International Journal of Modern Nonlinear Theory and Application, 2012, 1, 51-54
http://dx.doi.org/10.4236/ijmnta.2012.12007 Published Online June 2012 (http://www.SciRP.org/journal/ijmnta) 51
The Non-Equidistant Grey GRM (1, 1) Model and
Its Application
Ruibiao Zou1, Zouxin Mou2, Wei Yi2
1College of Sciences, Hunan Agriculture University, Changsha, China
2Sany Heavy Industry Co. Ltd., Changde, China
Email: rbzou@163.com, yinanchang0@126.com
Received April 5, 2012; revised May 4, 2012; accepted May 15, 2012
ABSTRACT
Applying the modeling method of Grey system and accumulated generating operation of reciprocal number for the
problem of lower precision as well as lower adaptability in non-equidistant GM (1, 1) model, the calculation formulas
were deduced and a non-equidistant GRM (1, 1) model generated by accumulated generating operation of reciprocal
number was put forward .The grey GRM (1, 1) model can be used in non-equal interval & equal in terv al time series and
has the characteristic of high precision as well as high adaptability. Example validates the practicability and reliability
of the proposed model.
Keywords: Background Value; Grey Model GRM (1, 1) Generated by Accumulated Generating Operation of
Reciprocal Number; Non-Equal Interval; Accumulation Generation Operatio n; Grey System
1. Introduction
The grey model should be deemed as an important ele-
ment of Grey System Theory. Since the Grey System
Theory was firstly put forward by Professor Deng Julong,
the grey model has been widely used in many fields [1].
Because of its research characteristics of “small sample”
& “poor information” and its advantages of simplicity &
practicability, the model GM (1, 1) has occupied an im-
portant position during the test data processing & testing
and the online monitoring [1-6]. The grey system model
is mostly based on the equidistance sequence; however,
the original data obtained in the practical work is mostly
based on the non-equidistant sequence; therefore, the
establishment of non-equidistant sequence model has the
certain practical and theoretical significances. For the
non-negative discrete po int range , its one-order ac-
cumulated generating sequence should be mono-
tonic; if the sequence is fitted by a curve , the curve
should be also monotonic, so that t he grey model GM (1, 1)
can be forecasted. If is decreased monotonically,
due to the monotonic of , its value of model
should be monotonic; if is used as the predictive
value of accumulated decreasing generating reducing
original sequence, there would be the unreasonable error
of calculation. For the original sequence with the mono-
tonic decreasing trend , the definition of reverse
accumulated generating operation was put forward in the
literature [7], so as to establish the grey model GOM (1,

0
X

1
X

1
X

0
XX
X

0
X

1

1

1
X
1) based on the reverse accumulated generating operation;
the definition of reciprocal generating operation was put
forward in the literature [8], so as to establish the grey
model GRM (1, 1) based on the reciprocal generating
operation; the grey model GRM (1, 1) was improved in
the literature [9], so as to establish the improved mod-
el—grey model CGRM (1, 1) based on the reciprocal
generating operation. The generated sequence with
the monotonic decreasing trend can be use for the models
established based on the reverse accumulated generating
operation and the reciprocal generating operation; the
value of model of can be gained after
is fitted by the monotonic curve; when the value of mod-
el is reduced as the predictive v alue of , there
would not be any unreasonable error of calculation pro-
duced by means of the modeling method as the same as
the traditional accumulated and decreasing generating
methods, so as to improve the modeling precision. In the
literature [2], the spacing between sequ ences was used as
the multiplier to establish the non- equidistance model
GM (1, 1); this method could give tacit consent to the
linear relationship between the data difference and the
time difference; however, it was difficult to ensure
whether the construction of model could be consistent
with the actual condition. In the literature [3], the coeffi-
cient of standard deviation of sequence was decreased by
means of the functional transformation method, so that
the original sequence could be converted into the new
data sequence, so as to estimate the model parameters,

