Test of Complete Spatial Randomness on Networks

A PROJECT SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY

Xinyue Chang

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED AND COMPUTATIONAL MATHEMATICS

YANG LI

Firstly, I would like to thank my advisor Professor Yang Li for his incredible support, guidance, and encouragement on my project and graduate study. I would also like to thank Professor Barry James and Professor Haiyang Wang for serving as my committee members and their time reading my project report. Last but not least, I am very grateful to Professor Kang James for her valuable suggestions and comments in the statistical seminar class.

i Abstract

Test of complete spatial randomness (CSR) is an essential part of spatial analysis and regarded as a minimal prerequisite to any serious attempt to model an observed point pattern. It has been investigated, discussed and veriﬁed in planar region by researchers for more than 40 years. Recently more and more data of spatial point processes on networks have been collected. This project aimed to apply CSR test method to any spatial point pattern on the network. The study started with the derivation of the cumulative distribution function (CDF) of inter-event distances between two locations randomly distributed on a grid network. We then carried out a test procedure based on Monte Carlo simulation. The procedure was proposed when considering both inter- event distances and nearest-neighbor distances. It was found that this method worked well when the process was constrained on a network. Finally, the car accident pattern on Minnesota major roads network was tested by both inter-event distances method and nearest-neighbor distances method.

ii Contents

Acknowledgements i

Abstract ii

List of Tables vi

List of Figures vii

1 Introduction 1 1.1 Background and Organization ...... 1 1.2 Deﬁnitions ...... 3 1.2.1 Complete Spatial Randomness ...... 3 1.2.2 Spatial Point Processes on Networks ...... 3 1.2.3 Inter-event Distances ...... 4 1.2.4 CSR Test Based on Inter-event Distances ...... 4 1.2.5 Nearest-neighbor Distances ...... 5 1.2.6 CSR Test Based on Nearest-neighbor Distances ...... 5

2 CDF of Inter-event Distances on a Grid Network under CSR 6 2.1 Preliminary ...... 6 2.2 CDF of Inter-event Distances if t < 1...... 9 2.2.1 Cumulative Distribution Function ...... 9 2.2.2 Simulation ...... 12 2.3 CDF of Inter-event Distances if t > 1...... 13 2.3.1 Cumulative Distribution Function ...... 13

iii 2.3.2 Simulation ...... 18

3 CSR Test Based on Inter-event Distances 20 3.1 CSR Test Implementation ...... 20 3.2 Simulations ...... 21 3.2.1 Random Process ...... 22 3.2.2 Cluster Process ...... 24 3.2.3 Regular Process ...... 25 3.2.4 Conclusion ...... 27

4 CSR Test Based on Nearest-neighbor Distances 28 4.1 CSR Test Implementation ...... 28 4.2 Simulations ...... 29 4.2.1 Random Process ...... 30 4.2.2 Cluster Process ...... 31 4.2.3 Regular Process ...... 32 4.2.4 Conclusion ...... 33

5 Car Crash Point Pattern on the Minnesota Major Roads 34 5.1 Dataset ...... 34 5.2 Implementation ...... 36 5.2.1 CSR Test Based on Inter-event Distances ...... 37 5.2.2 CSR Test Based on Nearest-neighbor Distances ...... 37 5.3 Result and Analysis ...... 37

6 Conclusion 39

References 40

Appendix A. Glossary and Acronyms 41 A.1 Glossary ...... 41 A.2 Acronyms ...... 41

iv Appendix B. Code 43 B.1 R Code ...... 43 B.1.1 Random Process ...... 43 B.1.2 Cluster Process ...... 44 B.1.3 Regular Process ...... 47 B.1.4 Random Network ...... 48 B.1.5 Car Crash Pattern on the MN Roads ...... 50 B.2 Python Code ...... 52

v List of Tables

A.1 Acronyms ...... 41

vi List of Figures

2.1 An example of a regular grid network with m = n = 11...... 7 2.2 Four possible locations of two arbitrary points on the 5 × 5 grid . . . . . 8 2.3 The 11 × 11 grid network ...... 13 2.4 Simulation result and plot for CDF when t < 1 (blue is the theoretical function) ...... 13 2.5 The 11 × 11 grid network ...... 19 2.6 Simulation result and plot for CDF when t > 1 (blue is the theoretical function) ...... 19 3.1 Random point pattern on the grid network ...... 23 3.2 Envelope plot for random process on the grid network ...... 23 3.3 Random point pattern on a random network ...... 23 3.4 Envelope plot for random process on a random network ...... 23 3.5 Cluster point pattern on the grid network ...... 24 3.6 Envelope plot for cluster process on the grid network ...... 24 3.7 Cluster point pattern on a random network ...... 25 3.8 Envelope plot for cluster process on a random network ...... 25 3.9 Regular point pattern on the grid network ...... 26 3.10 Envelope plot for regular process on the grid network ...... 26 3.11 Regular point pattern on a random network ...... 27 3.12 Envelope plot for regular process on a random network ...... 27 4.1 Grid network and random point pattern ...... 30 4.2 Envelope plot for random process on grid network ...... 30 4.3 Random network and random point pattern ...... 30 4.4 Envelope plot for random process on random network ...... 30

vii 4.5 Grid network and cluster point pattern ...... 31 4.6 Envelope plot for cluster process on grid network ...... 31 4.7 Random network and cluster point pattern ...... 31 4.8 Envelope plot for cluster process on random network ...... 31 4.9 Grid network and regular point pattern ...... 32 4.10 Envelope plot for regular process on grid network ...... 32 4.11 Random network and regular point pattern ...... 32 4.12 Envelope plot for regular process on random network ...... 32 5.1 Location of fatal crashes in Minnesota in 2013 ...... 35 5.2 R Plot of Minnesota Major Roads Network...... 35 5.3 R Plot of the car crash pattern on the Minnesota major roads...... 35 5.4 Display of the car crash pattern on the Minnesota major roads in ArcGIS 35 5.5 R plot of a CSR point pattern on the Minnesota major roads ...... 36 5.6 Display of a CSR point pattern on the Minnesota major roads in ArcGIS 36 5.7 Envelope plot for CSR test for car crash pattern by inter-event method. 38 5.8 Envelope plot for CSR test of car crash pattern by nearest-neighbor method. 38

viii Chapter 1

Introduction

1.1 Background and Organization

Practical investigation in ecology, epidemiology, and transportation often involves ob- servation and study of spatial distribution of events. Researchers are interested in the classification of a spatial point pattern and need to know if it is complete spatial randomness (CSR) in the very beginning. Then the method of testing CSR for spatial point process draws researchers more and more attention. The techniques proposed for detecting non-randomness may be divided broadly into two groups, described respectively as quadrant methods and distance methods [1]. The power of randomness tests and, particularly tests based on nearest-neighbor distances, inter-point distances and estimators of moment measures have been investigated by ar- ticle [2]. Some papers have also tried to develop some other methods besides distances and quadrants. In paper [3], the author introduced testing spatial randomness based on angles between the vectors joining each sample point to its nearest neighbors. And a method of qualifying spatial pattern where sample point move to a regular arrangement which resembles a hexagonal lattice was discussed by [4]. To explore deeper the per- formance of the CSR test, paper [5] presents results confirmed by ecological data and illustrates that tests without edge-effect correction proposed by Diggle have a higher power for small sample sizes. The assumption of all these works is that spatial point events can be located any- where on the planar region. However, spatial points can only be located on edges of

1 2 a specific network in some practical scenarios. For example, car crash locations lie on roads, which are able to form a roads network. Then the CSR test should become different and complicated in the sense that inter-event distances are not Euclidean distance any more, and have to adjust to the geometry of network. Motivated by the concern, CSR tests based on inter-event distances and nearest-neighbor distances [6] are discussed under network scenario and verified to be applicable to the network point pattern in this thesis. The result is confirmed by three point processes simulated on a grid network and random network. In terms of the test method based on the inter-event distance, it would be precise and simple enough to implement if the theoretical cumulative distribution function (CDF) of CSR were known. Not surprisingly, there are already some fancy results from the most common cases of square or circular regions. For a square of unit side, the distribution function of inter-event distances is   πt2 − 8t3/3 + t4/2 0 ≤ t ≤ 1  2 4 2 1 2 H(t) = 1/3 − 2t − t /2 + 4(t − 1) 2 (2t + 1)/3  √  +2t2 arcsin(2t−2 − 1) 1 < t ≤ 2

For a circle of unit radius the corresponding expression is

p H(t) = 1 + π−1{2(t2 − 1) arccos(t/2) − t(1 + t2/2) 1 − t2/4} for all 0 ≤ t ≤ 2 [6]. If we consider the CSR point pattern on the network, the distances relying on the geometry of network would make a diﬀerence from the case of planar region. Compared to the CDF of distances in planar space, CDF of inter-event distances for CSR point pattern on the network hence become more serious and important for our network- orientated research. To be organized, this thesis will start by presenting how to derive the CDF of inter-event distances for the CSR point pattern on a grid network.

