Algorithms for Port of Entry
Inspection for WMDs
Fred S. Roberts
DyDAn Center
Rutgers University
1
Port of Entry Inspection Algorithms
•Goal: Find ways to intercept illicit
nuclear materials and weapons
destined for the U.S. via the
maritime transportation system
•Currently inspecting only small
% of containers arriving at ports
•Even inspecting 8% of containers in Port of
NY/NJ might bring international trade to a halt
(Larrabbee 2002)
2
Port of Entry Inspection Algorithms
•Aim: Develop decision support algorithms that
will help us to “optimally” intercept illicit
materials and weapons subject to limits on delays,
manpower, and equipment
•Find inspection schemes that minimize total
“cost” including “cost” of false positives and
false negatives
Mobile Vacis: truckmounted gamma ray
imaging system
3
Port of Entry Inspection Algorithms
•My work on port of entry inspection has gotten
me and our students and colleagues to some
remarkable places.
Me on a Coast Guard
boat in a tour of the
harbor in Philadelphia
4
5
Sequential Decision Making Problem
•Stream of containers arrives at a port
•The Decision Maker’s Problem:
•Which to inspect?
•Which inspections next based on previous results?
•Approach:
–“decision logics”
–combinatorial optimization methods
–Builds on ideas of Stroud
and Saeger at Los Alamos
National Laboratory
–Need for new models
and methods
6
Sequential Diagnosis Problem
•Such sequential diagnosis problems arise in many
areas:
–Communication networks (testing connectivity, paging
cellular customers, sequencing tasks, …)
–Manufacturing (testing machines, fault diagnosis,
routing customer service calls, …)
–Artificial intelligence/CS (optimal derivation strategies
in knowledge bases, best-value satisficing search, coding
decision trees, …)
–Medicine (diagnosing patients, sequencing treatments,
…)
7
Sequential Decision Making Problem
•Containers arriving to be classified into categories.
•Simple case: 0 = “ok”, 1 = “suspicious”
•Inspection scheme: specifies which inspections are
to be made based on previous observations
8
Sequential Decision Making Problem
•Containers have attributes, each
in a number of states
•Sample attributes:
–Levels of certain kinds of chemicals or
biological materials
–Whether or not there are items of a certain
kind in the cargo list
–Whether cargo was picked up in a certain port
9
Sequential Decision Making Problem
•Currently used attributes:
–Does ship’s manifest set off an “alarm”?
–What is the neutron or Gamma emission
count? Is it above threshold?
–Does a radiograph image come up positive?
–Does an induced fission test come up positive?
Gamma
ray
detector
10
Sequential Decision Making Problem
•We can imagine many other attributes
•This project is concerned with general algorithmic
approaches.
•We seek a methodology not tied to today’s
technology.
•Detectors are evolving quickly.
11
Sequential Decision Making Problem
•Simplest Case: Attributes are in state 0 or 1
•Then: Container is a binary string like 011001
•So: Classification is a decision function F that
assigns each binary string to a category.
011001
F(011001)
If attributes 2, 3, and 6 are present, assign container to
category F(011001).
12
Sequential Decision Making Problem
•If there are two categories, 0 and 1, decision
function F is a Boolean function.
Example:
F(000) = F(111) = 1, F(abc) = 0 otherwise
This classifies a container as positive iff it has
none of the attributes or all of them.
1=
13
Sequential Decision Making Problem
•Given a container, test its attributes until know
enough to calculate the value of F.
•An inspection scheme tells us in which order to
test the attributes to minimize cost.
•Even this simplified problem is hard
computationally.
14
Sequential Decision Making Problem
•This assumes F is known.
•Simplifying assumption: Attributes are
independent.
•At any point we stop inspecting and output the
value of F based on outcomes of inspections so
far.
•Complications: May be precedence relations in
the components (e.g., can’t test attribute a4 before
testing a6.
•Or: cost may depend on attributes tested before.
•F may depend on variables that cannot be
directly tested or for which tests are too costly. 15
Sequential Decision Making Problem
•Such problems are hard computationally.
•There are many possible Boolean functions F.
•Even if F is fixed, problem of finding a good
classification scheme (to be defined precisely
below) is NP-complete.
•Several classes of functions F allow for efficient
inspection schemes:
–k-out-of-n systems
–Certain series-parallel systems
–Read-once systems
–“regular” systems
16
–Horn systems
Sensors and Inspection Lanes
•n types of sensors measure presence or absence of the n
attributes.
•Many copies of each sensor.
•Complication: different characteristics of sensors.
•Entities come for inspection.
•Which sensor of a given type to
use?
•Think of inspection lanes and
queues.
