Protection Model of Security Systems Based on Neyman-Person Criterion

doi:10.4236/cn.2013.53B2104

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Communications and Network, 2013, 5, 578-583

http://dx.doi.org/10.4236/cn.2013.53B2104 Published Online September 2013 (http://www.scirp.org/journal/cn)

Protection Model of Security Systems Based on

Neyman-Person Criterion

Haitao Lv, Ruimin Hu, Jun Chen, Zheng He

National Engineering Research Center for Multimedia Software, Wuhan University,

Wuhan, China

Email: [email protected], [email protected]

Received May 2013

ABSTRACT

In this paper security systems deployed over an ar e a are rega r ded abstractly as a diagram of security network. We pro-

pose the Neyman-Pearson protection model for security systems, which can be used to determine the protection proba-

bility of a security system and find the weakest breach path of a security network. We present the weakest breach path

problem formulation, which is defined by the breach protection probability of an unauthorized target passing through a

guard field, and provide a solution for this problem by using the Dijkstra’s shortest path algorithm. Finally we study the

variation of the breach pro tection prob ability w ith the change of th e parameters of the model.

Keywords: Security System; Protection Probability; Security Network; Breach Protection Probability; Breach Path

1. Introduction

The society security problem has been attached impor-

tance by national governments. In order to maintain so-

cial public safety, many security systems have been con-

structed in cities in the world. With the rapid develop-

ment of information technology, especially the Internet

of Things and cloud computing, the security system is

getting more and more complex, which consists of the

intrusion alarm system, the video surveillance system,

the access control system, the explosion-proof security

check system, etc. Security systems are deployed at dif-

ferent positions in an area, which can communicate and

share data each other through the internet, and complete

protection tasks cooperatively. In this paper, a large and

complex security system is regarded as abstractly as a

diagram of a security network. As shown in Figure 1,

there is a security netwo rk consisted of some security

systems in the guard zone, where each of yellow filled

circles represents a security system, and every triangle

represents a protection target.

For a security network, depending on the protection

ranges and the protection coverage sch emes of security

systems, as well as the deployment-density of the guard

field, the protection coverage area may contain vulnera-

ble paths. The probability that an unauthorized target

traverses the region through a vulnerable path gives in-

sight about the level of security provided by the security

network. In this paper, the protection mo del of security

systems based on Neyman-Pearso n Criterion is proposed.

The protection probability at any position in a guard field,

which is provided by a security system, can be calculated

by the model. The weakest breach pa th of a security net-

work, which is defined by the breach protection proba-

bility of an unauthorized target passing through a guard

field, can be found.

Figure 1. The abstract diagram of a security network.

H. T. LV ET AL.

579

The r eminder of this paper is organized as follows: In

the next section, the related work about seurity systems is

introduced. In Section 3, the protection model is pro-

posed, the weakest breach path of a security network is

described and the method based on Dijkstra’s shortest

path algorithm to find the weakest breach path is put

forward. After presenting the details of the problems for-

mally, th e results ar e simulated and analyzed in Section 4.

Finally, conclusions are drawn in Section 5.

2. Related Work

In 1970’s, U.S. Department of Energy’s Sandia National

Laboratories [1] first introduced the basic concepts of the

Physical Protection System, from which th e security system

evolved. Subsequently, the U.S. Department of Energy

put forward a model named adversary sequence diagram

(ASD) [2], which was applied to the field of nuclear

facilities protection. ASD can recognize vulnerability of

physical protection systems by analyzing how hypothet-

ical enemies might achieve their objects through various

barriers. The path th at is most easily broken through is

considered weakest.

In 1981, Doyon [3] presen ted a probabilistic network

model for a system consisting of guards, sensors, and

barriers. He determined analytic representations for de-

termining probabilities of intruder apprehension in dif-

ferent zones between site entry and a target object. In

1997 Kobza and Jacobson [4] have presented probability

models for access security systems with particular appli-

cations to aviation security. In 1998, Hicks et al. [5] Pre-

sented a cost and performance analysis for ph ysical pro-

tection systems. He considered the systems-level perfor-

mance metric is risk, which is defined as follows.

( )()

1RiskpAp EC= ×−×





(1)

where

( )

is Probability of Attack,

( )

is Prob-

ability of System Effectiven ess, and C is Consequence.

After the events of September 11, 2001, public safety

becomes the issue co ncerned by countries in the world.