• Chapter 2 describes how to derive the cumulative distribution function of inter- event distances for the complete spatial point pattern on grid network.

• In Chapter 3, complete spatial randomness test method based on inter-event distances is discussed and summarized for implementation. 3 • In Chapter 4, complete spatial randomness test method based on nearest-neighbor distances is discussed and summarized for implementation.

• Chapter 5 applies the proposed feasible complete spatial randomness test to the car crash pattern of 2013 on Minnesota major roads network.

• Chapter 6 presents a ﬁnal discussion and conclusion of the work presented in the project.

1.2 Deﬁnitions

1.2.1 Complete Spatial Randomness

We consider a network denoted as a graph G = {V,E}, where the set of vertices V =

{v1, v2, ··· , vl}, and the set of edges E = {e1, e2, ··· , em}. For the sake of simplicity, G is assumed to be connected, which means there is a path between any pair of vertices.

Furthermore, |ei| is the length of edge ei. Let S denote the events from a spatial points pattern on G, S = {s1, s2, ··· , sn}, constrained to be on the edges. Note: In graph theory, if the vertices v0, v1, ··· , vk of the walk W = v0e1v1e2v2 ··· ekvk are distinct, then W is called a path [7]. Speaking of the graph considered, we can say the edges are weighted representing the distance between two end vertices. The deﬁnition of complete spatial randomness (CSR) can be extended to the network case as follows [6, 8]:

• For a density λ > 0 and a ﬁnite network G, the number of events of S, say |S|, Pm must follow a Poisson distribution with mean λ|E| where |E| = i=1 |ei|.

• Given the number of events, i.e., |S| = n, events are distributed uniformly on the network G. That is to say the n events of S form an independent random sample from the continuous uniform distribution on E.

1.2.2 Spatial Point Processes on Networks

A spatial network point pattern, which can be classiﬁed as random, regular, or clustered, is a set of locations distributed within an observed network. The diﬀerent point process model achieves a certain deviation from complete spatial randomness. 4 • Random process is a realization of complete spatial randomness;

• Cluster process has smaller average inter-event distances than CSR and diﬀerent intensity within the network;

• Regular process has greater average inter-event distances than CSR, and points are distributed regularly (inter-event distance is no less than a speciﬁed value δ) within the network.

1.2.3 Inter-event Distances

Here are some notations which will be used in the following chapters. n is the number of events in a spatial points pattern on the network G. tij is the shortest inter-event distance between point i and j, i < j, along the edges in spatial point pattern S. T

= {tij|i, j = 1, ··· , n, and i < j} is the collection of all inter-event distances from S. Clearly, number of elements in set T is |T | = n(n − 1)/2.

1.2.4 CSR Test Based on Inter-event Distances

Based on the deﬁnition, a test of complete spatial randomness addresses whether or not the observed point pattern could possibly be a realization of a homogeneous Poisson process. Hˆ (t) is the empirical distribution function (EDF) of all inter-event distances in T from an observed spatial point pattern S lying on network G,

1 −1 Hˆ (t) = n(n − 1) #(t ≤ t). 2 ij

Hi(t), (i = 1, 2, ··· , s), is the EDF of all inter-event distances in the ith independent simulated CSR point pattern on the same network G. The average function H¯ (t), upper envelope U1(t), and lower envelope L1(t) can be calculated as follows.

s 1 X H¯ (t) = H (t) , s i i=1

U1(t) = max{Hi(t)} ,

L1(t) = min{Hi(t)} . for all i = 1, ··· , s. 5 1.2.5 Nearest-neighbor Distances n = # of events in a spatial points pattern S on the network G. tij = inter-event network distance between point i and j, i < j, in pattern S on the network G.

Nearest-neighbor distance: ri = min{tij|1 ≤ j ≤ n, j 6= i}, i = 1, 2, ··· , n, R = {ri, i = 1, 2, ··· , n}. Clearly, number of elements in set R is |R| = n.

1.2.6 CSR Test Based on Nearest-neighbor Distances

Kˆ (t) is the empirical distribution function (EDF) of nearest-neighbor distances in set R from an observed spatial point pattern S lying on G.

−1 Kˆ (t) = n #(ri ≤ t)

Ki(t), (i = 1, 2, ··· , s), is the empirical distribution function (EDF) of nearest-neighbor distances in the ith independent simulated CSR point pattern on the same network G.

Then the average function K¯ (t), upper envelope U2(t), and lower envelope L2(t) can be calculated as follows. s 1 X K¯ (t) = K (t) s i i=1

U2(t) = max{Ki(t)} and L2(t) = min{Ki(t)} for all i = 1, 2, ··· , s Chapter 2

CDF of Inter-event Distances on a Grid Network under CSR

Testing of complete spatial randomness (CSR) on a network is the primary task in this project. For a spatial point process on a planar network, we are interested in the distribution of the locations of events if the underlying mechanism is completely random. If we are able to have the theoretical cumulative distribution function (CDF) of inter-event distances for a point pattern under CSR, say H(t), then the CDF of the observed pattern should be close to H(t) if the pattern is completely random. If there is a signiﬁcant diﬀerence, the observed pattern does not have the property of CSR and some further investigation should be carried out. In this chapter, I will discuss how to derive the CDF of inter-event distances for the CSR point pattern in a regular grid network.

2.1 Preliminary

As stated in chapter 1, theoretical distributions of inter-event distances are available for some simple cases in planar spatial point processes. For regions with complex bound- aries, it is in general impossible to derive the distribution function in a straightforward way. The same situation also occurs for spatial point process on a network. The inter-event distance depends on the geometric structure of the network. There may be multiple paths connecting two locations on a network. If the network is convoluted,

6 7

Figure 2.1: An example of a regular grid network with m = n = 11. it is extremely challenging to work out an exact theoretical distribution of inter-event distances. In this chapter, we will work on a grid network which has a regular geometric structure. A regular grid network consists of m horizontal lines and n vertical lines with the same spacing in both directions. Figure 2.1 shows an example with m = n = 11. without loss of generality, the spacing is assumed to be 1 in both horizontal and vertical directions. The inter-event distance between two locations is defined to be their shortest-path distance allowed by the geometry of the space in which the spatial point process is embedded. For a spatial point pattern on a two-dimensional plane, it is simply the p 2 2 Euclidean distance (x1 − x2) + (y1 − y2) between two locations (x1, y1) and (x2, y2). For spatial point processes on a network, it could be challenging to get the shortest-path distance. As shown in Figure 2.2 where two points are located on a 5 × 5 regular grid, there are four different cases. Two points can be both on horizontal lines; or both on vertical lines; or one on a vertical line and the other one on a horizontal line. We denote s1, s2 to be the locations of two arbitrary CSR points on an m × n regular grid network. 8 Furthermore, we define a location function for an arbitrary point, ( h if si is on a horizontal line D(si) = v if si is on a vertical line

For a simpler notation, we use si = h representing horizontal point and si = v representing vertical points in the probability expressions. The notation is assumed to not have the property of function.

Figure 2.2: Four possible locations of two arbitrary points on the 5 × 5 grid

To simplify the notation, we also define ri = (mi, ni) to be (i) the nearest vertex to the left of si if si = h; (ii) the nearest vertex below si if si = v. In addition, we define x to be the disance between s1 and r1, and y to be the distance between s2 and r2. Apparently, both x and y are between 0 and 1. Based on the definition of CSR (discussed in 1.2.1 Complete Spatial Randomness), two points s1 and s2 are distributed independently, therefore their related vertices r1 and r2 are also independent. Suppose we have an m × n grid, the distribution of related variables and distance functions for four cases are easy to obtain. In this grid network, the inter-event distance should be equivalent to the taxicab distance. According to the

Wikipedia, the taxicab distance, d1, between two vectors p, q in an n-dimensional real vector space with ﬁxed Cartesian coordinate system, is the sum of the lengths of the projections of the line segment between the points onto the coordinate axes. Therefore, in the plane, d1(p, q) = |p1 − q1| + |p2 − q2| where p = (p1, p2) and q = (q1, q2).

(1) D(s1) = h, D(s2) = h. m1, m2 ∼ DU(1, m − 1); n1, n2 ∼ DU(1, n); x, y ∼ UNIF(0, 1).

(a) If m1 6= m2, d(s1, s2) = |n1 − n2| + |m1 − m2| + x − y

(b) If m1 = m2, n1 6= n2, d(s1, s2) = |n1 − n2| + min(x + y, 2 − x − y) 9

(2) D(s2) = v, D(s2) = v. m1, m2 ∼ DU(1, m); n1, n2 ∼ DU(1, n−1); x, y ∼ UNIF(0, 1).

