National Institute of Statistical Sciences
Workshop on Statistics and Counterterrorism
G. P. Patil
November 20, 2004
New York University
1
Geoinformatic Surveillance System
Geoinformatic spatio-temporal
data from a variety of data
products and data sources with
agencies, academia, and industry
Masks, filters
Spatially
distributed
response
variables
Hotspot
analysis
Prioritization
Decision
support
systems
Masks, filters
Indicators, weights
2
Homeland
Security
Disaster
Management
Public
Health
Ecosystem
Health
Other Case
Studies
Statistical Processing: Hotspot Detection, Prioritization, etc.
Arbitrary Data Model, Data Format, Data Access
Application Specific De Facto Data/Information Standard
Standard or De Facto Data Model, Data Format, Data Access
Data Sharing, Interoperable Middleware
Agency Databases
Thematic Databases
Other Databases
3
4
The Spatial Scan Statistic


Move a circular window across the map.
Use a variable circle radius, from zero up
to a maximum where 50 percent of the
population is included.
5
A small sample of the circles used
6
Detecting Emerging Clusters



Instead of a circular window in two
dimensions, we use a cylindrical window in
three dimensions.
The base of the cylinder represents space,
while the height represents time.
The cylinder is flexible in its circular base and
starting date, but we only consider those
cylinders that reach all the way to the end of
the study period. Hence, we are only
considering ‘alive’ clusters.
7
West Nile Virus Surveillance in
New York City
2000 Data: Simulation/Testing of
Prospective Surveillance System
 2001 Data: Real Time Implementation
of Daily Prospective Surveillance

8
West Nile Virus Surveillance
in New York City
Major epicenter on Staten Island




Dead bird surveillance system: June 14
Positive bird report: July 16 (coll. July 5)
Positive mosquito trap: July 24 (coll. July 7)
Human case report: July 28 (onset July 20)
9
10
Hospital Emergency Admissions in
New York City




Hospital emergency admissions data from a
majority of New York City hospitals.
At midnight, hospitals report last 24 hour of
data to New York City Department of Health
A spatial scan statistic analysis is performed every
morning
If an alarm, a local investigation is conducted
11
Issues
12
Geospatial Surveillance
13
Spatial Temporal Surveillance
14
Syndromic Crisis-Index Surveillance
15
Hotspot Prioritization
16
17
National Applications











Biosurveillance
Carbon Management
Coastal Management
Community Infrastructure
Crop Surveillance
Disaster Management
Disease Surveillance
Ecosystem Health
Environmental Justice
Sensor Networks
Robotic Networks










Environmental Management
Environmental Policy
Homeland Security
Invasive Species
Poverty Policy
Public Health
Public Health and
Environment
Syndromic Surveillance
Social Networks
Stream Networks
18
Geographic Surveillance and Hotspot Detection for Homeland Security: Cyber Security and Computer Network Diagnostics
Securing the nation's computer networks from cyber attack is an important aspect of Homeland Security. Project develops diagnostic
tools for detecting security attacks, infrastructure failures, and other operational aberrations of computer networks.
Geographic Surveillance and Hotspot Detection for Homeland Security: Tasking of Self-Organizing Surveillance Mobile Sensor
Networks
Many critical applications of surveillance sensor networks involve finding hotspots. The upper level set scan statistic is used to guide
the search by estimating the location of hotspots based on the data previously taken by the surveillance network.
Geographic Surveillance and Hotspot Detection for Homeland Security: Drinking Water Quality and Water Utility Vulnerability
New York City has installed 892 drinking water sampling stations. Currently, about 47,000 water samples are analyzed annually. The
ULS scan statistic will provide a real-time surveillance system for evaluating water quality across the distribution system.
Geographic Surveillance and Hotspot Detection for Homeland Security: Surveillance Network and Early Warning
Emerging hotspots for disease or biological agents are identified by modeling events at local hospitals. A time-dependent crisis index
is determined for each hospital in a network. The crisis index is used for hotspot detection by scan statistic methods
Geographic Surveillance and Hotspot Detection for Homeland Security: West Nile Virus: An Illustration of the Early Warning
Capability of the Scan Statistic
West Nile virus is a serious mosquito-borne disease. The mosquito vector bites both humans and birds. Scan statistical detection of
dead bird clusters provides an early crisis warning and allows targeted public education and increased mosquito control.
Geographic Surveillance and Hotspot Detection for Homeland Security: Crop Pathogens and Bioterrorism
Disruption of American agriculture and our food system could be catastrophic to the nation's stability. This project has the specific aim
of developing novel remote sensing methods and statistical tools for the early detection of crop bioterrorism.
Geographic Surveillance and Hotspot Detection for Homeland Security: Disaster Management: Oil Spill Detection, Monitoring, and
Prioritization
The scan statistic hotspot delineation and poset prioritization tools will be used in combination with our oil spill detection algorithm to
provide for early warning and spatial-temporal monitoring of marine oil spills and their consequences.
Geographic Surveillance and Hotspot Detection for Homeland Security: Network Analysis of Biological Integrity in Freshwater
Streams
This study employs the network version of the upper level set scan statistic to characterize biological impairment along the rivers and
streams of Pennsylvania and to identify subnetworks that are badly impaired.
19
Center for Statistical Ecology and Environmental Statistics
G. P. Patil, Director
Hotspot Detection Innovation
Upper Level Set Scan Statistic
Attractive Features









