Grey Accumulation Generation Relational Analysis Model for Nonequidistance Unequal-Length Sequences and Its Application

Abstract

As research is required on nonequidistance unequal-length sequences, so grey accumulation generation relational analysis model based on grey exponential law (AGRA) for nonequidistance unequal-length (NDUL) sequences is put forward in this paper. The original data is accumulated generation firstly and the generation sequences are simulated. Then the generation rate is established as the ratio of the tangent slope and the mean of the simulation function. Furthermore, the dynamic similarity of change trend of the original time sequences is characterized by the proximity of generation rate sequences. Meanwhile, properties of AGRA model for nonequidistance unequal-length sequences are discussed. The new relational analysis model is available for equal interval sequences, nonequidistance sequences, sequences which have relationship before transformation and sequences which have relationship after accumulation; therefore, the AGRA model has expanded the scope of application of grey relational analysis. Lastly, factors which affect the amount of passenger cars in China are sorted using AGRA model for NDUL sequences. This application is presented to illustrate the effectiveness and practicality of the proposed model.

1. Introduction

Grey relational analysis (GRA) has emerged as an effective model to measure the degree of correlation between sequences by calculating their similarity or proximity. Some research on GRA and its application has been done, such as the improvement and development of the existing GRA [1–5], extension of objects of studying [6], parallelism and consistency [7], consistency for affine transformation [8], and similarity and proximity of GRA [9].

GRA has been employed in various fields where they have produced promising results due to not limited data distribution and quantity. On the theoretical side, GRA plays its advantage during combination of clustering [10], decision making [11–14], and evaluating [15]. There are also a range of practical or industrial applications employing GRA, for instance, in the period of Chinese regional economic discrepancy [16], affecting factors of service brand [17], company financial performance [18], software effort estimation, energy consumption [19], Chinese economic development and energy consumption [20], residential energy consumption and renewable energy utilization [21], sustainability indicator of renewable energy system [22], maximum peak discharge at upper stream [23], and track association algorithm for distributed multitarget system [24]. As a summary, Liu et al. review the research approaches of GRA modeling systematically [25]. Yin conducts a bibliometric study on publication and citation patterns of GRA published from 1996 to 2010 through a systemic search using the ISI web-based databases; results demonstrate that there has been a substantial increase in the number of peer-reviewed papers on GRA indexed by the ISI Web of Knowledge [26].

From the view of grey systems, although the objective system looks complex, it always has the overall function and certain inherent law. There is a method called grey generation which can explore the real law of the original data. Accumulation generation is an effective kind of grey generation, which whitens grey process and inherent development tendency of accumulation process. Original data may have no regular raw seemingly but emerge integral characteristics by accumulation generation [27]. Because the economic system, ecological system, and agricultural system could be treated as generalized energy system, and the accumulation of energy follows the exponential rule, so accumulation generation has extensive adaptability and has been applied into grey forecasting models. Grey forecasting models based on grey exponential law are employed in a range of applications [28–30].

The existing GRA model examines geometric similarity or proximity of data directly or after dimensionless without considering other kinds of law, and research on GRA for nonequidistance unequal-length (NDUL) sequences is required, so AGRA for NDUL sequences based on grey exponential law will be put forward in this paper. Generation rate for NDUL sequences will be established. Then the dynamic change trend similarity of the original time series will be characterized by the proximity of generation rate sequence. Properties of AGRA for NDUL sequences will also be studied. Lastly, an application will be researched using the model proposed in this paper.

The paper is structured as follows. In the next section, AGRA for NDUL sequences is provided. Properties of AGRA for NDUL sequences are researched in Section 3. An application using the model is furnished in Section 4. In Section 5, appropriate conclusions are drawn.

2. Establishment of Grey Generation Rate Relational Analysis Model for Nonequidistance Unequal-Length Sequences

2.1. Simulation of Accumulation Generation Sequence of Nonequidistance Unequal-Length Sequences

Definition 1. Let 𝑋⁽⁰⁾=(𝑥⁽⁰⁾(𝑡₁),𝑥⁽⁰⁾(𝑡₂),…,𝑥⁽⁰⁾(𝑡_𝑛)) be a time sequence, where Δ𝑡_𝑘=𝑡_𝑘−𝑡_𝑘−1≠const; then 𝑋⁽⁰⁾ is a nonequidistance sequence.

Definition 2. Let 𝑋₁⁽⁰⁾=(𝑥₁⁽⁰⁾(𝑡₁),𝑥₁⁽⁰⁾(𝑡₂),…,𝑥₁⁽⁰⁾(𝑡_𝑛)) and 𝑋₂⁽⁰⁾=(𝑥₂⁽⁰⁾(𝑡₁),𝑥₂⁽⁰⁾(𝑡₂),…,𝑥₂⁽⁰⁾(𝑡_𝑚)) be time sequences, and 𝑚≠𝑛; then 𝑋₁⁽⁰⁾ and 𝑋₂⁽⁰⁾ are unequal-length sequences.

Definition 3. Let 𝑋⁽¹⁾=(𝑥⁽¹⁾(𝑡₁),𝑥⁽¹⁾(𝑡₂),…,𝑥⁽¹⁾(𝑡_𝑛)) be the sequence of 1-AGO (one-step accumulating generation operator), where 𝑥⁽¹⁾𝑡_𝑘=^𝑘_𝑗=1𝑥⁽⁰⁾𝑡_𝑗Δ𝑡_𝑗,𝑘=1,2,…𝑛, (1) are generated from 𝑋⁽⁰⁾.