(a) If n1 6= n2, d(s1, s2) = |m1 − m2| + |n1 − n2| + x − y

(b) If n1 = n2, m1 6= m2, d(s1, s2) = |m1 − m2| + min(x + y, 2 − x − y)

(3) D(s1) = h, D(s2) = v. m1 ∼ DU(1, m − 1); m2 ∼ DU(1, m); n1 ∼ DU(1, n); n2 ∼ DU(1, n − 1); x, y ∼ UNIF(0, 1).

(a) If m2 > m1, n2 ≥ n1, d(s1, s2) = m2 − m1 − x + n2 + y − n1

(b) If m2 > m1, n2 < n1, d(s1, s2) = m2 − m1 − x + n1 − n2 − y

(d) If m2 ≤ m1, n2 < n1, then d(s1, s2) = m1 + x − m2 + n1 − n2 − y

(4) D(s1) = v, D(s2) = h. m1 ∼ DU(1, m); m2 ∼ DU(1, m − 1); n1 ∼ DU(1, n −

1); n2 ∼ DU(1, n); x, y ∼ UNIF(0, 1).

(a) If n2 > n1, m2 ≥ m1, d(s1, s2) = m2 − m1 + y + n2 − n1 − x

(b) If n2 > n1, m2 < m1, d(s1, s2) = m1 − m2 − y + n2 − n1 − x

(d) If n2 ≤ n1, m2 < m1, d(s1, s2) = m1 − m2 − y + n1 + x − n2

Here DU(a, b) denotes the discrete uniform distribution on {a, a + 1, . . . , b − 1, b};

UNIF(a, b) denotes the continuous uniform distribution on interval (a, b); and d(s1, s2) is the shortest-path distance between locations s1 and s2.

2.2 CDF of Inter-event Distances if t < 1

2.2.1 Cumulative Distribution Function

We start by calculating the CDF of inter-event distances less than one. We would like to obtain the CDF, P (d ≤ t) where t < 1, by conditioning on the four cases stated above. 10

(1) s1 = h, s2 = h. m1, m2 ∼ DU(1, m − 1); n1, n2 ∼ DU(1, n); x, y ∼ UNIF(0, 1).

(a) P (d ≤ t|s1, s2 = h, m1 6= m2) = P (|n1 − n2| + |m1 − m2| + x − y < t|s1, s2 =

h, m1 6= m2). Since t < 1 and x − y ∈ (−1, 1), n1 = n2 and |m1 − m2| = 1 must be satisﬁed. We have

P (d ≤ t|s1, s2 = h, m1 6= m2)

= P (n1 = n2|s1, s2 = h)P (|m1 − m2| = 1|s1, s2 = h, m1 6= m2)P (x − y ≤ t − 1) 1 2(m − 2) 1 = (1 + t − 1)2 n (m − 1)2 − (m − 1) 2 t2 = . n(m − 1)

To obtain P (d ≤ t|s1, s2 = h) by the Total Probability Theorem, P (m1 6=

m2|s1, s2 = h) = (m − 2)/(m − 1) is needed in later calculation.

(b) P (d ≤ t|s1, s2 = h, m1 = m2, n1 6= n2) = P (|n1 −n2|+min(x+y, 2−x−y) ≤

t|s1, s2 = h, m1 = m2, n1 6= n2). Since min(x + y, 2 − x − y) ∈ (0, 1), the

equation makes sense if n1 = n2. However, assumption of n1 6= n2 results in

a contradiction. Thus, P (d ≤ t|s1, s2 = h, m1 = m2, n1 6= n2) = 0.

P (d ≤ t|s1, s2 = h, n1 = n2, m1 = m2)

= P (|x − y| ≤ t|s1, s2 = h, n1 = n2, m1 = m2) = P (x − y ≤ t) − P (x − y ≤ −t) 1 1 1 = − t2 + t + − (1 − t)2 2 2 2 = −t2 + 2t.

Also, P (m1 = m2, n1 = n2|s1, s2 = h) = 1/(m−1)n will be used in obtaining

P (d ≤ t|s1, s2 = h)

By the Total Probability Theorem, t2 m − 2 1 1 P (d < t|s , s = h) = + 0 + (−t2 + 2t) 1 2 n(m − 1) m − 1 m − 1 n −t2 + (2m − 2)t = n(m − 1)2 11

(2) s2 = v, s2 = v. m1, m2 ∼ DU(1, m); n1, n2 ∼ DU(1, n − 1); x, y ∼ UNIF(0, 1).

Obviously, we can get P (d ≤ t|s1, s2 = v) by just exchanging m and n in P (d ≤

t|s1, s2 = h) −t2 + (2n − 2)t P (d ≤ t|s , s = v) = 1 2 m(n − 1)2

(3) s1 = h, s2 = v. m1 ∼ DU(1, m − 1); m2 ∼ DU(1, m); n1 ∼ DU(1, n); n2 ∼ DU(1, n − 1); x, y ∼ UNIF(0, 1).

(a) P (d ≤ t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1) = P (m2 − m1 + n2 − n1 − x + y <

t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1). Because t < 1, m2 − m1 ≥ 1, and

−x + y ∈ (−1, 1). Therefore, m2 − m1 = 1, n2 = n1, and 1 − x + y < t must be satisﬁed. We have

P (d < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1)

= P (m2 − m1 = 1|s1 = h, s2 = v, m2 > m1)P (n2 = n1|s1 = h, s2 = v, n2 ≥ n1) P (x − y > 1 − t) m − 1 n − 1 1 1 = 1 − (− (1 − t)2 + 1 − t + ) m(m − 1)/2 n(n − 1)/2 2 2 2t2 = mn

2t2 (b) P (d < t|s = h, s = v, m > m , n < n ) = . P (m > m , n < n |s = 1 2 2 1 2 1 mn 2 1 2 1 1 h, s2 = v) = 1/4. 2t2 (c) P (d < t|s = h, s = v, m ≤ m , n ≥ n ) = . P (m ≤ m , n ≥ n |s = 1 2 2 1 2 1 mn 2 1 2 1 1 h, s2 = v) = 1/4. 2t2 (d) P (d < t|s = h, s = v, m ≤ m , n < n ) = . P (m ≤ m , n < n |s = 1 2 2 1 2 1 mn 2 1 2 1 1 h, s2 = v) = 1/4.

The conclusion of (b), (c), and (d) results from the equality among the four cases. It is easy to see that each of the four cases contributes to 1/4 of the whole condition

when s1 is horizontal and s2 is vertical. Thus, by the Total Probability Theorem,

1 2t2 1 2t2 1 2t2 1 2t2 2t2 P (d < t|s = h, s = v) = + + + = 1 2 4 mn 4 mn 4 mn 4 mn mn 12

(4) s1 is vertical, s2 is horizontal. Obviously, we can get P (d < t|s1 = v, s2 = h) by

exchanging m and n in P (d < t|s1 = h, s2 = v).

2t2 P (d < t|s = v, s = h) = 1 2 mn

For a point generated by CSR, the probability that it is located on a horizontal edge is

length of horizontal edges (m − 1)n P (h) = = length of horizontal edges + length of vertical edges (m − 1)n + (n − 1)m

(n − 1)m Similarly, the probability that the point lies on a vertical edge is P (v) = . (m − 1)n + (n − 1)m Combining all these four cases for two arbitrary points, we are able to get the cumulative distribution function of inter-event distances in an m × n grid network as follows.

P (d < t, t < 1) 2 2 =P (h) P (d < t|s1, s2 = h) + P (v) P (d < t|s1, s2 = v)

+ P (h)P (v)[P (d < t|s1 = h, s2 = v) + P (d < t|s1 = v, s2 = h)] −t2 + (2m − 2)t (m − 1)n 2 −t2 + (2n − 2)t (n − 1)m 2 = + n(m − 1)2 (m − 1)n + (n − 1m) m(n − 1)2 (m − 1)n + (n − 1)m 2t2 (m − 1)n (n − 1)m + 2 mn (m − 1)n + (n − 1)m (m − 1)n + (n − 1)m (4mn − 5m − 5n + 4)t2 + (4mn − 2m − 2n)t = (2mn − m − n)2

2.2.2 Simulation

As an illustration, I simulated 100 CSR point patterns consisting of 500 points each on a 11 × 11 grid network. The procedure of simulation for complete spatial random point pattern will be discussed in detail in the 3.3 Simulations. Once obtaining EDF of inter-event distances for each simulation, I was able to calculate the mean function, upper and lower envelopes which are deﬁned in the 1.2.4 CSR Test Based on Inter- event Distances. These three functions are plotted along with the theoretical result just derived. From Figure 2.4, we can tell that the simulated and theoretical mean functions are almost identical. 13 CDF 0.000 0.005 0.010 0.015 0.0 0.2 0.4 0.6 0.8 1.0 tt Figure 2.3: The 11 × 11 grid network Figure 2.4: Simulation result and plot for CDF when t < 1 (blue is the theoretical function)

2.3 CDF of Inter-event Distances if t > 1

2.3.1 Cumulative Distribution Function

The analysis for t > 1 is very similar to that of t < 1 in the previous section but much more complicated. I skip similar content with the t < 1 here.

