J. KSIAM Vol.12, No.3, 133 137, 2008 REFINABLE FUNCTIONS · j. ksiam vol.12, no.3, 133 137, 2008 on...

J. KSIAM Vol.12, No.3, 133–137, 2008

ON THE ASYMPTOTIC CONVERGENCE OF ORTHONORMAL CARDINALREFINABLE FUNCTIONS

RAE YOUNG KIM

DEPARTMENT OF MATHEMATICS, YEUNGNAM UNIVERSITY, 214-1, DAE-DONG, GYEONGSAN-SI, GYEONGSANGBUK-DO, 712-749, REPUBLIC OF KOREA

E-mail address: [email protected]

ABSTRACT. We prove an extended version of asymptotic behavior of the orthonormal cardi-nal refinable functions from Blaschke products introduced by Contronei et al [2]. In fact, weshow the orthonormal cardinal refinable function ϕk,q converges in Lp(R) (2 ≤ p ≤ ∞)to the Shannon refinable function as k → ∞ uniformly on a class QA,B of real symmetricpolynomials determined by positive constants A ≤ B.

1. INTRODUCTION

Recently, Contronei et al. [2] constructed an interesting class of orthonormal cardinal refin-able functions with rational symbols using Blaschke products. The rational symbols are of theform

Pk,q(z) :=(1 + z)2k+1q(z)

(1 + z)2k+1q(z)− (1− z)2k+1q(−z), z ∈ C, (1.1)

where

q(z) =N/2∑

j=0

α2j(1 + z)N−2j(1− z)2j , z ∈ C, (1.2)

is a real symmetric polynomial of degree N := 2n − 2k, and the corresponding orthonormalcardinal refinable functions ϕk,q are defined by the Fourier transform:

ϕk,q(w) :=∞∏

l=1

Pk,q(e−iw/2l). (1.3)

The study of refinable functions which are both orthonormal, and cardinal was addressed in[3, 6].

2000 Mathematics Subject Classification. 42C15, 42C40.Key words and phrases. Orthonormal function, Cardinal function, Refinable function, Shannon refinable

function.This work was supported by the Korea Research Foundation Grant funded by the Korean Government

(MOEHRD, Basic Research Promotion Fund) (KRF-2006-331-C00014).

133

134 RAE YOUNG KIM

We consider the classQA,B (0 < A ≤ B < ∞) of all real symmetric polynomials q of (1.2)such that

A ≤ |Q(w)| ≤ B for all w,

where Q is defined for w ∈ R as

Q(w) :=n−k∑

j=0

(−1)jα2j

(cos2

w

2

)n−k−j (sin2 w

2

)j.

Note that∣∣q (

e−iw)∣∣ = 22(n−k) |Q(w)|. We show that the refinable function ϕk,q converges

in Lp(R) (2 ≤ p ≤ ∞) to the Shannon refinable function ϕSH uniformly on q ∈ QA,B ask →∞, where

ϕSH(w) := χ[−π,π](w).The main result here is an extended version of [2, Thoerem 4.3]. We mention that the analogousasymptotic behaviors for other families of refinable functions are treated in [2, 4, 5] with similarproofs.

2. MAIN RESULT

For a real symmetric polynomial q(z) with

0 < A ≤ 2−2(n−k)|q(e−iw)| = |Q(w)| ≤ B < ∞, (2.1)

we can easily check that as k →∞ the symbol

Pk,q(e−iw) =1

1− i2k+1(tan w2 )2k+1q(−e−iw)/q(e−iw)

converges pointwise to the symbol

mSH(w) =

1, |w| < π/2,0, π/2 < |w| < π

of the Shannon refinable function ϕSH as k → ∞. We also note that for a fixed w with|w| < π/2 or π/2 < |w| < π the convergence is uniform on the class QA,B of all realsymmetric polynomial q satisfying (2.1).

Before the statement and proof of the main result, we need some technical lemmas. Fix apositive integer K, we define an auxiliary symbol

mK(w) =

1, |w| ≤ π2 ;

B

A

(cos2(2K+1)(w/2)

cos2(2K+1)(w/2) + sin2(2K+1)(w/2)

)1/2

, π2 ≤ |w| ≤ π

for the domination of Pk,q(e−iw).

136 RAE YOUNG KIM

Proof. (a) Fix w and choose l0 so that |w/2l0 | ≤ π/2. By Lemma 2.1 (c),∞∑

l=1

|Pk,q(e−iw/2l)− 1| =

l0∑

l=1

|Pk,q(e−iw/2l)− 1|+

∞∑

l=l0+1

|Pk,q(e−iw/2l)− 1|

≤ l0 +∞∑

l=l0+1

2B

πA

|w|2l

= l0 +2B

πA

|w|2l0

,

uniformly on k and on q ∈ QA,B . Therefore, the product ϕk,q(w) converges uniformly on kand on q ∈ QA,B for a fixed w.(b) Fix w /∈ ∪∞l=12

l(±π/2 + 2πZ) and let ε > 0. By (a) we can choose l1 (independent of kand q ∈ QA,B) so that

|ϕk,q(w)−l1∏

l=1

Pk,q(e−iw/2l)| < ε

and

|ϕSH(w)−l1∏

l=1

mSH(w/2l)| < ε.

Therefore, we have

|ϕk,q(w)− ϕSH(w)| ≤∣∣∣∣∣ϕk,q(w)−

l1∏

l=1

Pk,q(e−iw/2l)

∣∣∣∣∣

+

∣∣∣∣∣l1∏

l=1

Pk,q(e−iw/2l)−

l1∏

l=1

mSH(w/2l)

∣∣∣∣∣

+

∣∣∣∣∣l1∏

l=1

mSH(w/2l)− ϕSH(w)

∣∣∣∣∣

< 2ε +

∣∣∣∣∣l1∏

l=1

Pk,q(e−iw/2l)−

l1∏

l=1

mSH(w/2l)

∣∣∣∣∣ .

Note that w/2l /∈ ±π/2 + 2πZ for any l ≥ 1. Since Pk,q(e−iw/2l) → mSH(w/2l) as k →∞

for l = 1, 2, · · · , l1, we can choose k0 so that∣∣∣∣∣l1∏

l=1

Pk,q(e−iw/2l)−

l1∏

l=1

mSH(w/2l)

∣∣∣∣∣ < ε, k ≥ k0.

Therefore, for a.e. w, ϕk,q(w) → ϕSH(w) uniformly on q ∈ QA,B as k →∞.2

Now, we state and prove our main result. The case A = B = 1 reduces to Theorem 4.3 in[2].

ASYMPTOTIC CONVERGENCE OF ORTHONORMAL CARDINAL REFINABLE FUNCTIONS 137

Theorem 2.3. Let 0 < A < B < ∞.

(a) For 1 ≤ p < ∞, ||ϕk,q − ϕSH ||Lp(R) → 0 (k →∞) uniformly on q ∈ QA,B .(b) For 2 ≤ p′ ≤ ∞, ||ϕk,q − ϕSH ||Lp′ (R) → 0 (k → ∞) uniformly on q ∈ QA,B . In

particular, ϕk,q → ϕSH uniformly on R and on q ∈ QA,B

Proof. Choose K so large that K + 1/2 − log2 B/A > 1. We estimate the decay of ϕk,q fork ≥ K :

|ϕk,q(w)| =∏

l∈N

∣∣∣Pk,q(e−iw/2l)∣∣∣ ≤

∏

l∈NmK(w/2l)

= |ϕK(w)| ≤ C(1 + |w|)−K−1/2+log2 B/A ∈ L1(R) ∩ L2(R),

where we used Lemma 2.1. We now apply the Lebesgue dominated convergence theorem tosupq∈QA,B

|ϕk,q − ϕSH |p to get

supq∈QA,B

||ϕk,q − ϕSH ||Lp(R) ≤ || supq∈QA,B

|ϕk,q − ϕSH | ||Lp(R) → 0

as k → ∞. Therefore, ϕk,q → ϕSH in Lp(R) uniformly on q ∈ QA,B. The claim (b) followsfrom (a) by the Hausdorff-Young inequality:

||f ||Lp′ (R) ≤ ||f ||Lp(R), for 1 ≤ p ≤ 2,

where p′ is the exponent conjugate to p. 2

REFERENCES

[1] C.K. Chui, An Introduction to Wavelets, Academic Press, San Diego, 1992.[2] M. Cotronei, M.L. Lo Cascio, H.O. Kim, C.A. Micchelli and T. Sauer, Refinable functions from Blaschke

products, Rend. Mat. Appl. (7) 26 (2006), 267-290.[3] T.N.T. Goodman, C.A. Micchelli, Orthonormal cardinal functions, In Wavelets : Theory, Algorithms and

Applications, Chui, C. K., Montefusco, L. and Puccio, L., (eds.), Academic Press, (1994) 53-88.[4] H.O. Kim and R.Y. Kim, On asymptotic behavior of Battle-Lemarie scaling functions and wavelets, Appl.

Math. Lett. 20 (2007), 376–381.[5] H.O. Kim, R.Y. Kim and J.S. Ku, Wavelet frames from Butterworth filters, Sampl. Theory Signal Image

Process. 4 (2005), 231–250.[6] R.M. Lewis, Cardinal interpolating multiresolution, J. Approx. Theory, 76 (1994), 177-202.

J. KSIAM Vol.12, No.3, 139–151, 2008

DYNAMICS OF AN IMPULSIVE FOOD CHAIN SYSTEM WITH ALOTKA-VOLTERRA FUNCTIONAL RESPONSE

HUNKI BAEK

DEPARTMENT OF MATHEMATICS, KYUNGPOOK NATIONAL UNIVERSITY, DAEGU 702-701, SOUTH KO-REA


ABSTRACT. We investigate a three species food chain system with Lotka-Volterra type func-tional response and impulsive perturbations. We find a condition for the local stability of preyor predator free periodic solutions by applying the Floquet theory and the comparison theoremsand show the boundedness of this system. Furthermore, we illustrate some examples.

1. INTRODUCTION

In ecology, one of main goals is to understand the dynamical relationship between predatorand prey. Such relationship can be represented by the functional response which refers to thechange in the density of prey attached per unit time per predator as the prey density changes.One of well-known functional responses is a Lotka-Volterra functional response [1, 2] . Theprinciples of Lotka-Volterra models, conservation of mass and decomposition of the rates ofchange in birth and death processes, have remained valid until today and many theoreticalecologists adhere to their principles. For the reason, we need to consider a Lotka-Volterra typefood chain model, which can be described by the following differential equations:

x′(t) = x(t)(a− bx(t)− c1y(t)),y′(t) = y(t)(−d1 + c2x(t)− e1z(t)),z′(t) = z(t)(−d2 + e2y(t)),

(1.1)

where x(t), y(t) and z(t) are the densities of lowest-level prey, mid-level predator and toppredator at time t, respectively. The parameters a, b, c1, c2, d1, d2, e1 and e2 are positive con-stants.

As Cushing [5] pointed out that it is necessary and important to consider models with peri-odic ecological parameters or perturbations which might be quite naturally exposed (for exam-ple, those due to seasonal effects of weather, food supply, mating habits, hunting or harvestingseasons and so on.). Such perturbations were often treated continually. But, there are still someother perturbations such as fire, flood, etc, that are not suitable to be considered continually.

2000 Mathematics Subject Classification. 34A37,34D23,34H05,92D25.Key words and phrases. Food chain system, Lotka-Volterra functional response, impulsive differential equation,

Floquet thoery.

139

140 HUNKI BAEK

These impulsive perturbations bring sudden change to the system. Let’s think of the mid-levelpredator in (1.1) as a pest and the top predator as a natural enemy of it. There are many waysto beat pests. For examples, harvesting on pest, spreading pesticides, releasing natural enemiesand so on. Such tactics are discontinuous and periodical. With the idea mentioned above, inthis paper, we consider the following food chain system with a proportion periodic impulsivepoisoning for all species and periodic constant impulsive immigration of the top predator atdifferent fixed time.

x′(t) = x(t)(a− bx(t)− c1y(t)),y′(t) = y(t)(−d1 + c2x(t)− e1z(t)),z′(t) = z(t)(−d2 + e2y(t)),

t 6= nT, t 6= (n + τ − 1)T,

x(t+) = (1− p1)x(t),

y(t+) = (1− p2)y(t),

z(t+) = (1− p3)z(t),

t = (n + τ − 1)T,

x(t+) = x(t),

y(t+) = y(t),

z(t+) = z(t) + q,

t = nT,

(x(0+), y(0+), z(0+)) = (x0, y0, z0).

(1.2)

where 0 ≤ τ, p1, p2, p3 < 1 and T is the period of the impulsive immigration and q is the size ofimmigration. Such model is an impulsive differential equation whose theory and applicationswere greatly developed by the efforts of Lakshmikantham and Bainov et al. [9].

In recent years, models with sudden perturbations have been intensively researched[7, 10,11, 12, 13, 14, 18, 19, 20, 21, 22]. The authors in [10, 11] have studied the local stabilityfor a two species food chain system with Lotka-Volterra functional response and impulsiveperturbations.

In the next section, we introduce some notations and lemmas used in this paper. In section3, we find a condition for the local stabilities of a lower-level prey and mid-level predator freeperiodic solution and a mid-level predator free periodic solution by applying the Floquet theoryand the comparison theorems. In section 4, we illustrate some examples. Finally, we give aconclusion.

2. PRELIMINARIES

First, we shall introduce a few notations and definitions together with a few auxiliary resultsrelating to the comparison theorems, which will be useful for our main results.

Let R+ = [0,∞) and R3+ = xx = (x(t), y(t), z(t)) ∈ R3 : x(t), y(t), z(t) ≥ 0. Denote

N the set of all of nonnegative integers, R∗+ = (0,∞) and f = (f1, f2, f3)T the right hand of

AN IMPULSIVE FOOD CHAIN SYSTEM 141

the first three equations in (1.2). Let V : R+×R3+ → R+. Then V is said to be in a class V0 if

(1) V is continuous on(nT, (n + 1)T ]× R3+ and lim

(t,y)→(nT,x)t>nT

V (t,y) = V (nT+,x) exists.

(2) V is a local Lipschitzian in x.

Definition 2.1. For V ∈ V0, we define the upper right Dini derivative of V with respect to theimpulsive differential system (1.2) at (t,x) ∈ (nT, (n + 1)T ]× R3

+ by

D+V (t,x) = lim suph→0+

1h

[V (t + h,x + hf(t,x))− V (t,x)].

The solution of the system (1.2) is a piecewise continuous function x : R+ → R3+, x(t) is

continuous on (nT, (n + 1)T ], n ∈ N and x(nT+) = limt→nT+ x(t) exists. The smoothnessproperties of f guarantee the global existence and uniqueness of solutions of the system (1.2).(See [9] for the details).

Now, we give the basic properties of two impulsive differential equations. First, we considerthe following impulsive differential equation:

x′(t) = x(t)(a− bx(t)), t 6= nT, t 6= (n + τ − 1)T,

x(t+) = (1− p1)x(t), t = (n + τ − 1)T,

x(t+) = x(t), t = nT,

x(0+) = x0.

(2.1)

The system (2.1) is a periodically forced system. It is easily obtain that

x∗(t) =aη exp(a(t− (n + τ − 1)T ))

b(1− η + η exp(a(t− (n + τ − 1)T ))), (n + τ − 1)T < t ≤ (n + τ)T, (2.2)

is a positive periodic solution of (2.1), where η = (1−p1) exp(aT )−1exp(aT )−1 . Thus, we obtain the

following Lemma from [15].

Lemma 2.2. [15] The following statements hold.(1) If aT + ln(1− p1) > 0, then limt→∞ |x(t)− x∗(t)| = 0 for all solutions x(t) of (2.1) withx0 > 0.(2) If aT + ln(1− p1) ≤ 0, then x(t) → 0 as t →∞ for all solutions x(t) of (2.1).

Next, we consider the impulsive differential equation as follows:

z′(t) = −d2z(t), t 6= nT, t 6= (n + τ − 1)T,

z(t+) = (1− p3)z(t), t = (n + τ − 1)T,

z(t+) = z(t) + q, t = nT,

z(0+) = z0.