•Besides efficient inspection
schemes, could decrease costs by:
–Buying more sensors
–Change allocation of containers to sensor lanes.
17
Binary Decision Tree Approach
•Sensors measure presence/absence of attributes.
•Binary Decision Tree:
–Nodes are sensors or categories (0 or 1)
–Two arcs exit from each sensor node, labeled
left and right.
–Take the right arc when sensor says the
attribute is present, left arc otherwise
18
Binary Decision Tree Approach
•Reach category 1 from the
root only through the path
a0 to a1 to 1.
•Container is classified in
category 1 iff it has both
attributes a0 and a1 .
•Corresponding Boolean
function F(11) = 1, F(10) =
F(01) = F(00) = 0.
Figure 1
19
Binary Decision Tree Approach
•Reach category 1 from
the root by:
a0 L to a1 R a2 R 1 or
a0 R a2 R1
•Container classified in
category 1 iff it has
a1 and a2 and not a0 or
a0 and a2 and possibly a1.
•Corresponding Boolean
function F(111) = F(101)
= F(011) = 1, F(abc) = 0
otherwise.
Figure 2
20
Binary Decision Tree Approach
•This binary decision
tree corresponds to the
same Boolean function
F(111) = F(101) =
F(011) = 1, F(abc) = 0
otherwise.
However, it has one less
observation node ai. So,
it is more efficient if all
observations are equally
costly and equally likely.
Figure 3
21
Binary Decision Tree Approach
•Even if the Boolean function F is fixed, the
problem of finding the “optimal” binary decision
tree for it is very hard (NP-complete) if optimal
means smallest number of sensor nodes.
•For small n = number of attributes, can try to
solve it by brute force enumeration.
Port of Long Beach
•Even for n = 4, not practical. (n = 4 at Port of
Long Beach-Los Angeles)
22
Binary Decision Tree Approach
Promising Approaches:
•Heuristic algorithms, approximations to optimal.
•Special assumptions about the Boolean function F.
•For “monotone” Boolean functions, integer
programming formulations give promising
heuristics.
•Stroud and Saeger enumerate all
“complete,” monotone Boolean functions
and calculate the least expensive
corresponding binary decision trees.
•Their method practical for n up to 4,
23
not n = 5.
Binary Decision Tree Approach
Monotone Boolean Functions:
•Given two strings x1x2…xn, y1y2…yn
•Suppose that xi  yi for all i implies that
F(x1x2…xn)  F(y1y2…yn).
•Then we say that F is monotone.
•Then 11…1 has highest probability of being in
category 1.
24
Binary Decision Tree Approach
Incomplete Boolean Functions:
•Boolean function F is incomplete if F can be
calculated by finding at most n-1 attributes and
knowing the value of the input string on those
attributes
•Example: F(111) = F(110) = F(101) = F(100) =
1, F(000) = F(001) = F(010) = F(011) = 0.
•F(abc) is determined without knowing b (or c).
•F is incomplete.
25
Binary Decision Tree Approach
Complete, Monotone Boolean Functions:
•Stroud and Saeger: algorithm for enumerating
binary decision trees implementing complete,
monotone Boolean functions.
•Feasible to implement up to n = 4.
•n = 2:
–There are 6 monotone Boolean functions.
–Only 2 of them are complete, monotone
–There are 4 binary decision trees for
calculating these 2 complete, monotone
Boolean functions.
26
Binary Decision Tree Approach
Complete, Monotone Boolean Functions:
•n = 3:
–9 complete, monotone Boolean functions.
–60 distinct binary trees for calculating them
•n = 4:
–114 complete, monotone Boolean functions.
–11,808 distinct binary decision trees for
calculating them.
–(Compare 1,079,779,602 BDTs for all Boolean
functions)
27
Binary Decision Tree Approach
Complete, Monotone Boolean Functions:
•n = 5:
–6894 complete, monotone Boolean functions
–263,515,920 corresponding binary decision
trees.
•Combinatorial explosion!
•Need alternative approaches; enumeration not
feasible!
•(Even worse: compare 5 x 1018 BDTs
corresponding to all Boolean functions)
28
Cost Functions
•Stroud-Saeger method applies to more
sophisticated cost models, not just cost =
number of sensors in the BDT.
•Using a sensor has a cost:
–Unit cost of inspecting one item with it
–Fixed cost of purchasing and deploying it
–Delay cost from queuing up at the sensor
station
•Preliminary problem: disregard fixed and delay
29
costs. Minimize unit costs.
Cost Functions
•Simplification so far: Disregard characteristics
of population of entities being inspected.
•Only count number of observation (attribute)
nodes in the tree.
