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Copyright © John Lindsay, 2015

Spatial Data

John Lindsay

Fall 2015

- Jensen and Jensen Chapter 8

- Over the next two lectures, we'll discuss:
- Descriptive Statistics
- Descriptive Spatial Statistics
- Spatial Autocorrelation
- Point Pattern Analysis
- Quadrat Analysis
- Nearest-Neighbour Analysis

- Directional Analysis

- Measures of
central tendency - Mode, median, and mean (\(\overset{-}x\))
- \(\overset{-}x = \frac {\underset{i=1}{\overset{N}{\Sigma}} x} {N}\)

- Measures of
dispersion - Variance (\(s^2\))
- Standard deviation (\(s\))
- \(s^2 = \frac {\underset{i=1}{\overset{N}{\Sigma}} (x_i - \overset{-}x)^2} {N - 1}\)
- \(s = \sqrt \frac {\underset{i=1}{\overset{N}{\Sigma}} (x_i - \overset{-}x)^2} {N - 1}\)

- Skewness
- Measure of the
asymmetry of a distribution

- Measure of the
- Kurtosis
- Measure of the
peakedness of a distribution

- Measure of the

Mean Centre - Measure of central tendency that can be used to determine the centre of a distribution plotted in geographic coordinates.

Standard Distance - Measure of dispersion of geographically distributed data.

- The first law of geography: “everything is related to everything else, but near things are more related than distant things.” (Tobler, 1970)
- This simple statement forms the basis for a great deal of
geographical analysis and is concept underlying the idea of
spatial autocorrelation . - Synonymous with the concept of
spatial dependence in geostatistics

- Correlation of a variable with itself through space.
- Correlation versus spatial autocorrelation

- Actually bad news and good news
- Bad for statistical reasons
- Good because, “if geography is worth studying at all, it must be because phenomena do not vary randomly through space” (O'Sullivan and Unwin, 2003, pg. 28)
- Essential for spatial modelling through
Interpolation

- Three possibilities:
Clustered (positive autocorrelation): nearby locations are likely to be similar to one another.Random (autocorrelation near zero): no spatial effect is discernible, and observations seem to vary randomly through spaceDispersed (negative autocorrelation): observations from nearby observations are likely to be different from one another.

- Moran's \(I\) measures the interdependence in spatial distributions.
- Used with interval/ratio level data
- Used to detect spatial trends
- -1 ≤ \(I\) ≤ 1
- \(I\) = -1 = dispersed
- \(I\) = 0 = random
- \(I\) = +1 = clustered

\(I = \frac {N}{\underset{i=1}{\overset{N} \Sigma} \underset{j=1}{\overset{N} \Sigma} w_{ij}} \frac {\underset{i=1}{\overset{N} \Sigma} \underset{j=1}{\overset{N} \Sigma} w_{ij} (x_i - \overset{-} x) (x_j - \overset{-} x)}{\underset{i=1}{\overset{N} \Sigma} {(x_i - \overset{-} x)^2}}\)

Where \(\overset{-} x\) is the mean of variable \(x\); \( x_i \) is the
value at \(i\); \(j\) is a neighbour of \(i\); \( w_{ij} \) is the weight between neighbours \(i\) and \(j\).

- Mapped point data often exhibit distinct patterning.
- Patterns result from the spatial component of a control on the phenomenon.
- Understanding the pattern can help with understanding the controlling forces on the phenomenon.

- The patterns that we're interested in with
Point Pattern Analysis (PPA) result from the locations of individual points and not on their attributes, for which spatial autocorrelation is more relevant. Quadrat Analysis andNearest-Neighbour Analysis the the two most common methods for PPA

- A
quadrat is a user-defined geographic area, usually a square or rectangle, used to measure the distribution of a spatial phenomenon. Quadrat analysis can be used to test whether the phenomenon is uniformly distributed.- The
Chi Square test is used with quadrats.

- The value of Chi Square is compared with a table of critical values to determine whether the points are statistically significantly different from a uniformly distribution.
- You should be thinking about the MAUP about now!
- The size, shape, and number of quadrats will impact the results of the quadrat analysis.

- NNA is used in GIS to determine whether point sets are random or non-random.
- If a point set is found to be non-random then we are left with the task of determining what controls the distribution.
- For each point in the set, find the distance to the closest neighbour.

\(d_e = \frac 1 {2 \sqrt{N/A}} = \frac 1 {2 \sqrt{p}} \)

- where \(d_e\) is the expected density (assuming random distribution); \(N\) is the number of points; \(A\) is the study area; \(p\) is the point density.

\(NNR = \frac {Dist_{Obs}} {Dist_{Ran}} = \frac {d_a} {d_e} \)

- where \(NNR\) is the
nearest-neighbour ratio ; \(Dist_{Obs}\) is the mean NN distance; \(Dist_{Ran}\) is the expected distance for a random distribution.

- Warning: Our estimates of the point density is dependent on our definition of the study area.
- If we change the extent of the study area, we change the results.

Not so clustered | Very clustered |
---|---|

- NNA is also sensitive to the
non-uniformity of underlying space . - NNA assumes that points are free to locate anywhere.
- Consider the gap in stream channel heads below. It's the result of Lake Ontario.

- Geographers distinguish between
directional (0°-360°) andaxial (a.k.a. oriented; 0°-180°) data.- Wind is directional; a road is axial.

- Directional and axial data can be plotted using
Rose Diagrams , which are like circular histograms.

\(\overset{-}\theta = tan^{-1}(\frac{\overset{N}{\underset{i=1}{\Sigma}}{sin \theta_i}} {\overset{N}{\underset{i=1}{\Sigma}}{cos \theta_i}}) \)

- where \(\overset{-}\theta\) is the
mean direction , derived from the vector resultant.

\(\overset{-}R = \frac 1 N \sqrt{(\overset{N}{\underset{i=1}{\Sigma}}{cos \theta_i})^2 + ({\overset{N}{\underset{i=1}{\Sigma}}{sin \theta_i}})^2} \)

- where \(\overset{-}R\) is the standardized length of the vector resultant,
called the
mean resultant length , and is a measure of dispersion. - 0 ≤ \(\overset{-}R\) ≤ 1, where values near 1 indicate small angular dispersion and vice versa.

- Axial (oriented) data cannot easily be treated as vectors because there is nothing to distinguish one end of the line from the other.
- An angle of 179° is very close to one of 1°.
- To solve this double all the angles, calculate the statistics with the
doubled data, and then halve the angles to get the mean direction, mean resultant
length, etc.
- 45° × 2 = 90°
- 225° × 2 = 450° = 450° - 360° = 90°