(1) Both horizontal. m1, m2 ∼ DU(1, m − 1), n1, n2 ∼ DU(1, n), x, y ∼ UNIF(0, 1).

(a) When m1 6= m2,

P (d < t|s1, s2 = h, m1 6= m2)

=P (|n1 − n2| + |m1 − m2| + x − y < t|s1, s2 = h, m1 6= m2)

=P (|n1 − n2| + |m1 − m2| = dte|s1, s2 = h, m1 6= m2)P (x − y < t − dte)

+ P (|n1 − n2| + |m1 − m2| = btc|s1, s2 = h, m1 6= m2)P (x − y < t − btc)

+ P (|n1 − n2| + |m1 − m2| ≤ btc − 1|s1, s2 = h, m1 6= m2). 14 1 Let us deﬁne a function called Ch(x) as following,

1 Ch(x) = P (|n1 − n2| + |m1 − m2| = x|s1, s2 = h, m1 6= m2) t(x) X = P (|n1 − n2| = x − i)P (|m1 − m2| = i) i=s(x) t(x) X 2(m − 1 − i) 2(n − x + i) 2(m − 1 − x) 1 = · − I(t(x) = x) · (m − 1)(m − 2) n2 (m − 1)(m − 2) n i=s(x) Pt(x) 4 (m − 1 − i)(n − x + i) 2(m − 1 − x) = i=s(x) − I(t(x) = x) . (m − 1)(m − 2)n2 n(m − 1)(m − 2)

where s(x) = max{1, x − (n − 1)}, t(x) = min{x, m − 2}, and I(t(x) = x) is an indicator function. Also, we have 1 1 P (x − y < t − btc) = − (t − btc)2 + t − btc + 2 2 1 P (x − y < t − dte) = (1 + t − dte)2. 2

1 Thus, we are able to write the conditional probability in terms of Ch(x),

P (d < t|s1, s2 = h, m1 6= m2) (2.1) btc−1 1 1 1 X = C1(dte)(1 + t − dte)2 + C1(btc)(− (t − btc)2 + t − btc + ) + C1(j). 2 h h 2 2 h j=1

Note: If t is an integer, P (d < t|s1, s2 = h, m1 6= m2) = P (|n1 − n2| + |m1 −

m2| = t|s1, s2 = h, m1 6= m2)P (x − y < 0) + P (|n1 − n2| + |m1 − m2| ≤ 1 1 Pt−1 1 t − 1) = 2 Ch(t) + j=1 Ch(j). Generalize this result to the form of 2.1, it is 1 1 2 equal to 2.1 after removed the second term. That is 2 Ch(dte)(1 + t − dte) + Pbtc−1 1 j=1 Ch(j). Also, P (m1 6= m2|s1, s2 = h) = (m − 2)/(m − 1) is required to derive the probability of distance less than t conditioning on both points are horizontal.

(b) When m1 = m2 and n1 6= n2,

P (d < t|s1, s2 = h, m1 = m2, n1 6= n2)

=P (|n1 − n2| = btc)P (min(x + y, 2 − x − y) < t − btc) + P (|n1 − n2| ≤ btc − 1) 15 2 Let us deﬁne a function called Ch(x) as following. 2 Ch(x) = P (|n1 − n2| = x|s1, s2 = h, m1 = m2, n1 6= n2)   0 if x > n − 1 = 2(n − x) if 1 ≤ x ≤ n − 1  n(n − 1) Also, we have

P (min(x + y, 2 − x − y) < t − btc) =P (x + y < t − btc) + P (x + y > 2 − t + btc) 1 1 = (t − btc)2 + 1 − − (2 − t + btc)2 + 2(2 − t + btc) − 1 2 2 =(t − btc)2

2 Thus, we are able to write the conditional probability in terms of Ch(x).

P (d < t|s1, s2 = h, m1 = m2, n1 6= n2) (2.2) btc−1 2 2 X 2 =Ch(btc)(t − btc) + Ch(j) j=1

Note: If t is an integer, P (d < t|s1, s2 = h, m1 = m2, n1 6= n2) = P (|n1 − Pt−1 2 n2| ≤ t − 1) = j=1 Ch(j), which is also 2.2 after removed the ﬁrst term. Also, P (m1 = m2, n1 6= n2|s1, s2 = h) = (n − 1)/n(m − 1) should be known

for calculating P (d < t|s1, s2 = h). (c) Since |x − y| ∈ [0, 1] and t is assumed to be greater than 1 in this section.

So P (d < t|s1, s2 = h, m1 = m2, n1 = n2) = P (|x − y| < t|s1, s2 = h, m1 =

m2, n1 = n2) = 1. In addition, P (m1 = m2, n1 = n2|s1, s2 = h) = 1/n(m − 1).

Therefore, by the Total Probability Theorem,

P (d < t|s1, s2 = h) (2.3)  btc−1  m − 2 1 1 1 X = C1(dte)(1 + t − dte)2 + C1(btc)(− (t − btc)2 + t − btc + ) + C1(j) m − 1 2 h h 2 2 h  j=1  btc−1  n − 1 X 1 + C2(btc)(t − btc)2 + C2(j) + n(m − 1)  h h  n(m − 1) j=1 16

(2) Both vertical. Clearly, P (d < t|s1, s2 = v) can be obtained by exchanging m and 1 2 n in P (d < t|s1, s2 = h). So let us deﬁne function Cv (x) and Cv (x) as following. Pt(x) 4 (n − 1 − i)(m − x + i) 2(n − 1 − x) C1(x) = i=s(x) − I(t(x) = x) v (n − 1)(n − 2)m2 m(n − 1)(n − 2)   0 if x > m − 1 C2(x) = 2(m − x) v if 1 ≤ x ≤ m − 1  m(m − 1) where s(x) = max{1, x − (m − 1)}, t(x) = min{x, n − 2}. Then we can have

P (d < t|s1, s2 = v) (2.4)  btc−1  n − 2 1 1 1 X = C1(dte)(1 + t − dte)2 + C1(btc)(− (t − btc)2 + t − btc + ) + C1(j) n − 1 2 v v 2 2 v  j=1  btc−1  m − 1 X 1 + C2(btc)(t − btc)2 + C2(j) + m(n − 1)  v v  m(n − 1) j=1

(3) s1 is horizontal, s2 is vertical. m1 ∼ DU(1, m−1), m2 ∼ DU(1, m), n1 ∼ DU(1, n),

n2 ∼ DU(1, n − 1), x, y ∼ UNIF(0, 1).

(a) When m2 > m1 and n2 ≥ n1,

P (d < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1)

=P (m2 − m1 + n2 − n1 − (x − y) < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1)

=P (m2 − m1 + n2 − n1 = dte)P (x − y > dte − t) + P (m2 − m1 + n2 − n1 = btc)

P (x − y > btc − t) + P (m2 − m1 + n2 − n1 ≤ btc − 1).

1 Let us deﬁne a function called Chv(x) as following. 1 Chv(x) = P (m2 − m1 + n2 − n1 = x|s1 = h, s2 = v, m2 > m1, n2 ≥ n1) t(x) X = P (m2 − m1 = i)P (n2 − n1 = x − i) i=s(x) t(x) X 2(m − i) 2(n − 1 − x + i) = m(m − 1) n(n − 1) i=s(x) 4 Pt(x) (m − i)(n − 1 − x + i) = i=s(x) , mn(m − 1)(n − 1) 17 where s(x) = max{1, x − (n − 2)}, t(x) = min{x, m − 1}. Also, we have

1 1 1 1 P (x − y > dte − t) = 1 − − (dte − t)2 + (dte − t) + = (dte − t)2 − (dte − t) + 2 2 2 2 1 P (x − y > btc) = 1 − (1 + btc − t)2 2

1 Thus, we are able to write the conditional probability in terms of Chv(t),

P (d < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1) (2.5) 1 1 1 =C1 (dte) (dte − t)2 − (dte − t) + + C1 (btc) 1 − (1 + btc − t)2 hv 2 2 hv 2 btc−1 X 1 + Chv(j). j=1

Note: If t is an integer, P (d < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1) =

P (m2 − m1 + n2 − n1 = t)P (x − y ≥ 0) + P ((m2 − m1 + n2 − n1 ≤ t − 1) = 1 1 Pt−1 1 Chv(t) 2 + j=1 Chv(t), which is also equal to 2.5 took oﬀ the second term. Also, we have P (m2 > m1, n2 ≥ n1)|s1 = h, s2 = v) = 1/4.

