Mean Square Error (MSE) of Decentralized Estimation

That the sensor messages

Decentralized Estimation

where ni is the sensor measurement noise and qi is quantization noise. Therefore the final estimator is

( )

When each m_i is transmitted to the FC through a nonperfect channel with finite power, bit error occurs. It will impact on the estimation t accuracy at the FC. The links between each sensor and the FC are modeled as a memoryless binary symmetric channel. Suppose the probability of bit error achieved by sensor i is p and _bⁱ is

It shows that the actual achieved MSE is at most a constant factor away from what is achievable with perfect sensor channels, provided that each sensor’s bit error rate (BER) is bounded above (2.14). Because the perfect MSE D is easier to derived, the upper bound of actual achieved MSE D′ in (2.15) will be sed to formulate the u

Chapter 3

Equation Chapter (Next) Section 1

Minimal Energy Decentralized

Estimation Based on Sensor Noise Variance Statistics

This chapter studies minimal-energy decentralized estimation in sensor network under BLUE fusion rule. While most of the existing related works [6, 7, 8] require the knowledge of instantaneous noise variances for energy allocation, the proposed approach instead relies on an associated statistical model. Subject to severe energy and bandwidth limitation, each sensor in this scenario is allowed to transmit only a quantized version of its raw measurement to the FC to generate a final parameter estimate. While quantized message with longer bit length provide improved data fidelity, the consumed transmission energy is however proportional to the bit loads.

As energy efficiency is a critical concern for sensor network design, the minimal-energy decentralized estimation problem which formulated in an optimal bit-loading setup has been recently considered.

One key feature common to the existing related works is that the energy

allocated to each sensor must be determined via instantaneous local sensor noise he BLUE principle. In order to improve the estimation performance against the variation of sensing

nditions, repeated update of the noise profile would be needed. This comes inevitably at the cost of more training overhead and extra energy consumption. One

or long-term) information of the noise characteristics.

alized estimation by exploiting long term noise variance information. A commonly used s used and the estimation performance is assessed through an MSE based metric average with respect to the considered

stribution. A closed-form expression of the overall MSE requirement is derived. The analysis of the energy-minimization problem is formulated in the form of convex optim

characteristics (the noise variance), if the fusion rule follows t

co

typical approach to resolving such a drawback is to exploit the partial (

This chapter attempts to provide a solution to minimal-energy decentr

statistical model [6, 7] for noise variance i

di

ization. The problem is then analytically solved.

The proposed optimal scheme shares several interesting aspects pertaining to those based on the instantaneous noise variance information. Sensors with bad channel quality (specified via the path distance to FC) are shut off to conserve energy, and for those active nodes the allocated energy is proportional to the individual channel gain. Simulation results show that the proposed optimal solution yields significant energy saving against the equal-bit allocation policy.

3.1 Average Mean Square Error of Decentralized Estimation

For a fixed set of noise variances σ_i²’s, the energy minimization problem

subject to an allowable parameter distortion γ (in terms of MSE) can be formulated

i is the consumed energy for transmitting the message m_i. (3.1) is equivalent to

( )

¹

der to obtain universal solution with averaged measurement noise conditions, the following optimization problem is considered:

( )

¹ optimization problem (3.3), the equivalent MSE performance metric in (3.2) is averaged with respect to the noise variance statistic characterized in

− z

(2.2).

average MSE performance measure. Since

To solve (3.3), a crucial step is to derive an analytic expression of the

2 1

zi ∼χ is a central i.i.d. Chi-Square distributed random variable with degrees-of-freedom equal to one[10, p-24]

2

The average MSE performance can be derived as

(

²

)

where β δ_i = +R²4^−bi 12. The following lemma, with proof given in Appendix A, provides a closed-form expression of the integral involved in the summation in (3.5).

Lemma 3.1 : With α> and 0 β_i > as defined in (3.5), we have 0

With (3.5) and Lemma 3.1, the optimization problem

2

Exact solution to problem (3.7) appear intractable since the target MSE is highly

is to derive an easy-to-tackle lower bound on

nonlinear in b . We will thus seek for the suboptimal alternatives which can otherwise admit simple analytic expression. The underlying approach toward this end

i

the target MSE metric, and then replace the MSE constraint in (3.7) by one which forces the lower bound to be above γ⁻¹. Such a procedure will considerably simplify the analysis without incurring any loss in the desired MSE performance. This is done with the aid of the next lemma with proof given in Appendix B.