Identifies arbitrarily shaped clusters
Data-adaptive zonation of candidate hotspots
Applicable to data on a network
Provides both a point estimate as well as a confidence set for the
hotspot
Uses hotspot-membership rating to map hotspot boundary
uncertainty
Computationally efficient
Applicable to both discrete and continuous syndromic responses
Identifies arbitrarily shaped clusters in the spatial-temporal domain
Provides a typology of space-time hotspots with discriminatory
surveillance potential
20
Candidate Zones for Hotspots


Goal: Identify geographic zone(s) in which a response is
significantly elevated relative to the rest of a region
A list of candidate zones Z is specified a priori
– This list becomes part of the parameter space and the zone
must be estimated from within this list
– Each candidate zone should generally be spatially connected,
e.g., a union of contiguous spatial units or cells
– Longer lists of candidate zones are usually preferable
– Expanding circles or ellipses about specified centers are a
common method of generating the list
21
Time
Scan Statistic Zonation for
Circles and Space-Time
Cylinders
Space
Cholera outbreak along a river flood-plain
Outbreak expanding in time
•Small circles miss much of the outbreak
•Large circles include many unwanted cells
•Small cy linders miss much of the outbreak
•Large cylinders include many unwanted cells
22
ULS Candidate Zones




Question: Are there data-driven (rather than a priori) ways of selecting
the list of candidate zones?
Motivation for the question: A human being can look at a map and
quickly determine a reasonable set of candidate zones and eliminate
many other zones as obviously uninteresting. Can the computer do the
same thing?
A data-driven proposal: Candidate zones are the connected
components of the upper level sets of the response surface. The
candidate zones have a tree structure (echelon tree is a subtree),
which may assist in automated detection of multiple, but
geographically separate, elevated zones.
Null distribution: If the list is data-driven (i.e., random), its variability
must be accounted for in the null distribution. A new list must be
developed for each simulated data set.
23
ULS Scan Statistic




Data-adaptive approach to reduced parameter space 0
Zones in 0 are connected components of upper level sets of the
empirical intensity function Ga = Ya / Aa
Upper level set (ULS) at level g consists of all cells a where
Ga  g
Upper level sets may be disconnected. Connected components are
the candidate zones in 0

These connected components form a rooted tree under set inclusion.
– Root node = entire region R
– Leaf nodes = local maxima of empirical intensity surface
– Junction nodes occur when connectivity of ULS changes with
falling intensity level
24
Upper Level Set (ULS) of Intensity
Surface
Intensity G
g
Z1
Z2
Z3
Hotspot zones at level g
(Connected Components of upper level set)
Region R
25
Changing Connectivity of ULS
as Level Drops
Intensity G
g
g
Z2
Z1
Z4
Z3
Z5
Z6
Region R
26
ULS Connectivity Tree
Intensity G
g
Z3
Z2
Z1
Schematic
intensity “surface”
A
g
Z4
Z5
Z6
B
C
N.B. Intensity surface is cellular (piece-wise constant), with only finitely many levels
A, B, C are junction nodes where multiple zones coalesce into a single zone
27
A confidence set of hotspots on the ULS tree. The
different connected components correspond to different
hotspot loci while the nodes within a connected
component correspond to different delineations of that
hotspot
Tessellated Region R
MLE
Junction Node
Alternative
Hotspot Delineation
Alternative
Hotspot Locus
28
Network Analysis of
Biological Integrity in
Freshwater Streams
29
New York City
Water Distribution
Network
30
NYC Drinking Water Quality
Within-City Sampling Stations
• 892 sampling stations
• Each station about 4.5 feet high and
draws water from a nearby water main
• Sampling frequency increased after 9-11
Currently, about 47,000 water samples
analyzed annually
• Parameters analyzed:






