Advances in Systems Science and Applications (2013) Vol.13 No.3 233-248

Ways of Fusing Different Types of Information and
How Systemic Yoyo Model is Applied in Complex

Systems Evaluation and Estimation
Xiaojun Duan1 and Yi Lin2

1Department of Mathematics and Systems Science National University of
Defense Technology Changsha 410073, PR China

2Department of Mathematics Slippery Rock University Slippery Rock, PA 16057
USA

Abstract

Continuing the works in literature[1], we show in this paper how different types
of information can be fused together consistently in order to produce accurate
evaluations and estimations for complex systems. The theoretical part of this
presentation is based on the standard statistical reasoning, while the ending part
constructs three case studies in order to validate the main thinking logic and
results obtained in literature[1] and in this paper.

It is shown that (1) for linear systems, when fusing data of different types,
the weights placed on the data have profound effects on the outcomes and the
achieved precisions, meaning that in this case, the unique optimal weight matrix
is determined by the precisions of the data (Gauss-Markov Theorem of linear
models); (2) for nonlinear models, when fusing heterogeneous sets of data with
varied scales of precision, the structure of the weight matrix is no longer uniquely
determined by the precisionsbut also related to the degree of model nonlinearity,
indicating that the classical Gauss-Markov Theorem of linear models no longer
holds true. At the same time, a specific method of determining the optimal
weighting factor and the relevant computational method for estimating the pa-
rameters are established.

Combined with the process of conserved information applied in systems evalu-
ation, we provide three case studies, including (1) how to quantitatively measure
prior knowledge and observational data so that prior knowledge can be considered
in obtaining much improved optimal systems evaluations; (2) how to excavate new
sources of observational data of processes so that the established models can be
validated jointly using process data collected under different test environments
and the directly measured information of the specific indices of concern in or-
der to improve the quality of systems evaluation and estimation and to obtain
model validation results of better accuracy; and (3) how to more effectively fuse
prior knowledge and heterogeneous sets of data. All of these case studies further
witness the epistemological validity of the information conservation existing in
the systemic recognition process beneath the systems model description, prior
knowledge, and observational data, and their transformational relationship, as
obtained in literature[1].



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 234

Because other than establishing the theory, particular procedures are provid-
ed, conclusions of this work can be directly employed in system evaluations and
estimations and related works. This work shows how systemic thinking can be
practically applied to benefit the efforts of system evaluation and model estima-
tions involved in various engineering projects.
Keywords Systemic yoyo model, data fusion, linear/nonlinear model, parameter
estimation, Gauss-Markov Theorem

1 Introduction

In literature[1], after clarifying the relationship between observational quantities
and the target indices to be measured, the systemic yoyo model[2] is established
for system evaluations and estimations (Fig.1), where the form of the model, ob-
servational data, and prior knowledge are the main sources of information useful
for the evaluation, estimation, and prediction of the performance index of the
system to be measured.

Fig.1 The systemic yoyo model for complex system evaluation

From analyzing the characteristics of and connections between the three main
sources of information, model descriptions, prior knowledge, and observational
data, for system evaluations and estimations, the following conservation law of
information for system analysis is obtained.

AeIM ×BeID × CeIP = a (1)

where A, B, and C are constants, IM stands for the information content de-
scribed by the model, ID the information content of the autoptic test data, and
IP the information content of the prior knowledge. The constant a should some-
how depict the minimum amount of information required to satisfy the given
precision (in the estimate of the model or parameters), where the precision is



235 Advances in Systems Science and Applications (2013) Vol.13 No.3

given in terms of the model accuracy and parameter estimation precision. For
the detailed expressions of IM , ID, and IP , see [1].

With this law of conservation is established, Duan and Lin use it to investigate
the evolution direction of the process of a system evaluation and estimation.