(b) When m2 > m1 and n2 < n1, the probability is same as (a). Also, P (m2 >

m1, n2 > n1|s1 = h, s2 = v) = 1/4.

P (m2 ≤ m1, n2 ≥ n1|s1 = h, s2 = v) = 1/4.

(d) When m2 ≤ m1 and n2 < n1, the probability is same as (a). Also, we know

P (m2 ≤ m1, n2 < n1|s1 = h, s2 = v) = 1/4.

Again, the conclusion of (b), (c), and (d) comes from the equality among the four cases. It is easy to see that each of the four cases contributes 1/4 to the whole condition and they are equivalent to each other. Therefore, by the Total 18 Probability Theorem,

P (d < t|s1 = h, s2 = v) (2.6) 1 = 4P (d < t|s = h, s = v, m > m , n ≥ n ) 4 1 2 2 1 2 1

=P (d < t|s1 = h, s2 = v, m2 > m1, n2 ≥ n1) 1 1 1 =C1 (dte) (dte − t)2 − (dte − t) + + C1 (btc) 1 − (1 + btc − t)2 hv 2 2 hv 2 btc−1 X 1 + Chv(j). j=1

(4) s1 is vertical, s2 is horizontal. Clearly, P (d < t|s1 = v, s2 = h) can be obtained

by just exchanging m and n in P (d < t|s1 = h, s2 = v). Actually, P (d < t|s1 = ◦ v, s2 = h) must be equal to P (d < t|s1 = h, s2 = v) because a 90 rotation of grid will not change the distribution of distances.

Finally, we are able to get P (d < t) for t > 1 by combining the four cases above. Again, (m − 1)n (n − 1)m we will use P (h) = and P (v) = . (m − 1)n + (n − 1)m (m − 1)n + (n − 1)m

P (d < t, t > 1) (2.7) (m − 1)n 2 (n − 1)m 2 =P (d < t|s , s = h) + P (d < t|s , s = v) 1 2 (m − 1)n + (n − 1m) 1 2 (m − 1)n + (n − 1)m (m − 1)n (n − 1)m + 2 P (d < t|s = h, s = v) (m − 1)n + (n − 1m) (m − 1)n + (n − 1)m 1 2 where P (d < t|s1, s2 = h), P (d < t|s1, s2 = v) and P (d < t|s1 = h, s2 = v) refer to 2.3, 2.4, and 2.6.

2.3.2 Simulation

Exactly same as what I did for t < 1, I did 100 CSR simulations on the 11 × 11 grid network in order to demonstrate the result obtained is reasonable enough. I plot three functions (mean function, upper and lower envelopes which are deﬁned in the 1.2.4 CSR Test Based on Inter-event Distances) from simulations together with my theoretical function in one ﬁgure. From Figure 2.5, we can tell that the mean function 19 and theoretical function (blue) are already coincident and my result is between upper and lower envelope exactly. 1.0 CDF 0.0 0.2 0.4 0.6 0.8 5 10 15 20 tt

Figure 2.5: The 11 × 11 grid network Figure 2.6: Simulation result and plot for CDF when t > 1 (blue is the theoretical function) Chapter 3

CSR Test Based on Inter-event Distances

As shown in the previous chapter, the theoretical cumulative distribution function (CDF) is complicated and diﬃcult to obtain even for the simplest grid network. In practice, however, we have various kinds of networks which are far more complicated than the regular grid network. It will not be feasible to use the theoretical CDF of the complete spatial random (CSR) point pattern as a criterion when testing for CSR. Instead a well-approximated true function derived from Monte Carlo simulations can be utilized to replace the analytic solutions. In this chapter, I will present how to implement the CSR test based on inter-event distances using Monte Carlo simulations in real applications.

3.1 CSR Test Implementation

Since the empirical distribution function converges to the cumulative distribution function when the sample size is large, H¯ (t) can be used as an approximation to the theoretical CDF of inter-event distances for CSR point pattern on the given network. Under the hypothesis of CSR, Hˆ (t), which represents the observed distribution function, should be close to H¯ (t) which is regarded as the “true” function. In other words, a plot of Hˆ (t) against H¯ (t) should be roughly linear. To assess the signiﬁcance or departures from linearity, upper bound and lower bound of Hi(t) are also evaluated and plotted against

20 21

H¯ (t). Hence, if Hˆ (t) is always bounded by U1(t) and L1(t) across all simulations, a hypothesis of CSR certainly cannot be rejected. On the other hand, if Hˆ (t) is more extreme than U1(t) or L1(t), it is very likely that the underlying data-generating mechanism is not CSR. In summary, we can implement the CSR test based on inter-event distances as follows.

(1) Calculate all unique inter-event distances tij, 1 ≤ i < j ≤ n, in the observed spatial point pattern on the network G and get Hˆ (t);

(2) Simulate s complete spatial random point patterns on the network G independently. To make the p-value for that EDF of CSR pattern is upper or lower envelope at most 0.1, the s is generally greater than 100;

(3) For each ith simulated CSR pattern, calculate all inter-event distances tij and get

Hi(t);

(4) Obtain H¯ (t), U1(t), and L1(t);

(5) Plot Hˆ (t), U1(t), and L1(t) against H¯ (t).

In spatial point process on the network, distance between a pair of points relies on the geometry of observed network, not the usual Euclidean distance used for planar region. Therefore, the main difficulty would be obtaining the network distance which is an essential part for our CSR test. Only if all distances are calculated, the comparison between the observed data and the data simulated from CSR can be analyzed in terms of the CDF to test different models. The igraph package of R computing environment turns out to be a very useful tool to deal with the problem of finding the shortest distance on a network. It is very convenient and efficient for visualizing stream network, point pattern, and calculating inter-event distances on network. It is also able to visualize simulated network point process.

3.2 Simulations

Theoretically, since the CSR test method is discussed for a general setup, the proposed test method should work for both network and planar regions. I did simulations for 22 three typical point processes to for demonstration before I applied this method to a real application.

3.2.1 Random Process

Based on the deﬁnition of CSR, complete spatial random process is a binomial process if the number of events n is ﬁxed. To generate complete spatial random n points on a network, a binomial process was simulated such that n points are added onto the edges one by one. Let m be the number of edges in network. The procedure of simulation for random process is summarized as follows.

|ei| (a) Choose one edge ei with probability of Pm ; i=1 |ei|

(b) The point is distributed uniformly on ei, i.e., d ∼ UNIF(0, |ei|) where d is the

distance to one vertex of ei;

I generated complete spatial random point patterns on two different types of networks. The first one is a regular grid network as the one shown in the previous chapter where all edges have the same length. The second one is a random network in which edges have different lengths and vertices are distributed irregularly. A random network is constructed by vertices distributed randomly and edges with connection probability of 0.5. To have a connected and planar network, some restrictions were added onto the generation procedure of the random network. For example, distance between every two points is no less than 0.5, and only considering a point’s connection to its nearest 3 points. Figures 3.1 and 3.3 show these two types of networks. 23 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 3.1: Random point pattern on the Figure 3.2: Envelope plot for random pro- grid network cess on the grid network 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 3.3: Random point pattern on a ran- Figure 3.4: Envelope plot for random pro- dom network cess on a random network

For the spatial point pattern on the grid network, the envelope plot in Figure 3.2 shows that Hˆ (t) is roughly equal to H(t) and lies between U1(t) and L1(t) throughout its range, which suggests an acceptance of CSR. For the random network, the envelope plot is Figure 3.4 also shows an acceptance of CSR which is similar to that of the grid network. We conclude that these data are compatible with completely random spatial distribution. This is not surprising since the data are indeed generated using CSR. 24 3.2.2 Cluster Process

For cluster processes, we generate the data using a two-step procedure. First, parent points are independently generated from the uniform distribution on the network (binomial process) [9]. We then generate some child points from each of the parents. The number of children follows a Poisson distribution, and the positions of children relative to their parents are normally distributed. After child points are generated, we remove all parent points. Thus the ﬁnal data set only consists of the children. The procedure for generating cluster process is summarized as follows.