Bacteria
Chlorine levels
pH
Inorganic and organic pollutants
Color, turbidity, odor
Many others
31
Network-Based Surveillance






Subway system surveillance
Drinking water distribution system
surveillance
Stream and river system surveillance
Postal System Surveillance
Road transport surveillance
Syndromic Surveillance
32
Syndromic Surveillance

Symptoms of disease such as diarrhea,
respiratory problems, headache, etc

Earlier reporting than diagnosed disease

Less specific, more noise
33
Syndromic Surveillance
(left) The overall procedure, leading from admissions records to
the crisis index for a hospital. The hotspot detection algorithm is then
applied to the crisis index values defined over the hospital network.
(right) The -machine procedure for converting an event stream into a
parse tree and finally into a probabilistic finite state automaton (PFSA).
34
Experimental Validation
28
29
23
24
Formal Language Events:
a – green to red or red to green
b – green to tan or tan to green
c – green to blue or blue to green
d – red to tan or tan to red
e – blue to red or red to blue
f – blue to tan or tan to blue
7
26
25
27
6
17
5
18
19
20
21
22
4
12
13
14
15
16
3
7
6
2
1
1
8
2
10
9
3
11
4
5
0
0
1
2
3
4
5
6
7
8
9
10
Pressure sensitive floor
a
Wall following
a
8
9
d
c
a
5
1
2
4
3
f
d
c
7
f
12
c
0
f
6
d
a
c
11
d
f
0
a, b, c, d, e, f
Random walk
f
10
d
c
Clockwise
Counter-Clockwise
a
Target Behavior
Analyze String Rejections
35
Emergent Surveillance Plexus (ESP)
Surveillance Sensor Network Testbed
Autonomous Ocean Sampling Network
Types of Hotspots


Hotspots due to multiple, localized, stationary
sources
Hotspots corresponding to areas of interest in a
stationary mapped field

Time-dependent, localized hotspots

Hotspots due to moving point sources
36
Ocean SAmpling MObile Network
OSAMON
37
Ocean SAmpling MObile Network
OSAMON Feedback Loop

Network sensors gather preliminary data

ULS scan statistic uses available data to
estimate hotspot

Network controller directs sensor
vehicles to new locations

Updated data is fed into ULS scan
statistic system
38
SAmpling MObile Networks (SAMON)
Additional Application Contexts

Hotspots for radioactivity and chemical or biological agents
to prevent or mitigate the effects of terrorist attacks or to
detect nuclear testing

Mapping elevation, wind, bathymetry, or ocean currents to
better understand and protect the environment

Detecting emerging failures in a complex networked system
like the electric grid, internet, cell phone systems

Mapping the gravitational field to find underground
chambers or tunnels for rescue or combat missions
39
Sensor Devices
Mote, Smart Dust: Small, flexible, low-cost sensor node
RF Component of Alcohol Sensor
Miniaturized Spec Node Prototype
Giner’s Transdermal Alcohol Sensor
40
Scalable Wireless Geo-Telemetry
with Miniature Smart Sensors
Geo-telemetry enabled sensor nodes deployed by a UAV into a wireless ad hoc mesh
network: Transmitting data and coordinates to TASS and GIS support systems
41
Architectural Block Diagram of Geo-Telemetry Enabled
Sensor Node with Mesh Network Capability
42
Standards Based Geo-Processing
Model
43
UAV Capable of Aerial Survey
44
Data Fusion Hierarchy for Smart Sensor
Network with Scalable Wireless
Geo-Telemetry Capability
45
Wireless Sensor Networks
for Habitat Monitoring
46
Target Tracking in Distributed
Sensor Networks
47
Video Surveillance and Data Streams
48
Video Surveillance and Data Streams
Turning Video into Information
Measuring Behavior by Segments
• Customer Intelligence
• Enterprise Intelligence
• Entrance Intelligence
• Media Intelligence
• Video Mining Service
49
Deterministic Finite Automata (DFA)
b
a
b
start
c
a
c
b
Directed Graph (loops & multiple edges permitted) such that:
• Nodes are called States
• Edges are called Transitions
• Distinguished initial (or starting) state
• Transitions are labeled by symbols from a given finite alphabet,  = {a, b, c, . . . }
• The same symbol can label several transitions
• A given symbol can label at most one transition from a given state (deterministic)
50
Deterministic Finite Automata (DFA)
Formal Definition
b
a
b
start
c
a
b
c
Quadruple (Q, q0 , ,  ) such that:
• Q is a finite set of states
•  is a finite set of symbols, called the alphabet
• q0Q is the initial state
•  : Q    Q  {Blocked} is the transition function:
  (q, a) = Blocked
if there is no transition from q labeled by a
  (q, a) = q'
if a is a transition from q to q'
51
DFA and Strings
b
a
b
start
c
c
a
b
Any path through the graph starting from the initial state
determines a string from the alphabet.
Example: The blue dashed path determines the string a b c a
Conversely, any string from the alphabet is either
blocked or determines a path through the graph.
Example: The following strings are blocked:
c, aa, ac, abb, etc.
Example: The following strings are not blocked:
a, b, ab, bb, etc.
The collection of all unblocked strings is called the language accepted or
determined by the DFA (all states are “final” in our approach)
52
Strings and Languages
 = (finite) alphabet
* = set of all (finite) strings from 
A language is any subset of *.
Not all languages can be determined by a DFA.
Different DFAs can accept the same language
Let i   
  (i-fold cartesian product).
i consists of all strings of length i.
Then, * decomposes as
 