Continuing what is obtained in literature[1], in this paper, we show that for lin-
ear systems, when different types of data are available for systems evaluation and
estimation, then the analysis outcomes and precisions achieved are greatly deter-
mined by the weights placed on the data. More specifically, the Gauss-Markov
Theorem, established on the method of least squares method, holds true. How-
ever, when nonlinear systems are involved, the classical Gauss-Markov Theorem
of linear models no longer holds. That is, the structure of the weight matrix is
not uniquely determined by the precisions of the available data. To this end, we
provide a specific method of determining the optimal weighting factor and the
relevant computational method for estimating the parameters.

After this theoretical exploration, combined with the conservation law of in-
formation of system evaluations and estimations, we construct three case studies
to show

(1)How to quantitatively measure prior knowledge and observational data so
that better evaluation and estimation results can be obtained using prior knowl-
edge;

(2)How to excavate the available observational data of processes so that the
process information collected under different test environments and directly mea-
sured data of the specific indices can be employed jointly to fine-tune the model,
leading to improved system evaluations, estimations, and more accurate test re-
sults, and

(3)How to make prior knowledge and heterogeneous sets of data work effec-
tively together. These case studies further verify the validity of the conservation
of information of the model information, prior knowledge, and observational data
in the recognition process of systems and the evolutionary relationship between
three main sources of information.

This paper is organized as follows: Section 2 looks at various ways one can
fuse information in his analysis of complex systems. Section 3 focuses on three
specific cases studies. And, the paper is concluded by Section 4.

2 Ways Information Fusion Takes Place in Processes of System Evaluation

With the requirement of precision given, we can optimize the process of a sys-
tem evaluation. That is such a problem as how to obtain the optimal evaluation
results when multiple types of models and multiple kinds of data are available.
This end can be analyzed by placing various weights on the multiple kinds of
data and prior information.



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 236

When dealing with information by combining heterogeneous data, the most
typical case is the fusion of such information that are of different types and various
precisions. When observational information is expressed by using a parametric
model, the problem of how to fuse the information together can be transformed
into that of estimating the parameters of some regression models. Here, by differ-
ent types of information, we mean such information that is of different functional
relationships with the parameters to be estimated so that their various orders of
derivatives are also different. If in our treatment we have to deal with different
types of information of varying precision, different weightings placed on these
information will have direct effect on the estimations of the parameters. Some-
times, the effects can also be quite significant. Hence, how to place weights on
the information that are of heterogeneous types and varied precisions becomes a
key technique for obtaining high accuracies in the parameter estimations.

As for the parameter estimations of linear regression models, Gauss-Markov
Theorem provides the most optimal weighting method for observational data of
different precisions [3]. As for nonlinear regression models, current publications
have assumed that all observational data have the same scale of precision. That
is, the random errors in these data are identically independently distributed [3].
To this end, our work in this section theoretically shows that when jointly deal-
ing with heterogeneous sets of data of varied scales of precision, the structure
of the weight matrix is no longer uniquely determined by the precision of the
observational data; instead, it also has something to do with the degree of non-
linearity of the model, which can be measured by different orders of derivative
functions. That is, the classical Gauss-Markov Theorem of linear models does not
hold true anymore. In the following, we will specifically address the problem of
how to determine the most optimal weighting factor, while providing the relevant
computational method for estimating the parameters.

2.1 The Optimally Weighted Information Fusion of Linear System Evaluation

According to the research on the mean square errors of parameter estimation-
s of linear systems, it is readily to show that the estimate corresponding to the
optimal weighting factor is the Bayesian estimation, which is Gauss-Markov The-
orem[1,6].

For the problem of estimating parameters βp×1, assume that there are the fol-
lowing two types of observational information:{

Ym×1 = Xm×pβp×1 + εm×1

Eε = 0;Cov(ε, ε) = σ2
1Im×m

(2)

and {
β̃k×1 = Zk×pβp×1 + ηk×1

η ∼ N(0, σ2
2Ik×k)

(3)



237 Advances in Systems Science and Applications (2013) Vol.13 No.3

satisfying EεηT = 0.
If we treat model (2) as the direct observational data, while (3) the prior

information on the parameters, then the Bayesian estimation of the parameters
is the solution of the following extremum problem:

min
β∈RP

σ−2
1 ∥ Y −Xβ ∥22 +σ−2

2 ∥ β̃ − Zβ ∥22 (4)

that is given as follows:

