3. Spatial Point Patterns

Question 1

Point processes

Answer 1

Stochastic processes associated with the location of individuals in space and the occurrence of events in time

Question 2

Point patterns

Answer 2

The partial realizations of point processes

Question 3

What are the three types of point patterns

Answer 3

1. completely random
2. regular
3. aggregated

Question 4

What does a completely random point pattern look like?

Answer 4

Points are close to each other in some portions of the plots, are farther away in other parts, and can have parts of constant distance. No consistent pattern.

Question 5

What does a regular point pattern look like?

Answer 5

Points are regularly spaced. No point has a very close or very far nearest neighbor.

Question 6

What does an aggregated point pattern look like?

Answer 6

Points are grouped at small distances from each other in separate portions of the plot

Question 7

What are the two approaches in analyzing point patterns?

Answer 7

1. Count based approach
2. Distance based approach

Question 8

What is the variance-to-mean ratio for a completely random point pattern?

Answer 8

1

Question 9

What is the variance-to-mean ratio for a regular point pattern?

Answer 9

<1

Question 10

What is the variance-to-mean ratio for an aggregated point pattern?

Answer 10

> 1

Question 11

What does the histogram of the frequency distribution of counts of a completely random point pattern look like?

Answer 11

The frequency distribution of counts decreases linearly from 0 to 4. The highest bar corresponds to 0 counts.

Question 12

What does the histogram of the frequency distribution of counts of a regular point pattern look like?

Answer 12

The frequency distribution of counts follow a bell curve with 1 being at the peak. Bell curve is over 0-3.

Question 13

What does the histogram of the frequency distribution of counts of a aggregated point pattern look like?

Answer 13

The highest bar is 0 counts, by far. There is a gap in the histogram. There are counts in the high numbers like 7 and 8.

Question 14

What type of distance is used for the distance-based approach?

Answer 14

Euclidean

Question 15

What are the three types of distances that can be analyzed in the distance based approach?

Answer 15

1. all distances
2. nearest-neighbor distances
3. distances between sampling location and nearest point for a given sample design

Question 16

What is the issue when analyzing a histogram showing the frequencies of all distances?

Answer 16

It is nearly impossible to distinguish the completely random point pattern from the regular point pattern

Question 17

Can the all distances be used to identify an aggregated point pattern?

Answer 17

Yes, we clearly see a peak in the small distances (near 0), and a second peak in the higher numbers, corresponding to the average distance between the clusters.

Question 18

What does the histogram of the frequency distribution of the nearest neighbors of a completely random point pattern look like?

Answer 18

Frequencies are evenly distributed over the first three classes

Question 19

What does the histogram of the frequency distribution of the nearest neighbors of a regular point pattern look like?

Answer 19

Frequency distribution is belled-shaped. The range of nearest neighbors is narrower than for the completely random point pattern.

Question 20

What does the histogram of the frequency distribution of the nearest neighbors of an aggregated point pattern look like?

Answer 20

The frequency distribution is dominated by the first class

Question 21

What does the histogram of the frequency distribution of the distances between sampling location and nearest point of a completely random point pattern look like?

Answer 21

The frequency distribution is skewed to the right and bimodal with a main peak and a secondary peak

Question 22

What does the histogram of the frequency distribution of the distances between sampling location and nearest point of a regular point pattern look like?

Answer 22

The frequency distribution is bell-shaped over a narrow range of distances

Question 23

What does the histogram of the frequency distribution of the distances between sampling location and nearest point of an aggregated point pattern look like?

Answer 23

Frequencies are lower and more evenly distributed. The range of distribution is wide.

Question 24

What can we do to make the differences more apparent in a graph that would cover the whole range of nearest-neighbor distances?

Answer 24

Diggle’s randomization testing procedure

Question 25

How does Diggle’s randomization testing procedure work?

Answer 25

Combining the cumulative relative frequency distribution of nearest-neighbor distances of the observed point pattern, with lower and upper envelopes obtained by simulating 99 independent partial realizations of a completely random process with same number of points on a plot with same area and perimeter.

Question 26

What does the envelope in Diggle’s randomization represent?

Answer 26

They are the lower and upper bounds of an approximate 99% acceptance region.

Question 27

What does it mean if the cumulative relative frequency distribution line is within the envelope in Diggle’s randomization?

Answer 27

It means that it is a completely random point pattern

Question 28

What does it mean if the cumulative relative frequency is below the envelopes?

Answer 28

This indicates a lack of small nearest-neighbor distances, so a regular point pattern.

Question 29

What does it mean if the cumulative relative frequency is above the envelopes?

Answer 29

This indicates an excess of small nearest-neighbor distances, so an aggregated point pattern.

Question 30

Which types of distances can Diggle’s randomization be used with?

Answer 30

• nearest-neighbors distances
• distances between sampling location and nearest point for a given sample design

Question 31

What is the first order intensity function used to find?

Answer 31

The expected number of points in a quadrat E[N(A)]

Question 32

What is the second order intensity function used to find?

Answer 32

The variance and the covariance.

Question 33

Shape of the intensity function for a completely random point process

Answer 33

Rectangle

Question 34

Shape of the intensity function for a regular point process

Answer 34

Bell curve

Question 35

Shape of the intensity function for an aggregated point process

Answer 35

Multiple bell curves

Question 36

What is the epicenter of an earthquake measured in?

Answer 36

2D: latitude, longitude

Question 37

What is the hypocenter of an earthquake measured in?

Answer 37

3D: latitude, longitude, depth

Question 38

What is the number of Euclidean distances between n points?

Answer 38

[ n(n-1) ] / 2

Question 39

How many nearest neighbor distances are there between n points?

Answer 39

n

Question 40

What is a histogram?

Answer 40

A frequency plot for continuous quantitative data