(a) Generate 20 parents following steps in complete spatial randomness simulation;

(b) For each parent, the number of its children follows Poisson distribution with mean 10. The position of each child relative to its parent is normally distributed, i.e., N(0, 0.52);

Figure 3.5: Cluster point pattern on the grid Figure 3.6: Envelope plot for cluster process network on the grid network 25 1.0 0.8 0.6 H1 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 average1

Figure 3.7: Cluster point pattern on a ran- Figure 3.8: Envelope plot for cluster process dom network on a random network

Figures 3.5 and 3.7 show realizations of cluster processes on a regular grid network and a random network. Figure 3.6 shows that Hˆ (t) is greater than H¯ (t) within the range and lies above U1(t) for very small values of H¯ (t). This is typical for cluster processes because there is an excess of short inter-event distances. The plot suggests a rejection of CSR in favor of clustering. Similarly, Figure 3.8 suggests the rejection of CSR because of the obvious departure from complete spatial randomness. In summary, if the probability of getting small distances is higher than that of CSR point pattern, the observed point patten should be more likely to be classiﬁed as a cluster process.

3.2.3 Regular Process

In order to generate events with regularity, I started with a homogeneous Poisson process Pm with n events. The density λ = n/ i=1 |ei|. Any two events separated by a distance of less than a speciﬁed value of δ are thinned. The probability of retention for a point x 26 on network is πδ2 n P (x is retained) = 1 − Pm i=1 |ei| πδ2 = exp n ln 1 − Pm i=1 |ei| πδ2 ≈ exp −nPm i=1 |ei| 2 = e−λπδ

Here is the procedure for simulation of regular process. n (a) Generate a CSR point pattern with density λ = Pm . i=1 |ei| (b) For each point x, x is retained with probability of e−λπδ2 as long as the distance between any two events is less than a speciﬁed value δ.

Figure 3.9: Regular point pattern on the Figure 3.10: Envelope plot for regular pro- grid network cess on the grid network

As shown in the Figures 3.10 and 3.12, Hˆ (t) is less than H¯ (t) and even lower than

L1(t) for small inter-event distances. The reason is that a regular process does not have inter-event distances smaller than a certain lower threshold. Thus the envelope plots claim the rejection of complete spatial randomness in favor of regularity. 27 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 3.11: Regular point pattern on a ran- Figure 3.12: Envelope plot for regular pro- dom network cess on a random network

3.2.4 Conclusion

In conclusion, we considered Monte Carlo tests for CSR for both regular grid network and random network. The results are similar for two types of networks. The conclusions based on envelope plots are consistent with the original assumptions. In other words, the proposed CSR test method based on inter-event distances works well for network case and is able to test any spatial point pattern on network correctly. Chapter 4

CSR Test Based on Nearest-neighbor Distances

Besides inter-event distance, nearest-neighbor distance is another sensible quantity which can be used to test for complete spatial randomness (CSR). Similar to method based on inter-event distance, the essential part of CSR test using nearest-neighbor distances is to calculate nearest-neighbor distance of each event. In this chapter, I will discuss test method based on the nearest-neighbor distances in the same way as previous chapter. So some repeated deﬁnitions and analysis will be omitted in this chapter.

4.1 CSR Test Implementation

Similar to the demonstration in last chapter, we should know that plot of Kˆ (t) against K¯ (t) should be roughly linear under the hypothesis of CSR. Besides the rough linear plot, if Kˆ (t) is always between U2(t) and L2(t) from signiﬁcant number of simulations, then a hypothesis of CSR certainly cannot be rejected. In summary, we can implement CSR test based on nearest-neighbor distance as follows.

(1) Calculate all nearest-neighbor distances ri in the observed spatial point pattern on the network G ⇒ Kˆ (t)

(2) Simulate s(s ≥ 100) complete spatial random point pattern on the network G independently. Same reason as that of inter-event distances method, s should be

28 29 at least 100.

(3) For each ith simulated CSR pattern, calculate all the nearest-neighbor distances

ri ⇒ Ki(t).

(4) Obtain K¯ (t), U2(t), and L2(t). Plot Kˆ (t), U2(t), and L2(t) against K¯ (t).

Again, the main diﬃculty would be obtaining the nearest-neighbor network distance which is an essential part for CSR test. I also used the igraph package of R to deal with the problem here. Firstly, we can calculate the inter-event distances from a point x to all its neighbors. Then the minimum of these inter-event distances is the nearest- neighbor distance of event x.

4.2 Simulations

Theoretically, since the CSR test method is discussed in a general case, the proposed test method should work for both network and planar region. To guarantee its correctness and feasibility, I did simulations for three typical point processes to verify the CSR test method based on nearest-neighbor distance before I applied this method to a real application. In this section, I follow the same models and steps described in section 3.2. The main diﬀerence between inter-event method and nearest-neighbor method is the amount of distance data obtained. In contrast with n(n − 1)/2 inter-event distances used for calculating EDF, we can only have n nearest-neighbor distances. To ensure the accuracy of the approximated cumulative distribution function (CDF), an observed spatial point pattern should be required to have large numbers of events. After several practice, at least 200 points should be simulated for the CSR point pattern such that the plot of EDF is smooth enough. 30 4.2.1 Random Process 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 4.1: Grid network and random point Figure 4.2: Envelope plot for random pro- pattern cess on grid network 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 4.3: Random network and random Figure 4.4: Envelope plot for random pro- point pattern cess on random network

Figure 4.2 shows that Kˆ (t) is roughly equal to K¯ (t) and lies between U2(t) and L2(t) through out its range, which suggests an acceptance of CSR. For spatial point pattern on the network generated randomly, envelope plot 4.4 shows an acceptance of CSR too. So we conclude that these data are compatible with completely random spatial distribution. 31 4.2.2 Cluster Process 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average1

Figure 4.5: Grid network and cluster point Figure 4.6: Envelope plot for cluster process pattern on grid network 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average1

Figure 4.7: Random network and cluster Figure 4.8: Envelope plot for cluster process point pattern on random network

Figure 4.6 shows that Kˆ (t) is greater than K¯ (t) within the range and lies above U2(t) for very small values of K¯ (t). So the plot claimed a rejection of CSR. Similarly, envelope plot 4.8 tends to reject CSR because of the obvious departure from complete spatial randomness. Since the probability of getting small nearest-neighbor distances is higher than that of CSR point pattern, the simulated point patten should be more likely classiﬁed as a cluster process. 32 4.2.3 Regular Process 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 4.9: Grid network and regular point Figure 4.10: Envelope plot for regular pro- pattern cess on grid network 1.0 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 average

Figure 4.11: Random network and regular Figure 4.12: Envelope plot for regular pro- point pattern cess on random network

As shown in the Figure 4.10, Kˆ (t) is less than K¯ (t) and even lower than L2(t) for small nearest-neighbor distances. Thus the envelope plot claims a rejection of complete spatial randomness. For the network generated randomly, the test result 3.12 also leads to a rejection of CSR and conclusion that the point pattern intends to have larger nearest- neighbor distances than that of CSR point pattern. So the simulated point pattern should be more likely classiﬁed as a regular point pattern. 33 4.2.4 Conclusion

In conclusion, each of the three testings for their corresponding typical point process results in the correct conclusion which is consistent with the original assumption. In other words, the proposed CSR test method based on nearest-neighbor distances works well for network case and is able to test any spatial point pattern on the network correctly. Chapter 5

Car Crash Point Pattern on the Minnesota Major Roads

In this chapter, I tested the car crash point pattern on the Minnesota Major Roads by the method based on inter-event distance and nearest-neighbor distance respectively. Both results suggested that car crashes on the Minnesota Roads tend to follow a cluster point process.

5.1 Dataset

My datasets are

• Locations of car accidents in Minnesota in 2013 from National Highway Traﬃc Safety Administration (NHTSA) and

• Major road network of Minnesota (U.S. highway, interstate highway, and Minnesota highway) from Minnesota Geospatial Commons.

The Figure 5.1 represents how the car crash dataset looks originally on the website. And the Figure 5.2 is the presentation of Minnesota major roads network in R environment.

34 35

Figure 5.1: Location of fatal crashes in Min- Figure 5.2: R Plot of Minnesota Major nesota in 2013 Roads Network.

Since some of the car accidents did not occur on the highway of Minnesota. In order to consider car crashes on major roads only, I detected and eliminated crash points not on the major road network. So we end up with 234 observed points on this speciﬁc network for testing, and these are shown in Figure 5.3 and Figure 5.4.

Figure 5.3: R Plot of the car crash pattern Figure 5.4: Display of the car crash pattern on the Minnesota major roads. on the Minnesota major roads in ArcGIS 36 5.2 Implementation

Considering implementation of the proposed CSR test methods from last two chapters, computing all the inter-event distances must be the issue which should be handled above all. Different from the network that can be generated by igraph package in R, Minnesota major roads data is stored in a shape file and made up of hundreds of road segments. So the inter-event distance along the Minnesota highway can not be calculated by igraph package in R in this case. Fortunately, ArcGIS is very good at processing shape files and calculating distance in terms of map data. Thus, I utilized ArcGIS to compute inter-event distances of car crash pattern and all my simulated CSR patterns on the Minnesota major roads. Besides computing inter-event distances, simulating CSR point pattern on the Min- nesota major roads is also an essential part of testing since simulations are the key to obtaining H¯ (t). For the simulations, I just followed the same steps described in last two chapters and got 100 independent sets consisting of 200 CSR points’ locations. Here is one set out of 100 independent CSR point pattern shown in the Figure 5.5 and Figure 5.6.