•Unit Cost Complication: How many nodes of
the decision tree are actually visited during
average container’s inspection? Depends on
“distribution” of containers. In our early models,
depends on probability of sensor errors and
30
probability of bomb in a container.
Cost Functions: Delay Costs
•Tradeoff between fixed costs and delay costs:
Add more sensors cuts down on delays.
•Stochastic process of containers arriving
•Distribution of delay times for inspections
•Use queuing theory to find average delay
times under different models
31
Cost Functions:
Unit Costs
Tree Utilization
•Complication: Assume cost depends on how many
nodes of BDT are actually visited during an “average”
container’s inspection. (This is sum of unit costs.)
•Depends on characteristics of population of entities
being inspected.
•I.e., depends on “distribution” of containers.
•In our early models, we assume we are given
probability of sensor errors and probability of bomb in
a container.
•This allows us to calculate “expected” cost of
32
utilization of the tree Cutil.
Cost Function used for Evaluating
the Decision Trees.
Later: Expect to analyze models for distribution of
attributes of containers and more sophisticated
analysis of expected cost of utilizing the tree,
bringing in delay costs.
33
Cost Functions
•Cost of false positive: Cost of additional
tests.
–If it means opening the container, it’s
very expensive.
•Cost of false negative:
–Complex issue.
–What is cost of a bomb going off in
Manhattan?
34
Cost Functions: Sensor Errors
•One Approach to False Positives/Negatives:
Assume there can be Sensor Errors
•Simplest model: assume that all sensors checking
for attribute ai have same fixed probability of
saying ai is 0 if in fact it is 1, and similarly
saying it is 1 if in fact it is 0.
•More sophisticated analysis later describes a
model for determining probabilities of sensor
errors.
•Notation: X = state of nature (bomb or no bomb)
Y = outcome (of sensor or entire inspection
35
process).
Probability of Error for The Entire Tree
State of nature is zero (X =
0), absence of a bomb
State of nature is one (X =
1), presence of a bomb
A
A
C
0
B
0
B
0
C
1
1
Probability of false positive
(P(Y=1|X=0))
for this tree is given by
0
1
1
Probability of false negative
(P(Y=0|X=1))
for this tree is given by
P(Y=1|X=0) = P(YA=1|X=0) * P(YB=1|X=0)
+ P(YA=1|X=0) *P(YB=0|X=0)* P(YC=1|X=0)
P(Y=0|X=1) = P(YA=0|X=1) +
P(YA=1|X=1) *P(YB=0|X=1)*P(YC=0|X=1)
Pfalsepositive
Pfalsenegative
36
Cost Function used for Evaluating
the Decision Trees.
CTot = CFalsePositive *PFalsePositive + CFalseNegative *PFalseNegative +
Cutil
CFalsePositive is the cost of false positive (Type I error)
CFalseNegative is the cost of false negative (Type II error)
PFalsePositive is the probability of a false positive occurring
PFalseNegative is the probability of a false negative occurring
Cutil is the expected cost of utilization of the tree.
37
Stroud Saeger Experiments
• Stroud-Saeger ranked all trees formed
from 3 or 4 sensors A, B, C and D
according to increasing tree costs.
• Used cost function defined above.
• Values used in their experiments:
– CA = .25; P(YA=1|X=1) = .90; P(YA=1|X=0) = .10;
– CB = 10; P(YC=1|X=1) = .99; P(YB=1|X=0) = .01;
– CC = 30; P(YD=1|X=1) = .999; P(YC=1|X=0) = .001;
– CD = 1; P(YD=1|X=1) = .95; P(YD=1|X=0) = .05;
– Here, Ci = unit cost of utilization of sensor i.
• Also fixed were: CFalseNegative, CFalsePositive, P(X=1)
38
Stroud Saeger Experiments: Our
Sensitivity Analysis
• We have explored sensitivity of the StroudSaeger conclusions to variations in values of
these three parameters.
• We estimated high and low values for these
parameters and did experiments with selection of
values from the interval of possible values.
39
Stroud Saeger Experiments: Our
Sensitivity Analysis
– CFalseNegative was varied between 25 million and 10
billion dollars
• Low and high estimates of direct and indirect costs
incurred due to a false negative.
– CFalsePositive was varied between $180 and $720
• Cost incurred due to false positive
(4 men * (3 -6 hrs) * (15 – 30 $/hr)
– P(X=1) was varied between 1/10,000,000 and
1/100,000
40
Stroud Saeger Experiments: Our
Sensitivity Analysis
n = 3 (use sensors A, B, C)
• We chose one of the values from the interval of
values and then explored the highest ranked tree
as the other two were chosen at random in the
interval of values. 10,000 experiments for each
fixed value.
• We looked for the variation in the top-ranked tree
and how the top-rank related to choice of
parameter values.