β̂B = (σ−2
1 XTX + σ−2

2 ZTZ)−1(σ−2
1 XTY + σ−2

2 ZT β̃) (5)

Therefore, the following conclusions can be shown readily[3-5]:
(1)Eβ̂B = β; that is the Bayesian estimate is unbiased; and
(2)MSE(β̂B) = tr(σ−2

1 XTX + σ−2
2 ZTZ)−1 < MSE(β̂LS) = σ2

1tr(X
TX)−1 that

is, by making use of appropriate prior knowledge, one can always improve the
estimation precision of the parameters, where tr(Ak×k) =

∑k
i=1 ai,i is the trace

of the matrix.
When fusing these data, the weights placed on the data have profound effects

on the outcomes and the achieved precisions. These conclusions indicate that
when fusing observations of different precisions, the unique optimal weight matrix
is determined by the precisions of the data, which in essence is still the Gauss-
Markov Theorem of linear models established on the least squares method.

2.2 The Optimally Weighted Information Fusion of Nonlinear System Evaluation

For nonlinear models, we show in theory that when fusing heterogeneous sets
of data with varied scales of precision, the structure of the weight matrix is
no longer uniquely determined by the precisions. That is, the classical Gauss-
Markov Theorem of linear models no longer holds true. At the same time, we
establish a method for determining the optimal weighting factor and the relevant
computational method for estimating the parameters.

For the sake of convenience, for the parameter θ that is to be estimated, we
assume that we have the prior information (3) for a linear model, where the
error satisfies EεηT = 0. So, the weighting problem of fusing these two types of
observational data can be reduced to the following minimization problem:

min
ρ∈R1,ρ>0

min
β∈RP

∥ Y − f(X,β) ∥22 +ρ ∥ β̃ − Zβ ∥22 (6)

Then, we have the following result:
Theorem1. Denote S(β) =∥ Y −f(β) ∥22 +ρ ∥ β̃−Zβ ∥22, S(β̂) = minβ S(β), C =∑m

i=1 ḟ
2
i +ρ

∑k
i=1 z

2
i , D =

∑m
i=1 ḟif̈i, ξ =

∑m
i=1 ḟiεi+ρ

∑k
i=1 ziηi, A = σ2

1

∑m
i=1 ḟ

2
i +



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 238

ρ2σ2
2

∑k
i=1 z

2
i , then under the assumed conditions (i) and (ii), the following esti-

mation holds true:

β̂ − β =C−1ξ + C−2
m∑
i=1

f̈iεiξ −
3

2
C−3Dξ2 + C−3

m∑
i,j=1

f̈if̈jεiεjξ

− 9

2
C−4D

m∑
i=1

f̈iεiξ
2 +

9

2
C−5D2ξ3

(7)

with the bias and mean square error approximated as follows:

E(β̂ − β) = −1
2σ

2
1C

−2D− 3
2C

−3Dρ(ρσ2
2 − σ2

1)
∑k

i=1 z
2
i

MSE(β̂) =C−2A+ 6C−4D2σ4
1 + 3C−4Aσ2

1

m∑
i=1

f̈2
i

+
135

4
C−6D2A2 − 36C−5AD2σ2

1

(8)

where the assumed conditions (i) and (ii) are given below:
(i) The derivative of f(t, β) with respect to the parameter β exists and is

continuous, and

lim
m→+∞

1

m

m∑
i=1

= (
df(ti, β)

dβ
)2 = Ω1(β) > 0 (9)

(ii) The second order derivative of f(t, β) with respect to β exists and is
continuous, and

lim
m→+∞

1

m

m∑
i=1

= (
d2f(ti, β)

dβ2
)2 = Ω2(β) (10)

Proof. Taking the series expansion of Ṡ(β̂) about the true value of the parameter
β produces