Figure 5.5: R plot of a CSR point pattern Figure 5.6: Display of a CSR point pattern on the Minnesota major roads on the Minnesota major roads in ArcGIS 37 5.2.1 CSR Test Based on Inter-event Distances

(1) Simulate 100 CSR point patterns consisting of 200 random points each on MN Major Roads Network in R.

(2) In ArcGIS, utilized “Feature to Line” to split road segment at intersections. Then generated spatial network dataset and created “OD Cost Matrix Layer” for calculating inter-event distances.

(3) Solved “OD Cost Matrix Layer” of ArcGIS to compute inter-event distances for all the 100 simulated point patterns and the car crash pattern.

(3) Calculate Hˆ (t), Hi(t), i = 1, 2, ··· , 100 in R. Plot Hˆ (t), U1(t), and L1(t) against H¯ (t).

5.2.2 CSR Test Based on Nearest-neighbor Distances

(1) Simulate 100 CSR point patterns consisting of 200 random points each on MN Major Roads Network in R.

(3) Solved “OD Cost Matrix Layer” of ArcGIS to compute inter-event distances for all the 100 simulated point patterns and the car crash pattern.

(3) Obtained Kˆ (t), Ki(t), i = 1, 2, ··· , 100 by taking the minimum of all the inter-

event distances of each point in R. Plot Kˆ (t), U2(t), and L2(t) against K¯ (t).

5.3 Result and Analysis

As shown in the Figure 5.7 and 5.8, plot of crash location pattern exceed upper plot formed by complete spatial random process throughout its range. What’s more, probability of having small inter-event distances in crash point pattern is much greater than that of random point pattern. So we come to the conclusion that car crash point pattern 38 on Minnesota major roads cannot be a complete spatial random process and tends to be a cluster process instead. 1.0 1.0 H1 H1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 average average

Figure 5.7: Envelope plot for CSR test for Figure 5.8: Envelope plot for CSR test of car car crash pattern by inter-event method. crash pattern by nearest-neighbor method.

It is not surprising that a conclusion of cluster process is reached since there is obviously a small cluster around the Twin Cities. But the mechanism underlying the car crash point pattern, like what the exact factor causing cluster is, is still unknown. Intuitively, a network density ρ(G) can be deﬁned as the total length of edges per unit area, and the density of a point pattern ρ(S) is the number of points per unit area. So if we can get ρ(G) = αρ(S) all over the spatial point pattern S on the network G, then the cluster point pattern should be formed by just following the density of network without any external factors. However, we have not got the area data of Minnesota which needs to be used in the analysis mentioned at this point. Therefore, this would probably lead to research exploring deeper into the car crash pattern of Minnesota in the future. Chapter 6

Conclusion

In the project “ CSR Test of Spatial Point Pattern on Network”, I succeed in obtaining cumulative distribution function (CDF) of inter-event distance for complete spatial random (CSR) point pattern on the m×n grid network by conditioning on locations of two arbitrary points. Also, I proposed two CSR test methods based on inter-event distance and nearest-neighbor distance respectively in terms of Monte Carlo simulations. Both of the methods were veriﬁed by three typical simulations on grid and networks generated randomly. Finally, car crash point pattern on Minnesota major roads network was tested by the two proposed methods to be a cluster process.

39 References

[1] Peter J. Diggle, Julian Besag, and J.Timothy Gleaves. Statistical analysis of spatial point patterns by means of distance methods. Biometrics, 32:659 – 667, 1976.

[2] B. D. Ripley. Test of randomness for spatial point patterns. J.R.Statist.Soc.B, 41(3):368 – 374, 1979.

[3] Renato Assuncao. Testing spatial randomness by means of angles. Biometrics, 50:531 – 537, June 1994.

[4] Joe N.Perry. Spatial analysis by distance indices. Journal of Animal Ecology, 64:303 – 314, 1995.

[5] Jacques Gignoux, Camille Duby, and Sebastien Barot. Comparing the performances of diggle’s tests of spatial randomness for small samples with and without edge-eﬀect correction: Application to ecological data. Biometrics, 55:156 – 164, March 1999.

[6] Peter J. Diggle. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns. CRC Press, Third Edition, 2014.

[7] John Clark and Derek Allan Holton. A First Look at Graph Theory. World Scientiﬁc Publishing Co. Pte. Ltd., 1991.

[8] Atsuyuki Okabe and Kokichi Sugihara. Spatial Analysis along Networks. John Wiley & Sons, 2012.

[9] Oliver Schabenberger and Carol A. Gotway. Statistical Methods for Spatial Data Analysis. Chapman & Hall/CRC, 2005.

40 Appendix A

Glossary and Acronyms

Care has been taken in this thesis to minimize the use of jargon and acronyms, but this cannot always be achieved. This appendix contains a table of acronyms and their meaning.

A.1 Glossary

• Total Probability Theorem – Given n mutually exclusive events A1, ··· ,An

whose probabilities sum to unity, then P (B) = P (B|A1)P (A1)+···+P (B|An)P (An),

where B is an arbitrary event, and P (B|Ai) is the conditional probability of B

assuming Ai.

A.2 Acronyms

Table A.1: Acronyms

Acronym Meaning CSR Complete Spatial Randomness CDF Cumulative Distribution Function EDF Empirical Distribution Function Continued on next page

41 42 Table A.1 – continued from previous page Acronym Meaning UNIF Continuous Uniform Distribution DU Discrete Uniform Distribution si = h D(si) = h, i.e., si is in the horizontal direction si = v D(si) = v, i.e., si is in the vertical direction Appendix B

Code

This appendix contains selected crucial code.

B.1 R Code

B.1.1 Random Process

#function for adding one random point to update the network AddOnePointOnNetwork <- function(g) { # add one point on the network # the point is added uniformly on the length of the edges #Conditioning on the number of points, poisson process is binomial process edges <- get.edges(g, E(g)) # g is a igraph length <- E(g)$weight all.length <- sum(length) n.edges <- ecount(g) new.vertex <- vcount(g) + 1 #number of vertices after addiing this point new.loc <- sample(1:n.edges, size = 1, prob = length / all.length) # new.loc is the id of the edge where the new point should be added to new.length <- length[new.loc] #length of this edge dist.from.start <- new.length * runif(1)

43 44 #choose location where the point add on this edge dist.to.end <- new.length - dist.from.start #distance to another vertex start <- edges[new.loc, 1] #one vertex of this edge end <- edges[new.loc, 2] #another vertex of this edge coord <- g$layout #layout of igraph is the coordinates of vertices x1 <- coord[start, 1] y1 <- coord[start, 2] x2 <- coord[end, 1] y2 <- coord[end, 2] x0 <- dist.from.start / new.length * (x2 - x1) + x1 y0 <- dist.from.start / new.length * (y2 - y1) + y1 #(x0, y0) is the coordinate of new point g <- add.vertices(g, 1) #add this new point to igraph as a vertex #add two new edges (undirected), weight is the distance g <- g + edges(start, new.vertex, end, new.vertex, weight = c(dist.from.start, dist.to.end)) g$layout <- rbind(coord, c(x0, y0)) #update layout of igraph #remove the edge which the point is added on g[start, end] <- FALSE return (list(g = g, num.v = vcount(g))) }

