FUSION PREDICTORS FOR DISCRETE-TIME LINEAR

SYSTEMS WITH MULTISENSOR ENVIRONMENT

Ha Ryong Song and Vladimir Shin

Department of Mechatronics, Gwangju Institute of Science and Technology

1 Oryong-Dong, Buk-Gu, Gwangju, 500-712, South Korea

Keywords: Discrete-time linear system, Kalman predictor, fusion formula, multisensor.

Abstract: New fusion predictors for linear dynamic systems with different types of observations are proposed. The

fusion predictors are formed by summing of the local Kalman filters/predictors with matrix weights

depending only on time instants. The relationship between them and the optimal predictor is discussed. High

accuracy and computational efficiency of the fusion predictors are demonstrated on the first-order Markov

process and the damper harmonic oscillator motion with multisensor environment.

1 INTRODUCTION

The integration and fusion of information from a

combination of different types of observed

instruments (sensors) are often used in the design of

high-accuracy control systems. Typical applications

that can benefit, the use of multiple sensors, are

industrial tasks, military command, mobile robot

navigation, multi-target tracking, and aircraft

navigation (see Hall, 1992; Bar-Shalom and Li,

1995). If it is decided that all local sensors observe

the same target, then the next problem is how to

combine the correspondence local estimates.

Several distributed fusion architectures were

discussed in Bar-Shalom (1990) and Bar-Shalom

and Campo (1986) and Li et al. (2004) and

algorithms for distributed estimation fusion have

been developed in Bar-Shalom and Campo (1986)

and Shin et al. (2004, 2006) and Zhou et al. (2006).

The Bar-Shalom and Campo fusion formula (FF) for

two-sensors systems has been generalized for an

arbitrary number of sensors in Shin et al. (2004,

2006). FF represents an optimal mean-square linear

combination of the local estimates with the matrix

weights satisfying the linear algebraic equations.

The explicit expression for the matrix weights has

been derived in Zhou et al. (2006). Application of

FF to some estimation and filtering problems was

proposed in Bar-Shalom and Campo (1986), Li et

al. (2004), and Shin et al. (2004, 2006). The main

purpose of this paper is development of fusion

predictors to forecast the future state of the linear

multisensor systems.

This paper is organized as follows. In Section 2,

we present the statement of the prediction problem

with multisensor environment and give its optimal

solution. In Section 3, we propose two fusion

predictors, which are derived by using the FF. In

Section 4, the fusion predictors are tested and

compared. Finally, Section 5 is the conclusion.

2 STATEMENT OF PROBLEM

KALMAN PREDICTOR

Consider a discrete-time linear dynamic system with

additive white Gaussian noise,

,0,1,k,vGxFx

kkkk1k

K=

+

=

+

(1)

where

n

k

x ℜ∈

is state vector, and

r

k

v ℜ∈

is white

Gaussian noise,

(

)

kk

Q0,N~v

.

Suppose that overall observation vector

m

k

Y ℜ∈

is composed of

N

different types of observation

subvectors (local sensors)

(N)

k

(1)

k

y,,y K

,

[

]

,yyY

T

(N)

k

(1)

kk

TT

L=

(2)

where

(i)

k

y

are determined by the equations

,y,wxHy

N

1

m

(N)

k

(N)

kk

(N)

k

(N)

k

m

(1)

k

(1)

kk

(1)

k

(1)

k

ℜ∈+=

M

(3)

119

Ryong Song H. and Shin V. (2007).

FUSION PREDICTORS FOR DISCRETE-TIME LINEAR SYSTEMS WITH MULTISENSOR ENVIRONMENT.

In Proceedings of the Second International Conference on Signal Processing and Multimedia Applications, pages 119-124

DOI: 10.5220/0002131201190124

Copyright

c

 SciTePress

where

(N)

k

(1)

k

w,,w K

are white Gaussian noises,

(

)

(i)

k

(i)

k

R0,~w N

,

mmm

N1

=++L

. The initial state is

modeled as a Gaussian random vector,

(

)

000

P,x~x N

.

The system and observation noises

k

v

and

,N1,...,i,w

(i)

k

=

and the initial state

0

x

are mutually

uncorrelated.