Ṡ(β̂) = Ṡ(β) + S̈(β)(β̂ − β) + 2−1...
S (β)(β̂ − β)2

Notice that Ṡ(β̂) = 0, Ṡ(β) = −2(
∑m

i=1 ḟiεi + ρ
∑m

i=1 ziηi), S̈(β) = 2C −
2
∑m

i=1 f̈iεi,
...
S (β) = 6D − 2

∑m
i=1

...
f iεi, So, ignoring all terms of third or higher

order derivatives produces

β̂ − β = C−1
{
ξ +

m∑
i=1

f̈iεi(β̂ − β)− 3

2
D(β̂ − β)2 +

1

2

m∑
i=1

...
f iεi(β̂ − β)2

}
= C−1ξ + C−2

m∑
i=1

f̈iεiξ −
3

2
C−3Dξ2 + C−3

m∑
i,j=1

f̈if̈jεiεjξ

− 9

2
C−4D

m∑
i=1

f̈iεiξ
2 +

9

2
C−5D2ξ3



239 Advances in Systems Science and Applications (2013) Vol.13 No.3

That is equ.(7) holds true.
Because the expected values of normal variables to odd powers are zero and ε

and η are independent, we obtain the first equation in equ. (8) by ignoring the
error terms of the fourth and higher orders and then calculating the expected
value such that

E(β̂−β)2 = C−2Eξ2+3C−4E(
m∑
i=1

f̈iεi)
2ξ2+

45

4
C−6D2Eξ4−12C−5D

m∑
i=1

f̈iεiξ
3 (11)

Because

Eξ2 = σ2
1

m∑
i=1

ḟ2i + ρ2σ2
2

k∑
i=1

z2i =̂A,

Eξ4 = 3σ4
1(

m∑
i=1

ḟ2i )
2 + 6ρ2σ2

1σ
2
2

m∑
i=1

ḟ2i

k∑
i=1

z2i + 3ρ4σ4
2(

m∑
i=1

z2i )
2 = 3A2,

E(

m∑
i=1

f̈iεi)
2ξ2 = 2D2σ4

1 + σ4
1

m∑
i=1

ḟ2
i

m∑
i=1

f̈2
i + ρ2σ2

1σ
2
2

m∑
i=1

f̈2
i

k∑
i=1

z2i

= 2D2σ4
1 + σ2

1A

m∑
i=1

f̈2
i ,

E

m∑
i=1

f̈iεiξ
3 = 3Dσ4

1

m∑
i=1

ḟ2i + 3ρ2σ2
1σ

2
2D

k∑
i=1

z2i = 3DAσ2
1,

substituting these equations into equ.(11) leads to

E(β̂−β)2 = C−2A+6C−4D2σ4
1 +3C−4Aσ2

1

m∑
i=1

f̈2i +
135

4
C−6D2A2−36C−5AD2σ2

1

That is the second equation in equ.(8). QED.
Theorem2. For MSE(β̂)(ρ) in Theorem 1, the solution to the following mini-
mization problem

min
ρ

MSE(β̂)(ρ) (12)

exists, satisfying min
ρ

MSE(β̂)(ρ) < minρMSE(β̂)(
σ2
1

σ2
2
).

Proof. Because lim
ρ→+∞

MSE(β̂)(ρ) = σ2
2(
∑k

i=1 z
2
i )

−1, MSE(β̂)(ρ) is infinitely dif-



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 240

ferentiable on [0,+∞), MSE(β̂)(ρ) has its minimum value on [0,+∞). Because

d

dρ
MSE(β̂)(ρ) = (

k∑
i=1

z2i )(−2C−3A+ 2ρC−2σ2
2 − 24C−5D2σ4

1

− 12C−5σ2
1

m∑
i=1

f̈2
i + 6ρC−4σ2

1σ
2
2

m∑
i=1

f̈2
i − 405

2
C−7D2A2

+ 135ρC−6D2Aσ2
2 + 180C−6AD2σ2

1 − 72ρC−5D2σ2
1σ

2
2)

(13)

we have

d

dρ
MSE(β̂)(0) = −(

k∑
i=1

z2i )(2(
m∑
i=1

ḟ2
i )