B.1.2 Cluster Process

#function for generating offspring from a parent in poisson cluster process GenerateOneChild <- function(g, parent) { #extract information of this parent: edge id, distance to two vertices, etc edge.index <- as.numeric(parent[1]) dist.from.start <- as.numeric(parent[2]) dist.to.end <- as.numeric(parent[3]) start <- as.numeric(parent[4]) end <- as.numeric(parent[5]) distance <- rnorm(1, mean = 0, sd = 0.5) 45 #children’s location around parent is normal distribution #if the offspring doesn’t locate outside the edge holding parent if ((dist.from.start + distance >= 0) & (dist.to.end - distance >= 0)) { child <- c(edge.index, dist.from.start + distance, dist.to.end - distance, start, end)} #if the generated length exceed dist.from.start if ((distance < 0) & (distance < -dist.from.start)) { one.dist <- -(dist.from.start + distance) edges <- incident(g, start) #find all the edge passing through start vertex except for the parent’s edge cand.edges <- edges[!(edges %in% edge.index)] #if there are more than one candidate edges, choose one randomly if (length(cand.edges) > 1) { new.edge <- sample(cand.edges, 1) } else { new.edge <- cand.edges } newlength = as.numeric(E(g)[new.edge]$weight) if (newlength < one.dist && length(cand.edges) != 0) { one.dist = newlength } temp.edges <- get.edges(g, new.edge) #get the candidate edge child <- c(new.edge, one.dist, newlength - one.dist, start, temp.edges[!(temp.edges %in% start)])} #if the generated length exceeds dist.to.end #same as last loop if (distance > dist.to.end) { one.dist <- distance - dist.to.end edges <- incident(g, end) cand.edges <- edges[!(edges %in% edge.index)] if (length(cand.edges) > 1) { 46 new.edge <- sample(cand.edges, 1) } else { new.edge <- cand.edges } newlength = as.numeric(E(g)[new.edge]$weight) if (newlength < one.dist && length(cand.edges) != 0) { one.dist = newlength } temp.edges <- get.edges(g, new.edge) child <- c(new.edge, one.dist, newlength - one.dist, end, temp.edges[!(temp.edges %in% end)])} #print(child) return(child) #print(length(child)) }

#function for poisson cluster process GeneratePoissonClusterOnNetwork <- function(g, N, S) { parents <- GenerateNPointsOnNetwork(g, N) children <- NULL #num.children <- rpois(1, S) + 1 for (i in 1 : nrow(parents)) { num.children <- rpois(1, S) + 1 #number of children is poisson distribution #print(num.children) ichildren <- GenerateNChildren(g, parents[i, ], num.children) children <- rbind(children, ichildren) } children <- as.data.frame(children) #print(children) children.graph <- AddChildrenOnNetwork(g, children) return(children.graph) } 47 B.1.3 Regular Process

#function for generating regular pattern GenerateRegularOnNetwork <- function(g, num.veritces, N, delta) { #num.vertices <- g$num.v meas <- sum(E(g)$weight) #length of all the edges #print(meas) lamda <- N / meas P <- exp(-lamda * pi * delta ^ 2) #probability of each point can be retained #print(P) #children <- NULL for (i in 1 : N) { graph <- AddOnePointOnNetwork(g) indicator <- sample(0 : 1, size = 1, prob = c(1 - P, P)) #check each point whether it should be retained or not if (indicator == 1) { num.points <- graph$num.v #print(num.points) newg <- graph$g if (num.points > num.vertices + 1) { new.distance <- shortest.paths(newg, v = num.points, to = c((num.vertices + 1) : (num.points - 1)), weights = E(newg)$weight) #print(min(new.distance)) if (min(new.distance) > 0.5) { g <- newg } } else { g <- graph$g } } } return(list(g = g, num.v = vcount(g))) 48 }

B.1.4 Random Network

#function for generating a random network GenerateRandomNetwork <- function(m, size, p) { x <- runif(1, 0, size) y <- runif(1, 0, size) coord <- cbind(x, y) #the first point, choose randomly #for each point generated, the distance to every other point is approporiate for (i in 2 : m) { shortest <- 0 while (shortest <= 0.5) { x <- runif(1, 0, size) y <- runif(1, 0, size) xy <- cbind(rep(x, i - 1), rep(y, i - 1)) mat <- xy - coord shortest <- min(mat[, 1]^2 + mat[, 2]^2) #print(shortest) } coord <- rbind(coord, c(x, y)) } #print(coord) dist.mat <- as.matrix(dist(coord)) #max.distance <- max(dist.mat) adj.mat <- matrix(0, nrow = m, ncol = m) near <- 3 #to make the network nicer, only connect the near points for (i in 1 : (m - near)) { #find the nearest 3 points dist <- dist.mat[i, i : m]#no repeat rank <- rank(dist) connect <- sample(0:1, size = near, replace = TRUE, prob = c(1 - p, p)) 49 for (j in 1 : near) { num <- 1 : (m - i + 1) n <- num[rank == (j + 1)] #get the place corresponding the jth nearest point if (adj.mat[i, (i - 1 + n)] == 0 && dist.mat[i, (i - 1 + n)] <= 8) { adj.mat[i, i - 1 + n] <- connect[j] adj.mat[i - 1 + n, i] <- connect[j] } } #if a point is isolated, connect to its nearest point if (max(adj.mat[i, ]) == 0) { nn <- (1 : (m - i + 1))[rank == 2] adj.mat[i, i - 1 + nn] <- 1 adj.mat[i - 1 + nn, i] <- 1 } } for (i in (m - near + 1) : m) { d <- dist.mat[i, ] r <- rank(d) #print(adj.mat[i, r == 2]) #only connect the nearest point if (adj.mat[i, r == 2] == 0) { adj.mat[i, r == 2] <- 1 adj.mat[r == 2, i] <- 1 #print(adj.mat[i, ]) } } random.network <- GenerateNetwork(coord, adj.mat) num.v <- random.network$num.v g <- random.network$g all.distance <- c() new.distance <- c() 50 for (i in 1 : (num.v - 1)) { new.distance <- shortest.paths(random.network$g, v = i, to = c((i + 1) : num.v), weights = E(g)$weight) all.distance <- append(all.distance, new.distance) } return(list(g = g, num.v = num.v, max = max(all.distance))) }

B.1.5 Car Crash Pattern on the MN Roads

#this script is designed to generate random points on road network library(sp) library(rgdal) library(igraph)

#read the shape file for major road of Minnesota alldata <- readOGR(dsn = "/Users/xinyuechang/Documents/summer research/major_road", layer = "mda_major_roads_cartographic") class(alldata) slotNames(alldata) data <- alldata[alldata$ROAD_CLASS != 0,] #only consider interstate highway and US highway #plot(data, border = "grey") coord <- coordinates(data) length <- data$LENGTH all.length <- sum(length) n <- length(coord)

GenerateOnePoint <- function() { road <- sample(1:n, size = 1, prob = length/all.length) #print(road) loc <- length[road]*runif(1) points <- coord[[road]] 51 line <- NULL for (i in 1 : length(points)) { line <- rbind(line, points[[i]]) } line <- as.matrix(line) d <- 0 for (i in 2 : nrow(line)) { new.d <- dist(rbind(line[i-1, ], line[i, ]), method = "euclidean", diag = FALSE, p = 2) d <- d + new.d if (d > loc) { #print(i) res <- d - loc E1 <- line[i-1,1] N1 <- line[i-1,2] E2 <- line[i,1] N2 <- line[i,2] E <- E2 + (res/new.d)*(E1 - E2) N <- N2 + (res/new.d)*(N1 - N2) break } } return(c(E,N)) } RANDOM <- NULL #n <- length(coord) for (k in 1 : 300) { random <- GenerateOnePoint() RANDOM <- rbind(RANDOM, random) } RANDOM <- as.matrix(RANDOM) row.names(RANDOM) <- c() 52 #points(RANDOM[,1],RANDOM[,2],pch = 8) # prepare UTM coordinates matrix utmcoor<-SpatialPoints(RANDOM, proj4string=CRS("+proj=utm +zone=15")) #utmdata$X and utmdata$Y are corresponding to UTM Easting and Northing. #zone= UTM zone # converting longlatcoor<-spTransform(utmcoor,CRS("+proj=longlat")) randomcoords <- as.matrix(cbind(longlatcoor$coords.x1, longlatcoor$coords.x2)) write.csv(randomcoords, file = "/Users/xinyuechang/Documents/summer research/newpointsdata/random100.csv",row.names = FALSE)

B.2 Python Code

#python script for generating inter-event distance of 100 random points sets #this was entered into python window of ArcGIS import arcpy import glob import os arcpy.env.workspace = "C:/Users/mathgrad/My Documents/ArcGIS/Projects/ MajorRoad2/MajorRoad2.gdb" INPUT = "C:/Users/mathgrad/summer research/newpointsdata/random[1-9].csv" for file in glob.glob(INPUT): arcpy.MakeXYEventLayer_management(file, "V1", "V2", "random_layer","","") arcpy.na.AddLocations("OD Cost Matrix", "Origins", "random_layer", "CurbApproach # 0", "300 points",append = "CLEAR") arcpy.na.AddLocations("OD Cost Matrix", "Destinations", "random_layer", "CurbApproach # 0", "300 points",append = "CLEAR") arcpy.na.Solve("OD Cost Matrix","HALT") filename = os.path.relpath(file,"C:/Users/mathgrad/summer research/newpointsdata") name = filename.split(’.’)[0] name = name + "d.csv" direct = "C:/Users/mathgrad/summer research/300RandomDistance/" 53 arcpy.CopyRows_management("OD Cost Matrix\Lines", direct + name) arcpy.Delete_management("random_layer","Layer")