• Very surprising results.
41
Frequency of Top-ranked Trees when
CFalseNegative and CFalsePositive are Varied
7000
1st
2nd
3rd
4th
5th
6000
Frequency
5000
4000
3000
2000
1000
0
0
10
20
30
40
50
60
Tree no.
•
•
10,000 randomized experiments (randomly selected values of CFalseNegative and
CFalsePositive from the specified range of values) for the median value of P(X=1).
The above graph has frequency counts of the number of experiments when a
particular tree was ranked first or second, or third and so on.
• Only three trees (7, 55 and 1) ever came first. 6 trees came second,
10 came third, 13 came fourth.
42
Frequency of Top-ranked Trees when
CFalseNegative and P(X=1) are Varied
8000
1st
2nd
3rd
4th
5th
7000
6000
Frequency
5000
4000
3000
2000
1000
0
0
10
20
30
40
50
60
Tree no.
• 10,000 randomized experiments for the median value of CFalsePositive.
• Only 2 trees (7 and 55) ever came first. 4 trees came second. 7
trees came third. 10 and 13 trees came 4th and 5th respectively.
43
Frequency of Top-ranked Trees when
P(X=1) and CFalsePositive are Varied
7000
1st
2nd
3rd
4th
5th
6000
Frequency
5000
4000
3000
2000
1000
0
0
10
20
30
40
50
60
Tree no.
• 10,000 randomized experiments for the median value of CFalseNegative.
• Only 3 trees (7, 55 and 1) ever came first. 6 trees came second. 10
44
trees came third. 13 and 16 trees came 4th and 5th respectively.
Stroud Saeger Experiments: Our
Sensitivity Analysis: 4 Sensors
• Second set of computer experiments: n = 4
(use sensors, A, B, C, D).
• Same values as before.
• Experiment 1: Fix values of two of CFalseNegative,
CFalsePositive, P(X=1) and vary the third through
their interval of possible values.
• Experiment 2: Fix a value of one of CFalseNegative,
CFalsePositive, P(X=1) and vary the other two.
• Do 10,000 experiments each time.
45
• Look for the variation in the highest ranked tree.
Stroud Saeger Experiments: Our
Sensitivity Analysis: 4 Sensors
• Experiment 1: Fix values of two of
CFalseNegative, CFalsePositive, P(X=1) and vary the
third.
46
CTot vs CFalseNegative for Ranked 1 Trees
(Trees 11485(9651) and 10129(349))
Only two trees ever were ranked first, and one, tree 11485, was
ranked first in 9651 out of 10,000 runs.
47
CTot vs CFalsePositive for Ranked 1 Trees (Tree
no. 11485 (10000))
One tree, number 11485, was ranked first every time.
48
CTot vs P(X=1) for Ranked 1 Trees (Tree
no. 11485(8372), 10129(488), 11521(1056))
Three trees dominated first place. Trees 10201(60), 10225(17) and49
10153(7) also achieved first rank but with relatively low frequency.
Stroud Saeger Experiments: Our
Sensitivity Analysis: 4 Sensors
• Experiment 2: Fix the values of one of
CFalseNegative, CFalsePositive, P(X=1) and vary the
others.
50
Frequency of First Ranked Trees when Two
Parameters (CFalseNegative and CFalsePositive) were Varied
Keeping P(X=1) Constant at Randomly Selected
Only 6 trees
Values.
5
2
ever ranked 1st
Trees coming first -9541 10129 10153 10201 11485 11521
x 10
Tree 11521
dominated
1.8
1.6
1.4
Frequency
1.2
1
0.8
0.6
0.4
0.2
0
0
2000
4000
6000
Tree number
8000
10000
12000
10,000 randomized experiments with randomly selected values of P(X=1)
 The experiments were repeated for 20 different randomly selected values of
P(X=1)
51
Frequency of First Ranked Trees when Two
Parameters (CFalseNegative and P(X=1)) were Varied
Keeping CFalsePositive Constant at Randomly Selected
Only 12 trees
Values.
14
4
xTrees
10 coming first -505
4695
5105
5129
7353
ever ranked 1st
9541 10129 10153 10201 10225 11485 11521
Tree 11521
dominated
12
Frequency
10
8
6
4
2
0
0
2000
4000
6000
Tree number
8000
10000
12000
10,000 randomized experiments with randomly selected values of CFalsePositive
The experiments were repeated for 20 different randomly selected values of
CFalsePositive
52
Frequency of First Ranked Trees when Two
Parameters (P(X=1) and CFalsePositive) were Varied
Keeping CFalseNegative Constant at Randomly
Selected Values.