Lemma 3.2 : The following inequality holds:

(

^{2−bi 12α}

)

^⎞,

2

Lemma 3.2 suggests that we can replace the MSE constraint in (3.7) by the following one without incurring any loss in the target MSE:

1

is one-to-on e will thus instead focus on the optim

Since ^{Q i}

( )

e and monotone decreasing, w

ization problem with a modified MSE performance constraint:

1

his optimization problem will lead to a simple closed-form solution.

3.2 Energy Density Factor of Sensor Nodes

T

We assume that each sensor sends messages to FC using a separate channel. This can be achieved by using a multiple access technique such as TDMA or FDMA. Each channel is corrupted by additive white Gaussian noise (AWGN) with power spectral density N₀/2:

distance between sensor i and the FC, and κ is the path loss exponent common to all

sensor-to-FC links. Suppose that message m_i has length b_i bit.

We will assume that energy Ei required for transmission of mi is proportional to QAM is used, the consumed energy at the ith sensor is defined as

the number of bits in the message. If

M-, 1 , QAM symbol, and P_b is the target bit error rate. With (3.13), the specification of the

nergy allocated to the ith sensor amounts to determining th bits b .

For a fixed set of noise variances

e e number of quantization

i

2

σi ’ ter distortion level

s, the energy minimization problem subject to an allowable parame γ (in terms of MSE) can be

rmulated as optimization problem and will moreover lead to a simple closed-form solution as show

form Solution

The final optimization problem is as follows

⎜ ⎟

∑ ∑

In (3.15), the cost function is linear and the constrain is convex. It is thus a conve

n below.

3.3 Problem Formulation and Optimal

Closed-1

der to solve problem(3.16), let us form the Lagrangian function as

(

1 1

)

The associated set of Karush-Kuhn-Tucker (KKT) [14] conditions is as followed:

=

∑

1 i≤ ≤ . This case should be precluded since otheN rwise all sensors will rem in silent. a We must haveλ> . It means that the M E constraint in (3.16) is active so that 0 S

by the next lemma with proof given in Appendix C.

3.4 Discussions of Optimal Solution

1. The target distortion level γ cannot be set unlimitedly small. It is otherwise bounded by the MSE attained by the benchmark estimate based on un-quantized real-valued sensor measurements (i.e., the case when b_i = ∞, 1≤ ≤ ). By i N setting b_i = ∞ in the average MSE formula specified in

allowable

(3.7), the minimal γ can be immediately determined as

1

. Since 0 , a necessary condition for valida (3.15) is therefore

≥ ⎢⎢⎣ ⎦

2 ≤ < ∞b_i ting the MSE constraint in

1 1

By definition of the constant c in Lemma 3.2 and with (3.28), the MSE attainable by the proposed method is lower bounded by

( )

(3.29) is indeed larger than the lower bound (3.27).

3. In (3.14), the energy density factor is proportional to the path loss , if the same bit error rate is assumed throughout a

corres sors deploye ay from e usually

with poor background channel gain. In this point, the proposed optimal solution (3.25) is intuitively attractive. The sensors with large

conserve energy. A similar energy conservation strategy via shutting off the ors with poor channel links is found in [6], in which a scenario with instantaneous noise variance available to the FC is considered.

. From(3.25), the assigned message length is invers

those active sensors. This is intuitively reasonable since sensors with better link rformance.

), the equal-bit The lower bound

w i d_i^κ

ll the links. The large values of w i pond to the sen d far aw the FC. They ar

w are turned off to i

sens

4 ely proportional to w for _i

conditions should be allocated with more bits to realize desired pe 5. Based on the inequality constraint for average MSE in (3.15

schemes maintaining the desired MSE can be obtained by solving

2 1 1

( )

gy saving when compared with equal-bit scheme (3.31).