*

i 1
i   0
1
2
53
Probabilistic Finite Automata (PFA)
b, 1
a, .8
b, .2
start
q0
c, .5
c, .6
a, .4
b, .5
A PFA is a DFA (Q, q0 , ,  ) with a probability attached to each
transition such that the sum of the probabilities across all transitions
from a given node is unity.
Formally, p: Q    [0, 1] such that
• p(q, a) = 0 if and only if  (q, a) = Blocked
•  p(q, a)  1 for all q  Q
a
Multiplying branch probabilities lets us assign a probability value
(q0, s) to each string s in *. E.G., (q0, abca)=(.8)1(.6)(.4)=.192
54
Properties of (q0, s)
• For fixed q0, (q0, s) is a measure on *
• Support of  is the language accepted by the DFA
• For fixed q0, (q0, s) is a probability measure on 
i
i
(  = strings of length i )
This probability measure is written as (i).
• Given a probability distribution w(i) across string lengths i,

 (q0 , s)   w(i)  (i ) (q0 , s)
i 0
defines a probability measure across *, called the
w-weighted probability measure of the PFA.
If all w(i) are positive, then the support of  is also the
language accepted by the underlying DFA.
55
Distance Between Two PFA
Let A and B be two PFAs on the same alphabet 
Let w(i) be a probability distribution across string lengths i
Let A and B be the w-weighted probability measures of A and B
Define the distance between A and B as the variational distance
between the probability measures A and B :
d(A, B) = || A  B ||
56
Crop Attack Decision Support System
Site Identification Module
Crops
Key Crop
Areas
NOAA
Weather
Threat
Locations
Signature Development Module
Plants
Infected
Non-infected
Sentinel
Hyperspectral
Imagery
Data
Processing
Anomaly
Report
Ground
Cameras
Air/Space
Platforms
Signature
Library
Ground
Truthing
57
Crop Biosurveillance/Biosecurity
58
Crop Biosurveillance/Biosecurity
Data Processing Module
Hyperspectral
Imagery
Image
Segmentation
(segmentation)
(hyperclustering)
of raster grid
Tessellation
Signature
Similarity
Map
Proxy Signal
(per segment)
Similarity
Index
(per segment)
Signature
Library
Disease
Signature
Hotspot/
Anomaly
Detection
59
Prioritization Innovation
Partial Order Set Ranking
We also present a prioritization innovation. It lies in the
ability for prioritization and ranking of hotspots based
on multiple indicator and stakeholder criteria without
having to integrate indicators into an index, using Hasse
diagrams and partial order sets. This leads us to early
warning systems, and also to the selection of
investigational areas.
60
HUMAN ENVIRONMENT INTERFACE
LAND, AIR, WATER INDICATORS
for land - % of undomesticated land, i.e., total land area-domesticated (permanent
crops and pastures, built up areas, roads, etc.)
for air - % of renewable energy resources, i.e., hydro, solar, wind, geothermal
for water - % of population with access to safe drinking water
1
2
3
5
13
22
39
45
47