1
X

0
X

1
X

1
X

1
X

0
X
C
opyright © 2012 SciRes. IJMNTA
R. B. ZOU ET AL.
52
and then to establish the model GM (1, 1), resulting in
the more complex calculation. In order to improve the
fitting and prediction accuracy of mode GM (1, 1), the
structure method of multi-background values was put
forward in the literature [4-6], so as to establish a variety
of non-equidistance models GM (1, 1). The improvement
method of background value of non-equidis- tance model
GM (1, 1) was put forward in the literature [4,5]; by
means of this method, the one-order accumulated gener-
ating sequence could be fitted via the homogeneous ex-
ponential function, so as to obtain the higher accuracy;
however, it could be seen from the form of albino differ-
ential equation solution of model GM (1, 1) that, the ex-
ponential form of one-order accumulated gen- erated
sequence should not be deemed as the non-ho- mogene-
ous exponential form, but the homogeneous exponential
form only after accumulated decreasing and reducing; if
the one-order accumulated generating sequence was fit-
ted via the homogeneous exponential func- tion, there
would be the certain error of calculation . In the literature
[6], the one-order accumulated generating sequence was
fitted via the non-homogeneous exponential function to
deduce the optimal back ground va lue calculatio n fo r mula,
so as to establish th e non-equidis- tance model GM (1, 1).
In this paper, the first component was used as the initial
condition of differential equation of grey model; there-
fore, on the basis of the literature [8, 9], the
non-equidistance model GM (1, 1) could be established
based on the reciprocal accumulating generating method,
which was characterized by the following: High accuracy;
good theoretical and practical values. The pra- cticality
and reliability of the established model could be shown
in the fatigue test data processing examples.
2. Non-Equidistance Model GRM (1, 1)
Based on the Reciprocal Accumulated
Generating Operation
Definition 1. Supposing that the sequence is set as
 





00 000000
12
,,,
m
x
tx tx t


X,
if 1iii , , then should
be referred to as the non-equidistance sequence. If
ttt const
 2, ,im

00
X




0
00
1
k
k
xt
x
t

, 1, 2,,km
then
 






00 00
12
,,,
m
x
tx tx tX
00
should be re-
ferred to as the reciprocal sequence of .
X
Definition 2. Supposing that the sequence is set as
 





11 11
12
,,,
m
x
txtxt

X




if




10
11
x
txt



0
11k
,



11
kk 1k
x
tx ttxt

 

1
, , 1, ,1km
then should be referred to as the first order recip-
rocal accumulated generating operation of non-equidis-
tance sequence .
X

00
X
Definition 3. Supposing that the original data sequence
is set as
 





00 000000
12
,,,
m
x
tx tx t
X (where,

00
j
x
t
1, 2,,jm
shall represent the observed
value of variable quantity at the moment
j
t, m shall
represent the number of data), then the sequence






00 0
12
,,,
m
x
tx tx t
should be referred to as
the non-equidistant sequence, i.e. the spacing 1
j
j
tt
should not be the con st a nt.
In order to establish the model, the one-order recipro-
cal accumulated generating operation shall be carried out
firstly for the original data, to generate a new sequence.
 





11 11
12
,,,
m
x
txtxt
X (1)
where,

1
j
x
t
1, 2,,jm
can meet the Defini-
tion 2, i.e.





 


 
01
11
11
01
2, ,
1
k
jjj
j
k
x
txtttk
xt
xt k

m
(2)
For based on the one-order reciprocal accumu-
lated generating operation, the non-equidistance model
GRM (1, 1) should be established as per the first order

1
X
differential equation set
 
11
d
d
xa
t
b (in which

1
should be the background value), and its albino differen-
tial equation should be:
 
11
d
d
xax b
t
(3)
Its differential form should be:




01
kk
x
taztb
(4)
In the formula






111
1
0.5
kk
ztxt xt
 k
,
,
0
00
1
k
k
xt
x
t

, 1, 2,,km


1
k
zt should be referred to as the background value
of non-equidistance grey model GRM (1, 1) based on the
reciprocal accumulated generating operation, i.e. the mean
value of accumulated gen erat i ng se quence.
If is the parameter of non-equidistance
model GRM (1, 1), then the least square of non-equidis-
tance grey model GRM (1, 1) based on the reciprocal ac-

ˆ,
abT
a
Copyright © 2012 SciRes. IJMNTA
R. B. ZOU ET AL. 53
cumulated generating operation should be estimated as:

1
ˆ
TT
aBBBY (5)
where,















11
21
11
32
11
1
11
2
11
2
11
2mm
xt xt
xt xt
xt xt














2
03
0
m


B


0
,
x
t
x
t
x
t






Y

,
then the time response of differential Equation (4) of non-
equidistance grey model should be:





1
10
1
ˆˆ
ˆˆˆk
at t
k
bb
xtxt e
aa


 



(6)

1, 2,,km
The fitted value of the recipro cal of original data gained
after reduction should be



 




01
001
,
ˆ
ˆ1
ˆ,2,3,,
k
at
k
k
xt k
b
xt xt e
akm
t





1
(7)
The value of model of original sequence based on the
Definition 1 should be


00
ˆk
x
t

1, 2,,km
The absolute error of fitted data should be





00 00
ˆ
kk
qtx tx t
k
.
(8)
The relative error of fitted data (%) should be

 