3.5 Numerical Simulation

N

(3.31)

Nume lations in the next section show that the proposed optimal scheme (3.25) yields significant ener

For a fixed set of energy density factors w_i, 1≤ ≤i , the performance is measured via the percentage of energy saving (PES) [6, 7]:

1 1 distributed random variable. The results are averaged over 50000 independent trials.

The total number of sensors is N=1500 underγ =0.005.

The Figure 3.1(a) shows the PES for 0.1≤ ≤α 1.6 and Figure 3.1(b) depicts the uted b in

comp (3.31) with fixed δ =0.8. That the PES exhibits two “jumps” can be rved. This accounts for the two level change of b as

obse α varies. Within each

tion of constant b , energy efficiency of the optimal solution α

dura improves as

increases (a large α corresponds to a more inhomogeneous sensing environ ent).

ilar phenomenon has been observed in the existing works relying dge [6, 7]. When the sensing condition

m We note that a sim

on instantaneous noise variance knowle

becomes more inhomogeneous, it is more likely that a large fraction of sensors suffers

effic oise variance description would reflect the long-term characteristic of the schemes [6, 7], this consistency is expected.

iency. Since the proposed solution (3.25) based on statistical n

Figure 3.1 : PES for fixed minimal noise variance threshold (δ =0.8)

We repeat the experiment by fixing α=0.4 and varying the minimal threshold δ . The results are shown in Figure 3.2. Obviously, the PES exhibits a counter

denc pared t re 3.1. For e ation o

ten y as com o Figu ach dur f constant b , the energy saving achieved by proposed optimal solution is lower as δ increases. This is reasonable because the large minimal noise variance threshold results in severe noise

corru ore sensor n

uf mation

ption in all sensor measurement. M odes should be turned on to provide a s ficient amount of infor for MSE reduction.

Figure 3.2 : PES for fixed noise variance variation (α=0.4)

3.6 Summary

This chapter provides a solution to the minimal-energy decentralized estimation problem by exploiting a statistical noise variance model. Based on a closed-form expression of the MSE performance measure averaged over the noise variance distribution, energy minimization is reformulated as convex optimization problem.

The proposed solution simply allocates energies to sensors with large channel gain nd shut off those suffering from poor link quality. Numerical simulation shows that

f reduc w

a

the proposed optimal solution is capable o ing about 80% energy consumption hen compared with the uniform-allocation scheme. The energy saving efficiency is particularly significant when the minimal measurement noise variance threshold is small or the variation factor is large.

Chapter 4

Equation Chapter (Next) Section 1

Minimal Mean Square Error

Decentralized Estimation Based on Sensor Noise Variance Statistics

Relying on partial noise variance knowledge in the form of the background distribution, the problem of minimizing total transmission energy under an allowable average distortion level is recently considered in [15]. This chapter considers the

o find the optimal bit load which minimizes the average distortion under a fixed total energy budget. The main contribution of the current work

counterpart problem: how t

can be summarized as follows:

i. While the design metric, the reciprocal of the average MSE is shown in [15]

to be highly nonlinear in the sensor bit load. Several analytic approximation relations are used to derive an associated tractable low bound.

ii. By maximizing this lower bound, the problem can be further formulated in the form of convex optimization which yields a closed-form solution.

The analytic results reveal that under limited energy budget, sensors with bad

link quality should be shut off toward utmost estimation accuracy, and energy allocated to those active nodes should be proportional to the individual channel gain.

A similar energy conservation policy is also found in the previous work [6, 7, 15].

Numerical simulations show the effectiveness of the proposed scheme which outperforms the uniform allocation strategy under an energy-limited environment.

4.1 Average Mean Square Error of Decentralized Estimation

The MMSE decentralized estimation which is counterpart problem of (3.1) can be formulated as function is averaged with respect to the noise variance statistic characterized in (2.2):

ET to solve problem (4.3), the first step is to find an analytic expression of the equivalent average MSE metric.