Only 7 trees
4
15
x 10
Trees coming first -9541 10129 10153 10201 10225 11485 11521
ever ranked1st
Tree 11521
dominated
Frequency
10
5
0
0.95
1
1.05
1.1
Tree number
1.15
1.2
4
x 10
10,000 randomized experiments with randomly selected values of CFalseNegative
The experiments were repeated for 20 different randomly selected values of
CFalseNegative
53
Conclusions from Sensitivity
Analysis
• Considerable lack of sensitivity to
modification in parameters for trees using 3 or
4 sensors.
• Very few optimal trees.
• Very few Boolean functions arise among
optimal and near-optimal trees.
54
Modeling Sensor Errors
•One Approach to Sensor Errors: Modeling
Sensor Operation
•Threshold Model:
–Sensors have different discriminating power
–Many use counts (e.g., Gamma radiation
counts)
–See if count exceeds
threshold
–If so, say attribute is present.
55
Modeling Sensor Errors
Threshold Model:
•Sensor i has discriminating power Ki,
threshold Ti
•Attribute present if counts exceed Ti
•Seek threshold values that minimize the
overall cost function, including costs of
inspection, false positive/negative
•Assume readings of category 0 containers
follow a Gaussian distribution and similarly
category 1 containers
•Simulation approach
56
Probability of Error for Individual
Sensors
• For ith sensor, the type 1 (P(Yi=1|X=0)) and type 2
(P(Yi=0|X=1)) errors are modeled using Gaussian
distributions.
– State of nature X=0 represents absence of a bomb.
– State of nature X=1 represents presence of a bomb.
– i represents the outcome (count) of sensor i.
– Σi is variance of the distributions
– PD = prob. of detection, PF = prob. of false pos.
Ki
P(i|X=0)
Ti
P(i|X=1)
i
Characteristics of a typical sensor
57
Modeling Sensor Errors
The probability of false positive for the ith sensor is computed as:
P(Yi=1|X=0) = 0.5 erfc[Ti/√2]
The probability of detection for the ith sensor is computed as:
P(Yi=1|X=1) = 0.5 erfc[(Ti-Ki)/(Σ√2)]
erfc = complementary error function erfc(x) = (1/2,x2)/sqrt()
The following experiments have been done using sensors A, B,
C and using:
KA = 4.37; ΣA = 1
KB = 2.9; ΣB = 1
KC = 4.6; ΣC = 1
We then varied the individual sensor thresholds TA, TB and TC
from -4.0 to +4.0 in steps of 0.4. These values were chosen since
they gave us an “ROC curve” for the individual sensors over a
complete range P(Yi=1|X=0) and P(Yi=1|X=1)
58
Frequency of First Ranked Trees for
Variations in Sensor Thresholds
18000
16000
14000
Frequency
12000
10000
8000
6000
4000
2000
0
0
10
20
30
40
50
60
Tree no.
• 68,921 experiments were conducted, as each Ti was varied through its
entire range. (n = 3)
• The above graph has frequency counts of the number of experiments when
a particular tree was ranked first. There are 15 such trees. Tree 37 had the
highest frequency of attaining rank one.
59
Modeling Sensor Errors
•A number of trees ranking first in other
experiments also ranked first here.
•Similar results in case of n = 4.
•4,194,481 experiments.
•244 different trees were ranked first in at least one
experiment.
•Trees ranked first in other experiments also
frequently appeared first here.
•Conclusion: considerable insensitivity to change
60
of threshold.
New Approaches to Optimum
Threshold Computation
• Extensive search over a range of thresholds
has some practical drawbacks:
– Large number of threshold values for every sensor
– Large step size
– Grows exponentially with the number of sensors
(computationally infeasible for n > 4)
• A non-linear optimization approach proves
more satisfactory:
– A combination of Gradient Descent and modified
Newton’s methods
61
Optimum Threshold Computation
• We implemented standard approaches including:
– Gradient Descent Method
– Newton’s Method in Optimization
• In Gradient Descent Method we have to adjust the
“step size” heuristically, while the Newton method
uses the Hessian matrix to adjust the step size
automatically
Though the computation of the
Hessian matrix is a little
expensive and tedious, the
method quickly converges in
fewer iterations than the
gradient descent method
 2 f

2
 Ta
 2 f

Hf     Tb Ta


 2 f
 T T
 n a
2 f
Ta Tb
2 f
Tb2
2 f
Tn Tb
2 f 

Ta Tn 
2 f 

Tb Tn 


2
 f 
Tn2 
62
Problems with Standard Approaches
• Gradient Descent Method: Setting the value of the
step size heuristically, since:
– Too small step size results in large number of
iterations to reach the minimum
– Too big step size results in skipping the minimum
• Newton’s Method:
– The convergence depends largely on the starting
point. This method occasionally drifts in the wrong
direction and hence fails to converge.