(2.3)

142 HUNKI BAEK

The system (2.3) is a periodically forced linear system. It is easy to obtain that

z∗(t) =

q exp(−d2(t− (n− 1)T ))1− (1− p3) exp(−d2T )

, (n− 1)T < t ≤ (n + τ − 1)T,

q(1− p3) exp(−d2(t− (n− 1)T ))1− (1− p3) exp(−d2T )

, (n + τ − 1)T < t ≤ nT,(2.4)

z∗(0+) = z∗(nT+) = q1−(1−p3) exp(−d2T ) , z∗((n + τ − 1)T+) = q(1−p3) exp(−d2τT )

1−(1−p3) exp(−d2T ) is apositive periodic solution of (2.3). Moreover, we can obtain that

z(t) =

(1− p3)n−1

(z(0+)− q(1− p3)e−T

1− (1− p3) exp(−d2T )

)exp(−d2t) + z∗(t),

(n− 1)T < t ≤ (n + τ − 1)T,

(1− p3)n

(z(0+)− q(1− p3)e−T

1− (1− p3) exp(−d2T )

)exp(−d2t) + z∗(t),

(n + τ − 1)T < t ≤ nT,

(2.5)

is a solution of (2.3). From (2.4) and (2.5), we get easily the following result.

Lemma 2.3. All solutions z(t) of (1.2) with z0 ≥ 0 tend to z∗(t). i.e., |z(t) − z∗(t)| → 0 ast →∞.

Thus, there are at least two periodic solutions (0, 0, z∗(t)) and (x∗(t), 0, z∗(t)) of the system(1.2). It is important to investigate the stabilities of these two periodic solutions because theyplay a major role in the impulsive system (1.2) like the equilibrium points of a system with noimpulsiveness. Moreover, if we regard the mid-level predator y(t) as a pest, then the periodicsolutions represent a kind of pest-free solutions.

We will use a comparison result of impulsive differential inequalities. We suppose thatg : R+ × R+ → R satisfies the following hypotheses:(H) g is continuous on (nT, (n+1)T ]×R+ and the limit lim(t,y)→(nT+,x) g(t, y) = g(nT+, x)exists and is finite for x ∈ R+ and n ∈ N.

Lemma 2.4. [9] Suppose V ∈ V0 and

D+V (t,x) ≤ g(t, V (t,x)), t 6= (n + τ − 1)T, t 6= nT,

V (t,x(t+)) ≤ ψ1n(V (t,x)), t = (n + τ − 1)T,

V (t,x(t+)) ≤ ψ2n(V (t,x)), t = nT,

(2.6)

where g : R+ × R+ → R satisfies (H) and ψ1n, ψ2

+ : R+ → R+ are non-decreasing for alln ∈ N. Let r(t) be the maximal solution for the impulsive Cauchy problem

u′(t) = g(t, u(t)), t 6= (n + τ − 1)T, t 6 nT,

u(t+) = ψ1n(u(t)), t = (n + τ − 1)T,

u(t+) = ψ2n(u(t)), t = nT,

u(0+) = u0,

(2.7)


defined on [0,∞). Then V (0+,x0) ≤ u0 implies that V (t,x(t)) ≤ r(t), t ≥ 0, where x(t) isany solution of (2.6).

We now indicate a special case of Lemma 2.4 which provides estimations for the solution of asystem of differential inequalities. For this, we let PC(R+,R)(PC1(R+,R)) denote the classof real piecewise continuous(real piecewise continuously differentiable) functions defined onR+.

Lemma 2.5. [9] Let the function u(t) ∈ PC1(R+,R) satisfy the inequalities

du

dt≤ f(t)u(t) + h(t), t 6= τk, t > 0,

u(τ+k ) ≤ αku(τk) + βk, k ≥ 0,

u(0+) ≤ u0,

(2.8)

where f, h ∈ PC(R+,R) and αk ≥ 0, βk and u0 are constants and (τk)k≥0 is a strictlyincreasing sequence of positive real numbers. Then, for t > 0,

u(t) ≤u0

( ∏

0<τk<t

αk

)exp

(∫ t

0f(s)ds

)+

∫ t

0

( ∏

0≤τk<t

dk

)exp

(∫ t

sf(γ)dγ

)h(s)ds

+∑

0<τk<t

( ∏τk<τj<t

dj

)exp

(∫ t

τk

f(γ)dγ)βk.

Similar result can be obtained when all conditions of the inequalities in the Lemma 2.4 and2.5 are reversed. Using Lemma 2.5, it is proven that all solution of (1.2) with x0 > 0 remainstrictly positive as follows:

Lemma 2.6. The positive orthant (R∗+)3 is an invariant region for the system (1.2).

Proof. Let (x(t), y(t), z(t)) : [0, t0) → R2 be a saturated solution of the system (1.2) with astrictly positive initial value (x0, y0, z0). By Lemma 2.5, we can obtain that, for 0 ≤ t < t0,

x(t) ≥ x(0)(1− p1)[tT

] exp(∫ t

0f1(s)ds

),

y(t) ≥ y(0)(1− p2)[tT

] exp(∫ t

0f2(s)ds

),

z(t) ≥ z(0)(1− p3)[tT

] exp(∫ t

0f3(s)ds

)q,

(2.9)

where f1(s) = a − bx(s) − cy(s), f2(s) = −d1 − e1z(s) and f3(s) = −d2. Thus, x(t), y(t)and z(t) remain strictly positive on [0, t0). ¤

144 HUNKI BAEK

3. MAIN THEOREMS

In this section, we study the stability of the lowest-level prey and mid-level predator free pe-riodic solution (0, 0, z∗(t)) and of the mid-level predator free periodic solution (x∗(t), 0, z∗(t)).First, we show that all solutions of (1.2) are uniformly ultimately bounded.

Theorem 3.1. There is an M > 0 such that x(t) ≤ M, y(t) ≤ M and z(t) ≤ M for all tlarge enough, where (x(t), y(t), z(t)) is a solution of the system (1.2).

Proof. Let (x(t), y(t), z(t)) be a solution of (1.2) and define u(t) = c2c1

x(t) + y(t) + e1e2

z(t). Itis easily shown that, for t 6= nT ,t 6= (n + τ − 1)T ,

du(t)dt

= −c2b

c1x2(t) +

e1a

e2x(t)− d1y(t)− c3d2

c4z(t). (3.1)

Also, it follows from choosing 0 < β0 < mind1, d2 that

du(t)dt

+ β0u(t) ≤ −c2b

c1x2(t) +

c2

c1(a + β0)x(t), t 6= nT, t 6= (n + τ − 1)T. (3.2)

Since the right-hand side of (3.2) is bounded from above by M0 = c2(a+β0)2

4bc1, we obtain that

du(t)dt

+ β0u(t) ≤ M0, t 6= nT, n 6= (n + τ − 1)T.

If t = nT , then u(t+) = u(t)+ e1e2

q and if t = (n+ τ − 1)T , then u(t+) ≤ (1− p)u(t), wherep = minp1, p2, p3. It is from Lemma 2.5 that

u(t) ≤ u0

( ∏

0<kT<t

(1− p)

)exp

(∫ t

0−β0ds

)

+∫ t

0

( ∏

0≤kT<t

(1− p)

)exp

(∫ t

s−β0dγ

)M0ds

+∑

0<kT<t

( ∏

kT<jT<t

(1− p)

)exp

(∫ t

kT−β0dγ

)e1

e2q

≤ u(0+) exp(−β0t) +M0

β0(1− exp(−β0t)) +

c3q exp(β0T )c4 exp(β0T )− 1

.

(3.3)

Since the limit of the right-hand side of (3.3) as t →∞ is

M0

β0+

e1q exp(β0T )e2 exp(β0T )− 1

< ∞,

it easily follows that u(t) is bounded for sufficiently large t. ¤

Theorem 3.2. (1)The periodic solution (0, 0, z∗(t)) is locally stable if aT + ln(1− p1) ≤ 0.(2)The periodic solution (0, 0, z∗(t)) is unstable if aT + ln(1− p1) > 0.


Proof. The local stability of the periodic solution (0, 0, z∗(t)) of the system (1.2) may be de-termined by considering the behavior of small amplitude perturbations of the solution. Let(x(t), y(t), z(t)) be a solution of the system (1.2), where x(t) = u(t), y(t) = v(t) andz(t) = w(t) + z∗(t). Then they may be written as

u(t)v(t)w(t)

= Φ(t)

u(0)v(0)w(0)

where Φ(t) satisfies

dΦdt

=

a 0 00 −d1 − e1z

∗(t) 00 e2z

∗(t) −d2

Φ(t)

and Φ(0) = I is the identity matrix. So the fundamental solution matrix is

Φ(t) =

exp(at) 0 00 exp(

∫ t0 −d1 − e1z

∗(s)ds) 00 exp(

∫ t0 e2z

∗(s)ds) exp(−d2t)

.

The resetting impulsive conditions of the system (1.2) become

u((n + τ − 1)T+)v((n + τ − 1)T+)u((n + τ − 1)T+)

=

1− p1 0 00 1− p2 00 0 1− p3

u((n + τ − 1)T )v((n + τ − 1)T )w((n + τ − 1)T )

and

u(nT+)v(nT+)w(nT+)

=

1 0 00 1 00 0 1

u(nT )v(nT )w(nT )

.

Note that the eigenvalues of

S =

1− p1 0 00 1− p2 00 0 1− p3

1 0 00 1 00 0 1

Φ(T )

are µ1 = (1 − p1) exp(aT ), µ2 = (1 − p2) exp(− ∫ T0 d1 + e1z

∗(s)ds) < 1 and µ3 =exp(−d2T ) < 1. The condition µ1 ≤ 1(> 1) is equivalent to the equation aT + ln(1− p1) ≤0(> 0). Therefore, by Floquet theory of impulsive differential equations, the periodic solution(0, 0, z∗(t)) is locally stable if aT +ln(1−p1) ≤ 0 and is unstable if aT +ln(1−p1) > 0. ¤

Theorem 3.3. The periodic solution (x∗(t), 0, z∗(t)) is locally asymptotically stable if aT +ln(1− p1) > 0 and

c2

b(aT +ln(1−p1))+ln(1−p2) < d1T +

e2q(1− (1− p3) exp(−d2T )− p3 exp(−d2τT ))d2(1− (1− p3) exp(−d2T ))

.

(3.4)

146 HUNKI BAEK

Proof. Assume that aT + ln(1 − p1) > 0 and the equation (3.4) hold. We apply the samemethod as Theorem 3.2 to the periodic solution (x∗(t), 0, z∗(t)) to determine its stability. So,we define x(t) = u(t) + x∗(t), y(t) = v(t), z(t) = w(t) + z∗(t). Then they may be written as

u(t)v(t)w(t)

= Φ(t)

u(0)v(0)w(0)

where Φ(t) satisfies

dΦdt

=

a− 2bx∗(t) −c1x∗(t) 0

0 −d1 + c2x∗(t)− e1z

∗(t) 00 e2z

∗(t) −d2

Φ(t)

and Φ(0) = I is the identity matrix. The resetting impulsive conditions of the system (1.2)become

u((n + τ − 1)T+)v((n + τ − 1)T+)u((n + τ − 1)T+)

=

1− p1 0 00 1− p2 00 0 1− p3

u((n + τ − 1)T )v((n + τ − 1)T )w((n + τ − 1)T )

and

u(nT+)v(nT+)w(nT+)

=

1 0 00 1 00 0 1

u(nT )v(nT )w(nT )

.

Further, the eigenvalues of

S =

1− p1 0 00 1− p2 00 0 1− p3

1 0 00 1 00 0 1

Φ(T )

are µ1 = (1− p1) exp(∫ T0 a− 2bx∗(t)dt), µ2 = (1− p2) exp

(∫ T0 −d1 + c2x

∗(t)− e1z∗(t)dt

)

and µ3 = (1 − p3) exp(−d2T ) < 1. Since x∗(t) =aη exp(at)

b(1− η + η exp(at))for 0 < t ≤ T , we

get

∫ T

0x∗(t)dt =

aη

b

∫ T

0

exp(at)1− η + η exp(at)

dt

=1b(ln(1− p1) + aT ),

(3.5)


where η =(1− p1) exp(aT )− 1

exp(aT )− 1. And since z∗(t) =

q exp(−d2t)1− (1− p3) exp(−d2T )

for 0 < t ≤

τT and z∗(t) =q(1− p3) exp(−d2t)

1− (1− p3) exp(−d2T )for τT < t ≤ T , we obtain that

∫ T

0z∗(t)dt =

∫ τT

0z∗(t)dt +

∫ T

τTz∗(t)dt

=q(1− (1− p3) exp(−d2T )− p3 exp(−d2τT ))

d2(1− (1− p3) exp(−d2T )).

(3.6)

From (3.5) and (3.6), we get that the conditions |µ1| < 1 and |µ2| < 1 are equivalent tothe equations aT + ln(1− p1) > 0 and (3.4), respectively. Therefore, it follows from FloquetTheory of impulsive differential equations that (x∗(t), 0, z∗(t)) is locally asymptotically stable.

¤

02

4 02

460

100

200

y

(a)

x

z

0 0.5 10

2

4

6

t

x

(b)

0 4 80

2

4

6

t

y

(c)

19950 19975 20000164

165

166

167

t

z

(d)

FIGURE 1. a = 1.0, b = 0.0002, c1 = 1.0, c2 = 0.3, d1 = 0.3, d2 =0.001, e1 = 0.05, e2 = 0.0005, p1 = 0.9, p2 = 0.1, p3 = 0.01, τ = 0.2, q = 2and T = 2. (a) The trajectory of the system (1.2). (b-d) Time series.

4. NUMERICAL EXAMPLES

In this section, we investigate numerical examples for the system (1.2).

148 HUNKI BAEK

Firstly, we consider the unperturbed system (1.1) which has no control terms. It is easy tosee that the unperturbed three species food chain system (1.1) has four non-negative equilibria:

510

1520 0

24

20

40

y

(a)

x

z

19950 19975 2000017

18

19

20

tx

(b)

0 50

2

4

6

t

y

(c)

19950 19975 2000041

42

43

44

45

t

z(d)

FIGURE 2. a = 1.0, b = 0.2, c1 = 0.5, c2 = 0.1, d1 = 0.4, d2 = 0.02, e1 =0.5, e2 = 0.002, p1 = 0.1, p2 = 0.3, p3 = 0.01, τ = 0.3, T = 3, q = 3. (a)The trajectory of the system (1.2). (b-d) Time series.

(1) The trivial equilibrium A(0, 0, 0).(2) The mid-level predator and top predator free equilibrium B(d1

b , 0, 0).(3) The top predator free equilibrium C(d1

c2, ac2−d1b

c1c2, 0) if ac2 − d1b > 0.

(4) The positive equilibrium E∗ = (x∗, y∗, z∗), where

x∗ =ae2 − d2c1

be2, y∗ =

d2

e2, z∗ =

ae2c2 − d2c1c2 − d1be2

be1e2and ae2c2 − d2c1c2 − d1be2 > 0.

Stabilities for the equilibria of the system (1.1) have been studied by Zhang and Chen[19].

Lemma 4.1. [19] (1) If positive equilibrium E∗ exists, then E∗ is globally stable.(2) If positive equilibrium E∗ dose not exist and C exists, then C is globally stable.(3) If positive equilibrium E∗ and C do not exist, then B is globally stable.

Throughout this section, we chose (x0, y0, z0) = (5, 5, 5) as an initial point. We considerthe following three cases:(A) a = 1.0, b = 0.0002, c1 = 1.0, c2 = 0.3, d1 = 0.3, d2 = 0.001, e1 = 0.05, e2 =0.0005, p1 = 0.9, p2 = 0.1, p3 = 0.01, τ = 0.2, T = 2, q = 2.


010

20 0

50

1000

50

y

(a)

x

z

19950 19975 2000017

18

19

20

t

x

(b)

0 4 80

50

100

t

y

(c)

19950 19975 2000041

42

43

44

45

t

z

(d)

050

100 0

50

1000

100

200

y

(e)

x 19950 19975 2000017

18

19

20

t

x

(f)

0 30

20

40

60

80

t

y

(g)

19950 19975 2000041

42

43

44

45

t

z

(h)

FIGURE 3. a = 1.0, b = 0.2, c1 = 0.5, c2 = 0.1, d1 = 0.4, d2 = 0.02, e1 =0.5, e2 = 0.002, p1 = 0.1, p2 = 0.3, p3 = 0.01, τ = 0.3, T = 3, q = 3. (a-d)The trajectory of the system (1.2) with the initial value (0.1, 100, 0.1). (e-h)The trajectory of the system (1.2) with the initial value (100, 100, 100).