The concep t of Physical Protection Syste m has been

changed. Some researchers from USA and Australia con-

side red that a ph ysical protection system is made up of

people, architectures and electronic devices. So the con-

cept of Security Sys te m was born. Many researcher s were

interested in assess the protection effectiveness of secu-

rity systems through risk analysis. In 2004, Fische r [6]

presented a very subj ectiv e risk analysis approach to rank

threats using a probability matrix, criticality matrix, and

vulnerability matrix. In 2006, Zhihua Chen [7] in Chi-

nese Peop l e ’s P ub lic Security Univ ersity h as proposed

performance evaluation index and evaluation methods of

crime prevention system for the assessment of the effec-

tiveness and vulnerability of crime prevention system. This

method is a qualitative assessment to the crime preven-

tion system based on management science. In 2007, Gar-

cia [8] gave an integrated approach f or designing physi-

cal security systems. The protection effectiveness of a

physical protection system was defined as the cumulative

probability of detection fro m th e star t of an adversary

path to th e point determined by the time available for

response. In 2009, Jonathan Pollet and Joe [9] Cummins

put forward a performance assessment framework of the

Security Systems, which considered not only the charac-

teristics of the system, also the ris k outsid e the system.

In recent years, some researcher s considered that the re

were enormous uncertainty in the vulnerability evalua-

tion of security systems, and they put forward some me-

thods to reduce unc er tain ty. In 2011, Xu peida [10]

thought that each individual component of the security

system w as modeled, and he used the Dempster-Shafer

(D-S) evidence theory to analyze potential threats. Some

literatures also proposed methods such as bounded inter-

vals [11], exoge n ous dynamics [12], games of imperfect

information [13-15], to characterize uncertainty in vul-

nerability analysis.

3. Protection Model and the Weakest Breach

Path Problem Formulation

3.1. Neyman-Pearson Protectio n Model

In our research, th er e is a basic assumption that is a secu-

rity system can eliminate any threat as long as a threat is

detected. If a security system finds a threat, it will sound

alarm. So each security system has its own false alarm

rate, and it is regarded abstractly as the process of deci-

sion. The optimal decision rule that maximizes the detec-

tion probability subject to a maximum allowable false

alarm rate α that is given by the Neyman-Pearson lemma

[16]. Two hypotheses that represent th e ab sen ce and pr e s-

ence of an unauthorized object are set up. The model

computes the likelihood ratio of the respective probabili-

ty density functions, and compares it against a threshold

which is configured, and the false alarm constraint is sa-

tisfied. The process of a security system finding threats

can be considered as the process signal reception. Sup-

pose that an unauthorized object is a passive signal re-

ception th a t happens in the presence of additive white

Gaussian noise (AWGN) with zero mean and variance

, as well as path-loss with path loss exponent

. Every

breach protection decision is based on the processing of

L data samples. If samples are collected fast enough, the

distance between a security system and a object can be

considered constant throughout the observation period.

Let

be the Euclidean distance between the grid point

and the security system

. Based on Neyman-Pearson

Criterion with false alarm rate

, the pro tection probabil-

ity of an unauthorized object at grid point

by the se-

H. T. LV ET AL.

580

curity system

is defined as follows.

( )

()

vi vi

p Ld

αγ

−−

=−Φ Φ−−

. (2)

Where

( )

xΦ

is the cumulative distribution function

of the zero-mean, unit-variance Gauss i an random varia-

ble at point

γσ

=. (3)

controls the per-datum signal-to-noise power r atio

where the security system transmits information with

power

, and A is a cons tant, which is regarded as sig -

nal propagation losses, emergency res o urces, infor mation

communication, etc.

3.2. Grid-Based Guard Field

In ord er to simplify the formulations, we consider the

guard field as a cross-connected grid. A sa mple field

model is pre s ented in Figure 2.

The guard field model consists of the grid points and

two auxiliary nodes which are the starting and the desti-

nation points. The aim of the target is to go through the

guard field from the starting point that represents the

insecure side to the destination point that represents the

secure side. The horizontal axis is divided into N − 1 and

the vertical axis is divided into M − 1 equal parts . Thus,

there are

NM×

grid points plus the starting and desti-

nation points. For the sake of simplifying the notation,

instead of us ing two dimensional grid point indices

( )

where

0,1, ,1

xN= −

and

0,1, ,1

yM= −

we utilize a kind of one dimensional grid point index

which is calculated as

v yN x= ++

. The index of the

starting point is defin ed as

0v=

, a nd th e index of the

destination point is

1v NM= +

. We use the connection

matrix

( )( )

vw NM NM

+× +

∈

to represent the connections

Figure 2. A sample field model constructed to find the vul-

nerable pat h for the guar d field whe re the le ngth is 8 m, the

width is 4 m, and the grid size is 1m (N = 8, M = 4).

of the grid points. The matrix

,vw

is defi n ed as defin ed

in Equation (4).