−2σ2
1 +

93

2
(

m∑
i=1

ḟ2
i )

−5D2σ4
1+

12(

m∑
i=1

ḟ2
i )

−5
m∑
i=1

ḟ2
i

m∑
i=1

f̈2
i σ

4
1) < 0

Also, because lim
ρ→+∞

d
dρMSE(β̂)(ρ) = 0, and when ρ → +∞, each term starting

from the third in equ.(13) is a higher order infinitesimal when compared to the
previous two terms; and when ρ > σ2

1σ
−2
2 , the sum of the previous two terms is

greater than zero there is ρ0 > 0 so that d
dρMSE(β̂)(ρ) > 0 when ρ ∈ [ρ0,+∞).

Therefore, the solution of min
ρ

MSE(β̂)(ρ) satisfies ρ̂ ∈ (0, ρ0).

And because

d

dρ
MSE(β̂)(

σ2
1

σ2
2

) =
σ4
1

2C5

k∑
i=1

z2i (33D
2 − 12(

m∑
i=1

ḟ2i +
σ2
1

σ2
2

k∑
i=1

z2i )
m∑
i=1

f̈2i ) ̸= 0,

ρ = σ2
1σ

−2
2 is not a solution of min

ρ
MSE(β̂)(ρ). Hence, Theorem 2 is proven.

QED.
Remarks:
(1) Theorem 2 indicates that for nonlinear models, because their least squares

estimates are generally biased, when fusing observational data of varied scales of
precision, the weight matrix obtained by using Gauss-Markov Theorem, which is
derived out of the least squares estimation for linear models, is no longer opti-
mal. The optimal weights can be obtained by solving the minimization problem
in equ.(12).

(2) If the prior knowledge equ. (3) is a nonlinear model and the first and
second order derivatives of the model are the same as those of nonlinear function
f(X,β), then the optimal weight can be approximated by ρ = σ2

1σ
−2
2 .



241 Advances in Systems Science and Applications (2013) Vol.13 No.3

2.3 The Parameter Estimation of Heterogeneous Data Fusion

When solving problem (6), one can simply follow the following iterative method:
Step 1: For an initial weight value ρ0 = σ2

1σ
−2
2 , solve the following minimization

problem

min
β∈R1

∥ Y − f(X,β) ∥2 +ρ0 ∥ β̃ − Zβ ∥2 (14)

to obtain its solution β̂(1);
Step 2: Calculate the mean square error MSE(β̂(1))(β̂(1), β) of the estimated

parameter at β̂(1);
Step 3: Solve the minimization problem min

ρ>0
MSE(β̂(1))(β̂(1), β) to obtain ρ1;

and
Step 4: Repeat Steps 1-4 with the initial value ρ0 replaced by ρ1 until the

estimated value of the parameter becomes stable.

Fig.2 The three samples and their relationships to the information flow of the
yoyo model

3 Case Studies

In this section, we will use case studies to illustrate the systemic yoyo model and
the evolutionary process naturally existing in system evaluations and estimations.
Fig.2 shows the connection of these examples and their individual relationships
with the information flow of the systemic yoyo model. We will mainly consider
the scenario of supplementing prior information. Because of the shortage of ob-
servational data, the convergence of the system evaluation and estimation process
is very slow or becomes stagnant after converging to a certain degree. However,
after additional prior information becomes available, the stagnated process will
continue to converge; and the speed of the resumed convergence is dependent on
the quality of the newly supplied prior information and how consistent it is with
the observational data.

We will look at three examples to respectively illustrate the following: (1)
how to quantitatively measure both the prior and data information; (2) how to



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 242

excavate a new source of observational process data so that the quality of the
ultimate system evaluation and estimation is improved; and (3) how to fuse prior
information with heterogeneous sets of data effectively.