(B) a = 1.0, b = 0.2, c1 = 0.5, c2 = 0.1, d1 = 0.4, d2 = 0.02, e1 = 0.5, e2 = 0.002, p1 =0.1, p2 = 0.3, p3 = 0.01, τ = 0.3, T = 3, q = 3.(C) a = 2, b = 0.0001, c1 = 1.0, c2 = 0.3, d1 = 0.3, d2 = 0.01, e1 = 0.05, e2 = 0.0025, p1 =0.01, p2 = 0.02, p3 = 0.001, τ = 0.3, T = 5, q = 4.

For (A), it follows from Theorem 3.2 that the periodic solution (0, 0, z∗(t)) is locally stablesince aT + ln(1− p1) = −0.3026 ≤ 0. Also, we infer from Lemma 4.1 that the unperturbedsystem (1.1) has a globally stable top predator free equilibrium C(1, 0.9997, 0), but no positiveequilibria. The behavior of the trajectories of the system (1.2) exhibited in Figure 1 shows thatz(t) is synchronizing with the periodic solution z∗(t) for sufficiently large t.

For (B), it is easy to see from Theorem 3.3 that the periodic solution (x∗(t), 0, z∗(t)) islocally asymptotically stable. Further, from Lemma 4.1, we see that the unperturbed system(1.1) has a global stable positive equilibrium E∗ = (0.79, 0.1, 11.77). The trajectory of thesystem (1.2) is illustrated in Figure 4. In this case, we conjecture that the periodic solution(x∗(t), 0, z∗(t)) may be globally stable under the same condition of Theorem 3.3 because manytrajectories with different initial values show stable phenomena. (See Figure 3 )

For (C), it is from Lemma 4.1 that the unperturbed system (1.1) also has a globally stabletop predator free equilibrium C(1, 2, 0). However, Figure 4, which is the phase portraits ofthe system (1.2) for the case (C), suggests that the system (1.2) may be permanent thoughwe cannot determine whether the system (1.2) is stable or not because it does not satisfy thecondition (3.4).

150 HUNKI BAEK

0100

200 020

400

20

40

y

(a)

x

z

19500 20000 195000

50

100

150

200

t

x

(b)

19500 20000 195000

10

20

30

40

t

y

(c)

19500 20000 19500150

152

154

156

158

t

z

(d)

FIGURE 4. a = 2, b = 0.0001, c1 = 1.0, c2 = 0.3, d1 = 0.3, d2 = 0.01, e1 =0.05, e2 = 0.0025, p1 = 0.01, p2 = 0.02, p3 = 0.001, τ = 0.3, T = 5, q = 4.(a) The trajectory of the system (1.2). (b-d) Time series.

5. CONCLUSION

In this paper, we have studied a three species food chain system with Lokta-Volterra func-tional response and impulsive perturbations. We have found a condition for the local stabilitiesof a lower-level prey and mid-level predator free periodic solution and a mid-level predatorfree periodic solution by applying the Floquet theory and the comparison theorems and haveproven the boundedness of this system. In addition, we have given numerical examples andsuggested that the prey or predator free solutions may be globally stable and the system (1.2)may be permanent.

REFERENCES

[1] S. Anita, Analysis and control of age-dependent population dynamics, Kluwer Academic Publishers, 2000.[2] F. Brauer and C. Castillo-Chavez, Mathematical models in population biology and epidemilogy, Texts in

applied mathematics 40, Springer-Verlag, 2001.[3] J. B. Collings, The effects of the functional response on the bifurcation behavior of a mite predator-prey

interaction model, J. Math. Biol., 36(1997), 149-168.[4] C. Cosner, D.L. DeAngelis, Effects of spatial grouping on the functional response of predators, Theoretical

Popuation Biology, 56(1999), 65-75.[5] J. M. Cushing, Periodic time-dependent predator-prey systems, SIAM J. Appl. Math., 32(1977), 82-95.[6] H. I. Freedman and R. M. Mathsen, Persistence in predator-prey systems with ratio-dependent predator

influence, Bulletin of Math. Biology, 55(4)(1993), 817-827.


[7] S.-B. Hsu and T.-W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math.,55(3)(1995), 763-783.

[8] K. Kitamura, K. Kashiwagi, K.-i. Tainaka, T. Hayashi, J. Yoshimura, T. Kawai and T. Kajiwara, Asymmet-rical effect of migration on a prey-predator model, Physics Letters A, 357(2006), 213-217.

[9] V Lakshmikantham, D. Bainov, P.Simeonov, Theory of Impulsive Differential Equations, World ScientificPublisher, Singapore, 1989.

[10] B. Liu, Y. Zhang and L. Chen, Dynamic complexities in a Lotka-Volterra predator-prey model concerningimpulsive control strategy, Int. J. of Bifur. and Chaos, 15(2)(2005), 517-531.

[11] B. Liu, Z. Teng and W. Liu, Dynamic behaviors of the periodic Lotka-Volterra competing system withimpulsive perturbations, Chaos, Solitons and Fractals, 31(2007),356-370.

[12] B. Liu, Z. Teng and L. Chen, Analsis of a predator-prey model with Holling II functional response concerningimpulsive control strategy, J. of Comp. and Appl. Math., 193(1)(2006), 347-362

[13] B. Liu, Y. J. Zhang, L. S. Chen and L. H. Sun, The dynamics of a prey-dependent consumption modelconcerning integrated pest management, Acta Mathematica Sinica, English Series, 21(3)(2005), 541-554.

[14] X. Liu and L. Chen, Complex dynamics of Holling type II Lotka-Volterra predator-prey system with impul-sive perturbations on the predator, Chaos, Solitons and Fractals, 16(2003), 311-320.

[15] P. Georgescu and G. Morosanu, Impulsive perturbations of a three-trophic prey-dependent food chain sys-tem, Mathematical and Computer Modeling(2008), doi:10.1016/j.mcm.2007.12.006.

[16] S. Ruan and D. Xiao, Global analysis in a predator-prey sytem with non-monotonic functional response,SIAM J. Appl. Math., 61(4)(2001), 1445-1472.

[17] E, Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, SIAM J. Appl. Math., 59(5)(1999),1867-1878.

[18] W. Wang, H. Wang and Z. Li, Chaotic behavior of a three-species Beddington-type system with impulsiveperturbations, Chaos Solitons and Fractals, 37(2008), 438-443.

[19] S. Zhang and L. Chen, Chaos in three species food chain system with impulsive perturbations, Chaos Soli-tons and Fractals, 24(2005), 73-83.

[20] S. Zhang and L. Chen, A Holling II functional response food chain model with impulsive perturbations,Chaos Solitons and Fractals, 24(2005), 1269-1278.

[21] S. Zhang, D. Tan and L. Chen, Dynamic complexities of a food chain model with impulsive perturbationsand Beddington-DeAngelis functional response, Chaos Solitons and Fractals, 27(2006), 768-777.

[22] Y. Zhang, B. Liu and L. Chen, Extinction and permanence of a two-prey one-predator system with impulsiveeffect, Mathematical Medicine and Biology, 20(2003), 309-325.

J. KSIAM Vol.12, No.3, 153–160, 2008

ON SIZE-BIASED POISSON DISTRIBUTION AND ITS USE INZERO-TRUNCATED CASES

KHURSHID AHMAD MIR†1

1 DEPARTMENT OF STATISTICS, GOVT. DEGREE COLLEGE (BOYS), BARAMULLA, JAMMU AND KASHMIR,INDIA


ABSTRACT. A size-biased Poisson distribution is defined. Its characterization by using a re-currence relation for first order negative moment of the distribution is obtained. Different esti-mation methods for the parameter of the model are also discussed. R-Software has been usedfor making a comparison among the three different estimation methods.

1. INTRODUCTION

The probability function of the Poisson distribution is given as

P (X = x) = e−ααx/x!, x = 0, 1, 2..... (1)David and Johnson [5] defined the decapitated Poisson distribution with probability function

as

P1 (X = x) = e−ααx/x!(1− e−α), x = 1, 2..... (2)Murakami [9] discussed the maximum likelihood estimator of the model (2). David and

Johnson [5] studied the estimator of the model (2) based on the sample moments. They alsoderived the maximum likelihood estimator (MLE) of α, its asymptotic variance and efficiencyby the method of moments. Placket [10] put forward a similar estimate of α in order to showthat it is highly efficient. Tate and Goen [12] obtained minimum variance unbiased estimationand Cohen ([3],[4]) provided the estimation of the model (2) from the sample that are truncatedon the right. Ayesha and Ahmad [1] studied the inverse ascending factorial moments and theestimation of the parameter of hyper-Poisson distribution using negative moments. Munir andRoohi [8] have discussed the characterization of the Poisson distribution. A brief list of authorsand their substantial works can be seen in Johnson and Kotz [6] and Johnson, Kotz and Kemp[7].

In this paper, the size-biased Poisson distribution (SBPD) is defined and the characterizationof the model is obtained by using a recurrent relation for its first order negative moment. The

2000 Mathematics Subject Classification. 62E15.Key words and phrases. Negative moments, Size-based Poisson Distribution, Maximum Likelihood Estimation,

Baye’s Estimator, R-Software.

153

154 K. A. MIR

estimates have been obtained by employing the moments, maximum likelihood and Bayesianmethod of estimation. In order to make a comparative analysis among the three estimationmethods for the parameter of the size-biased Poisson distribution (SBPD), one of the standardsoftware packages R- Software is used which is meant for data analysis and graphics. It isfreely available on internet.Its resemblance with the S-PLUS software makes it more useful.(See http://cran-project.org and Bates [2]).

2. SIZE-BIASED POISSON DISTRIBUTION(SBPD)

The size-biased Poisson distribution is obtained by taking the weights of the Poisson distri-

bution (1) as x. Then,we have∞∑

x=0xP (X = x) = α, which gives the probability function of

size-biased Poisson distribution as

P2 (X = x) = e−ααx−1/ (x− 1)!, α > 0, x = 1, 2....., (3)

The moment generating function of the distribution (3) is given by

Mx (t) = e−αe(t+etα) (4)By using the relation (4), the mean and variance of the distribution are given as

µ/1 = 1 + α (5)

µ2 = α (6)

3. RECURRENCE RELATION

In this section, we use a property of hyper-geometric series function and give an alternatemethod of deriving the recurrence relation for the negative moment of size-biased Poissondistribution.

Theorem 1. : Suppose X has a size-biased Poisson distribution with parameter α , then forA>1 the relation

E(X + A)−1 =1α− A

αE(X + A− 1)−1 (7)

holds.

Proof. Since X is a size-biased Poisson variate with parameter α , then

E(X + A)−1 =∞∑

x=1

1(x + A)

P2 (X = x)

= e−α (A + 1)−1 1 F1 [A + 1;A + 2;α] (8)

where1F1 [A + 1;A + 2;α] = 1 +

(A + 1)(A + 2)

α +(A + 1) (A + 2)(A + 2) (A + 3)

α2.......

ON SIZE-BASED POISSON DISTRIBUTION 155

Replacing A by A-1, we get

E(X + A− 1)−1 = e−αA−1 1F1 [A;A + 1;α] (9)

The R.H.S of the above relation gives the negative moment for simple Poisson model (1).Using the identity ( see Rainville [11] page 124)

b 1F1 [a; b;x] = b 1F1 [a− 1; b;x] + x 1F1 [a; b + 1;x] ,

for a = A + 1, b = A + 1,and x = α, we get

(A + 1) 1 F1 [A + 1;A + 1;α] = (A + 1) 1 F1 [A;A + 1;α] + α 1F1 [A + 1;A + 2;α](10)

Also,1F1 [A + 1;A + 1;α] = eα (11)

Using (8),(9) and (11) in (10), we get the result. ¤

4. CHARACTERIZATION

In this section, the recurrence relation derived in theorem 1 is used for the characterizationof the size-biased Poisson distribution.

Theorem 2. : If X is a random variable taking the positive-integer values and the relationE(X + A)−1 = 1

α − Aα E(X + A − 1)−1 for A>1 is true, then X is characterized by a

size-biased Poisson distribution.

Proof. Since for A>1, we have

E(X + A)−1 =1α− A

αE(X + A− 1)−1

∞∑

x=1

1(x + A)

P2 (X = x) =1α− A

α

∞∑

x=1

1(x + A− 1)

P2 (X = x)

=1α− 1

αP2 (X = 1)− A

α

∞∑

x=2

1(x + A− 1)

P2 (X = x)

=1α− 1

αP2 (X = 1)− A

α

∞∑

x=1

1(x + A)

P2 (X = x + 1) .

By simple computation, we get

∞α

∑

x=1

1(x + A)

P2 (X = x) = 1− P2 (X = 1)−A

∞∑

x=1

1(x + A)

P2 (X = x + 1) . (12)

156 K. A. MIR

Since∞∑

x=1P2 (X = x) = 1, which gives

∞∑

x=2

P2 (X = x) = 1− P2 (X = 1) =∞∑

x=1

P2 (X = x + 1) ,

(12) becomes

α

∞∑

x=1

1(x + A)

P2 (X = x) =∞∑

x=1

x

(x + A)P2 (X = x + 1) ,

∞∑

x=1

αP2 (X = x)− xP2 (X = x + 1)(x + A)

= 0.

Since αP2 (X = x)− xP2 (X = x + 1) is either >= or <0, then in each case, we get

αP2 (X = x) /(x + A) = xP2 (X = x + 1) /(x + A),

thus

P2 (X = x + 1) =α

xP2 (X = x) .

Putting x = 1, 2, 3.....x− 1, we get

P2 (X = 2) = αP2 (X = 1) ,

P2 (X = 3) =α

2P2 (X = 2) =

α2

2!P2 (X = 1) , ......

P2 (X = x) =αx−1P2 (X = 1)

(x− 1)!From equation(3) for x = 1, P2 (X = 1) = e−α.Therefore,

P2 (X = x) = αx−1e−α/ (x− 1)!,which is the probability function of size-biased Poisson distribution.This completes the proof.

¤

5. ESTIMATION METHODS

In this section, we discuss the basic three estimation methods for the parameter of the size-biased Poisson distribution and verify their efficiencies.

5.1. METHOD OF MOMENTS. In the method of moments, replacing the population mean

µ/1 = 1 + α by the corresponding sample mean x = 1

n

n∑i=1

xi , we get

α = x− 1 (13)


5.2. METHOD OF MAXIMUM LIKELIHOOD. Let X1, X2......Xn be a random samplefrom the size-biased Poisson distribution, then the corresponding likelihood function is givenas

L = e−nαα

n∑i=1

xi−n/

nΠ

i=1(xi − 1)!, x = 1, 2... (14)

= e−nααy−n/nΠ

i=1(xi − 1)!, where y =

n∑

i=1

xi. (15)

The log likelihood function of (15) can be written as

log L = −nα + (y − n) log α−n∑

i=1

log((xi − 1)!)

The corresponding likelihood equation is given as

∂ log L

∂α= −n +

(y − n)α

On equating the above derivative equal to zero,we get the maximum likelihood estimate as

α = x− 1.

This coincides with the moment estimate.

5.3. BAYESIAN METHOD OF ESTIMATION. We assume that before the observationswere made, our knowledge about the parameter α was only a vague one. Consequently, thenon-informative vague prior g(α) proportional to 1

α is applicable to a good approximation.Thus

g(α) =1α

, α > 0. (16)

The posterior distribution from (15) and (16) is given as

Π(α/y) =αy−n−1 e−nα

∞∫0

αy−n−1 e−nαdα

The Bayes estimator of α becomes

158 K. A. MIR

α =

∞∫

0

αΠ(α/y) dα

=

∞∫0

αy−n e−nαdα

∞∫0

αy−n−1 e−nαdα

= y − n/n = x− 1,

which coincides with mle and moment estimate.In order to find out the more general estimate of α, we consider the more general prior of

α which is given by the gamma distribution with known hyper-parameters a, b>0 having thedensity function as

f (α) = abe−aααb−1/Γb;α ≥ 0 (17)Using (15) and (17), the Bayes estimator of α comes out to be

α = y + b− n/n + a (18)

For a=b=0, the estimator coincides with mle and moment estimate. This shows that theBaye’s estimate α serves as a general estimate which can be used for fitting purposes to a reallife data.