By equation (3.5) and lemma 3.1, problem (4.3) can be equivalently rewritten as

( ) ( )

Exact solution to the considered optimization (4.4) appears formidable to tackle ormulation which is more tractable is proposed and an analytic solution can be obtained. By the

1

Max 2

4 ^bi

i

π ^⋅

∑

⁼ α δ β+ ⁻

because the cost function is highly nonlinear in b . An alternative f_i

following approximation to Q(.) function [16, p115]

( ) ( )

and some straightforward manipulations, the cost function can be approximated by

( ) ( )

The main advantage of (4.6) is that it can lead to an associated lower bound in a more tractable form. Thought maximizing this lower bound we can eventually obtain a closed-form optimal solution. By the inequality equation:

1 1 1 2

( ) ( ) ( ) ( )

We will thus focus on maximizing the lower bound

(4.8)

The cost function is simple in (4.9). It can lead to an analytic solution of the ptimization problem.

gy Density Factor of Sensor Nodes

We assume that each sensor sends messages to FC using a separate channel. This achieved by a m

density N₀/2:

o

4.2 Ener

can be using ultiple access technique such as TDMA or FDMA. Each channel is corrupted by additive white Gaussian noise (AWGN) with power spectral

ˆ_{i i} 2 _i ,

m =d^−κ m +v_i (4.10)

is the received

where m ˆ_i message at FC and vi is the AWGN. The signal power received at the FC is assumed to be inversely proportional to d_i^κ where d is the _i distance between sensor i and the FC, and κ is the path loss exponent common to all sensor-to-FC links.

amplitude modulation with a constellation size 2^bi. The consumed energy is [4, 5, 6, 17]

(

^{2bi 1 , 1}

)

i i

E =w − ≤ ≤i N. (4.11)

The energy density factor wi is defined as

2 rate assumed common to all sensor-to-FC links.

With (4.11), the specification of the energy allocated to the ith sensor amounts to etermining the number of quantization bits bi.For a fixed s

d et of noise variances

2

σi ’s, the MSE minimization problem subject to an allowable energy level E_T can be formulated as

replace the total energy constraint in (4.13) by the following one without violating the over

=

With the aid of (4.14) and by performing a change of variable with the optimization problem then becomes

all energy budget requirement:

( )

is relaxed to be a nonnegative real number so as to render the problem tractable. ile the optimal real-valued B is computed, _i

the associated bit loads can be obtain through upper integer rounding. The major advantage of the alternative problem formulation is that it admits the form of convex

ptimization and can lead to a simple closed-form solution. It is shown section.

4.3 Problem Formulation and Optimal Closed-form Solution

The finial optimization problem is as followed

o in the next

In order to solve problem (4.16), let us form the Lagrangian as

( )

The associated set of KKT conditions [14] is as followed:

1 .

The condition (4.18) leads to

1 1 .

0, 1 i N

bi = ≤ ≤ . This case should be precluded since all sensors are turned off. From (4.19) and (4.21), λ can be obtained:

,λ and μ_i’s should be determined to fulfill the desired constraints.

1 2 N

1 1

t average distortion level then equals

∑ ∑

4.4 Discussions of Optimal Solution

1. The minimal average MSE is attained when all the raw sensor measurements

=⎜ ⋅

⎜⎜

∑

with infinite-precision (i.e., b_i =0, 1≤ ≤i N ) are available to the FC. Hence, by setting in the mean MSE formula specified (4.4), we have the following performance bound

Formula (4.28) reveals the impacts of the noise model parameters α and δ on the estimation performance. It is easy to see from (4.28) that minim MSE increases with

the al α. This implies the estimation accuracy degrades as the sensing environm es more and more inhomogeneous (corresponding to large

ent becom

α). Furthermore it can be checked that MSEmin also increases with the δ . This is reasonable since large δ

minimal noise power threshold implies

poor measurement quality of all sensor data and a less accurate p rameter estimate. Although these facts are inferred based on

a

the idealized distortion

with 2. The energy

links with the same ). Large value of wi correspond to sensors deployed far away from the FC, usually with poor background channel gains. In this point, the proposed optimal solution is intuitively attractive. The senso

the (K₁-1)th largest w_i’s are turned off to conserve energy. A similar energy is also found in [6, 7, 15].

s is inversely proportional to sensor data quantization.

density factor wi is proportional to the path loss d (assuming all _i^κ κ

rs associated with

conservation strategy via shutting off sensors with poor channel links

From (4.26), the assigned message length for those active node w . This is intuitively reasonable since _i sensors with better link conditions should be allocated with more bits (energy) to improve the estimation accuracy.

