Modelling Spheres of Interaction

Interacting partners

• individual interactions
• interaction in groups
• interaction between groups

Interacting systems

• Point Patterns
• Networks
• Systems

Some terminology: in case we have an random point pattern, Complete Spatial Randomness or an Independent Random Process prevails. The condition are:

• equal probability: any event has an equal probability of being in any position
• independence: the positioning of any event is independent of the positioning of any other event

Some terminology: in case we have an random point pattern, Complete Spatial Randomness or an Independent Random Process prevails. The condition are:

• equal probability: any event has an equal probability of being in any position
• independence: the positioning of any event is independent of the positioning of any other event

CSR means that the process is random not the resulting point pattern!

In point pattern analyses we test against CSR and different forms of specific processes to learn more about our pattern.

A structured point pattern violates CSR conditions:

1. First-order effects influence the probability of events being in any position of the region --> to trace such influences we investigate the intensity function (~ density) of the points

2. In case second-order effects are present, points are not independent from one another --> to trace such influences we investigate the distance distributions of the points

• a stationary point process has a constant point density function.
• a homogeneous point process is stationary and isotropic. --> CSR
• a stationary Poisson point process is the reference model in many test for CSR

Events are independent

• a non-stationary (Poisson) point process has an inhomogeneous intensity function --> e.g. caused by a covariate.

Events are not independent

• the Cox process is an inhomogeneous Poisson process with a random intensity function. (Approach: create random pattern, create points using Poisson and covariate of random intensity)
• a Gibbs process involves influence from other points and models an explicit interaction between points (mainly for inhibition). In the case of a hard core Gibbs process, points avoid each other up to a certain threshold and they ignore each other.
• a Strauss process has a constant influence within a certain distance threshold.
• a Neyman–Scott process is used to create clustered point pattern by creating random cluster centres that create "offspring" points

Simple measures: mean, standard deviation, intensity (~ density)

mc <- cbind(sum(meg_pp$x/meg_pp$n),
            sum(meg_pp$y/meg_pp$n)
            )

stdist <- sqrt(sum((meg_pp$x-mean(meg_pp$x))^2 +
                   (meg_pp$y-mean(meg_pp$y))^2) /
               meg_pp$n
               )

plot(meg_pp)
points(mc)
library(plotrix)
draw.circle(x = mc[1], y = mc[2], radius = stdist, border = "red")

Global intensity = Number of points per area

## A = a * b
area.sqm <- diff(meg_pp$window$xrange) * diff(meg_pp$window$yrange)
area.sqkm <- area.sqm*10^-6
# area <- area/1000000
area.sqkm

## [1] 431.3529

## calculate intensity
intensity <- meg_pp$n/area.sqkm
intensity

## [1] 0.6189827

Local intensity

qc.meg <- quadratcount(X = meg_pp)

plot(qc.meg);
points(meg_pp, pch = 20, cex = .5, col = rgb(.2,.2,.2,.5))

Local intensity

Local intensity

qt_meg <- quadrat.test(meg_pp)
qt_meg

## 
##  Chi-squared test of CSR using quadrat counts
##  Pearson X2 statistic
## 
## data:  meg_pp
## X2 = 274.48, df = 24, p-value < 2.2e-16
## alternative hypothesis: two.sided
## 
## Quadrats: 5 by 5 grid of tiles

Local intensity

plot(qt_meg)

Kernel density estimation

Kernel density estimation

meg_dens <- density(x = meg_pp,
                    sigma = 2500)

plot(raster(meg_dens))

Kernel density estimation

Kernel density estimation

We assume that the intensity of the point process is a function of the covariate (Z):

\[\lambda(u) = \rho(Z(U))\]

\(\lambda(u)\) can be regarded as location selection function --> it causes an inhomogeneous probability of points to be located in same areas of the study region

Calculation in R is straightforward

cov_meg <- rhohat(object = "YOUR POINT PATTERN",
                     covariate = "YOUR COVARIATE RASTER",
                     bw = 100
                  )
pred_cov_meg <- predict(cov_meg)
cov_compare <- meg_dens - pred_cov_meg

Kernel density estimation

Kernel density estimation

Definition

• A networks are objects, in which elements (vertices) are connected by edges.

• Networks are models, mapping certain facets of the real world.

• Network theory has roots in geography and in social sciences

Network theory is based on graph theory

• Rivers, Knappett, Evans
• Cyclades in Bronze Age
• fixed radius network

• Rivers, Knappett, Evans
• distance decay function

• Rivers, Knappett, Evans
• Cyclades in Bronze Age
• entropy model using double constrains

Construction rules for some neighbourhood graphs

degree(n4)
##  [1] 5 3 6 5 5 3 6 3 5 3
closeness(n4)
##  [1] 0.07142857 0.06666667 0.08333333 0.07692308 0.07692308 0.05882353
##  [7] 0.08333333 0.06250000 0.07692308 0.06666667
betweenness(n4)
##  [1] 3.0000000 0.6666667 5.1666667 2.5000000 3.9166667 0.0000000 5.1666667
##  [8] 0.0000000 3.9166667 0.6666667
edge_betweenness(n4)
##  [1] 3.666667 2.833333 2.833333 2.000000 3.666667 4.166667 2.500000
##  [8] 3.750000 2.666667 3.500000 4.083333 2.500000 1.833333 2.833333
## [15] 3.750000 1.833333 3.083333 4.083333 3.666667 3.083333 3.500000
## [22] 4.166667

ABM comprises

• an actors
• an envirionment and
• a process

Example Heuneburg

• indigenous people
• merchants

Reasoning for certain relationships

The process

Actors can:

• move
• trade
• accumulate wealth

Some results

But is it useful?

What do you think?

Introducing some complexity

Some more results

Interpretation for the Heuneburg