Prediction (or fixed-lead prediction) is the

estimation of the state at future time

0s,sk ≥+

beyond the observation interval, that is, based on

data up to an earlier time

k

,

(

)

{}

.k0,...,i,YY,YxEx

ˆ

ik][0,k][0,sk

ksk

===

+

(4)

The Kalman predictor (KP). The optimal predictor

KP

kskksk

x

ˆ

x

ˆ

++

≡

and its error covariance

KP

kskksk

PP

++

≡

are

given by the Kalman predictor equations:

,1,2,...s,GQGQ

~

,Q

~

FPFP

,x

ˆ

Fx

ˆ

T

1sk1sk1sk1-sk

1sk

T

1sk

KP

k1-sk

1sk

KP

ksk

KP

k1-sk

1sk

KP

ksk

==

+=

=

−+−+−++

−+−+

+

−+

+

−+

+

(5)

with initial conditions

KP

kk

KP

kk

P,x

ˆ

determined by the

standard Kalman filter (KF) equations(Bar-Shalom

et al. 2001, Lewis 1981). Note that the optimal

predictor

KP

ksk

x

ˆ

+

represents the centralized predictor,

which processing the

overall observations

k][0,

Y

simultaneously.

Many advanced systems now make use of a large

number of sensors in practical applications ranging

from aerospace and defense, robotics automation

systems, to the monitoring and control of process

generation plants. Recent developments in integrated

sensor network systems have further motivated the

search for decentralized signal processing algorithms.

An important practical problem in the above systems

is to find a fusion estimate to combine the

information from various local estimates to produce

a global (fusion) estimate.

In next Section, we propose two new fusion

predictors for multisensor discrete-time dynamic

systems (1), (3).

3 TWO FUSION PREDICTORS

The derivation of the fusion predictors is based on

the assumption that the

overall observation vector

k

Y

combines the local (individual) sensors

(N)

k

(1)

k

y,,y K

, which can be processed separately.

According to (1) and (3), we have

N

unconnected

dynamic subsystems (

N,1,i K

=

) with the state

k

x

and local sensor

(i)

k

y

:

,wxHy

,vGxFx

(i)

kk

(i)

k

(i)

k

kkkk1k

+=

+

=

+

(6)

where

i

is the fixed-number of subsystem. Then the

optimal mean-square local filtering

(

)

(i)

k][0,k

(i)

kk

yxEx

ˆ

=

and prediction

(

)

(i)

k][0,sk

(i)

ksk

yxEx

ˆ

+

=

estimates are

determined by the recursive Kalman filtering

equations,

{

}

k0,...,j,yy

(i)

j

(i)

k][0,

==

(Bar-Shalom et al.

2001). We have

,Q

~

FPFP

,,...2,1s,x

ˆ

Fx

ˆ

1sk

T

1sk

(ii)

k1-sk

1sk

(ii)

ksk

(i)

k1-sk

1sk

(i)

ksk

−+−+

+

−+

+

−+

+

+=

==

(7)

[]

,PHLIP

,RHPHHPL

,PP,Q

~

FPFP

,x

ˆ

HyLx

ˆ

x

ˆ

,xx

ˆ

,x

ˆ

Fx

ˆ

(ii)

k1k

(i)

1k

(i)

1kn

(ii)

1k1k

1

(i)

1k

(i)

1k

(ii)

k1k

(i)

1k

(i)

1k

(ii)

k1k

(i)

1k

0

(ii)

00

k

T

k

(ii)

kk

k

(ii)

k1k

(i)

k1k

(i)

k

(i)

k

(i)

1k

(i)

k1k

(i)

1k1k

0

00kk

k

k1k

TT

+

++

−

++

+

++

+

+++

+

−=

+=

=+=

−+=

=

(8)

where

n

I

is an

nn

×

identity matrix, and

(ii)

kk

P

and

(ii)

ksk

P

+

are the filtering and prediction local error

covariances, respectively, i.e.,

(

)

..0,1,2,..j,x

ˆ

xx

~

,x

~

,x

~

covP

(i)

kjk

jk

(i)

kjk

(i)

kjk

(i)

kjk

(ii)

kjk

=−=

=

+

+++

(9)

Thus we have

N

local Kalman estimates

,x

ˆ

,x

ˆ

(i)

ksk

(i)

kk +

and the corresponding error

covariances

(i)

ksk

(i)

kk

P,P

+

for

N.1,...,i =

Using these

local estimates and covariances we propose two

fusion prediction algorithms.