0
00 00
00
ˆ() ()
100
i
kk
k
k
xt xt
et
xt

. (9)
The average value of relative errors of fitted data col-
umn should be

1
m
i
k
ek
. (10)
After the analog values, predictive values, errors and
other data can be gained, the non-equidistance model
GRM (1, 1) shall be inspected as well [1 ].
3. Application Examples
P. G. Forrest studied how the temperature would impact
on the fatigue strength under the action of symmetrical
cyclic load on many long-life materials. The experimen-
tal data on the fatigue strength of titanium alloy with the
change of temperature should be shown at Table 1,
which should be deemed as the non-equidistance se-
Table 1. Change relation of Ti alloy fatigue strength (σ–1)
along with temperature (T).
T 100 130 170
σ–1 560 557.54 536.10
T 210 240 270
σ–1 516.10 505.60 486.1
T 310 340 380
σ–1 467.4 453.8 436.4
quence. In this paper, the data indexed in the literature [2,
3] were applied, while the model was established by
means of the method indexed in this paper, so as to ob-
tain
0.00098951a


1 0
1.7852
k
xte
, ,
.
0.0017647b

.00098951 1001.783
t

ˆ4
The fitted value of original data is:
1
ˆ
= [560, 557.7302, 538.7297, 517.8231, 500.2108,
485.5801, 469.0376, 453.0846, 437.6491].
The absolute erro r of fitted data is
k
qt = [0, –0.19022, –2.6297, –1.7231, 5.3892,
0.51988, –1.6376, 0.71544, –1.24913].
The relative error of fitted data (%) is:
k
et = [0, –0.034117, –0.49053, –0.33387, 1.0659,
0.10695, –0.35037, 0.15766, –0.28623].
The average value of relative errors of fitted data col-
umn is 0.31396%.
In the literature [2], the original data was pre-proc-
essed with
50 50tT ,


01400 50
X, so
that the maximum relative error of modeling was up to
4.86%, and the average relative error of modeling was up
to 3.19%. In the literature [3], the model was established
by means of the functional transformation method, so
that the average relative error of modeling was up to
0.6587%. In the literature [5], the one-order accumulated
generating sequence was fitted via the homogeneous ex-
ponential function, so that the average relative error of
modeling was up to 0.9765%. As a result, it could be
seen that there should be the adaptive and scientific me-
thods indexed in this paper.
4. Conclusion
In this paper, applying the Grey System Theory with the
first component of original data was used as the initial
condition of grey differential equation, the non-equidis-
tance model GRM (1, 1) based on the reciprocal accu-
mulated generating operation was put forward; the MAT-
LAB program was prepared. The model is suitable for
both the modeling of equidistance sequence and the
Copyright © 2012 SciRes. IJMNTA
R. B. ZOU ET AL.
Copyright © 2012 SciRes. IJMNTA
54
modeling of non-equidistance sequence with the charac-
teristics of high precision, strong adaptability, etc. The
correctness and validity of this model could be shown in
the test data processing examples; due to its important
practical and theoretical significances, this model should
be widely used.
REFERENCES
[1] Y. X. Luo, L. T. Zhang and M. Li, “Grey Systems and
Applications in Mechanical Engineering,” National Uni-
versity of Defense Technology Press, Changsha, 2001.
[2] Y. X. Luo and J. R. Zhou, “Non-Equidistance GM (1, 1)
Model and Its Application in Fatigue Experimental Data
Processing and On-Line Control,” Journal of Mechanical
Strength, Vol. 18, No. 3, 1996, pp. 60-63.
[3] Y. X. Luo, X. Wu and M. Li, “Function-Transfer Method
of Parameters Estimation of Grey GM (1, 1) Model and
Its Application,” Journal of Mechanical Strength, Vol. 24,
No. 3, 2002, pp. 450-452.
[4] W. Z. Dai and J. F. Li, “Modeling Research on Non-
Equidistance GM (1, 1) Model,” Systems Engineering-
Theory & Practice, Vol. 25, No. 9, 2005, pp. 89-93.
[5] F. X. Wang, “Improvement on Unequal Interval Gray
Forecast Model,” Fuzzy Information and Engineering,
Vol. 6, No. 1, 2006, pp. 118-123.
[6] Y. M. Wang, Y. G. Dang, Z. X. Wang, “The Optimiza-
tion of Background Value in Non-Equidistant GM (1, 1)
Model,” Chinese Journal of Management Science, Vol.
16, No. 4, 2008, pp. 159-162.
[7] Z. M. Song and J. L. Deng, “The Accumulated Generat-
ing Operation in Opposite Direction and Its Use in Grey
Model GOM (1, 1),” Systems Engineering, Vol. 19, No. 2,
2001, pp. 66-69.
[8] B. H. Yang and Z. Q. Zhang, “The Grey Model Has Been
Accumulated Generating Operation in Rcciprocal Num-
ber and Its Application,” Mathematics in Practice and
Theory, Vol. 33, No. 10, 2003, pp. 21-25.
[9] H. Zhou and X. G. Wang, “A Improvement of the Grey
Model GRM (1, 1) Generated by Accumulattion Gener-
ating Operation of Reciprocal Number,” Transactions of
Shenyang Ligong University, Vol. 27, No. 4, 2008, pp.
84-86.