6. NUMERICAL EXPERIMENTS AND DISCUSSIONS

It is very difficult to compare the theoretical performances of different estimators proposedin the previous section. Therefore, we perform extensive simulations to compare the perfor-mances of different methods of estimation mainly with respect to their biases and the meansquared errors (MSE’s), for different sample sizes and different parametric values. Regardingthe choice of values of (a, b) in Baye’s estimator (α), there was no information about theirvalues except that they are real and positive numbers. Therefore, 25 combinations of values of(a, b) were considered for a, b = 1, 2, 3, 4, 5 and those values of a, b were selected for whichthe Baye’s estimator has minimum variance. It was found that for a=b=2, the Baye’s estima-tor has minimum variance and χ2 values between the simulated sample frequencies and theestimated Baye’s frequencies were least.

6.1. AVERAGE RELATIVE ESTIMATES AND AVERAGE RELATIVE MEAN SQUAREDERRORS OF α. For the sample sizes n = 15, 20, 30, 50, 100 and different values of the pa-rameter α = 0.5, 1.0, 2.0, 2.5 and for each combination of n and α, we generate a sample ofsize n from the size-biased Poisson distribution and estimate α by different methods of esti-mation. We report the average values of

(αα

)and the corresponding average MSE’s within

brackets.. All the reported results are based on 10, 000 replications. The results are presented


in table 1. From the table it is clear that the average biases and the average MSE’s decreaseas sample size increases. It indicates that all the methods of estimation provide the asymptoti-cally unbiased and the consistent estimators. It is also observed that the average biases and theaverage MSE’s of

(αα

)depend on α . On comparing the performances of all the methods, it is

clear that as far as the minimum bias is concerned the Baye’s works the best in almost all thecases.

TABLE 1. AVERAGE RELATIVE ESTIMATES AND AVERAGE RELA-TIVE MEAN SQUARED ERRORS OF α

n Method α = 0.5 α = 1.0 α = 2.0 α = 2.515 Baye′s

MLE1.180(0.230)

1.364(0.678)

1.221(0.435)1.383(0.871)

1.331(0.857)1.455(1.624)

1.356(0.97)1.465(1.4166)

20 Baye′sMLE

1.132(0.141)1.314(0.448)

1.161(1.88)1.285(0.479)

1.223(0.343)1.317(0.675)

1.244(0.424)1.338(0.791)

30 Baye′sMLE

1.084(0.075)1.218(0.248)

1.100(0.100)1.191(0.242)

1.129(0146)1.197(0.293)

1.145(0.178)1.213(0.357)

50 Baye′sMLE

1.048(0.038)1.134(0.131)

1.054(0.045)1.112(0.117)

1.077(0.068)1.123(0.145)

1.082(0.077)1.125(0.157)

100 Baye′sMLE

1.022(0.016)1.065(0.060)

1.027(0.020)1.056(0.053)

1.035(0.027)1.058(0.061)

1.038(0.030)1.061(0.066)

6.2. FITTING OF SIZE-BIASED POISSON DISTRIBUTION MODEL. The two differ-ent varieties of Mulberry Ichinose and Kokuso-20 having different leaf spot disease intensitywere chosen for the study in a local Kashmir Sericulture division. Three trees of each varietywere selected at random. From each tree, three branches were selected randomly and then fromeach branch, the spots were recorded from all the leaves. The leaves with no spot were referredas disease free and named as grade zero (0 grade). The leaves having 1 to 5, 6 to 10, 11 to15, 16 to 20 and more than 20 spots were graded as 1,2,3,4 and 5 grades respectively. In ourstudy, the leaves of zero grades were not found. The data for two varieties of Mulberry Ichi-nose and Kokuso-20 are listed in tables 2 and 3, respectively. A comparison is made betweendifferent methods of estimation for the parameter of the size-biased Poisson distribution and itwas found that the Baye’s estimator constitutes a better fit against MLE or moment estimator.

ACKNOWLEDGMENTS

The author is highly thankful to the referee and the editor for their constructive suggestions.

REFERENCES

[1] Ayesha Roohi and Munir Ahmad, Estimation of Characterization of the parameter of Hyper- Poisson distri-bution using negative moments , Pakistan Jounal of Statistics,19(2003), 99-105.

[2] Bates, D.M, Using open source to teach mathematical statistics.http://www.stat.wisc.edu/˜bates/JSM(2001).pdf.

160 K. A. MIR

TABLE 2

Leaf SpotGrade Obseved Frquency

Expected FrequencyMLE Baye’sα α

1 18 17.5 17.92 15 14.7 14.953 10 9.86 9.914 14 13.94 13.995 13 14 13.25Total 70 70 70χ2 0.094 0.0063

TABLE 3

Leaf SpotGrade Obseved Frquency

Expected FrequencyMLE Baye’sα α

1 37 36.42 36.922 16 15.92 15.973 15 14.93 14.964 8 7.64 7.915 8 9.09 8.24Total 84 84 84χ2 0.142 0.0083

[3] Cohen, A.C, An extension of a truncated Poisson distribution, Biometrics, 16(1960a), 446- 450.[4] Cohen, A.C, Estimation in a truncated Poisson distribution when zeroes and some ones are missing, Journal

of American Statistical Association 55(1960b),342- 348.[5] David, F.N and Johnson, N.L, The truncated Poisson, Biometrics, 8(1952), 275- 285.[6] Johson, N.L and kotz, S, Discrete distribution in Statistics, John Wiley,(1969).[7] Johson, N.L and kotz, S and Kemp, A.W, Univariate discrete distributions, John Wiley and Sons(1992).[8] Munir Ahmad and Ayesha Roohi , Characterization of the Poisson Probability distribution, Pakistan Journal

of Statistics Vol. 20(2)(2004), 301-304.[9] Murakami, M, Censored sample from truncated Poisson distribution, J.g the College of arts and sciences,

Chiba Urmssily, 3(1964),263-268.[10] Plackett, R.L, The truncated Poisson distributions Biometrics, 9(1953), 185-188.[11] Rainville, E.D, Special functions. Chelsa publishing company,Bronnx N(1960).[12] Tate, R.F. and Goen, R.L, MVUE for the truncated Poisson distribution, Annals of Mathematical Statistics, 29

(1958),755- 765.

J. KSIAM Vol.12, No.3, 161–169, 2008

MULTIPLICITY AND NONLINEARITY IN THE NONLINEAR ELLIPTICSYSTEM

TACKSUN JUNG1 AND Q-HEUNG CHOI2†

1DEPARTMENT OF MATHEMATICS, KUNSAN NATIONAL UNIVERSITY, KUNSAN 573-701, KOREA


2DEPARTMENT OF MATHEMATICS, INHA UNIVERSITY, INCHEON 402-751, KOREA


ABSTRACT. We investigate the existence of solutions u(x, t) for perturbations of the ellipticsystem with Dirichlet boundary condition

Lξ + µg(ξ + 2η) = f in Ω,

Lη + νg(ξ + 2η) = f in Ω,(0.1)

where g(u) = Bu+−Au−, u+ = maxu, 0, u− = max−u, 0, µ, ν are nonzero constantsand the nonlinearity (µ + 2ν)g(u) crosses the eigenvalues of the elliptic operator L.

1. INTRODUCTION

Let Ω be a bounded domain in Rn with smooth boundary ∂Ω and let L denote the differentialoperator

L =∑

1≤i,j≤n

∂

∂xi(aij

∂

∂xj),

where aij = aji ∈ C∞(Ω). In [2] the authors investigate multiplicity of solutions of thenonlinear elliptic equation with Dirichlet boundary condition

Lu + g(u) = f(x) in Ω,

u = 0 on ∂Ω,(1.1)

where the semilinear term g(u) = bu+−au− and L is a second order linear elliptic differentialoperator and a mapping from L2(Ω) into itself with compact inverse, with eigenvalues −λi,each repeated according to its multiplicity,

0 < λ1 < λ2 < λ3 ≤ · · · ≤ λi ≤ · · · → ∞.

2000 Mathematics Subject Classification. 35J05, 35J25, 35J55.Key words and phrases. Elliptic system, eigenvalue problem, Dirichlet boundary condition, uniqueness, multi-

ple solutions.This work was supported by Inha University Research grant.† Corresponding author.

161

162 TACKSUN JUNG AND Q-HEUNG CHOI

Here the source term f is generated by the eigenfunctions of the second order elliptic operatorwith Dirichlet boundary condition.

In [5, 7, 8], the authors have investigated multiplicity of solutions of (1.1) when the forcingterm f is supposed to be a multiple of the first eigenfunction and the nonlinearity−(bu+−au−)crosses eigenvalues. In [4], the authors investigated a relation between multiplicity of solutionsand source terms of (1.1) when the forcing term f is supposed to be spanned two eigenfunctionφ1, φ2 and the nonlinearity −(bu+ − au−) crosses two eigenvalues λ1, λ2.

In this paper we investigate the existence of solutions u(x, t) for perturbations of the ellipticsystem with Dirichlet boundary condition

Lξ + µ(B(ξ + 2η)+ −A(ξ + 2η)−) = f in Ω,

Lη + ν(B(ξ + 2η)+ −A(ξ + 2η)−) = f in Ω,

ξ = 0, η = 0 on ∂Ω,

(1.2)

where u+ = maxu, 0, u− = max−u, 0, µ, ν are nonzero constants and the nonlinearity(µ + 2ν)(B(ξ + 2η)+ −A(ξ + 2η)−) crosses the eigenvalues of the elliptic operator L.

Equation (1.1) and the following type nonlinear equation with Dirichlet boundary conditionwas studied by many authors:

Lu = b[(u + 2)+ − 2] in Ω,

u = 0 on ∂Ω.(1.3)

In [9] Lazer and McKenna point out that this kind of nonlinearity b[(u + 2)+ − 2] canfurnish a model to study traveling waves in suspension bridges. So the nonlinear equationwith jumping nonlinearity have been extensively studied by many authors. For fourth ellipticequation Tarantello [14] , Micheletti and Pistoia [11][12] proved the existence of nontrivialsolutions used degree theory and critical points theory separately. For one-dimensional caseLazer and McKenna [10] proved the existence of nontrivial solution by the global bifurcationmethod. For this jumping nonlinearity we are interest in the multiple nontrivial solutions of theequation. Here we used variational reduction method to find the nontrivial solutions of problem(1.2).

The organization of this paper is as following. In section 2, we have a concern with a relationbetween multiplicity of solutions and source terms of a nonlinear elliptic equation when thenonlinearity crosses eigenvalues. We investigate the uniqueness and multiplicity of solutionsfor the single nonlinear elliptic equation. In section 3, we investigate the uniqueness and theexistence of multiple solutions u(x, t) for the elliptic system with Dirichlet boundary conditionwhen the nonlinearity (µ + 2ν)(Bu+ −Au−) crosses the eigenvalues of the elliptic operator.

2. A SINGLE NONLINEAR ELLIPTIC EQUATION

We have a concern with a relation between multiplicity of solutions and source terms of anonlinear elliptic equation

Lu + bu+ − au− = f in L2(Ω). (2.1)

THE NONLINEAR ELLIPTIC SYSTEM 163

Here we suppose that the nonlinearity −(bu+ − au−) crosses eigenvalues. We consider threecases: The nonlinearity crosses no eigenvalue; the nonlinearity crosses the eigenvalue λ1; thenonlinearity crosses the eigenvalues λ1, λ2.

Let us denote an element u, in H0, as u =∑

hjφj and we define a subspace H of H0 as

H = u ∈ H0 :∑

|λj |h2j < ∞.

Then this is a complete normed space with a norm ‖u‖ = (∑ |λmn|h2

mn)12 . If f ∈ H0 and a, b

are not eigenvalues of L, then every solution in H0 of Lu + bu+ − au− = f belongs to H (cf.[2]).

Case 1) The nonlinearity crosses no eigenvalueWe suppose that the nonlinearity −(bu+ − au−) crosses no eigenvalue, that is, a, b < λ1.

By the contraction mapping principle we have the following uniqueness theorem.

Theorem 2.1. Let a, b < λ1 and f ∈ H0. Then equation (2.1) has a unique solution in H .

Case 2) The nonlinearity crosses the eigenvalues λ1, λ2.We suppose that the nonlinearity −(bu+ − au−) crosses two eigenvalues λ1, λ2, i.e., a <

λ1 < λ2 < b < λ3. We have a concern with a relation between multiplicity of solutions andsource terms of a nonlinear elliptic equation

Lu + bu+ − au− = f in L2(Ω). (2.1)

Here we suppose that f is generated by two eigenfunctions φ1 and φ2.Let V be the two dimensional subspace of L2(Ω) spanned by φ1, φ2 and W be the or-

thogonal complement of V in L2(Ω). Let P be an orthogonal projection L2(Ω) onto V . Thenevery element u ∈ H is expressed by

u = v + w,

where v = Pu, w = (I − P )u. Hence equation (2.1) is equivalent to a system

Lw + (I − P )(b(v + w)+ − a(v + w)−) = 0, (2.2)

Lv + P (b(v + w)+ − a(v + w)−) = s1φ1 + s2φ2. (2.3)

Lemma 2.1. For fixed v ∈ V , (2.2) has a unique solution w = θ(v). Furthermore, θ(v) isLipschitz continuous (with respect to L2 norm) in terms of v.

The proof of the lemma is similar to that of Lemma 2.1 of [3].By Lemma 2.1, the study of multiplicity of solutions of (2.1) is reduced to that of an equiv-

alent problemLv + P (b(v + θ(v))+ − a(v + θ(v))−) = s1φ1 + s2φ2 (2.4)

defined on the two dimensional subspace V spanned by φ1, φ2.We note that if v ≥ 0 or v ≤ 0 then θ(v) ≡ 0.Since the subspace V is spanned by φ1, φ2 and φ1(x) > 0 in Ω, there exists a cone C1

defined byC1 = v = c1φ1 + c2φ2 : c1 ≥ 0, |c2| ≤ kc1


for some k > 0 so that v ≥ 0 for all v ∈ C1 and a cone C3 defined by

C3 = v = c1φ1 + c2φ2 : c1 ≤ 0, |c2| ≤ k|c1|so that v ≤ 0 for all v ∈ C3.

We define a map Φ : V → V given by

Φ(v) = Lv + P (b(v + θ(v))+ − a(v + θ(v))−), v ∈ V. (2.5)

Then Φ is continuous on V, since θ is continuous on V and we have the following lemma (cf.Lemma 2.2 of [3]).

Lemma 2.2. Φ(cv) = cΦ(v) for c ≥ 0 and v ∈ V .

Lemma 2.2 implies that Φ maps a cone with vertex 0 onto a cone with vertex 0. We set thecones C2, C4 as follows

C2 = c1φ1 + c2φ2 : c2 ≥ 0, c2 ≥ k|c1|,C4 = c1φ1 + c2φ2 : c2 ≤ 0, c2 ≤ −k|c1|.

Then the union of four cones Ci (1 ≤ i ≤ 4) is the space V.We investigate the images of the cones C1 and C3 under Φ. First we consider the image of

the cone C1. If v = c1φ1 + c2φ2 ≥ 0, we have

Φ(v) = L(v) + P (b(v + θ(v))+ − a(v + θ(v))−)= −c1λ1φ1 − c2λ2φ2 + b(c1φ1 + c2φ2)= c1(b− λ1)φ1 + c2(b− λ2)φ2.

Thus the images of the rays c1φ1 ± kc1φ2(c1 ≥ 0) can be explicitly calculated and they are

c1(b− λ1)φ1 ± kc1(b− λ2)φ10 (c1 ≥ 0).

Therefore Φ maps C1 onto the cone

R1 =

d1φ00 + d2φ10 : d1 ≥ 0, |d2| ≤ k

(b− λ2

b− λ1

)d1

.

The cone R1 is in the right half-plane of V and the restriction Φ|C1 : C1 → R1 is bijective.We determine the image of the cone C3. If v = −c1φ1 + c2φ2 ≤ 0, we have

Φ(v) = L(v) + P (b(v + θ(v))+ − a(v + θ(v))−)= Lv + P (av)= −c1(−λ1 + a)φ1 + c2(−λ2 + a)φ2.