( )

10, 1,

10 0

111

v wvw

ifv wNMandxxyyD

if vand y

cifwNMandyM

otherwise

 <<+ −−∈

 = =



= =+ =−





(4)

{{ 1,0,1}{ 1,0,1}}{(0,0)}D= −×−− which is the set of

possible difference-tuples of the two-dimensional grid

point indices excluding

vw=

Using the grid-based field model above, the protection

probabilities can be computed for every grid point through

the Neyman-Pearson Protection Model of security sys-

tems.

3.3. The Weakest Breach Path Problem

The weakest breach path problem can be defined as fin ding

the permutation of a subset of grid points

{,,, }

V vvv= with which an object traverses from

the starting point to the destination point with the least

probability of be in g detected. The nodes

v−

and

are connected to each other where 1,1

c−=. The miss

probability p of the most vulnerable pathVis defined as

follows.

( )

p pn

∈



= −





∑. (5)

Where vi

p is the breach protection probability asso-

ciated with the grid point

vV∈

, n is the number of

The weakest breach path can be find by solving the

following optimization problem as defined in Equation

(6).

denotes the edg e which originates from the ith

node and sinks in the jth node, s is the starting node and

d is the destination node and C is as defined in Equation

(4). In this formulation , the aim is to maximize the miss

probability P defined in Equation (6). By using the

logarithm function the optimization problem defined in

Equation (7) can be changed to a linear program, where

the aim is to find the minimum value.

( )

()( )

( )( )

max 1

1;1;1, 2,,

01, 2,,,

v ij

sj id

s jCidC

ij ki

ij Cki C

pxsubject to

xx iNM

ifith andjth nodes areonthepath andc

xotherwise

∈

∈∈

−

= = ∀ =

−= ∀ =



=



∑

∑∑



(6)

( )

minlog(1 )

i ij

ij C

∈

−−

∑

(7 )

Guard Fi el dS ECURE SIDE

INSECURE S IDES TARTING P O INT

DE STINA TION P O INT

H. T. LV ET AL.

581

4. Simulation and Analysis

The grid-based field can be regard abstractly as a graph,

so Dijkstr a’s shortest path algorithm can be employed to

solve the weakest breach path problem too. The protec-

tion probability asso ciated with the grid points cannot be

used as we ights of the grid points. Consequently, the

weights of the grid points need be converte d to a new

measure dv, which is defined as log(1 )

dp=−−

. This

algorithm finds the path with the smallest negative loga-

rithm value that is equal to be the weakest breach path. A

sample security systems co verag e graph and the weakest

breach path is shown in Figure 3. Using the two-dimen-

sional field model and adding the protection prob ab ility

as the third axis.

Valleys and hills of protection probabilities are shown

in Figure 3. The weakest breach path follows the valleys

because the valleys denote the low protection probabili-

ties.

4.1. Effects of Parameters on the Protection

Probability

In this subsection, the effects of the Neyman-Pearson

Protection Model parameters on the breach protection

probability are analyzed. The deployment of security sys-

tems is random with uniform distribution. The parame-

ters are shown in Table 1. The figures, which are pre-

sented in the following are the averages of 100 runs, de-

pict how the environmental properties and th e toleran c e

to the false alarms affect the vulnerability of a security

network.

Ten security systems are deplo yed in a field where the

parameters are same as in Ta ble 1. The effect of the false

Figure 3. A sample of a guard field and vulnerable path

where the length i s 101 m, the widt h is 60 m, and grid si ze is

1 m. Twenty security systems are deployed in this field

randomly. The Neyman-Pearson Protection Model is confi-

gured with L = 100, R = 9, α = 0.01, η = 2, γ = 20 db. The

breach probability is 0.0639.

alarm rate, α, on the breach protection probability P is

shown in Figure 4, which essen tially r epr esents the net-

work operating characteristics. With greater tolerance to

false alarms, the P performance improves, and hence the

protection range becomes larger. Sufficiently high signal

noise ratio is necessary for an acceptable level of breach

protection probability, whi ch is relatively insensitive to

the false alarm rate.