3.1 The Fusion of Local Predictors

(FLP Algorithm)

The fusion predictor

FLP

ksk

x

ˆ

+

of the state

sk

x

+

based on

the overall sensors (2) is constructed from the local

predictors

(i)

ksk

x

ˆ

+

by using FF (Shin et al. 2004, 2006

and Zhou et al. 2006):

,Ia,x

ˆ

ax

ˆ

n

N

1i

(i)

sk,

N

1i

(i)

ksk

(i)

sk,

FLP

ksk

==

∑∑

==

+

(10)

where

(N)

sk,

(1)

sk,

a,,a K

are

nn

×

time-varying matrix

weights determined from the mean-square criterion,

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

120

.minx

ˆ

a-xEJ

(i)

sk,

a

2

N

1i

(i)

ksk

(i)

sk,sk

FLP

ksk

→

⎟

⎠

⎞

⎜

⎝

⎛

=

∑

=

+

(11)

The following Theorem and Corollary completely

define the fusion predictor

FLP

ksk

x

ˆ

+

and its error

covariance,

(

)

FLP

ksk

FLP

ksk

FLP

ksk

x

~

,x

~

covP

+++

=

.

Theorem: Let

(N)

ksk

(1)

ksk

x

ˆ

,,x

ˆ

++

K

are the local Kalman

predictors of an unknown state

sk

x

+

. Then

(a) The weights

(N)

sk,

(1)

sk,

a,,a K

satisfy the linear

algebraic equations

[]

,Ia,0PPa

n

N

1i

(i)

sk,

N

1i

(iN)

ksk

(ij)

ksk

(i)

sk,

==−

∑∑

==

++

(12)

and they can be explicitly written out in the

following form

()

;N1,...,i,PPa

1

N

1h,

1)(

h)(

ksk

N

1j

)1(

(ij)

ksk

(i)

sk,

=

⎟

⎠

⎞

⎜

⎝

⎛

=

−

=

−

+

=

−

+

∑∑

l

(13)

(b) The local cross-covariances

(

)

jiN;1,...,ji,,x

~

,x

~

covP

(j)

ksk

(i)

ksk

(ij)

ksk

≠==

+++

(14)

satisfy the following recursions:

,.1,2,..s,Q

~

FPFP

1sk

T

1sk

(ij)

k1sk

1sk

(ij)

ksk

=+=

−+−+

−+

+

(15)

()

(

)

()

0

(ij)

00

T

(j)

k

(j)

kn

1k

T

1k

(ij)

1k1k

1k

(i)

k

(i)

kn

(ij)

kk

PP,HLI

Q

~

FPFHLIP

=−×

+⋅−=

−−

−

(16)

with the gains

(i)

k

L

determined by (8).

(c) The fusion error covariance

FLP

ksk

P

+

is given by

.aPaP

N

1ji,

(j)

sk,

(ij)

ksk

(i)

sk,

FLP

ksk

T

∑

=

++

=

(17)

Corollary: If

(N)

ksk

(1)

ksk

x

ˆ

,,x

ˆ

++

K

are unbiased local

Kalman estimates then the fusion predictor

FLP

ksk

x

ˆ

+

in

(10) is unbiased

.

The proofs of Theorem and Corollary are given in

Appendix.

Thus the local Kalman filtering estimates (8), and

the recursive fusion equations (10)-(17) completely

define FLP algorithm.

In particular case at

2N

=

, FF (10)-(13) reduces

to the Bar-Shalom and Campo formula:

()

.PPPPD

,DPPa

,x

ˆ

ax

ˆ

ax

ˆ

(21)

ksk

(12)

ksk

(22)

ksk

(11)

ksk

sk,

1

sk,

(12)

ksk

(11)

ksk

(2)

sk,

1

sk,

(21)

ksk

(22)

ksk

(1)

sk,

(2)

ksk

(2)

sk,

(1)

ksk

(1)

sk,

FLP

ksk

++++

−

++

−

++

+

−−+=

−=

+=

(18)

Further, in parallel with FLP we offer the other

algorithm for fusion prediction.

3.2 The Prediction of Fusion Filter

(PFF Algorithm)

This algorithm consists of two parts. The first part

fuses the local Kalman filtering estimates

(N)

kk

(1)

kk

x

ˆ

,...,x

ˆ

using FF. We have

,Ib,x

ˆ

bx

ˆ

n

N

1i

(i)

k

N

1i

(i)

kk

(i)

k

FF

kk

==

∑∑

==

(19)

where the weights

(N)

k

(1)

k

b,,b K

satisfy the linear

algebraic equations (Shin et al. 2006)

[

]

,Ib,0PPb

n

N

1i

(i)

k

N

1i

(iN)

kk

(ij)

kk

(i)

k

==−

∑∑

==

(20)

or explicitly expressed from the formula (Zhou et al.