Thus the images of the rays −c1φ00± kc1φ2 (c1 ≥ 0) can be explicitly calculated and they are

−c1(−λ1 + a)φ1 ± kc1(−λ2 + a)φ2 (c1 ≥ 0).


Thus Φ maps the cone C3 onto the cone

R3 =

d1φ1 + d2φ2 : d1 ≥ 0, d2 ≤ k|λ2 − a

λ1 − a||d1|

.

The cone R3 is in the right half-plane of V and the restriction Φ|C3 : C3 → R3 is bijective. Wenote that R1 ⊂ R3 since a < λ1 < λ2 < b < λ3.

Theorem 2.2. If f belongs to R1, then equation (2.1) has a positive solution and a negativesolution.

Lemma 2.2 means that the images Φ(C2) and Φ(C4) are the cones in the plane V. Beforewe investigate the images Φ(C2) and Φ(C4), we set

R′2 =

d1φ1 + d2φ2 : d1 ≥ 0,−k

∣∣∣∣λ2 − a

λ1 − a

∣∣∣∣ d1 ≤ d2 ≤ k

∣∣∣∣λ2 − b

λ1 − b

∣∣∣∣ d1

,

R′4 =

d1φ1 + d2φ2 : d1 ≥ 0,−k

(λ2 − b

λ1 − b

)d1 ≤ d2 ≤ k

(λ2 − a

λ1 − a

)d1

.

We note that all the cones R′2, R3, R′

4 contains R1. R3 contain R1, R′2, R′

4.To investigate a relation between multiplicity of solutions and source terms in the nonlinear

equationLu + bu+ − au− = f in H, (2.6)

we consider the restrictions Φ|Ci(1 ≤ i ≤ 4) of Φ to the cones Ci. Let Φi = Φ|Ci , i.e.,

Φi : Ci → V.

For i = 1, 3, the image of Φi is Ri and Φi : Ci → Ri is bijective.

Lemma 2.3. For every v = c1φ1 + c2φ2, there exists a constant d > 0 such that

(Φ(v), φ1) ≥ d|c2|.For the proof see [2].From now on, our goal is to find the image of Ci under Φi for i = 2, 4. Suppose that γ is

a simple path in C2 without meeting the origin, and end points (initial and terminal) of γ lieon the boundary ray of C2 and they are on each other boundary ray. Then the image of oneend point of γ under Φ is on the ray c1(b − λ1)φ1 + kc1(b − λ2)φ2, c1 ≥ 0 (a boundary rayof R1) and the image of the other end point of γ under Φ is on the ray −c1(−λ1 + a)φ1 +kc1(−λ2 + a)φ10, c1 ≥ 0 (a boundary ray of R3). Since Φ is continuous, Φ(γ) is a path in V.By Lemma 2.2, Φ(γ) does not meet the origin. Hence the path Φ(γ) meets all rays (startingfrom the origin) in R′

2.Therefore it follows from Lemma 2.3 that the image Φ(C2) of C2 contains R′

2.Similarly, we have that the image Φ(C4) of C4 contains R′

4.If a solution of (2.1) is in IntC1, then it is positive. If a solution of (2.1) is in IntC3, then it

is negative. If it is in Int(C2∪C4), then it has both signs. Therefore we have the main theoremof this section.


Theorem 2.3. Let a < λ1 < λ2 < b < λ3. Let v = c1φ1 + c2φ2. Then we have the followings.(i) If f ∈ Int R1, then equation (2.1) has a positive solution, a negative solution, and at leasttwo solutions changing sign.(ii) If f ∈ ∂R1, then equation (2.1) has a positive solution, a negative solution, and at leastone solution changing sign.(iii) If f ∈ Int(R3\R1), then equation (2.1) has a negative solution and at least one solutionchanging sign.(iv) If f ∈ ∂R3, then equation (2.1) has a negative solution.

Case 2) The nonlinearity crosses the eigenvalue λ1

We suppose that the nonlinearity −(bu+ − au−) crosses the eigenvalues λ1, i.e., a < λ1 <b < λ2. Then it is easy to prove the following theorem.

Theorem 2.4. Let a < λ1 < b < λ2 and f = αφ1. Then we have the followings.(i) If α > 0, then equation (2.1) has a positive solution and a negative solution.(ii) If α < 0, then equation (2.1) has no solution.

3. MULTIPLE SOLUTIONS FOR THE ELLIPTIC SYSTEM

Let Ω be a bounded domain in Rn with smooth boundary ∂Ω and let L denote the differentialoperator

L =∑

1≤i,j≤n

∂

∂xi(aij

∂

∂xj),

where aij = aji ∈ C∞(Ω). In this section we investigate the existence of solutions u(x, t) forperturbations of the elliptic system with Dirichlet boundary condition

Lξ + µg(ξ + 2η) = f in Ω,

Lη + νg(ξ + 2η) = f in Ω,

ξ = 0, η = 0 on ∂Ω,

(3.1)

where g(u) = Bu+ − Au−, u+ = maxu, 0, u− = max−u, 0, µ, ν are nonzero constantsand the nonlinearity (µ + 2ν)g(u) crosses the eigenvalues of the elliptic operator L.

Here we assume that −7 < µ− ν < −3.We suppose that the nonlinearity (µ + 2ν)g(u) crosses no eigenvalue of L, that is, (µ +

2ν)A, (µ + 2ν)B < λ1. By the contraction mapping principle we have the following unique-ness theorem.

Theorem 3.1. Let µ, ν be nonzero constants and 2 + µν 6= 0. Assume that (µ + 2ν)A, (µ +

2ν)B < λ1. and f ∈ H0. Then elliptic system (3.1) has a unique solution (ξ, η).

Proof. From problem (3.1) we get that Lξ−f = µν (Lη−f). By Theorem 2.1, for any F ∈ H0

the problemLu = F in Ω,

u = 0 on ∂Ω,(3.2)


has a unique solution. If u1−µν

is a solution of L(ξ − µν η) = (1− µ

ν )f , then the solution (ξ, η)of problem (3.1) satisfies

ξ − µ

νη = u1−µ

ν. (A)

On the other hand, from problem (3.1) we get the equation

L(ξ + 2η) + (µ + 2ν)g(ξ + 2η) = 3f in Ω,

ξ = 0, η = 0 on ∂Ω.(3.3)

Put w = ξ + 2η. Then the above equation is equivalent to

Lw + (µ + 2ν)g(ξ + 2η) = 3f in Ω,

w = 0 on ∂Ω.(3.4)

When (µ+2ν)A, (µ+2ν)B < λ1, by Theorem 2.1 the above equation has a unique solution,say w1. Hence we get the solutions (ξ, η) of problem (3.1) from the following systems:

ξ − µ

νη = u1−µ

ν,

ξ + 2η = w1.(3.5)

Since 2 + µν 6= 0, system (3.5) has a unique solution (ξ, η). ¤

Theorem 3.2. Let µ, ν be nonzero constants and 2 + µν 6= 0. Assume that µ + 2ν)A < λ1 <

λ2 < (µ + 2ν)B < λ3 and f = c1φ1 + c2φ2. Then we have the followings.(i) If f ∈ Int R1, then system (3.1) has a positive solution, a negative solution, and at leasttwo solutions changing sign.(ii) If f ∈ ∂R1, then system (3.1) has a positive solution, a negative solution, and at least onesolution changing sign.(iii) If f ∈ Int(R3\R1), then system (3.1) has a negative solution and at least one solutionchanging sign.(iv) If f ∈ ∂R3, then system (3.1) has a negative solution.

Proof. (i) From problem (3.1) we get that Lξ − f = µν (Lη − f). By Theorem 2.1, for any

F ∈ H0 the problemLu = F in Ω,

u = 0 on ∂Ω(3.6)

has a unique solution. If u1−µν

is a solution of L(ξ − µν η) = (1− µ

ν )f , then the solution (ξ, η)of problem (3.1) satisfies

ξ − µ

νη = u1−µ

ν. (A)

On the other hand, from problem (3.1) we get the equation

L(ξ + 2η) + (µ + 2ν)g(ξ + 2η) = 3f in Ω,

ξ = 0, η = 0 on ∂Ω.(3.7)


Put w = ξ + 2η. Then the above equation is equivalent to

Lw + (µ + 2ν)g(w) = 3f in Ω,

w = 0 on ∂Ω,(3.8)

where g(w) = Bw+ −Aw− and the nonlinearity (µ + 2ν)g(w) crosses the eigenvalues of theelliptic operator L. When (µ + 2ν)A < λ1 < λ2 < (µ + 2ν)B < λ3 and f ∈ Int R1, byTheorem 2.3 (i) the above equation has a positive solution wp, a negative solution wn, and atleast two solutions changing sign wc1 , wc2 .

Hence we get the solutions (ξ, η) of problem (3.1) from the following systems:

ξ − µ

νη = u1−µ

ν

ξ + 2η = wp

(3.9)

ξ − µ

νη = u1−µ

ν

ξ + 2η = wn

(3.10)

ξ − µ

νη = u1−µ

ν

ξ + 2η = w1

(3.11)

ξ − µ

νη = u1−µ

ν

ξ + 2η = w1.(3.12)

Since 2 + µν 6= 0, system (3.9) has a unique solution (ξ1, η1) with ξ1 + 2η1 > 0. From (3.10)

we get the solution (ξ2, η2) with ξ2 +2η2 < 0. From (3.11), (3.12) we get the solution (ξ3, η3),(ξ4, η4), where ξi + 2ηi(i = 1, 2) are changing sign.

Therefore system(3.1) has at least four solutions.By using the similar method as in the proof of (i), we have (ii), (iii), (iv). ¤

We suppose that the nonlinearity (µ+2ν)g(u) crosses the eigenvalues λ1, i.e., (µ+2ν)A <λ1 < (µ + 2ν)B < λ2. By using the similar method as in the proof of Theorem 3.2, we havethe following theorem.

Theorem 3.3. Let µ, ν be nonzero constants and 2 + µν 6= 0. Assume that (µ + 2ν)A < λ1 <

(µ + 2ν)B < λ2 and f = αφ1. Then we have the followings.(i) If α > 0, then system (3.1) has at least two solutions (ξ1, η1), (ξ2, η2) with ξ1 + 2η1 > 0,ξ2 + 2η2 > 0.(ii) If α < 0, then system (3.1) has no solution.

ACKNOWLEDGMENTS

The authors appreciate very much the referee’s kind corrections.


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J. KSIAM Vol.12, No.3, 171–190, 2008

VIDEO TRAFFIC MODELING BASED ON GEOY /G/∞ INPUT PROCESSES

SANG HYUK KANG1 AND BARA KIM2

1DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING, UNIVERSITY OF SEOUL, SEOUL 130-743, KOREA


2DEPARTMENT OF MATHEMATICS, KOREA UNIVERSITY, SEOUL 136-701, KOREA


ABSTRACT. With growing applications of wireless video streaming, an efficient video trafficmodel featuring modern high-compression techniques is more desirable than ever, because thewireless channel bandwidths are ever limited and time-varying. We propose a modeling andanalysis method for video traffic by a class of stochastic processes, which we call ‘GeoY /G/∞input processes’. We model video traffic by GeoY /G/∞ input process with gamma-distributedbatch sizes Y and Weibull-like autocorrelation function. Using four real-encoded, full-lengthvideo traces including action movies, a drama, and an animation, we evaluate our modelingperformance against existing model, transformed-M /G/∞ input process, which is one of mostrecently proposed video modeling methods in the literature. Our proposed GeoY /G/∞ modelis observed to consistently provide conservative performance predictions, in terms of packetloss ratio, within acceptable error at various traffic loads of interest in practical multimediastreaming systems, while the existing transformed-M /G/∞ fails. For real-time implementationof our model, we analyze G/D/1/K queueing systems with GeoY /G/∞ input process to upperestimate the packet loss probabilities.

1. INTRODUCTION

With the advance of video compression technologies, next generation ubiquitous networksare expected to accommodate heavily compressed streaming video traffic on wireless and/orwireline channels. An efficient video traffic model featuring modern high compression tech-niques is more desirable than ever, because the wireless channel bandwidths are ever lim-ited and time-varying. Quality of service management requires sophisticated resource controlschemes based on up-to-date video traffic models which are verified with real video traces withthe state-of-the-art of video compression techniques such as MPEG4 or H.264 [1].

It has been shown that packetized video traffic streams have both long-range dependenceand short-range dependence properties [2, 3, 4, 5]. Therefore, a video modeling method based

2000 Mathematics Subject Classification. 93B05.Key words and phrases. video traffic modeling; GeoY /G/∞ input processes; packet loss ratio; large deviation

theory.The first author’s work was supported by Seoul R&BD program (GS070154).1 Corresponding author.

171

172 KANG AND KIM

on either of long-range dependence or short-range dependence is not expected to gurantee themodeling performance for full length video streams over practical communication systems. Amodeling method, ‘transformed-M/G/∞ input process’ has recently been proposed [3] with au-tocorrelation function ρ(k) = e−β

√k as a compromise between long-range dependence models

with ρ(k) = e−β log k and Markovian or short-range dependence models with ρ(k) = e−βk.As shown in the performance evaluation section of [3], the transformed-M/G/∞ input processmodel outperforms traditional existing models. From a practical application point of view,however, its drawback is that it is not tractable in queueing analysis due to the transforma-tion of the marginal distribution. Therefore, it is not readily applicable as real time algorithmssuch as real-time call admission control, real-time packet scheduling, and so on. Also, in ourextensive simulations, the transformed-M/G/∞ input process model does not estimate exactqueueing performance in all situations.

In this paper, we propose a class of stochastic process which we call ‘GeoY /G/∞ input pro-cess’ characterizing both long-range and short-range dependences and yet tractable in queueinganalyses for real-time applications. In order to model the number of video packets in a fixed-size time slot, we propose GeoY /G/∞ input process with gamma-distributed batch sizes Y andWeibull-like autocorrelation function. The model parameters are determined by matching themarginal statistics and autocorrelation of video frame size to the corresponding statistics of themodel. A desirable video traffic model is expected to be mathematically tractable so that itcan be real-time implementable as a practical packet scheduler. In this context, we analyze adiscrete-time single server, infinite-buffer queue with GeoY /G/∞ input process by large devi-ation theory. We obtain an upper bound of loss ratio with a form simple enough to be real-timeimplementable. The derived upper bound of loss ratio helps to guarantee quality of service be-cause it gives conservative packet loss levels in real systems such as routers, scheduling servers,etc.

The rest of this paper is organized as follows. In Section II we introduce our proposedGeoY /G/∞ input processes. In Section III we present the procedure for parameter matchingbetween real video traces and GeoY /G/∞ input processes in terms of autocorrelation functionand marginal distribution. Discrete-time single server queue with GeoY /G/∞ input processesis analyzed in Section IV. Finally, a conclusion is drawn in Section V.

2. GEOY /G/∞ INPUT PROCESSES

In this section, we introduce our proposed stochastic process, which we call ‘GeoY /G/∞input process.’ Consider a discrete-time queueing system where bt+1 denotes the number ofarrivals at the start of time slot [t, t + 1), t = · · · ,−1, 0, 1, · · · . Let us now call the arrivalprocess bt a ‘GeoY /G/∞ input process’ when it is given as the busy server process of adiscrete-time GeoY /G/∞ system, as an extension of M/G/∞ input process [3, 7, 8].

For a GeoY /G/∞ system, the arrival is according to a batch Bernoulli process where thebatch size Yt ∈ 0, 1, · · · in time slot t forms an i.i.d. random process Yt; t = · · · ,−1, 0, 1, · · · .During t-th time slot, Yt = yt new customers arrive into the GeoY /G/∞ system; then, customer

VIDEO TRAFFIC MODELING 173

i (i = 1, . . . , yt) is presented to its own server who begins service from the next slot with ser-vice duration σt,i ∈ 1, 2, · · · time slots. We take σt,i to be i.i.d. rv’s.

We have the number of busy servers (or, equivalently, number of customers) bt at time slott, which is given by

bt =t∑

s=−∞

Ys∑

i=1

1[σs,i > t− s], (1)

where 1[·] is the indicator function.The correlated process bt; t = · · · ,−1, 0, 1, · · · is easily shown to be stationary and

ergodic. To characterize the marginal statistics, the probability generating function (PGF) of bt

is represented as∞∑

i=0

ziP[bt = i] =∞∏

k=0

X (akz + (1− ak)) (2)

where ak ≡ P[σ > k] and X(z) is the PGF of batch size, such that

X(z) ≡∞∑

i=0

ziP[Y = i].

