## Lecture 10 - Spatial Arrangement

OUTLINE:

1. Point Patterns
2. Analysis Procedures
3. Area Patterns
4. Linear Patterns

Definition

Spatial Arrangement: "placement, ordering, concentration, connectedness or dispersion of multiple objects within a confined geographic space" (De Meer, 1997)

I) Point Patterns

a) Point Density: The number of points per unit area.

Common measures of density are population, housing and trees. Useful for comparison between areas and over time.

b) Point Distribution: the pattern of points in space. Four patterns are possible:

1. uniform: point density for each subarea is the same as the point density in other subareas.
2. regular: points are spaced at a constant distance throughout a region.
3. random: points are scattered throughout a region with no predetermined pattern.
4. clustered: points are grouped together in numerous bunches.

II) Analysis Procedures

a) Quadrat analysis: uniform patterns are defined through the analysis of regularly spaced sub-regions called quadrats. Assumption is that a uniform number of points will be found in each quadrat. The expected value (E) occurring in each quadrat is the total number of points divided by the total number of quadrats. The Sum of the squared difference between the observed number of points (O) and the expected number of point (E) divided by (E) yields a X 2 value that can be used to test whether or not the distribution is uniform.

X2 = Sum of (O – E)2 /E

Large X2 values indicate that the distribution is not uniform and that there may be some underlying process causing the non-uniformity.

b) Variance-mean index: relationship between frequency of subarea variability and the average number of points in each subarea (VMR = var/mean). The resultant can also be used as a X2 value to test for significance.

c) Nearest neighbor analysis: analysis of the distance between neighbors by calculating the mean nearest neighbor distance between each possible pair of near neighbor points. Comparison of the NNA index can be made with respect to the regular, random and clustered pattern. This forms a continuum between:

a perfectly clustered set of points with an NNA ~ 0,

a random distribution with a NNA of (1/(2D1/2)) and

a perfectly dispersed pattern with an NNA of (1.07453/D1/2), where D = point density

III) Area Patterns

Areas of polygons of a particular type are a measure as a percentage of the total area of the region being studied.

Join count statistics can be generated to determine the frequency that polygons with certain characteristics share boundaries. This count statistic can be compared to random frequencies of adjacency to determine whether or not co-occurrence of polygons with these characteristics is greater than that which can be attributed to chance.

IV) Linear Patterns

Analysis of line patterns has included line spacing, distribution of line length, and line orientation.

Connectivity of linear features: An important aspect of linear features is their ability to form networks. A network is a series of connected linear features through which a "resource" can flow (i.e. cars, people, data, ... etc.).

Measuring Connectivity: the gamma index and the alpha index.

Gamma index (g ) measures linkage. It is defined by the equation:

g = L/Lmax = L/3(V-2) where V is the number of nodes. The index ranges from 0 to 1.0.

Alpha index (a ) measures circuitry, the degree to which nodes are connected by circuits or alternative routes. It is the ratio of the existing number of circuits to the maximum possible number. No circuits exist if the number of links is equal to the number of nodes - 1. A circuit is present when L > V-1. The maximum number of circuits is the maximum number of links 3(V-2) minus the minimum number of links in a minimumly connected network (V-1).

a = actual # of circuits/ max # of circuits[(L - V) + 1] / (2V - 5)

Gravity Model: In our previous discussion of circuits we treated all routes as equally likely to be taken. In reality certain nodes may be more attractive than others (city center, lake) and therefore may draw more flow toward them. A simple model for the interaction between nodes is

Lij = K (PiPj/d2)

where Lij is the interaction between nodes i and j, Pi is the magnitude of node i Pj and is the magnitude of node j, d is the distance between nodes and K is a constant that relates to the object being studied (i.e. population).

Routing and allocation

Definition: Analysis of the movement of phenomena over a network.

Very useful for allocating resources over a network (snow plows, school buses, etc.) or finding the most efficient route given distance, impedance and other constraints (turn penalties, congestion, etc.). Can also be used for traffic analysis and trip generation. We have a whole course devoted to this topic.