• Solution: combination of gradient descent and
Newton’s methods
63
A Combined Method
• The Hessian matrix H f(τ) might not be a wellconditioned, positive definite matrix
• We explored alternative approaches to
computing positive definite approximations to
H f(τ)
– Modified Cholesky Algorithms
•If the Hessian matrix H f(τ) is ill-conditioned, we
take small steps towards the minimum using the
gradient descent method until it becomes well
conditioned
64
Results: Threshold Optimization
• Costs of false positive CFalsePositive and false negative
CFalseNegative and prior probability of occurrence of a bad
container, P(X=1), were fixed as means of the min and max
values given by Stroud and Saeger (same as we used in
earlier experiments)
• We were able to converge to a (hopefully-close-to-minimum)
cost every time with a modest number of iterations. For
example:
– For 3 sensors, it took an average of 0.081 seconds (as opposed to
0.387 seconds using extensive search) to converge to a cost for all
114 trees
– For 4 sensors, it took an average of 0.196 seconds (as opposed to
more than 2 seconds using extensive search) to converge to a cost
for all 66,936 trees
• In each case, min cost attained with new algorithm was
lower, and often much lower, than that attained with
65
extensive search.
Results: Threshold Optimization
Tree costs at optimum thresholds
500
Combined Optimization
Extensive search
450
400
Total Cost
350
300
250
200
150
100
0
20
40
60
80
100
Tree Number
Many times the minimum obtained using the
optimization method was considerably less
than the one from the extensive search
technique.
66
New Idea: Searching through a
Generalized Tree Space
• We expand the space of trees from those
corresponding to Stroud and Saeger’s “Complete”
and “Monotonic” Boolean Functions to “Complete
and Monotonic BDTs”, because…
• Unlike Boolean functions, BDTs may not consider all
sensor outputs to give a final decision
• Advantages:
– Allow more potentially useful trees to participate in the
analysis
– Help define an irreducible tree space for search operations
– Move focus from Boolean Functions to Binary Decision
Trees
67
Revisiting Monotonicity
•
Monotonic Decision Trees
– A binary decision tree will be called monotonic
if all the left leaves are class “0” and all the
right leaves are class “1”.
•
a
Example:
b
a b c F(abc)
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
0
1
1
0
1
0
1
0
b
c
1 0
a
1
0
0
c
c a
c
1
0
0 11 0
c
b
c
b
c
a a 1
0 11 0
b
0
a
b
0
1
1
c
a
a 1
1 0 0 1
a
0
b
a
0
1
c
b
0 a
1
c
0 1
b
1
0
a
a
b 1
1 0 0 1
All these trees correspond to same monotonic Boolean function68
Only one is a monotonic BDT.
Revisiting Completeness
• Complete Decision Trees
– A binary decision tree will be called complete if every
sensor occurs at least once in the tree and, at any nonleaf node in the tree, its left and right sub-trees are not
identical.
• Example:
a
a b c F(abc)
0
0
0
0
1
1
1
1
0
0
1
1
0
0
1
1
0
1
0
1
0
1
0
1
0
1
1
1
0
1
1
1
b
c
0
a
c
1 b
1
c
1
0 1
b
0
b
1
1
0 1
a
c
b
0
a
c
1 b
1
c 1
0 1
b
1
c
0
b
1
1
c 1
0 1
69
The CM Tree Space
complete, monotonic BDTs
Distinct BDTs
Trees From CM
Boolean
Functions
Complete,
Monotonic
BDTs
2
74
4
4
3
16,430
60
114
4
1,079,779,602
11,808
66,936
No. of
attributes
70
Tree Neighborhood and Tree Space
• Define tree neighborhood by giving operations for
moving from one tree in CM Tree Space to another.
• Structure-based methods
• Classifications-based methods
• We choose structure-based neighborhood methods
because the cost of a BDT depends more on its
structure
• Small changes in tree structure don’t significantly
affect cost, and…
• All BDTs with the same Boolean functions may differ
a lot in cost
71
Search Operations in Tree Space
• Split
Pick a leaf node and replace it with a sensor that is
not already present in that branch, and then insert
arcs from that sensor to 0 and to 1.
a
b
0
a
c
c
d 1
b
SPLIT
0
d 10 1
0
1
c
c
d 1
d 1 b 1
0
1 0
1
72
Search Operations
• Swap
Pick a non-leaf node in the tree and swap it with its
parent node such that the new tree is still
monotonic and complete and no sensor occurs
more than once in any branch.