From (2), the mean and variance of bt are given by, respectively,

E[b] = E[Y ]E[σ] (3)

and

Var[b] = E[b] + (Var[Y ]− E[Y ])∞∑

i=0

P[σ > i]2. (4)

The covariance structure of bt; t = · · · ,−1, 0, 1, · · · is given by

Γ(k) ≡ cov [bt, bt+k]

= E[Y ]∞∑

i=k

P[σ > i] + (Var[Y ]− E[Y ])∞∑

i=0

P[σ > i]P[σ > i + k], (5)

where k = 0, 1, · · · , and the autocorrelation function (ACF) is defined as

ρ(k) ≡ Γ(k)Γ(0)

=Γ(k)Var[b]

. (6)

We also define the forward recurrence time, σ, associated with the service time σ such that

P[σ = r] ≡ P[σ ≥ r]E[σ]

, r = 1, 2, · · · ,

and definevt ≡ − ln P[σ > t]. (7)

174 KANG AND KIM

TABLE 1. Summary of the MPEG4-encoded VBR traces used in the study(QCIF, Qp=5)

Movie Length Number of Mean frame Variance of(time) frames size [packets] frame size [packets2]

Terminator 2 2:16:02 195,909 134.84 3829.4The Fifth Element 2:05:49 181,195 157.10 4529.1

The English Patient 2:41:18 199,998 91.0 2238.9Shrek 1:29:58 129,566 128.65 4356.5

3. VIDEO TRAFFIC MODELING BY GEOY /G/∞ INPUT PROCESS WITHGAMMA-DISTRIBUTED BATCHES Y

In this section, we propose a video traffic modeling scheme with GeoY /G/∞ input processwith gamma-distributed batches, Y . In our study, we examined real video traces includingaction movies, a drama, and an animation as in Table 1. Each video is full-length in quartercommon intermediate format (QCIF) format at 24 frames/sec, that is most widely-used formatin wireless video transmission. We use public domain Microsoft MPEG4 Visual ReferenceSoftware version 2 FDMA1-2.3-001213, to encode/decode video data. In our experiments,we encode each frame as an I frame so that we obtain as many samples as possible to getexact statistics from a real video trace. The quantization step (Qp) is set to be 5, which iscorrespond to about 200 Kbps video traffic that is suitable for current 3-rd generation (3G)wireless channels. In the table, the mean and standard deviation of the frame size is representedin [packets] where the packet size is 24 Bytes as is the 192-bit payload size in General PacketRadio Service (GPRS) [6].

Our modeling with GeoY /G/∞ input process is by matching the moments and autocorre-lation function of bt with those of the sequence of frame size. We note that our proposedmodel can be applied to modern Group of Picture (GOP) based video compression with I, Pand B frames by taking the modeling unit as a GOP size instead of a frame size.

From Table 1, we match the mean and variance of frame size with E[b] and Var[b], re-spectively. We then match the autocorrelation function as Weibull-like characteristics ρ(k) =e−βkα

(0 < α < 1, β > 0) in terms of least mean squared error. The fitting parameters areshown in Table 2 for our proposed method together with ρ(k) = e−βM

√k from transformed-

M/G/∞ input process, or ‘t-M/G/∞,’ [3].We show the curve fitting results for autocorrelation function (ACF) with The Fifth Element

in Fig. 1. As expected, the curve ρ(k) = e−0.078k0.42from GeoY /G/∞ better fits to the real

trace than ρ(k) = e−0.043√

k from t-M/G/∞, in terms of mean squared error.Given measured E[b], Var[b], α, and β, we now determine the distribution of the service

time, σ, for GeoY /G/∞ input process. Let us define the distribution of σ

C(k) ≡ P[σ > k]. (8)


TABLE 2. Parameter estimation for autocorrelation function (ACF)

Movie GeoY /G/∞ t-M/G/∞ [2]α β βM

Terminator 2 0.48 0.065 0.055The Fifth Element 0.42 0.078 0.043

The English Patient 0.55 0.040 0.056Shrek 0.61 0.013 0.030

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−0.2

0

0.2

0.4

0.6

0.8

1

1.2

Lag [frames]

Aut

ocor

rela

tion

func

tion

(AC

F)

The Fifth Element

real trace GeoY/G/inft−M/G/inf

FIGURE 1. Autocorrelation function fitting for movie The Fifth Element

Manipulate (5) and (6) to obtain

(E[Y ] + Var[Y ])∞∑

k=n

C(k)

= Var[b]e−βnα+

∞∑

k=0

C(n + k) Var[Y ] (1− C(k)) + E[Y ]C(k) . (9)

We calculate the distribution of σ, C(n), by successive iteration according to the followingsteps:

176 KANG AND KIM

• Step 1: (Initialization) For n = 0, 1, · · · , calculate

Cold(n) ⇐ e−βnα − e−β(n+1)α

1− e−β(10)

Here the left arrow ‘⇐’ means assignment of the value on right hand side the term onleft hand side.

• Step 2: Calculate

E[Y ] ⇐ E[b]∑∞k=0 Cold(k)

Var[Y ] ⇐ Var[b]− E[b]∑∞k=0 Cold(k)2

+ E[Y ]

• Step 3: For n = 0, 1, · · · , calculate

Cnew(n) ⇐ Var[b](e−βnα − e−β(n+1)α)

+ A(n)−A(n + 1)E[Y ] + Var[Y ]

(11)

where

A(n) ≡∞∑

k=0

Cold(n + k) Var[Y ] (1− Cold(k)) + E[Y ]Cold(k) .

• Step 4: Assure that Cnew(0) = 1.• Step 5: If Cold(n) and Cnew(n) are close to each other within a criterion, then C(n) ⇐

Cold(n) and stop. Otherwise, update Cold(n) ⇐ Cnew(n) for n = 0, 1, · · · and go toStep 2.

All that is left now is to find the distribution of the batch size Y . From extensive studies withvarious distribution functions, we observe best modeling performance with gamma-distributedY for our proposed GeoY /G/∞ input process model. Note that we have already obtained E[Y ]and Var[Y ] when calculating the distribution of σ in the above successive iteration. We proposethat

P[Y ≤ y] =γ(η, (y + 1)/θ)

Γ(η), (y = 0, 1, · · · ) (12)

where the scale parameter θ and shape parameter η are given by

θ =Var[Y ]E[Y ]

, (13)

η =E[Y ]

θ. (14)

Here γ(a, z) is a lower incomplete gamma function and Γ(a) is a complete gamma function.In summary, we determine model parameters GeoY /G/∞ input process with gamma-distributed

batch according to the following steps:(i) Match ρ(k) with real trace in terms of minimum mean squared error to obtain α and β.


0 100 200 300 400 500 60010

−6

10−5

10−4

10−3

10−2

10−1

100

s [packets]

Pro

b(fr

ame−

size

> s

)The Fifth Element

real trace t−M/G/inf GeoY/G/inf

FIGURE 2. Complementary frame-size distribution for The Fifth Element

(ii) Measure the mean and variance of frame size to be matched with E[b] and Var[b], respec-tively.(iii) Calculate the distribution of σ, C(k), together with E[Y ] and Var[Y ].(iv) Finally, calculate the parameters θ and η for gamma distribution for Y .

The marginal distribution of the frame size is illustrated in Fig. 2 where the complemen-tary frame-size distribution is shown for The Fifth Element. With QCIF-encoded video traces,The Fifth Element, the tail of the distribution drops rapidly yielding the maximum frame size,fmax = 506 [packets], rather than Pareto tail [3]. This is expected because QCIF-encodedvideo frames are highly compressed, which are commonly used in transmission on limited-bandwidth wireless channel. In Fig. 2, dashed line is from our proposed GeoY /G/∞ inputprocess. It is observed that GeoY /G/∞ input process gives a more slowly decaying tail thanthe real trace. We only use gamma transformation for t-M/G/∞ input process due to rapidly-dropping tail. It is noted that for both models, the fitting is based on matching the mean andvariance between models and real traces.

Finally, we evaluate modeling performance in terms of packet loss ratio (PLR). A summaryof the simulation results to two significant digits is given in Table 3 for each of the four videotraces at various offered traffic loads: heavy load (U = 0.7), moderate load (U = 0.5), andlight load (U = 0.3). With The Fifth Element, we use U = 0.4 for light load, because nopacket losses are observed with the real trace for U = 0.3. For the modeling results, we

178 KANG AND KIM

generate synthesis traces and apply them into a G/D/1/K queueing system to estimate packetloss probabilities. We conduct the discrete-event simulation at the frame level.

The buffer size is varied from 10 to 3000. It is observed that increasing buffer barely providesimprovement in performance, which is remarkable especially for higher load. In contrast,reducing the load (i.e., increasing bandwidth) improves the packet loss ratio significantly.

In the heavy load regime, both t-M/G/∞ and GeoY /G/∞ models provide acceptable pre-dictions of PLR, with GeoY /G/∞ being more accurate. In the moderate regime, GeoY /G/∞model yields more accurate estimations than t-M/G/∞ model which underestimates PLR byabout 50buffer size of 1000 [packets]. In the light load regime, GeoY /G/∞ model overesti-mates PLR, while t-M/G/∞ model underestimates it and is overly sensitive to the buffer sizeas shown with Terminator 2. Also, it is desirable for a model to give overestimated, i.e., con-servative PLR rather than underestimated PLR, because implementing real systems based onunderestimated PLR may not guarantee quality of service. Overall, GeoY /G/∞ model is ob-served to consistently provide conservative performance predictions within acceptable error atvarious traffic loads. It is difficult to analyze queueing systems with t-M/G/∞ input processes.GeoY /G/∞ model is mathematically tractable as shown in the next section

4. ANALYSIS OF QUEUEING SYSTEMS WITH GEOY /G/∞ INPUT PROCESSES

It is desirable to show that our proposed GeoY /G/∞ Input Processes is applicable in real-time implementation is practical video streaming stream on the communication systems. Wenow consider a discrete-time single-server queue with infinite buffer and constant release rateof c packets/slot under first-come first-served discipline, of which the arrival is GeoY /G/∞input process. The goal of this section is to obtain an upper bound of the packet loss ratio in aform simple enough to be real-time implementable.

Let qt denote the number of cells remaining in the buffer by the end of slot [t− 1, t), and letbt+1 denote the number of new cells which arrive at the start of time slot [t, t + 1). Then thebuffer content sequence qt, t = · · · ,−1, 0, 1, · · · evolves according to the Lindley recursion

qt+1 = [qt + bt+1 − c]+, t = · · · ,−1, 0, 1, · · · (15)

for some initial condition q.qt; · · · ,−1, 0, 1, · · · is uniquely determined by equation (15) if E[Y ]E[σ] < c, and it

is stationary. The queueing system will reach statistical equilibrium if E[Y ]E[σ] < c. Thestationary and ergodic input process bt; t = 0, 1, · · · is reversible sequence and the steadystate buffer content is given by

q0 =st supSt − ct, t = 0, 1, · · · (16)

withS0 = 0; St = b−1 + b−2 + · · ·+ b−t, t = 1, 2, · · · . (17)

We note that, for each t = 0, 1, · · · ,bt = b

(0)t + b

(a)t , (18)


TABLE 3. Modeling Performance : Mean packet loss ratio at three differentloads, U

Buffer U = 0.3 U = 0.5 U = 0.7

size [pkts] real t-M/G/∞ GeoY /G/∞ real t-M/G/∞ GeoY /G/∞ real t-M/G/∞ GeoY /G/∞10 3.6E-4 3.1E-5 3.1E-3 1.5E-2 8.3E-3 1.8E-2 5.6E-2 5.2E-2 5.7E-250 3.5E-4 2.0E-5 2.8E-3 1.5E-2 7.4E-3 1.8E-2 5.5E-2 4.9E-2 5.7E-2100 3.5E-4 1.4E-5 2.6E-3 1.5E-2 6.9E-3 1.7E-2 5.4E-2 4.7E-2 5.6E-2500 2.9E-4 2.8E-6 1.7E-3 1.4E-2 5.4E-3 1.4E-2 5.0E-2 4.2E-2 5.1E-2

1000 2.1E-4 3.5E-7 1.3E-3 1.3E-2 4.5E-3 1.3E-2 4.7E-2 3.8E-2 4.8E-22000 1.2E-4 0 8.2E-4 1.2E-2 3.5E-3 1.0E-2 4.2E-2 3.4E-2 4.4E-23000 8.5E-5 0 6.0E-4 1.2E-2 2.9E-3 8.8E-3 3.9E-2 3.1E-2 4.0E-2

(a) Terminator 2

Buffer U = 0.4(∗) U = 0.5 U = 0.7


1000 1.2E-3 4.0E-4 3.7E-3 7.6E-3 3.0E-3 9.9E-3 4.1E-2 3.3E-2 4.1E-21500 1.1E-3 3.5E-4 3.2E-3 7.1E-3 2.7E-3 8.9E-3 3.9E-2 3.1E-2 3.9E-22000 1.1E-3 3.2E-4 2.8E-3 6.7E-3 2.4E-3 8.1E-3 3.7E-2 3.0E-2 3.7E-23000 1.0E-3 2.8E-4 2.2E-3 6.1E-3 2.1E-3 6.9E-3 3.4E-2 2.7E-2 3.5E-2

(b) The Fifth Element

Buffer U = 0.3 U = 0.5 U = 0.7



(c) The English Patient

Buffer U = 0.3 U = 0.5 U = 0.7



(d) Shrek

where the rv’s b(0)t and b

(a)t describe the contributions to the number of customers in the system

at the beginning of time slot [t, t + 1) from those present at t = 0 and from the new arrivals,respectively. We have

b(a)t =

t∑

s=1

Ys∑

i=1

1[σs,i > t− s] (19)

180 KANG AND KIM

and

b(0)t =

0∑s=−∞

Ys∑

i=1

1[σs,i > t− s]. (20)

Invoking large deviation theory, we define

Λt(θ) ≡ 1vt

ln E[eθt(St−ct)

], θ ∈ R. (21)

whereθt ≡ θ

vt

t.

where vt = − ln P[σ > t] for which it is assumed that limt→∞ vt = ∞. The remaining of thissection is for finding the limit

Λ(θ) ≡ limt→∞Λt(θ). (22)

NoteSt =st St(in distribution), t = 0, 1, 2, · · · ,

whereS0 = 0, St = b1 + · · ·+ bt, t = 1, 2, · · · ,

HenseΛt(θ) ≡ 1

vtln E

[eθt(St−ct)

], θ ∈ R.

Theorem 1. Assume that(1) limt→∞ vt/t = 0,(2) E[σ2] < ∞ (⇔ E[σ] < ∞⇔ ∑∞

s=1 E[(σ − s)+] < ∞),(3)

∑∞t=1 P[σ > t]α < ∞ for 0 < α < 1,

(4) vt/t; t = 1, 2, · · · is monotone decreasing,(5) Y has an exponential moments, i.e., E[eθY ] < ∞ for θ > 0,(6) E[σ] < ∞, and(7) limt→∞ P[σ > t]/P[σ > t] = ∞.