Fig.3 The relationship between the accuracy of post fusion parameter estimation
and sample size
(The left represents posterior fusion estimation accuracy (measured by informa-
tion); the right the posterior fusion accuracy improvement (measured by gain in
information))

Table 1 The relation between the Fisher information gain and the increase in
sample size

Increase of sample size 1 2 3 4 5 6 7

Informa-
(1)Without prior 29.2893 12.9757 7.7350 5.2786 3.8965 3.0284 2.4411
(2)Fuse prior,w = 0.4 9.3127 6.0794 4.3675 3.3328 2.6511 2.1741 1.8249

tion gain (3)Fuse prior,w = 1 3.8965 3.0284 2.4411 2.0220 1.7106 1.4716 1.2836
Increase of sample size 8 9 10 11 12 13 14

Informa-
(1)Without prior 2.0220 1.7106 1.4716 1.2836 1.1325 1.0089 0.9062
(2)Fuse prior,w = 0.4 1.5601 1.3537 1.1892 1.0555 0.9451 0.8527 0.7744

tion gain (3)Fuse prior,w = 1 1.1325 1.0089 0.9062 0.8199 0.7464 0.6833 0.6287
Increase of sample size 15 16 17 18 19

Informa-
(1)Without prior 0.8199 0.7464 0.6833 0.6287 0.5809
(2)Fuse prior,w = 0.4 0.7075 0.6496 0.5992 0.5551 0.5161

tion gain (3)Fuse prior,w = 1 0.5809 0.5389 0.5017 0.4686 0.4390

Example1. Let us look at the information measurement of the prior and obser-
vational data.
Take the prior parameter variance to be sigma0 = 50 and the sample variance
sigma1 = 100. Let us vary the sample size from 1 to 20 and consider three s-
cenarios: no prior information is fused, and prior information is fused with the



243 Advances in Systems Science and Applications (2013) Vol.13 No.3

consistency weights w = 1 and w = 0.4, respectively. Figure 3 shows the rela-
tionship between the accuracy of the post-fusion parameter estimation and the
sample size.

The Fisher information gain is shown in Table 1. Evidently, when the thresh-
old of fusion accuracy is fixed at 1.5, one needs to repeat his test ten times if he
does not fuse any additional prior information; if he fuses additional prior infor-
mation and sets its weight at w = 0.4 (that means the consistency between the
additional prior information and the test data is measured by weight 0.4), then
he needs to repeat the test 9 times; and if he fuses additional prior information
with weight set at w = 1 (that means the additional prior information has very
good consistency with the test data), then he only needs to repeat the test 6
times.

This result indicates that with correct fusion of prior information, the same
requirement of precision can be met with a fewer number of times the test is
repeated.
Example2. In this case study, we will see how we can speed up the convergence
of our recognition of the underlying system by making use of process information
and data.

The precision evaluation of active homing radar [6] is a complex recognition
process of systems. The impact error of clustered warhead missiles with active
homing radar is mainly composed of the measurement error of the Radar naviga-
tion system, the instrumental error of the inertial navigation system (INS), the
method error of the terminal guidance, the error of the distribution, and some
random error. Let us take the impact error of an active homing radar system
[6] as our system evaluation performance index and compare the outcomes of
the following two methods: one is to fuse the different observational information,
directly observed index data, and some indirectly observed data; and the other
the point estimate method [7] for the impact error. The main difference here
lies in that the later, more traditional method uses only the final impact point
error information so that the evaluation outcomes are strongly influenced by the
random error of the impact points, while the former method validates the pro-
cedure error model with systematic error and the characteristic of the random
error by making using of all the observational information. That is how the for-
mer method provides more robust evaluation results.

In order to avoid analyzing a heterogeneous population created by different
testing states, we will transform these different testing states to standard full
range process testing states so that the consequent analysis will become manage-
able. The impact errors of different testing states are denoted as follows: △LRadar

stands for the assessed impact deviation caused by the error of the radar mea-
surement in the standard overall operational test; ∆L̃Radar the measured impact



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 244

deviation caused by the error of Radar measurement in a substitute test.
For a missile with active homing radar, its warhead-target relative positional

x-, y-, and z-errors are mainly caused by its radar’s measurement error. What
an actual radar measures includes the range R, azimuth angle A, and elevation
angle E. These measurements satisfy the following transformational connection
with the measured warhead-target relative x-, y-, and z-positions:

x = R cosE cosA

y = R sinE

z = R cosE sinA

(15)

That is, the performance index (that is to be evaluated), the relative positional
error of warhead and target, is a function of the observational quantities (range
R, azimuth angle A, and elevation angle E). For details, see Fig.4.

