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STAT6016 Spatial Data Analysis

创建日期：2024-08-12 17:27:15

所属分类：统计学statistics

标签： STAT6016

Spatial Data Analysis

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STAT6016 Spatial Data Analysis

Background and problem setting

Consider Hong Kong as a 2D rectangular spatial set

S = {(x, y) : 113.831 ≤ x ≤ 114.409, 22.153 ≤ y ≤ 22.546},

where x and y denote the longitude and latitude measures, respectively. We may define the “area”

of a sub-region R of Hong Kong to be ∫R dx dy. Thus, the total “area” of Hong Kong is given by

(114.409? 113.831)(22.546? 22.153) = 0.227154. The (x, y)-coordinates of the locations of the 34

incidents are given in the dataset, which form a sample S = {S1, . . . , S34} of a point process on S .

Hong Kong is divided into 18 districts. In this study we assume a rectangular shape for each

district, whose four boundaries are given in the spreadsheet “18 districts” of data.xlsx, measured in

longitude and latitude. The spreadsheet also provides the area and population density (in million per

unit area) of each district. Define a covariate V (s), for s ∈ S , by

V (s) = population density of district d if s is located in district d.

Thus, V (s) has a constant value for all s belonging to the same district.

Assume that S follows a Cox point process model driven by a random intensity Λ satisfying

ln Λ(s) = lnWα + βV (s), s ∈ S ,

where β ∈ R is an unknown coefficient and Wα denotes a gamma (α, α) random variable with an

unknown parameter α > 0 and density function

fW (w|α) = α

αwα?1e?αw

Γ(α)

, w > 0.

Note that Wα has mean 1 and variance 1/α.

Complete TASKS 1–4 with the help of a convenient computer software of your choice.

TASK 1 (4%)

Display Hong Kong as a rectangle on a 2D graph, with x- and y-axes representing longitude

and latitude, respectively.

Show on the same graph (i) the boundaries of the 18 districts, and (ii) the 34 points in S .

TASK 2 (8%)

Show that the proposed Cox point process has a parametric intensity given by

ρ(s|β) = eβV (s), s ∈ S ,

and a parametric conditional intensity given by

λ(s,S|α, β) = e

βV (s)(α + #S)

α +

∫

eβV (s)ds

, s ∈ S , S ∈ E (exponential space).

[Hint: derive first the density f(S) of the Cox point process, then use the fact that the conditional

intensity is given by f({s} ∪ S)/f(S).]

With the help of a numerical optimisation package, calculate the maximum conditional

pseudo-likelihood estimates (α?Cox, β?Cox) of (α, β).

Based on the above results, estimate the expected number of incidents occurring in each of

the 18 districts.

TASK 3 (8%)

Assuming a Poisson point process model for S with intensity ρ(s|β) = eβV (s), calculate the

maximum likelihood estimate β?Poi of β.

Generate 10000 parametric bootstrap samples from the Poisson point process model with

intensity ρ(·|β?Poi).

Based on each bootstrap sample, calculate the maximum conditional pseudo-likelihood

estimate α Cox of α, following the same procedure as described in TASK 2. Plot the

empirical distribution function of the 10000 replicates of α Cox thus obtained.

For α > 0, the Cox point process has a reweighted pair correlation function equal to a constant

1+1/α > 1, suggesting a tendency of “clusterings” of the violent incidents. As α→∞, the reweighted

pair correlation function converges to 1 and Wα converges to a non-random constant 1. In the limiting

case α =∞, the Cox point process reduces to a Poisson point process with intensity ρ(s|β), and the

violent incidents become independently located.

TASK 4 (5%)

With the help of the parametric bootstrap distribution found in TASK 3, suggest a

procedure for testing the null hypothesis (H0) that the violent incidents are independently

located, against the alternative (H1) that the incidents have a clustering tendency.

Report a p-value for the test.

[Hint: You may consider using α?Cox as a test statistic, and explain why it is an appropriate choice.]

Points to note

In the main text of your report, show and explain your steps, and display formulae in their

conventional mathematical form. Do not explain anything using computer code.

Attach your computer code to your report as an appendix. Include brief comments on lines

which involve complicated operations.

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COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 CIVL2010 ECMT2150 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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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