2006):

()

;N1,...,i,PPb

1

N

1h,

1)(

h)(

kk

N

1j

)1(

(ij)

kk

(i)

k

=

⎟

⎠

⎞

⎜

⎝

⎛

=

−

=

−

=

−

∑∑

l

(21)

where the local cross-covariances

(ij)

kk

P

are

determined by (8) and (16).

In the second part we predict the fusion filtering

estimate

FF

kk

x

ˆ

using one-step prediction:

.x

ˆ

x

ˆ

,FA,x

ˆ

A

x

ˆ

FFx

ˆ

Fx

ˆ

FF

kk

PFF

kk

1s

0j

j1sksk,

FF

kk

sk,

PFF

k2sk

2sk1sk

PFF

k1sk

1sk

PFF

ksk

≡==

===

∏

−

=

−−+

−+

−+−+

−+

+

L

(22)

Remark 1 (Estimation accuracy): Experimentally,

FLP and PFF have very close accuracy, as in Section

4. Unfortunately, now we do not have a rigorous

proof or disproof of this result.

Remark 2 (Computational complexity): In general,

the both results, namely, linear equations (12), (20)

and expressions (13),(21) are equivalent, being the

implicit and explicit forms of the solution,

respectively. However, from the computational point

of view, when the number of sensors

N

is large or

the local cross-covariance matrices

(ij)

ksk

P

+

are ill-

conditioned, the linear equations may be more

preferable than the explicit expressions.

To predict the state

sk

x

+

using the FLP we need

to compute the matrix weights

(N)

jk,

(1)

jk,

a...,,a

for each

FUSION PREDICTORS FOR DISCRETE-TIME LINEAR SYSTEMS WITH MULTISENSOR ENVIRONMENT

121

lead

s1,2,...,j =

in contrast to the PFF, wherein the

weights

(N)

k

(1)

k

b...,,b

are computed only once, since

they do not depend on the leads

1.s ≥

Therefore the

FLP is more complex than the PFF, especially for

the large leads

1.s >>

Remark 3 (Real-time implementation): We may note,

that the local filter gains

(i)

k

L

, the error cross-

covariances

(ij)

ksk

(ij)

kk

P,P

+

, and the weights

(i)

k

(i)

sk,

b,a

may

be pre-computed, since they do not depend on the

current observations

.Y

k

But only on the noises

statistics

k

Q

and

(i)

k

R

, and the system matrices

(i)

kkk

H,G,F

, which are part of the system model (1),

(3). Thus, once the observation schedule has been

settled, the real-time implementation of the fusion

predictors FLP and PFF requires only the

computation of the local Kalman estimates

(N)

kk

(1)

kk

x

ˆ

,,x

ˆ

K

and final fusion predictors

FLP

ksk

x

ˆ

+

and

PFF

ksk

x

ˆ

+

.

Remark 4 (Parallel implementation): The local

Kalman estimates

(N)

kk

(1)

kk

x

ˆ

,,x

ˆ

K

are separated for

different sensors. Therefore, they can be

implemented in parallel for various types of

observations

.N1,...,i,y

(i)

k

=

4 EXAMPLES

4.1 Prediction for a Scalar

Multi-sensor System

Consider a scalar system described by

,k,0,1,k,vaxx

Tkk1k

K=+=

+

(23)

,N,1,2,i,wxy

(i)

kk

(i)

k

K=+=

(24)

where

() ()

(

)

.σ,x~x,r0,~w,q0,~v

2

000i

(i)

kk

NNN

This represents the model which takes N sensor

modes. The parameters are subject to

0.9,a

=

,0.2q =

.4N1,σ,0.5x,20k

2

00T

====

The

optimal Kalman predictor, and two fusion predictors

FLP and PFF were used to estimate

.1s,x

sk

≥

+

The

noise statistics were taken as follows:

.5.0r,5.1r.8,1r2.0,r

4321

====

Table 1: Comparison of MSEs at N=3,4.