Fig.4 The relationship between radar measurements R, A, and E, and the
warhead-target relative x-, y-, and z-positions

According to the Gaussian Law of error propagation, combined with equ. (15),
we can obtain the transfer relation from the x-, y-, and z-errors to the R-, A-,
and E-errors below: △x

△y

△z

 =

 ∂x
∂R

∂x
∂A

∂x
∂E

∂y
∂R

∂y
∂A

∂y
∂E

∂z
∂R

∂z
∂A

∂z
∂E

 △R

△A

△E


=

 cosE cosA −R cosE sinA −R sinE cosA
sinE 0 R cosE

cosE sinA R cosE cosA −R sinE sinA

 △R

△A

△E


(16)

where △x, △y, and △z stand for the warhead-target relative positional errors of
the standard testing state, and △R, △A, and △E the errors in the radar mea-
surements R, A, and E of the whole testing state.



245 Advances in Systems Science and Applications (2013) Vol.13 No.3

Firstly, we use the observational data of actual tests to obtain the errors △̃R,
△̃A, and △̃E of Radar measurements. Secondly, we analyze the iterative pro-
cess of the errors in the measured range and angles. In the following, we use a
Monopulse Radar system as our example to specifically analyze the influencing
factors on the errors in radar measured range and angles. Based on the analysis
on the sources of errors in radar measurements R, A, and E [6], and the law of
error synthesis, we can obtain the main errors in Radar measured ranges and
angles as follows:

U2
Angle = △2

Rader +△2
Target +△2

Enviroment + ...

= △2
ThermaNoise +△2

PhaseUnbalance +△2
AngularGlint+

△2
DynamicLag +△2

ClutterInterference + ...

U2
Distance = △2

Rader +△2
Target +△2

Enviroment + ...

= △2
ThermaNoise +△2

AngularGlint +△2
DynamicLag+

△2
ClutterInterference + ...

If we look at Radar measured ranges, we see that the systematic error is mainly
the dynamic lag error. The time-dependent random error mainly includes the
error of clutter interference, thermal noise, and distance glint.

Now, let us consider the methods of computation for the impact errors of active
homing Radar systems, as mentioned earlier, under two different testing states:
One is to fuse different observational information, including the direct observa-
tional index data and indirect observational data(the concrete fusion model refers
to subsection 2.2 and 2.3), and the other the point estimate method of impact
errors [7]. Our simulated Radar measurements are the range R, azimuth angle
A, and elevation angle E. Other than analyzing the single point measurements at
the impact moments [7], we also provide a method on how to combine observed
process quantities.

We respectively model the errors in Radar measured range and angle signals,
and the speed and acceleration of the actual measured ranges and angles for the
two testing states: the simulated substitute tests and the whole process tests.
Assume that the random error term includes the independent errors of clutter
interference, thermal noise, and distance glint. For the systematic deviation, we
mainly consider the error caused by dynamic lag.

By comparing the point estimate method of impact points and that of combin-
ing process information that is used to calculate the impact point error caused
by radar errors, the outcomes are listed in Table 2. The point estimate method
does not employ the process data of the active homing radar. Instead, it only
applies the observational data of the last moments to calculate the impact point
deviation caused by Radar errors. When the sample size is small, the outcome



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 246

Table 2 The impact error estimate comparisons between the two methods con-
sidered (significance level α= 0.01)

Unit (meter)
Cross impact Cross impact Longitudinal Longitudinal
error point error confidence impact error impact error
estimation interval point estimation confidence interval

Real impact error
-12.93 [-15.72,-10.13] 18.78 [18.09,19.47]for standard

overall test
Substitute test

-13.96 [-49.43,21.51] 20.23 [-29.56,70.02]point conversion
(traditional method)

Substitute test

-12.56 [-15.33,-9.80] 18.33 [17.61,19.05]
conversion method

fusing with
indirect observational

information

of this method is greatly affected by random factors. On the other hand, the
method developed in this research righteously employs the physical background
information and observed process data so that interval estimates can be directly
produced from the estimation of the parameters. Comparing to the point estima-
te method, our method reduces the effect of random errors and makes the system
evaluation and estimation process converge more quickly.
Example3. Let us now look at how to place optimal weights for nonlinear mod-
els.