a
b
0
a
c
c
d 1
b
SWAP
0
d 10 1
0
1
c
d
d 1
c 10 1
0
1
73
Search Operations
• Merge
Pick a parent node of two leaf nodes and make it a
leaf node by collapsing the two leaf nodes below it,
or pick a parent node with one leaf node, collapse
both the parent node and its one leaf node, and
shift the sub-tree up in the tree by one level.
a
b
0
c
c
a
a
d 1
d 1 0 1
b
MERGE
0
c
d
d 1 0
0 1 0 1
b
c
c
d 1
0 1 0 1
0 1
74
Search Operations
• Replace
Pick a node with a sensor occurring more than
once in the tree and replace it with any other
sensor such that no sensor occurs more than once
in any branch.
a
b
0
a
c
c
d 1
b
REPLACE
0
d 1 0 1
0
1
c
c
b 1
d 1 0 1
0
1
75
76
Tree Neighborhood and Tree Space
• Define tree neighborhood by using these four
operations for moving from one tree in CM
Tree Space to another.
• Irreducibility
– Theorem: Any tree in the CM tree space can be
reached from any other tree by using these
neighborhood operations repetitively
– An irreducible CM tree space helps “search” for
the cheapest trees using neighborhood operations
77
Tree Space Traversal
•
Greedy Search
1. Randomly start at any tree in the CM tree
space
2. Find its neighboring trees using the above
operations
3. Move to the neighbor with the lowest cost
4. Iterate until we find a minimum
–
The CM Tree space is highly multi-modal (more
than one minimum)!
–
Therefore, we implement a stochastic search
algorithm with simulated annealing to find the
best tree
78
Tree Space Traversal
• Stochastic Search
– Randomly start at any tree in CM space
– Find its neighboring trees, and evaluate each one for its
total cost
– Select next move according to a probability distribution
over the neighboring trees
• To deal with the multimodality of the tree space, we
introduce Simulated Annealing:
– Make more random jumps initially, gradually decrease the
randomness and finally converge at the overall minimum
79
Tree Space Traversal
• Stochastic Search
– Randomly start at any tree in CM space
– Find its neighboring trees, and evaluate each one for its total cost
– Select move according to a probability distribution over
neighboring trees proportional to the ratio of the costs of two trees
– Specifically, if we are at the ith tree τi, then the probability of going
to its kth neighbor, denoted τik, is given by
Pki 
 f ( i )
ni
f ( ik ) 
1t
  f ( i ) f ( ij ) 
1t
j 1
– where f(τj) and f(τij) are the costs of trees τi and τij, respectively
and ni is the number of trees in the neighborhood of τij
– so-called “temperature” t is initiated to one and lowered in discrete
unequal steps after every m hops until we reach a minimum 80
Results: Searching CM Tree Space
• We were able to perform experiments for 3, 4 and 5
sensors, successfully
• Results show improvement compared to the extensive
search method. E.g., for 4 sensors (66,936 trees)
– 100 different experiments were performed
– Each experiment was started 10 times randomly at some tree
and chains were formed by making stochastic moves in the
neighborhood, until we find a local minimum
– Only 4890 trees were examined on average for every
experiments
– Global minimum was found 82 times while the second best
tree was found 10 times
– The method found trees that were less costly than those found
by earlier searches of BDTs corresponding to complete,
81
monotonic Boolean functions.
Genetic Algorithms-based Approach
• Structure-based neighborhood moves allow
very short moves only. Therefore,…
• Techniques like Genetic Algorithms and
Evolutionary Techniques may suggest ways
for getting more efficiently to better trees,
given a population of good trees
82
Genetic Algorithms-based Approach
• Started implementing genetic algorithmsbased techniques for tree space traversal
• Basically, we try to get “better” trees from the
current population of “good” trees using the
basic genetic operations on them:
– Selection
– Crossover
– Mutation
• Here, “better” decision trees correspond to
lower cost decision trees than the ones in the
current population (“good”).
83
Genetic Algorithms-based Approach
• Therefore, as new trees are generated, the size
of the genePool keeps increasing.
• Selection:
– Select an initial population of trees, bestPop,
randomly out of the CM tree space
– Always keep a population of N best trees for
further crossover and mutation operations
• Crossover:
– Performed between every pair of trees in bestPop.
– For each crossover operation between two trees τi
and τj, we randomly select nodes and exchange the
subtrees
84
Genetic Algorithms-based Approach
• We randomly perform crossover operation k
times between a pair of trees
• However, we impose certain restrictions to
ensure that resultant trees lie in CM tree space85
Genetic Algorithms-based Approach
• Mutation:
– Performed after every m generations of the
algorithm
– We do two types of mutations:
• Generating all the neighboring trees of the current best
population of trees using the four operations used in the
stochastic search method and put them into the gene
pool
• Replacing some fraction of the trees of bestPop with
random samples from the CM tree space which are not
in the gene pool
– Only ~1600 trees had to be examined to obtain the
10 best trees for 4 sensors!