N=3 Type of Fusion Predictors

k KP FLP PFF

0 1.00000 1.00000 1.00000

1 1.04623 1.04602 1.04623

2 0.96947 0.98315 0.98314

3 0.95727 0.96658 0.96657

4 0.95417 0.96131 0.96131

5 0.95330 0.95962 0.95962

M

10 0.95295 0.95918 0.95918

N=4 Type of Fusion Predictors

k KP FLP PFF

0 1.00000 1.00000 1.00000

1 1.04623 1.04602 1.04623

2 0.95045 0.96037 0.96050

3 0.94361 0.94967 0.94966

4 0.94257 0.94753 0.94753

5 0.94239 0.94718 0.94718

M

10 0.94235 0.94735 0.94735

Table 1 illustrates the mean-square errors (MSEs)

KP

ksk

P

+

,

FLP

ksk

P

+

, and

PFF

ksk

P

+

at the lead

10s =

and

.4,3N

=

Note that the MSEs

FLP

ksk

P

+

and

PFF

ksk

P

+

are very

close and reduced from

3N =

to

.4N =

Moreover,

the differences between the optimal KP and fusion

predictors are negligible, especially for steady-state

regime at

10.k ≥

The numerical simulations were

performed using a computer with the following

specification: Intel® Pentium® 4 CPU 3.0GHz 1G

RAM. The CPU time for evaluation of the estimate

PFF

ksk

x

ˆ

+

is 4.9 times less than for

FLP

ksk

x

ˆ

+

. This is due to

the fact that the PFF’s weights

(i)

k

b

do not depend

on leads

10,...,1s

=

in contrast to the FLP’s weights

(i)

sk,

a

.

4.2 The Damper Harmonic Oscillator

Motion

System model of the harmonic oscillator is

considered in Lewis (1986):

,4t0,v

1

0

x

2

10

x

tt

2

n

t

≤≤

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎦

⎤

⎢

⎣

⎡

−−

=

αω

&

(25)

where

[

]

T

t2,t1,t

xxx =

, and

t1,

x

is position, and

t2,

x

is velocity,

t

v

is zero-mean white Gaussian noise

with intensity q,

(

)()

stqδvvE

st

−

=

,

()

.P,x~x

000

N

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

122

Assume that the observation system contains two

sensors which are observing the position

t1,

x

. Then

we have

,

w

x

01

y

(2)

t

(1)

t

(2)

t

(1)

t

⎥

⎦

⎤

⎢

⎣

⎡

+

⎥

⎦

⎤

⎢

⎣

⎡

=

⎥

⎦

⎤

⎢

⎣

⎡

=

(26)

where

)1(

t

w

and

)2(

t

w

are uncorrelated white Gaussian

noises with zero-mean and intensities

1

r

and

2

r

,

respectively.

After discretization of the system and observation

models (25), (26) three predictors were applied: KP,

FLP and PFF. The performance of the fusion

predictors was expressed in the terms of

computation load (CPU time) and loss in estimation

accuracy (MSE) with respect to the optimal KP. The

model parameters, noises statistics, initial conditions,

and lead were taken to

[] []

.10s,1.02.0diagP,0.00.0x

,1r,2r,1q,16.0,64.0

0

T

0

21

2

n

===

=====

αω

Figure 1: KP, FLP and PFF MSEs for position.

Figure 2: KP, FLP and PFF MSEs for velocity.

In Figs.1 and 2 we show the MSEs for position

(

)

1

x

,

,PP

KP

ksk1,

KP

1

+

= ,PP

FLP

ksk1,

FLP

1

+

= ,PP

PFF

ksk1,

PFF

1

+

=

and

analogously for velocity

()

2

x

PFF

2

FLP

2

KP

2

P,P,P

,

respectively.

The analysis of results in Figs. 1 and 2 shows that

the fusion predictors FLP and PFF have very close

accuracy, i.e.,

.2,1i,PP

PFF

i

FLP

i

=≈

Moreover, the

differences between both fusion MSEs

PFF

i

FLP

i

P,P

,

and optimal one

KP

i

P

are negligible, especially for

steady-state regime. The CPU times for KP, FLP,

and FPP are equaled to

,(s)016.0T,(s)094.0T,0.015(s)T

PFFFLPKP

===

respectively. Thus these combined effects provide

the best balance between the computational

efficiency and desired prediction

accuracy for the

PFF.

5 CONCLUSION

In this paper, we present two fusion predictors (FLP

and PFF) for discrete-time linear systems with

multisensor environment. Both of these predictors

represent the optimal linear combination of an

arbitrary number of local Kalman filters or

predictors. Each local filter (predictor) is fused by

the MMSE criterion. Experimentally the FLP and

PFF algorithms have very close accuracy. In view

of the computational complexity, however, the PFF

more efficient than the FLP. The examples

demonstrate the efficiency and high-accuracy of the

proposed predictors.