Assume

f(t, β) = 1 + (5 + tβ)0.1,

y(t) = f(t, β + ε(t),

ε(t)
iid∼N(0, 0.012),

β̃ = β + η, η ∼ N(0, 0.052)

Let t = 0.01 ∗ j, j = 1, ..., 100, and the true value of β is 8. Let us generate 50
observational data {yi(t)}1001 , β̃i, i=1,...,50. Then, when ρ = 0.012/0.052, let us
solve the minimization problem (6) 50 times, producing the mean square error
0.0394. And when ρ = 2.33∗0.012/0.052, the mean square error reaches the min-
imum value 0.0372. However, in theory, the mean square error is 0.0361. This
end indicates that for a nonlinear system, the optimal choice of weights for fus-
ing information not only depends on observational precisions but also the model
curvature involved.

All the three examples above verify the transformational relationship of sys-
tem model information, prior knowledge, and observational data in the recog-
nition process of the underlying system, where the model information can be



247 Advances in Systems Science and Applications (2013) Vol.13 No.3

strengthened through the usage of process data, and when the observational da-
ta is insufficient, the shortage in information can be made up by supplementing
additional prior knowledge.

4 Summary

Continuing literature[1], in this paper we studied how to fuse heterogeneous sets
of data together consistently so that better results can be obtained for system
evaluations and estimations. It is shown that if the types of data considered are
few and the available observational data is insufficient, one can consider obtain-
ing process information and additional prior knowledge. It is because the limited
amount of observational data could make the process of system evaluation and
estimation converge extremely slowly or stop converging completely after reach-
ing a certain degree. With process information or additional prior knowledge
added, the convergence will continue. The speed of the resumed convergence is
dependent on how the process information is applied, how good quality the prior
knowledge is and how consistent the newly adopted prior information is with
the available observational data. This end has been well illustrated by the case
studies considered in Section 3. When there is only a small sample available for
a specific system evaluation and estimation, this work provides the theoretical
guideline for how to excavate other sources of information and how newly adopt-
ed information should be fused with what is available.

Acknowledgements
This work is supported by the Natural Science Foundation of China (60974124),
the Program for New Century Excellent Talents in University and the Project-
sponsored by SRF for ROCS, SEM in China, the key lab open foundation for
Space Flight Dynamics technique (SFDLXZ-2010-004).

References

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cation in evaluation and estimation of complex systems”, Kybernetes: The
International Journal of Cybernetics, Systems, and Management Science,
Vol.40 No.1/2, pp.262-274

[2] Y. Lin. (2008), Systemic yoyos: Some Impacts of the Second Dimension,
Taylor and Francis, New York.

[3] S. S. Mao. (1999), Bayesian Statistics. Beijing: China Statistics Publishing
House.



Xiaojun Duan, Yi Lin:Ways of Fusing Different Types of Information and How Systemic... 248

[4] D. M. Bates and D. G. watts. (1997), Nonlinear Regressive Analysis and Its
Application, Translator: Bocheng Wei. Beijing: China Statistics Publishing
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[5] X. P. Zhang, J. H. Zhang, and H. W. Xie (2003), “A few discussion of sam-
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[6] D. C. Wang, J. H. Ding, W. D. Chen (2006), Radar Measurement Technique
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[7] G. Wang, X. J. Duan, Z. M. Wang (2009), “Conversion method of impact
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Corresponding author
Xiaojun Duan can be contracted at: xjduan@nudt.edu.cn