86
Discussion and Future Work
• Very few optimal trees; optimality insensitive to
changes in parameters.
• The extensive search techniques become practically
infeasible beyond a very small number of sensors
• The new threshold optimization algorithms provide
faster ways to arrive at a low tree cost; cost is lower
and often much lower than in extensive search
• The irreducible tree space helps us to “search” for
the best trees rather than evaluating all the trees for
their cost
• The stochastic search algorithm allows us to search
for optimum inspection schemes beyond 4 sensors
87
successfully
Discussion and Future Work
• Future work: Because of the rapid growth in
number of trees in CM Tree Space when the
number of sensors grows, it is necessary to try
to reduce the number of trees we need to
search through.
• A notion of tree equivalence could be
incorporated when the number of sensors go
beyond 5 or 6
• We hope that incorporating this into our model
will enable us to extend our model to a large
88
number of sensors
Tree Equivalence and Tree
Irreducibility
• A decision tree in CM tree space can be
exactly equivalent to another decision tree in
the CM tree space:
• This property is called transposition and the
two trees are called transposes of each others
89
Tree Equivalence and Tree
Irreducibility
• The problem is that the size of the equivalence
class (number of transposes for a given tree)
increases exponentially with the number of
sensors.
• Therefore, we need to define a “canonical”
representation of the equivalence class and
deal with the equivalence class as a whole
rather than treating each individual tree
separately
90
Tree Irreducibility
• Another problem is that in addition to being
complete and monotonic, a tree should also be
irreducible (no non-useful sensor). For
example:
a
b
c
d
0 1 0 c
0 1
b
b
c
a
d
0 d0 c
0 1 0 1
c
a
c
0 1 0 d
0 1
d
d
0 c0 c
0 1 0 1
91
Discussion and Future Work
•Future work: Need for more complicated
cost models; bringing in costs of delays
92
Discussion and Future Work
•Future work: More than two values of an
attribute
(present, absent, present with
probability > 75%, absent with probability
at least 75%)
(ok, not ok, ok with probability > 99%,
ok with probability between 95% and
99%)
•Future work: In the Boolean function model:
inferring the Boolean function from
observations (partially defined Boolean
functions)
93
Discussion and Future Work
•Future Work: Explain why conclusions are so
insensitive to variation in parameter values.
•Future Work: Explore the structure of the
optimal trees and compare the different optimal
trees.
•Future Work: Develop methods for
approximating the optimal tree.
Pallet vacis
94
Closing Remark
•Recall that the “cost” of inspection includes the
cost of failure, including failure to foil a terrorist
plot.
•There are many ways to lower the total “cost”
of inspection:
Use more efficient
orders of inspection.
Find ways to inspect
more containers.
Find ways to cut down
on delays at inspection lanes.
95
Research Team
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
Tayfur Altiok, Rutgers, Ind. & Systems Engineering
Saket Anand, Rutgers, ECE graduate student
Endre Boros, Rutgers, Operations Research
Elsayed Elsayed, Rutgers, Ind. & Systems Engineering
Liliya Fedzhora, Rutgers, Operations Res. grad. student
Paul Kantor, Rutgers, Schl. of Infor. & Library Studies
Abdullah Karaman, Rutgers Ind. & Syst. Eng. grad. student
David Madigan, Rutgers, Statistics
Richard Mammone, Rutgers, Center for Advanced Information Processing
Ben Melamed, Rutgers Business School
Sushil Mittal, Rutgers, ECE graduate student
S. Muthukrishnan, Rutgers, Computer Science
Saumitr Pathek, Rutgers ECE graduate student
Richard Picard, Los Alamos, Statistical Sciences Group
Fred Roberts, Rutgers, DIMACS Center
Kevin Saeger, Los Alamos, Homeland Security
Christina Schroepfer, Rutgers, Ind. & Syst. Eng. Grad. student
Phillip Stroud, Los Alamos, Systems Engineering and Integration Group
Minge Xie, Rutgers, Statistics
Hao Zhang, Rutgers Ind. & Systems Eng., graduate student
96
Collaborators on Work Described:
• Saket Anand
• David Madigan
• Richard Mammone
• Sushil Mittal*
• Saumitr Pathak
Research Support:
• Office of Naval Research
• National Science Foundation
• New support: Domestic Nuclear Detection Office
Los Alamos National Laboratory:
• Rick Picard
• Kevin Saeger
• Phil Stroud
97