As shown in Figure 4, the false alarm rate α has a

great effect on the breach protection probability P, and as

α increases, the breach probability decreases, which re-

flects the protection probability v

p of an unauthorized

object at grid

increases.

Although α is very in fluentia l on br each protection

probability, η does not have an appreciable impact when

the signal noise ratio is small. When the values of γ

become large, η significantly increases the breach protec-

tion probability as shown in Fig ure 5.

This is due to the fact that the protection probability is

inversely proportional to the distance on the order of η.

The effect of η is very significant when

≤

As shown in Figu re 6, wh en the signal noise ratio γ

incre a ses, the breach protection probability decreases,

Table 1. Parameter values used in the simulations for Ney-

man-Pearson Protection Model.

Parameters Value

Length of the field 51 m

Width of the fileld 41 m

Grid Size 1 m

Numbers of Security Systems 10

α 0.1

η 2

γ 20 db

L 100

Figure 4. The effect of α on th e b rea ch p rot ecti on probability.

020 40 60 80 100

0.92

0.94

0.96

0.98

Protec tion P robabl i ty

Most V ul nerabl e P ath

Secure S i de

Insecure Si de

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

B reac h Prot ec tion Probabil i t y

=20db

=5db

=60db

H. T. LV ET AL.

582

Figure 5. The effect of η on the breach protection probability,

Figure 6. The effect of γ on the breach protection probability .

which indicates that the protection probability of the se-

curity netwo rk improves and the protection performance

increases. If targets are closer to security systems, signal

noise ratio has more influence on the protection probabil-

ity. When th e parameter

≥

, the effect of the signal

noise ratio γ becomes very small.

4.2. Effects of Number of Security Systems on

the Protection Probability

While analyzing the required number of security systems

for a given breach probability, one case of random dep-

loyment is considered. The case is assumed that security

systems are uniformly distributed alon g bo th the vertical

and horizontal axes . The effect of numbers of security

system in a field on the breach protection prob ability is

shown in Figure 7.

As the density of security systems increases in a field,

the breach protection probability tends to stabilize, which

approximates the zero. The results suggest that there is a

Figure 7. The effect of numbers of security systems on the

breach protection probability.

saturation point after which randomly placing more secu-

rity systems does not significantly impact the breach pro-

tection probability of a secu rity network in a field.

When the signal noise ratio is same, α affects the

breach protection probability more than

(see Figures 4

and 5), so the false alarm rate α is more influential here

too. The rapid d ecreas e in the breach protection probabil-

ity can be explained by the fact as the density of security

systems is saturated in a field, grid points are cover ed

with high protection probabilities. Consequently, at the

beginning, an additional security system decreases the

breach protection probability considerably, however, once

the saturation is reached, the affection of numbers of

security systems is not so large anymore.

5. Conclusion

In this paper, we put forward the Neyman-Pearson pro-

tection model of security systems that can be employed

to find the weakest breach path of a security network. In

order to f in d the weakest breach path, we apply Dijkstra’s

shortest path algorithm by using the negative log of the

breach protection probability as the grid point weights.

Finally, the effect of para met ers of N eyman-Pearson pro-

tection model on breach protection probability is studied

by MATALAB simulation. The simulation experiments

show that the false alarm rate is the most influential pa-

rameter on the breach protection prob ability.

6. Acknowledgements

Thanks for the assistance from National Science Founda-

tion of China (No. 61170023), the Major Nat ional Science

and Technology Special Projects of China (2010ZX

03004-003-03, 2010ZX03004-001-03), National Nature

Science Foundation of China (No. 60832002). The au-

thors would like to thank Ruimin Hu. IEEE Member, pro-

12345678910

0.05

0.1

0.15

0.2

0.25

0.3

0.35

B reac h Protect i on Probabil i t y

=20db

=5db

=60db

051015 2025 30 35 40 4550

0.2

0.22

0.24

0.26

0.28

0.3

0.32

0.34

0.36

B reac h Protect i on Probabil i t y

η=2

η=3

η=4

010 20 30 4050 60 70 80 90 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Numbers of S ecurity S yst em s

B reac h Prot ec tion Probabil i t y

=0.9

=20

=0.9

=20

=0.9

=60

H. T. LV ET AL.

583

fessor of School of Computer, Wuhan University, Ren

Pin teaching assistant Department of Electrical Engi-

neering and Computer Science, Northwestern University

USA, for their thoughtful comments.

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