REFERENCES

Bar-Shalom, Y., 1990. Multitarget-Multisensor Tracking:

Advanced Applications

, Artech House, Norwood, MA.

Bar-Shalom, Y., Campo, L., 1986. The effect of the

common process noise on the two-sensor used track

covariance, IEEE Trans. Aerospace and Electronic

Systems

, Vol. 22, No. 11, pp. 803-805.

Bar-Shalom, Y., Li, X.R., Kirubarajan, T., 2001.

Estimation with Applications to Tracking and

Navigation

, John Wiley & Sons, New York.

Bar-Shalom, Y., Li, X. Rong, 1995.

Multitarget-

Multisensor Tracking: Principles and Techniques

,

YBS Publishing.

Hall, D. L., 1992. Mathematical Techniques in

Multisensor Data Fusion,

Artech House, London.

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123

Lewis, F.L., 1986. Optimal Estimation with an

Introduction to Stochastic Control Theory, John Wiley

& Sons, New York.

Li, X.R., Zhu, Y.M., Wang, J., Han, C.Z., 2004. Optimal

linear estimation fusion – Part I: Unified fusion rules,

IEEE Trans. Inf. Theory

, vol. 49, no. 9, pp. 2192-2208.

Shin, V.I., Lee, Y., Choi, T., 2004. Suboptimal Linear

Filtering and Generalized Millman’s Formula, Proc.

IASTED Inter. Conf. Signal and Image

Process,

Honolulu, Hawaii, USA, pp. 369-374.

Shin, V., Lee, Y., Choi, T.-S., 2006. Generalized

Millman's formula and its applications for estimation

problems,

Signal Processing, vol.86, No.2, pp. 257-

266.

Zhou, J., Zhu, Y., You, Z., Song, E., 2006. An efficient

algorithm for optimal linear estimation fusion in

distributed multisensor systems,

IEEE Trans. Syst.,

Man, Cybern.

, vol.36, no.5, pp. 1000-1009.

APPENDIX

Proof of Theorem and Corollary:

(a) Equations (12) and expression (13)

immediately follow as a result of application of the

general FF (Shin et al. 2006 and Zhou et al. 2006) to

the optimization problem (11).

(b) Equation for the local error takes the form

.vGx

~

Fx

ˆ

-xx

~

1sk1sk

(i)

k1-sk

1sk

(i)

ksk

sk

(i)

ksk

−+−+

+

−+

+

+==

Then equation for the cross-covariance (15)

associated with the

(i)

ksk

x

~

+

and

(j)

ksk

x

~

+

follows from the

standard propagation equation for

(

)

.x

~

x

~

EP

T

(j)

ksk

(i)

ksk

(ij)

ksk +++

=

Equation (16) was given in Shin et al. (2006).

(c) Using (10) the fusion error covariance can be

rewritten as

(

)

()

.aPax

~

ax

~

aE

x

ˆ

axx

ˆ

axE

)x

ˆ

-(x)x

ˆ

-(xEP

N

1ji,

(j)

sk,

(ij)

ksk

(i)

sk,

N

1j

T

(j)

ksk

(j)

sk,

N

1i

(i)

ksk

(i)

sk,

T

N

1j

(j)

ksk

(j)

sk,sk

N

1i

(i)

ksk

(i)

sk,sk

TFLP

ksk

sk

FLP

ksk

sk

FLP

ksk

T

∑∑∑

∑∑

=

+

=

+

=

+

=

+

=

+

=

⎭

⎬

⎫

⎩

⎨

⎧

=

⎪

⎭

⎪

⎬

⎫

⎪

⎩

⎪

⎨

⎧

⎥

⎦

⎤

⎢

⎣

⎡

−⋅

⎥

⎦

⎤

⎢

⎣

⎡

−=

⋅=

This completes

the proof of Theorem.

If the local predictors

(i)

ksk

x

ˆ

+

are unbiased, i.e.,

(

)

(

)

sk

(i)

ksk

xEx

ˆ

E

+

=

, then we have

(

)

(

)

()()

.xExEax

ˆ

Eax

ˆ

E

N

1i

sksk

N

1i

(i)

sk,

(i)

ksk

(i)

sk,

FLP

ksk

∑∑

=

++

=

++

=

⎥

⎦

⎤

⎢

⎣

⎡

==

Corollary is proved.

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