semivariance

If the variogram form does not depend on direction, the spatial process is isotropic.

If it does depend on direction, it is anisotropic.

Multiple semi-variograms for different directions. Note changing parameter is the range.

Surface map of semivariance shows values more similar in NW-SE direction and more different in SW-NE direction.

Kriging then uses semivariogram model results to define weights used for interpolating values where no data exists.

The result is called the “Best Linear Unbiased Predictor”. The basic form is

Where the λi assign weights to neighboring values according to semivariogram modeling that defines a distance-decay relation within the range, beyond which the weight goes to zero.

- Several variations of Kriging:
- Simple (assumes known mean)
- Ordinary (assumes constant mean, though unknown) [our focus this week]
- Universal (non-stationary mean)
- Cokriging(prediction based on more than one inter-related spatial processes)
- Indicator (probability mapping based on binary variable) [you will see in the lab work]
- Block (areal prediction from point data)
- And other variations …

Example of two types of Kriging for California O3:

Ordinary Kriging (Detrended, Anisotropic)

-continuous surface

Indicator Kriging

- probability isolines

What if point locations are centroids of polygons and the value Z(si) represents aggregation within polygon i?

With polygon data, we can still define neighbors as some function of Euclidean distance between polygon centroids, as we do for point-level data,

but now we have other ways to define neighbors and their weights …

Defining spatial “Neighborhoods”

Raster or Lattice:

Rook

Queen

- 1st order

Queen

- 2nd order

- Spatial Regression Modeling as a method for both
- assessing the effects of covariates and…
- smoothing a response variable