Then, for each θ 6= 1 in R, the limit Λ(θ) is given by

Λ(θ) =

(E[Y ]E[σ]− c)θ, if θ < 1∞, if θ > 1 (23)

Proof of Theorem 1: From Lemma 1 through 9 in the Appendix, we have, for each θ ∈ R,

Λb(θ) ≡ limt→∞

1vt

ln E[exp(

vt

tθSt)

]=

E[Y ]E[σ]θ, if θ < 1

∞, if θ > 1

SinceΛ(θ) = Λb(θ)− cθ,

(23) holds. ¤Our goal is to approximate P[q0 > n] in an asymptotic manner. Let

Λ∗(z) ≡ supθ∈R

(θz − Λ(θ)), (z ∈ R). (24)


Invoking [8], we have an upper bound

lim supn→∞

1βnα

ln P[q0 > n] ≤ −γ∗, (25)

whereγ∗ = sup

y>0min(f(y), g(y)) (26)

with the notation

f(y) = supθ>0

lim infn→∞

[infx>y

βnα

β(nx)α(θx− Λ(θ))

](27)

andg(y) = Λ∗(0)/yα. (28)

Let m = E[b] = E[Y ]E[σ]. We have

f(y) =

(c−m)1−α α−α

(1−α)1−α , if 0 < y ≤ α1−α(c−m)

y1−α + (c−m)y−α, if y > α1−α(c−m)

(29)

To compute min(f(y), g(y)), we have(Case I) if y < (c−m)α(1− α)

1−αα , min(f(y), g(y)) = (c−m)1−α α−α

(1−α)1−α ,

(Case II) if y > (c−m)α(1− α)1−α

α , min(f(y), g(y)) = y−α(c−m).Consequently, we obtain

γ∗ = (c−m)1−α α−α

(1− α)1−α. (30)

On the other hand, we have a lower bound

−γ∗ ≤ lim infn→∞

1βnα

ln P[q∞ > n], (31)

where γ∗ = infy>0 Λ∗(y)/yα. It is calculated as

γ∗ = infy>0

[1yα· sup

θ∈Rθy − Λ(θ)

]

= infy>0

1yα

(y + c−m)

= (c−m)1−α α−α

(1− α)1−α. (32)

Finally, we haveγ(c) ≡ γ∗ = γ∗ (33)

as a function of output link rate, and, from (25) and (31),

limn→∞

1βnα

ln P[q∞ > n] = −γ(c). (34)

182 KANG AND KIM

The above equation can be utilized as an overestimation of packet loss ratio for a queueingsystem with GeoY /G/∞ input processes, when we implement video transmission systems. Anoverestimation of PLR gives a conservative result which guarantees given quality of servicelevels in terms of packet loss ratio.

5. CONCLUSION

We have proposed and analyzed a GeoY /G/∞ input process to model video traffic. Ourproposed model effectively characterize the video traffic in the aspect of marginal distributionand autocorrelation and yet tractable in queueing analysis. With four full-length video tracesincluding action movies, a drama, and an animation, we evaluate the modeling performance interms of packet loss ratio. GeoY /G/∞ model is observed to consistently provide conservativeperformance predictions within acceptable error at various traffic loads. For real-time imple-mentation of our model, we analyze G/D/1/K queueing systems with GeoY /G/∞ input processto upper estimate the packet loss probabilities.

REFERENCES

[1] F. Fitzek and M. Reisslein, “MPEG-4 and H.263 video traces for network performance evalu-ation,” IEEE Network, pp. 40 – 54, Nov./Dec. 2001. Traces available at http://www-tkn.ee.tu-berlin.de/research/trace/trace.html and http://trace.eas.asu.edu

[2] J. Beran, R. Sherman, M. S. Taqqu, and W. Willinger, “Long-range dependence in variable-bit-rate videotraffic,” IEEE Trans. on Commun., vol.43, no.2/3/4, pp. 1566 – 1579, Feb/Mar/Apr. 1995.

[3] M. M. Krunz and A. M. Makowski, “Modeling video traffic using M/G/∞ input processes: a compromisebetween Markovian and LRD Models,” IEEE Journal on Selected Areas in Communications, vol. 16, no. 5,pp. 733 – 748, June 1998.

[4] K. Nagarajan and G. T. Zhou, “Self-similar traffic sources: modeling and real-time resources allocation,”Statistical Sig. Processing 2001, pp. 74 – 76, 2001.

[5] K. Nagarajan and G. T. Zhou, “A new resource allocation scheme for VBR video traffic source,” Proc. 34thAsilomar Conference on Signals, Systems, and Computers, pp. 1245 – 1249, Pacific Grove, CA, Oct. 2000.

[6] ETSI, ”GSM 03.64 Overall description of GPRS radio interface, Stage 2,” vol. 2.1.1, May 1997.[7] M. Parulekar and A. M. Makowski, “Tail probabilities for M/G/∞ input process (I): Preliminary asymptotics,”

Queueing Systems, vol. 27, pp. 271 – 296, 1997.[8] M. Parulekar and A. M. Makowski, “M/G/∞ input processes: A versatile class of models for network traffic,”

Proc. IEEE INFOCOM ’97, vol.2, pp. 419 – 426, April 1997.

APPENDIX A. APPENDIX

Lemma 1. For t = 1, 2, · · · ,(i) St = S

(0)t + S

(a)t where S

(0)t ≡ ∑t

i=1 b(0)i and S

(a)t ≡ ∑t

i=1 b(a)i

(ii)

ln E[eθS

(0)t

]=

∞∑

s=1

ln E[

E[eθ((σ−s)+∧t)

]Y]

(35)


(iii)

ln E[eθS

(a)t

]=

t∑

s=1

ln E[

E[eθ(σ∧s)

]Y]

. (36)

where (x ∧ y) = min(x, y).Proof of Lemma 1: (i) is trivial. For (iii), refer to [7]. (ii) is shown as follows:

S(0)t =

t∑

r=1

0∑s=−∞

Ys∑

i=1

1[σs,i > r − s]

=0∑

s=−∞

Ys∑

i=1

t∑

r=1

1[σs,i > r − s]

=0∑

s=−∞

Ys∑

i=1

t−s∑

l=1−s

1[σs,i > l]

=0∑

s=−∞

Ys∑

i=1

min(σ + s− 1)+, t

=∞∑

m=1

Ym∑

i=1

((σ −m)+ ∧ t

). ¤

Lemma 2. If θ ≤ 0,

limt→∞

1vt

ln E[eθtS(a)t ] = E[Y ]E[σ]θ. (37)

Proof of Lemma 2: For each t = 1, 2, · · · , we have θt ≤ 0 and

1vt

ln E[eθtS(a)t ] =

1vt

t∑

s=1

ln E[

E[eθt(σ∧s)

]Y]

≥ t

vtln E

[E

[eθtσ

]Y]

= θ · ln E[E[eθtσ]Y ]θt

.

For a nonnegative rv X , it is known that limθ↑0 ln E[eθX ]/θ = E[X]. Thus we have

lim inft→∞

1vt

ln E[eθtS

(a)t

]≥ θE[σ1 + · · ·+ σY ]

= θE[Y ]E[σ].

184 KANG AND KIM

Let M be a positive integer. For t > M ,

1vt

ln E[eθtS(a)t ] =

1vt

t∑

s=1

ln E[

E[eθt(σ∧s)

]Y]

≤ 1vt

t∑

s=M

ln E[

E[eθt(σ∧M)

]Y]

=t−M + 1

vtln E

[E

[eθt(σ∧M)

]Y]

=t−M + 1

t· θ · ln E[E[eθt(σ∧M)]Y ]

θt

=t−M + 1

t· θ · ln E[eθt[(σ1∧M)+···+(σY ∧M)]]

θt

Thus we have

lim supt→∞

1vt

ln E[eθtS(a)t ] ≤ θE[Y ]E[σ ∧M ].

Letting M →∞ leads to

lim supt→∞

1vt

ln E[eθtS(a)t ] ≤ θE[Y ]E[σ]. ¤

Lemma 3. If (i) θ ≤ 0, (ii) limt→∞ vt/t = 0, and (iii) E[σ2] < ∞, then

limt→∞

1vt

ln E[eθtS(0)t ] = 0. (38)


Proof of Lemma 3: For each t = 1, 2, · · · ,

− 1vt

ln E[eθtS

(0)t

]= − 1

vt

∞∑

s=1

ln E[

E[eθt((σ−s)+∧t)

]Y]

≤ − 1vt

∞∑

s=1

E[ln E

[eθt((σ−s)+∧t)

]Y]

= − 1vt

∞∑

s=1

E[Y ] ln E[eθt((σ−s)+∧t)

]

≤ − 1vt

∞∑

s=1

E[Y ]E[θt((σ − s)+ ∧ t)

]

=E[Y ](−θt)

vt

∞∑

s=1

E[(σ − s)+ ∧ t

]

≤ E[Y ](−θt)vt

∞∑

s=1

E[(σ − s)+

]

→ 0 (as t →∞). (39)

from the fact that E[σ2] < ∞⇐⇒ E [(σ − s)+] < ∞. ¤Lemma 4. If (i) 0 < θ < 1, (ii)

∑∞t=1 P[σ > t]1−θ < ∞, and (iii) vt/t, t = 1, 2, · · · is

monotone decreasing, then

limt→∞

E[eθt(σ∧t)]− 1θt

= E[σ]. (40)

Proof of Lemma 4: Let M > 1. Since P[σ > x] ≥ E[σ]P[σ > bxc], for x ≥ 0 and t such thateθt ≤ M ,

P[σ > x]eθtx1[x < t] ≤ E[σ]P[σ > bxc]eθtbxceθt1[x < t]

= E[σ]e−vbxcebxcθteθt1[x < t]

= E[σ]e−vbxcebxcθvt/teθt1[x < t]

≤ E[σ]e−vbxceθvbxceθt1[x < t]

≤ E[σ] (P[σ > bxc])1−θ M.

Since∫∞0 E[σ] (P[σ > bxc])1−θ Mdx < ∞, by Lebesgue’s dominated convergence theorem

(LDCT)

limt→∞

∫ ∞

0P[σ > x]eθtx1[x < t]dx =

∫ ∞

0P[σ > x]dx

= E[σ]. (41)

186 KANG AND KIM

On the other hand,

E[eθt(σ∧t)]− 1θt

= E[∫ σ∧t

0eθtxdx

]

= E[∫ t

01[σ > x]eθtxdx

]

=∫ t

0P[σ > x]eθtxdx. (42)

By (41) and (42), the proof is completed. ¤Lemma 5. If (i) 0 < θ < 1, (ii) vt/t, t = 1, 2, · · · is monotone decreasing, and (iii) Y hasan exponential moment, i.e., E[eθY ] < ∞ for θ > 0, then

lim supt→∞

1vt

ln E[eθtS

(a)t

]≤ E[Y ]E[σ]θ. (43)

Proof of Lemma 5: For each t = 1, 2, · · · ,

1vt

ln E[eθtS

(a)t

]=

1vt

t∑

s=1

ln E[

E[eθt(σ∧s)

]Y]

≤ t

vtln E

[E

[eθt(σ∧t)

]Y]

= θ1θt

ln E[

E[eθt(σ∧t)

]Y]

= θ · E[eθt(σ∧t)]− 1θt

· ln E[E[eθt(σ∧t)]Y

]

E[eθt(σ∧t)]− 1. (44)

By Lemma 4, E[eθt(σ∧t)] → 1 as t →∞. Hence

limt→∞

ln E[E[eθt(σ∧t)]Y

]

E[eθt(σ∧t)]− 1= lim

x→1

ln E[xY ]x− 1

= E[Y ]. (45)

By Lemma 4, (44) and (45), the proof is completed. ¤Lemma 6. If θ > 0,

lim inft→∞

1vt

ln E[eθtS

(a)t

]≥ E[Y ]E[σ]θ. (46)


Proof of Lemma 6: Let M be a positive integer. For t ≥ M ,

1vt

ln E[eθtS

(a)t

]=

1vt

t∑

s=1

ln E[

E[eθt(σ∧s)

]Y]

≥ 1vt

t∑

s=M

ln E[

E[eθt(σ∧M)

]Y]

=t−M + 1

vtln E

[E

[eθt(σ∧M)

]Y]

≥ t−M + 1vt

E[Y ] ln E[eθt(σ∧M)

]

=t−M + 1

tE[Y ] · θ · ln E

[eθt(σ∧M)

]

θt.

Hencelim inft→∞

1vt

ln E[eθtS

(a)t

]≥ θE[Y ]E[σ ∧M ].

Letting M →∞ completes the proof. ¤Lemma 7. If (i) 0 < θ < 1, (ii) vt/t, t = 1, 2, · · · is monotone decreasing in the limit, (iii)Y has an exponential moment, and (iv) E[σ] < ∞, then

limt→∞

1vt

ln E[eθtS

(0)t

]= 0. (47)

Proof of Lemma 7: Since, for each t = 1, 2, · · · , 1vt

ln E[eθtS(0)t ] ≥ 0, it suffices to show

lim supt→∞

1vt

ln E[eθtS

(0)t

]= lim sup

t→∞1vt

∞∑

s=1

ln E[


]Y]

= 0.

First we note thatE


]≥ 1

andE


]≤ E

[eθt(σ∧t)

]→ 1 (as t →∞).

Since

limx↓1

ln E[xY ]x− 1

= E[Y ],

for 1 < x < 1 + δ with sufficiently small δ, we have ε > 0 such that

ln E[xY ] < (E[Y ] + ε)(x− 1).

Thus, for sufficiently large t,

ln E[


]Y]

< (E[Y ] + ε)(


]− 1

),

188 KANG AND KIM

and it suffices to show

1vt

∞∑

s=1

(E


]− 1

)→ 0 as t →∞.

We have

1vt

∞∑

s=1

E[eθt((σ−s)+∧t) − 1

]=

1vt

∞∑

s=1

θtE

[∫ (σ−s)+∧t

0eθtx

]dx

=1vt

∞∑

s=1

θt

∫ t

0P[σ − s > x]eθtxdx

=θt

vt

∫ t

0

∞∑

s=1

P[σ > x + s]eθtxdx

=θt

vt

∫ t

0

( ∞∑

s=0

P[σ > x + s]− P[σ > x]

)eθtxdx

=θ

t

(∫ t

0P[σ > x]eθtxdx−

∫ t

0P[σ > x]eθtxdx

)

=θ

t

(∫ ∞

0P[σ > x]1[x < t]eθtxdx−

∫ ∞

0P[σ > x]1[x < t]eθtxdx

)

→ 0 as t →∞, (48)

because, by LDCT,

limt→∞

∫ ∞

0P[σ > x]1[x < t]eθtxdx = E[σ]E[σ] < ∞

and

limt→∞

∫ ∞

0P[σ > x]1[x < t]eθtxdx = E[σ] < ∞. ¤

Lemma 8. If (i) θ > 1 and (ii)

limt→∞

P[σ > t]P[σ > t]

= ∞,

then

limt→∞

1vt

ln E[eθtSt

]= ∞. (49)


Proof of Lemma 8: From Lemma 1,

1vt

ln E[eθtSt

]≥ 1

vt

∞∑

s=1

ln E[


]Y]

+1vt

ln E[

E[eθt(σ∧t)

]Y]

=1vt

∞∑

s=0

ln E[


]Y]

≥ E[Y ]vt

∞∑

s=0

ln E[eθt((σ−s)+∧t)

]

=E[Y ]vt

∞∑

s=0

ln(

1 + θt

∫ t

0P[σ > s + x]eθtxdx

)

≥ E[Y ]vt

∞∑

s=0

ln(

1 + θtP[σ > s + t]∫ t

0eθtxdx

)

=E[Y ]vt

∞∑

s=0

ln(1 + P[σ > s + t]eθtt − P[σ > s + t]

)

≥ E[Y ]vt

∞∑

s=0

[ln

(1 + P[σ > s + t]eθtt

)− P[σ > s + t]

]

≥ E[Y ]vt

∞∑

s=0

ln(1 + P[σ > t]eθtt)P[σ > t]eθtt

P[σ < s + t]eθtt − E[σ]vt

P[σ > t]

= E[Y ]E[σ]vt

ln(1 + P[σ > t]eθtt)P[σ > t]eθtt

P[σ > t]eθtt − E[σ]vt

P[σ > t]

= E[Y ](θ − 1)E[σ]P[σ > t]eθtt

ln(P[σ > t]eθtt)ln(1 + P[σ > t]eθtt)

P[σ > t]eθtt− E[σ]

vtP[σ > t]

→ ∞ as t →∞, (50)

by the following Lemma 9. ¤Lemma 9. If we have sequence an and bn such that bn > 0, an → ∞, and an/bn → ∞ asn →∞, then

limn→∞

an

bn

ln(1 + bn)ln(1 + an)

= ∞,

and, hence,

limn→∞

an

bn

ln(1 + bn)ln an

= ∞.

Proof of Lemma 9: If bn ≤ 1, then

an

bn


≥ ln 2an

ln(1 + an).