3. In order to prevent sensors from exhausting energy quickly, one natural way is to impose an additional peak energy constraint:

(

^{2bi 1}

)

^{, 1 .}

i P

w − ≤E i N

requirement (4.29), there does not seem to exist a closed-form optimal solution. As a simple suboptimal

de inde set

≤ ≤ (4.29)

In optimization problem (4.16), with extra inequality

alternative, we can first identify the infeasible no x

{

^{i wi}

( )

^{EP, K1 i N}

}

Γ = 2^biopt − >1 ≤ ≤ from (4.26) and then instead fix the energy associated with each of these nodes to be E_P. The resultant solution is thus

(4.30) The actual solution can be obtained by using the iterative proced

[18] with (4.30) as an initialization point. The algorithm to derive the optimal analytical solution is followed.

(1) Solve the problem without individual power constraints (4.16) to obtain the solution (4.26).

ures reported in

Set the index set ^{Γ =}

{

^{i wi}

(

^{2biopt − >1}

)

^{EP, K1≤ ≤i N}

}

^.

Rem the design variable space.

(3) Repeat the first and second steps until Γ is empty in the first step.

To prove that the algorithm leads to the global optimum, we need only to prove when we that in the second step we do not lose optimality of b_i^opt for i∈ Γ

set b_i^opt =log2

(

1+E_P w_i

)

for i∈ Γ.

We compare the sim

4.5 Numerical Simulation

ulated performance of proposed optimal solution (4.26) against the uniform energy allocation scheme with bit load determined through

(

^{2bi 1}

)

^{, 1}

i T

w − =E N ≤ ≤i N. (4.31)

In (4.31), b_i is computed via lower integer rounding so that the resultant total energy can be kept below ET. It leads to

In each independent run we simply choose w_i =d_i^κ , where κ =2 and 0.5 0.3

i i

d = + Z with Zi ∼χ1²

( )

z being i.i.d.. The total numbe

the following experiments we set the numb N=200, and consider

three . different levels of total energy

correspond to the low, medium, and high energy cases.

With fixed δ = , Figure 4.1 shows the computed average MSE as 2 α varies 0.5 to 8. With fixed 2

from α= , Figure 4.2 shows the average MSE as δ varies 0.5 to 8. Both figures show that the estimation accuracy improves as E from

increases. It is expect alloc

environm

T

ed. The proposed solution (4.26) outperforms uniform energy ation (4.32), especially when E_T is small. It is more effective in an energy-limited

ent. The simulated average MSE increases with both α and β .

10⁰

0 1 2 3 4 5 6 7 8

10^-2 10^-1

noise variance variation α

average MSE Figure 4.1 : Average MSE for fixed minimal noise variance threshold ( )

1 2 3 4 5 6 7 8

Figure 4.2 : Average MSE for fixed noise variance variation (α = ) 2

4.6 Su

This chapter provides a solution to th ma ation problem

expre

distribution, MSE minimization is reformulated as convex optimization problem. The analytic closed-form solution reveals the energy saving policy. The proposed solution channel gain and shut off those suffering from poor link quality. Numerical simulation shows that the estimation accu

hus is ore effective in an energy-limited environment.

mmary

e mini l-MSE decentralized estim by exploiting a statistical noise variance model. Based on a closed-form ssion of the MSE performance measure averaged over the noise variance

simply allocates energies to sensors with a large

racy improves as total energy increases. The proposed solution outperforms uniform energy allocation especially when the total used energy is small, and t

m

Chapter 5

Equation Chapter (Next) Section 1

Minimal Energy Decentralized

r Noise

n esign, the minimal-energy decentralized estimation problem which is formulated in an optimal bit-loading setup has been recently considered. In order to improve the estimation performance against the variation of sensing conditions, repeated update of the noise

n esign, the minimal-energy decentralized estimation problem which is formulated in an optimal bit-loading setup has been recently considered. In order to improve the estimation performance against the variation of sensing conditions, repeated update of the noise

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