51
52
59
61
64
77
78
81
RANK COUNTRY
LAND
AIR
WATER
Sweden
Finland
Norway
Iceland
Austria
Switzerland
Spain
France
Germany
Portugal
Italy
Greece
Belgium
Netherlands
Denmark
United Kingdom
Ireland
69.01
76.46
27.38
1.79
40.57
30.17
32.63
28.34
32.56
34.62
23.35
21.59
21.84
19.43
9.83
12.64
9.25
35.24
19.05
63.98
80.25
29.85
28.10
7.74
6.50
2.10
14.29
6.89
3.20
0.00
1.07
5.04
1.13
1.99
100
98
100
100
100
100
100
100
100
82
100
98
100
100
100
100
100
61
Hasse Diagram
(all countries)
1
2
3
8
9
13
17
22
4
10
45
15
25
26
36
46
6
12
23
28
43
5
48
11
14
18
21
32
27
39
47
7
29
41
50
31
56
20
40
35
51
54
19
38
33
42
52
16
53
60
24
44
55
65
68
76
71
72
34
49
66
69
30
73
37
82
80
114
86
102
88
112
113
57 58 61 62 63 64 67 74 75 77 78 79 83 84 85 93 94 96 98 99 101 104 111 131
59
81
70
89
95
100
97
87
107
90
103
105
135
117
91
106
116
119
92
108
110
109
118
122
130
115
120
121
127
124
133
123
125
126
129
138
140
132
134
128
137
139
141
136
62
Hasse Diagram
(Western Europe)
Norway
Sweden
Finland
Austria
Switz.
Ireland
Italy
Portugal
Greece
Spain
Den.
France
UK
Belgium
Germany
Neth.
63
Ranking Partially Ordered Sets – 5
Linear extension decision tree
Poset
(Hasse Diagram)
e
a
b
c
d
f
b
a
c
e
b
b
b
e
d
d
e
d
c
d
e
f
d
d
e
c
e
c
f
d
a
e
f
d e
d
f
e
d
a
c
c
f
e
f
e f f e f e f f e f e f e
Jump Size: 1 3 3 2 3 5 4 3 3 2 4 3 4 4 2 642
f
f
f
Cumulative Rank Frequency Operator – 5
An Example of the Procedure
In the example from the preceding slide, there are a total of 16 linear
extensions, giving the following cumulative frequency table.
Rank
Element
1
2
3
4
5
6
a
9
14
16
16
16
16
b
7
12
15
16
16
16
c
0
4
10
16
16
16
d
0
2
6
12
16
16
e
0
0
1
4
10
16
f
0
0
0
0
6
16
65
Each entry gives the number of linear extensions in which the element (row
label) receives a rank equal to or better that the column heading
Cumulative Rank Frequency Operator – 6
An Example of the Procedure
Cumulative Frequency
16
a
b
c
d
e
f
12
16
8
4
0
1
2
3
4
5
6
Rank
The curves are stacked one above the other and the result is a
66
linear ordering of the elements: a > b > c > d > e > f
Cumulative Rank Frequency Operator – 7
An example where F must be iterated
2
F
F
Original Poset
(Hasse Diagram)
f
a
b
c
a
a
f
f
e
e
b
b
ad
ad
c
g
h
c
e
g
h
d
g
67
h
Incorporating Judgment
Poset Cumulative Rank Frequency Approach
• Certain of the indicators may be deemed more
important than the others
• Such differential importance can be accommodated by
the poset cumulative rank frequency approach
• Instead of the uniform distribution on the set of linear
extensions, we may use an appropriately weighted
probability distribution  , e.g.,
 ( )  w0  w1n1 ( )  w2n2 ( )   wp np ( )
68
69
70
71
72
Space-Time Poverty Hotspot Typology

Federal Anti-Poverty Programs have had
little success in eradicating pockets of
persistent poverty