190 KANG AND KIM

Since limn→∞ an/ ln(1 + an) = ∞, we may assume that bn > 1 for all n. It is easy to seethat, for each z > 1, ln(1 + x)/ ln(1 + zx) is an increasing function of x. Hence, if an > 1

an

bn


≥ an

bn

ln(1 + 1)ln(1 + an/bn)

and, hence,

lim infn→∞

an

bn


≥ ln 2an/bn

ln(1 + an/bn)→∞. ¤

J. KSIAM Vol.12, No.3, 191–199, 2008

AN ITERATIVE DISTRIBUTED SOURCE METHOD FOR THE DIVERGENCE OFSOURCE CURRENT IN EEG INVERSE PROBLEM

JONGHO CHOI1, CHNAG-OCK LEE2†, AND HYUN-KYO JUNG1

1SCHOOL OF ELECTRICAL ENGINEERING, SEOUL NATIONAL UNIVERSITY, SOUTH KOREA

E-mail address: cjhaha, [email protected] OF MATHEMATICAL SCIENCES, KAIST, SOUTH KOREA


ABSTRACT. This paper proposes a new method for the inverse problem of the three-dimensionalreconstruction of the electrical activity of the brain from electroencephalography (EEG). Com-pared to conventional direct methods using additional parameters, the proposed approach solvesthe EEG inverse problem iteratively without any parameter. We describe the Lagrangian cor-responding to the minimization problem and suggest the numerical inverse algorithm. Therestriction of influence space and the lead field matrix reduce the computational cost in this ap-proach. The reconstructed divergence of primary current converges to a reasonable distributionfor three dimensional sphere head model.

1. INTRODUCTION

Electroencephalography (EEG) is one of noninvasive tools to measure the potential differ-ence on the scalp surface, which is the result of movement of ions, the primary current, withinactivated regions in the brain. EEG has several strong aspects as a tool of exploring brain ac-tivity; for example, its time resolution is very high, i.e., on the level of a single millisecond,compared to other methods of looking at brain activity, such as PET and fMRI which have timeresolution between seconds and minutes. EEG measures the brain’s electrical activity directly,while other methods record changes in blood flow (e.g., SPECT, fMRI) or metabolic activity(e.g., PET) which are indirect markers of brain electrical activity.

The reconstruction of the primary current distribution using EEG data which is the electricpotential measured at sensors on the scalp surface is the main goal of EEG research and calledthe inverse problem of EEG. Its solution requires the repeated simulation of the electric poten-tial distribution in the head by a given primary current, which is called the forward problem.

It has been known that there are primary current distributions which induce the same EEGdata [1]. This is an indication of the fact that the solution to the inverse problem is not unique.

2000 Mathematics Subject Classification. 35R30, 65N21, 92C55.Key words and phrases. Divergence of source current, EEG inverse problem, Finite element method, Lagrangian

approach.† Corresponding author. This author’s work was supported by the SRC/ERC program (R11-2002-103) of

MOST/KOSEF.

191

192 J. CHOI, C.-O. LEE, AND H.-K. JUNG

The non-uniqueness of the inverse problem implies that assumptions on the model as wellas anatomical and physiological a priori knowledge should be taken into account to obtain aunique solution. In order to make the solution unique, additional conditions are imposed onthe solution. e.g., small number of current dipoles or spatial smoothness of the primary currentdistribution.

The equivalent dipole method [2] is based on the moving dipole model. In this methodprimary current in the brain are approximated by a small number of current dipoles, and theirlocations and moments are estimated by fitting to the EEG data. In BESA (Brain ElectricSource Analysis) [3] and MUSIC (Multiple Signal Classification) [4], the locations of dipolesare assumed to be fixed during certain time interval, and they are determined from the EEG datameasured repeatedly during that time interval. These approaches can be easily and intuitivelyunderstood by clinicians. Moreover, since no restriction is imposed on the dipoles, this methodseems adequate so long as dipoles are expected to be localized. However a key problem is thecorrect estimation of the number of dipoles. In these model, only the cross product of locationand moment of dipole is determined.

Alternatively continuous distribution is discretized as an array of numerous dipoles sited atthe activated region. Since the model allows a large degree of freedom, there are infinite num-bers of primary current distribution that reconstruct the measured EEG data. In order to makethe solution unique, for example, spatial smoothness of the primary current distribution is usedin the Minimum Norm Method (MNM) and the Low Resolution Electromagnetic Tomography(LORETA) [5]. In such distributed source models, however, the additional constraint has noconnection with the electro-physiological phenomena and it is quite difficult to evaluate theadequacy of this constraint.

One conventional iterative method, the Focal Underdetermined System Solution (FOCUSS),has another problem in convergence. The range of source distribution is getting smaller withiterations and eventually converges to a point distribution. Therefore FOCUSS is required todetermine the proper number of iterations.

As one of iterative distributed source methods, the adjoint state approach in continuous casewas proposed in [6] where the source currents are obtained. However the method can not avoidthe intrinsic ill-posedness of the EEG inverse problem. In this paper we modify this approach toreconstruct the divergence of source current instead of source current itself and implement themethod for EEG inverse problem in discrete case. The reconstructed source current convergesto a reasonable distribution for three dimensional sphere head model.

2. FORWARD PROBLEM FORMULATION

2.1. Maxwell equations. We begin with introduction of notations. Let E and D be the electricfield and electric displacement, respectively, ρ the electric free charge density, ε the electricpermeability and J the electric current density. By µ we denote the magnetic permeability andby H and B the magnetic field and induction, respectively.

In the low frequency band (below 2000Hz) the temporal derivatives can be neglected in theMaxwell equations of electrodynamics [7]. Therefore, the electric and magnetic fields can be

AN ITERATIVE DISTRIBUTED SOURCE METHOD FOR EEG INVERSE PROBLEM 193

described by the quasi-static Maxwell equations

∇ · D = ρ, (2.1a)∇× E = 0, (2.1b)∇× B = µJ, (2.1c)∇ ·H = 0. (2.1d)

The electric field is expressed as a negative gradient of a scalar electric potential u,

E = −∇u. (2.2)

The electric current density is generally divided into two part, the so-called primary current,Jp and secondary current σE,

J = Jp + σE, (2.3)

where σ denotes the 3×3 conductivity tensor.

2.2. Electric forward problem. Since the divergence of curl of a vector is zero, taking thedivergence of (2.1c) and using (2.2) and (2.3) give the Possion equation

∇ · (σ∇u) = ∇ · Jp in Ω . (2.4)

Let us assume that all fields of biological origin are quasi-static [8]. This allows us to establishthe relationship between the electric potential u and the primary current Jp that is movementof ions within the dendrites of the large pyramidal cells of activated regions in the cortex of thehuman brain through (2.4). It describes the electric potential distribution in the head domain Ωdue to the primary current Jp in the brain. For the forward problem which is assumed that theprimary current and the conductivity distribution in the head are given, the equation has to besolved for the unknown electric potential distribution u with the boundary condition

σ∂u

∂n= σ∇u · n = 0 on S = ∂Ω (2.5)

and an additional reference point with given potential, i.e.,

uref = 0.

3. INVERSE PROBLEM FORMULATION

The inverse EEG problem aims to reconstruct the primary current distribution by only useof EEG data in the conductivity head model. However, since Eq. (2.4) establishes that theelectric potential field u is generated by the divergence of primary current, we reconstruct thedivergence of primary current Ip = ∇ · Jp which is referred as the current source density [9].


3.1. Mathematical formulation. Let N be the number of EEG sensors. We measure the dif-ference of electric potential at several locations x1, x2, . . . , xN on the scalp and at a referencelocation x0. Let m1,m2, . . . , mN be those measured EEG data. In the EEG problem, wewant to estimate the divergence of primary current Ip to produce the electric potential as closeas possible to the measured EEG data. This means that if we compute the electric potentialdistribution u for such Ip with Eq. (2.4), then the error function

ϕ(u) =12

N∑

i=1

(u(xi)− u(x0)−mi)2

achieves a minimum at u. We form the Lagrangian as

L(u,w, Ip) = ϕ(u) +∫

Ω(∇ · (σ∇u)− Ip)w dx,

where w is the Lagrangian multiplier. We then use the integration by part and the boundarycondition (2.5) to obtain

L(u,w, Ip) = ϕ(u)−∫

Ωσ∇u · ∇w dx−

∫

ΩIpw dx . (3.1)

This form of the Lagrangian is suitable for computing Gateaux derivatives with respect to Ip.From the Gateaux derivative of the Lagrangian with respect to Ip in (3.1), we obtain

∂L∂Ip

= −w.

This equation says that in order to find the divergence of primary current that yields a potentialdistribution that minimizes ϕ(u), we should follow the descent direction w at every point x ofthe interested region.

Using the integration by part again with imposed boundary condition for w such that σ∇w ·n = 0 yields the form that is suitable for taking Gateaux derivatives with respect to u:

L(u,w, Ip) = ϕ(u) +∫

Ω∇ · (σ∇w)u dx−

∫

ΩIpw dx .

Taking the Gateaux derivatives of L with respect to the function u, we obtain

δL =N∑

i=1

(u(xi)− u(x0)−mi)(δu(xi)− δu(x0)) +∫

Ω∇ · (σ∇w)δu dx .

This must be equal to 0 for all variations δu of u, hence the adjoint state equation is

∇ · (σ∇w) + f = 0 in Ω,∇w · n = 0 on S,

(3.2)

where f(x) =∑N

i=1(u(xi)− u(x0)−mi)(δ(x− xi)− δ(x− x0)) with the usual Dirac deltafunction δ(x).


FIGURE 1. Head model

3.2. Restriction to influence space. Let the influence space be the activated region where theprimary current can be produced. Physically, the primary current cannot be produced at theregion out of the brain and the signal from deep part of the brain is not detected with EEGsensor. Therefore we consider only the cortex surface region CSε inside the cortex surfacewhose distance from the cortex surface is less than ε as described in Fig. 1, where ε dependson the measurability of EEG machine. Then the Lagrangian form (3.1) is expressed withcharacteristic function χcsε as

L(u,w, Ip) = ϕ(u)−∫

Ωσ∇u · ∇w dx−

∫

ΩIpχcsεw dx . (3.3)

As considering the restricted influence space, from (3.3), we have

∂L∂Ip

= −wχcsε . (3.4)

3.3. Algorithm: the adjoint state approach. We give a description for the algorithm to solvethe inverse EEG problems that is the constrained minimization problem as follows:

(1) Start with an estimate I(0)p and set i = 0.

(2) Solve the state equation (2.4) with Ip = ∇ · Jp to obtain u(i).(3) Solve the adjoint state equation (3.2) with u(i) to obtain w(i).(4) With u(i) and w(i), determine the gradient ∂L

∂Ipby (3.4) and I

(i+1)p using the steepest

descent method.(5) If I

(i+1)p is not close to I

(i)p , then go to step 2 and set i = i + 1, else stop.

The main advantage of this adjoint state approach is to find Ip instead of Jp. It is well knownthat finding Jp is severely ill-posed. On the contrary, finding Ip gives more stable results.


4. NUMERICAL ASPECTS

For the numerical solution of the method, we choose a finite-dimensional subspace with di-mension M , the number of nodes in Ω and a standard finite element basis φ1,. . . ,φM . Applyingthe finite element techniques to Eq. (2.4) with Ip = ∇ · Jp yields an M ×M system of linearequations

AU = Ip . (4.1)Eq. (3.2) yields a similar M ×M system of linear equations to (4.1).

4.1. Lead field matrix. In each iteration of the proposed algorithm, we solve the state equa-tion (2.4) for u and the adjoint state equation (3.2) for w. Additionally, in the modified steepestdecent method to determine how far we go to the negative gradient direction we solve the stateequation at least twice [10]. Therefore, we solve the state equation three times and the adjointequation once in each iteration of the algorithm.

To reduce the computational cost, we suggest computing the lead field matrix [11] denotedby B, which maps Ip, the vector corresponding to Ip to UEEG, the vector of electric potentialat the sensors,

BIp = UEEG. (4.2)When the restriction matrix R is a mapping from the potential vector onto the sensors such that

RU = UEEG (4.3)

with only one nonzero entry with the value 1 in each row, B is computed as

B = RA−1.

by the relations (4.1) and (4.3). To find B, since we A is symmetric, we solve

ABT = RT . (4.4)

If the lead field matrix B is precomputed, then the estimated electric potential UEEG at EEGsensors, which is needed in the state equation and the steepest decent method to determine theline search is computed by (4.2) instead of solving the linear system (4.1). It means that weonly need to solve the adjoint state equation for w in each iteration of the inverse method.

4.2. Numerical inverse algorithm. The proposed method for inverse EEG problem to recon-struct the divergence of the primary current is described by following steps:

(1) Compute the stiffness matrix A and the lead field matrix B with a given conductivity.(2) Start with an estimated divergence of primary current I

(0)p and set i = 0.

(3) Obtain UEEG from (4.2) with I(i)p .

(4) Solve the adjoint state equation (3.2) to obtain w with UEEG and EEG data.(5) Determine a new value I

(i+1)p with the line search α and the gradient −w using the

steepest decent method:I(i+1)p = I(i)

p + αw.

Here we compute Ueeg from (4.2) to find the line search α; see [10].


TABLE 1. Computation time (sec) of with/without B

Computation with B without BMeshing 21.1

Precomputation A 0.6B 0.3 0

Inverse iteration (# of iterations) 803.2 (121) 2023.1 (121)Total time 825.0 2044.7

(6) If I(i+1)p is not close to I

(i)p , then go to step 3 and set i = i + 1, else stop.

In the inverse algorithm, the step 1 is called the precomputation step and the steps 2 to 6 arecalled the inverse iteration.

As a stop condition for the inverse iteration in the step 6, we use the relative error of theestimated divergecne of primary current:

‖I(i+1)p − I

(i)p ‖

‖I(i)p ‖

< 10−3. (4.5)

The inverse method with the precomputed B is more efficient than the inverse method withoutB. Because with B, we additionally solve (4.4) in the step 1, but in the steps 3 and 5 weobtain UEEG by only the product of B and Ip instead of solving (4.1). Table 4.2 shows thecomputational costs of two methods with h = 1/32 and the stop condition (4.5).

4.3. Numerical simulation. The numerical simulation is performed with a three dimensionalsphere domain divided into three layers with radius 0.6, 0.8, 1.0 and the conductivities 0.33,0.01, 0.43 representing the brain, the skull, and the scalp, respectively; see Fig. 1. The corticalsurface is the boundary of brain. The sphere head model is approximated by meshes with 2,199nodes and 11,415 tetrahedrons [12]. We employ the finite element method to implement themethod in discrete case and the modified steepest decent method as an update algorithm [10].Let we further assume that EEG sensors directly correspond to nodes on the boundary of thehead model.

Let Nb and Nε be the number of nodes on boundary and nodes on CSε for ε = 1/20,respectively. If we assume that we measured EEG data on all boundary nodes and the influencespace is small enough to make Nε is less than Nb then the number of EEG data is greater thanthe number of unknowns. Therefore it becomes an over-determined problem which gives aunique solution in the least squares sense.

The threshold process is defined as Ip = 0 if

Ip < max(Ip)− (100− β)(max(Ip)−min(Ip))100

where β is the given percentage of threshold.


(a) (b)

FIGURE 2. (a) True Ip (b) Reconstructed Ip with threshold 85%

The computation time for inverse iterations with the stop condition depends on the initialestimated Ip. If any a priori information for initial guess is not given, we start with Ip = 0for all nodes in the influence space. When the divergence of true primary current distributionin the cortex surface is given as Fig. 2 (a), computational time from all zero distribution to theresult in Fig. 2 (b) is 63 seconds with 50 iterations. The result with 85% threshold is shown inFig. 2 (b). The node with the maximum value in Fig. 2 (b) is corresponding to the node withthe maximum value in Fig. 2 (a). The numerical tests are performed using MATLAB 2008a onMicrosoft Windows XP with Intel Core 2 Duo (2.3G MHz CPU clock rate) and 2G RAM.

5. CONCLUSION

We have introduced a new iterative distributed source method for EEG inverse problem usingthe Lagrangian approach. It reconstructs the divergence of source current instead of sourcecurrent itself. The restriction of influence space is required to determine focal and realisticsolution and we suggest the computation of the lead field matrix in the numerical algorithm. Inthe numerical simulation with the sphere model, it was shown that the reconstructed solutionconverges to a reasonable distribution. The approach proposed in this paper is able to be appliedto other inverse problems.

REFERENCES

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