Can spatial-temporal patterns of poverty
hotspots provide clues to the causes of
poverty and lead to improved locationspecific anti-poverty policy ?
73
Covariate Adjustment
Known Covariate Effects (age, population size, etc.)
Ya  count in cell a
Ya
Poisson(a Aa ) where a  unknown relative risk for cell a
Aa  known numerical covariate adjustment
Hotspot Hypothesis Testing Model
H 0 : a are equal for all cells a (constant relative risk)
H1 : a take two distinct values, an elevated value in an unknown zone Z
and a smaller value outside Z
List of candidate zones Z (ULS approach)
All connected components of upper level sets
of the adjusted cellular surface Ya / Aa
74
Covariate Adjustment
Given Covariates, Unknown Effects
Ya
Poisson(a Aa ) where a  unknown relative risk for cell a
Aa  unknown covariate adjustment
GLM Model
X a  vector of known covariate values for cell a
β  vector of unknown covariate effects
Model: log( Aa )  XTa β or log(a Aa )  a  log(a )  XTa β
Hotspot Hypothesis Testing Model
H 0 : a are equal for all cells a (constant relative risk)
H1 : a take two distinct values, an elevated value in an unknown zone Z
and a smaller value outside Z
List of candidate zones Z (ULS approach)
All connected components of upper level sets
of the adjusted cellular surface Ya / Aa
Here the model must be fitted under the null hypothesis before
determining the adjustments Aa and the candidate zones Z
75
Incorporating Spatial Autocorrelation
Ignoring autocorrelation typically results in:
 under-assessment of variability
 over-assessment of significance (H0 rejected too frequently)
How can we account for possible autocorrelation?
GLMM (SAR) Model
Ya = count in cell a
Ya distributed as Poisson
a = log(E[Ya])
The Ya are conditionally independent given the a
The a are jointly Gaussian with a Simultaneous AutoRegressive (SAR) specification
Here, a  E[a ]
a  a   Wab (a  a )   a
b
 a are iid N (0,  2 )
Wab is a spatial weight expressing the "degree of association"
between cells a and b (Take Waa  0 and Wa  1)
Thus, the residual a  a for cell a is a deflated (by  ) weighted average
of the residuals for neighboring cells plus a disturbance term  a
76
Incorporating Spatial Autocorrelation
SAR Model: a  a    Wab (a  a )   a
b
Matrix Form: η  μ   W( η  μ)  ε
η  μ  (I   W )1 ε
η

MVN μ,  (I   W) (I   W) 
2
T
1

Unknown Parameters: a ,  ,  2
Special Cases:  2  0  classical (iid) spatial scan
( is not identifiable here)
  0,  2  0  overdispersed classical scan
77
Incorporating Spatial Autocorrelation
GLMM (SAR) Model
Ya Poisson(exp( a ))  Poisson(exp( a ) exp( a   a ))  Poisson(a Aa )

where η MVN μ,  (I   W ) (I   W ) 
2
T
1

Hotspot Hypothesis Testing Model
H 0 : a are equal (to  ) for all cells a (constant relative risk)
H1 : a take two distinct values, an elevated value in an unknown zone Z
and a smaller value outside Z
List of candidate zones Z (ULS approach)
All connected components of upper level sets
of the adjusted cellular surface Ya / Aa where Aa  exp(a   )
Here the model must be fitted under the null hypothesis (a = ) before
determining the adjustments Aa and the candidate zones Z
78
Spatial Autocorrelation Plus Covariates
In the SAR model,
where η
Ya

Poisson(exp(a ))
MVN E[ η],  (I   W )T (I   W ) 
2
1
,
express the mean of η as
E[ η]  μ  Xβ
and formulate the Hotspot Hypothesis Testing Model in terms
of the constant term μ in this expression.
79
CAR Model
The entire formulation is similar for Conditional AutoRegressive (CAR) specs
except that the form of the variance-covariance matrix of  is changes.
In the CAR model, Ya
η
Poisson(exp( a )) where
MVN  E[ η],  2 (I   W* ) 1 A*  and A* is diagonal.
However, parameters in CAR and SAR have very different interpretations.
In CAR, the conditional variances are
*
Var( a | b , b  a)   2 Aaa
which (strangely) do not depend on the autocorrelation parameter  .
In SAR, the conditional variances are

Var( a | b , b  a )   2 1   2  b Wba2

This expression is intuitively appealing since the conditional variances
are decreasing functions of  2 and are smallest for cells a with many
strongly associated neighbors (relatively large Wba2 for many b).
80
Geoinformatic Surveillance System
Geoinformatic spatio-temporal
data from a variety of data
products and data sources with
agencies, academia, and industry
Masks, filters
Spatially
distributed
response
variables
Hotspot
analysis
Prioritization
Decision
support
systems
Masks, filters
Indicators, weights
81
Homeland
Security
Disaster
Management
Public
Health
Ecosystem
Health
Other Case
Studies
Statistical Processing: Hotspot Detection, Prioritization, etc.
Arbitrary Data Model, Data Format, Data Access
Application Specific De Facto Data/Information Standard
Standard or De Facto Data Model, Data Format, Data Access
Data Sharing, Interoperable Middleware
Agency Databases
Thematic Databases
Other Databases
82
Descargar

NIST Presentation - National Institute of Statistical Sciences