The background value optimized nonequidistance GM (1,1) model in the literature [18] will be used to simulate accumulation generation sequences. The differential equation is 𝑑𝑥⁽¹⁾(𝑡)𝑑𝑡+𝑎𝑥⁽¹⁾(𝑡)=𝑏. (2)

The discrete form is 𝑥⁽⁰⁾𝑡_𝑘Δ𝑡_𝑘+𝑎𝑧⁽¹⁾𝑡_𝑘=𝑏Δ𝑡_𝑘. (3) If 𝑎=(𝑎,𝑏)^𝑇 is a sequence of parameters and 𝑥𝑌=⁽⁰⁾(2)Δ𝑡₂𝑥⁽⁰⁾(3)Δ𝑡₃⋮𝑥⁽⁰⁾(𝑛)Δ𝑡_𝑛,𝐵=−𝑧⁽¹⁾(2)Δ𝑡₂−𝑧⁽¹⁾(3)Δ𝑡₃⋮⋮−𝑧⁽¹⁾(𝑛)Δ𝑡_𝑛,𝑧⁽¹⁾(𝑘)=Δ𝑡_𝑘²𝑥⁽⁰⁾𝑡_𝑘ln𝑥⁽⁰⁾𝑡_𝑘−ln𝑥⁽⁰⁾𝑡_𝑘−1+𝑥⁽⁰⁾𝑡₁Δ𝑡_𝑘−𝑥⁽⁰⁾𝑡_𝑘²𝑥⁽⁰⁾𝑡_𝑘/𝑥⁽⁰⁾𝑡_𝑘−1^{(𝑡1−𝑡𝑘)/Δ𝑡𝑘}×Δ𝑡_𝑘²𝑥⁽⁰⁾(𝑡_𝑘)−𝑥⁽⁰⁾(𝑡_𝑘−1)⁻¹. (4)

Then the parameters of the model based on the least squares method are (𝑎,𝑏)^𝑇=𝐵^𝑇𝐵⁻¹𝐵^𝑇𝑌. (5)

The solution, also known as time response function, of the whitenization equation 𝑑𝑥⁽¹⁾(𝑡)/𝑑𝑡+𝑎𝑥⁽¹⁾(𝑡)=𝑏 is given by 𝑥⁽¹⁾(𝑡)=𝑐𝑒^{−𝑎(𝑡−𝑡1)}+𝑏𝑎, (6) where 𝑐=𝑥⁽¹⁾𝑡₁−𝑏𝑎. (7)

Its discrete form is 𝑥⁽¹⁾𝑡_𝑘=𝑥⁽¹⁾𝑡₁−𝑏𝑎𝑒^{−𝑎(𝑡𝑘−𝑡1)}+𝑏𝑎. (8)

2.2. Generation of Equidistance Equal-Length Sequences from Nonequidistance Unequal-Length Sequences

Suppose that 𝑋₀⁽⁰⁾=(𝑥₀⁽⁰⁾(𝑡¹₀),𝑥₀⁽⁰⁾(𝑡²₀),…,𝑥₀⁽⁰⁾(𝑡^𝑞0₀)) and 𝑋_𝑖⁽⁰⁾=(𝑥_𝑖⁽⁰⁾(𝑡¹_𝑖),𝑥_𝑖⁽⁰⁾(𝑡²_𝑖),…,𝑥_𝑖⁽⁰⁾(𝑡^𝑞𝑖_𝑖)) (𝑖=1,2,…,𝑚) are nonnegative sequences, and 𝑛=max{𝑞_𝑖}(𝑖=0,1,2,…,𝑚); then equidistance equal-length sequences 𝑋^′₀=(𝑥^′₀(1),𝑥^′₀(2),…𝑥^′₀(𝑛)) and 𝑋^′_𝑖=(𝑥^′_𝑖(1),𝑥^′_𝑖(2),…,𝑥^′_𝑖(𝑛)) (𝑖=0,1,2,…,𝑚) can be obtained by time response equation in Section 2.1 based on 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾. The time intervals of new sequences are one and lengths are 𝑛.

2.3. Establishment of Grey Generation Rate Relational Analysis Model Based on Generated Sequences

The slope of the tangent of 𝑋⁽¹⁾ at time 𝑡 can be got as (9) by the function of 𝑋⁽¹⁾ and the meaning of slope: ℎ(𝑡)=𝑑𝑥⁽¹⁾(𝑡)𝑑𝑡=−𝑎𝑐𝑒^{−𝑎(𝑡−𝑡1)}. (9)

The mean value of 𝑋⁽¹⁾ between [1,𝑛] can be calculated as 𝑋⁽¹⁾=1𝑛−1^𝑛₁𝑥⁽¹⁾(𝑡)=1𝑑𝑡𝑛−1^𝑛₁𝑐𝑒^{−𝑎(𝑡−𝑡1)}+𝑏𝑎=𝑐𝑑𝑡()𝑎𝑒𝑛−1^{−𝑎(1−𝑡1)}−𝑒^{−𝑎(𝑛−𝑡1)}+𝑏𝑎. (10)

The slope is the measure of the tilt of the linear. The slope of the tangent of 𝑋⁽¹⁾ at time 𝑡 is the derivative value of 𝑋⁽¹⁾ at time 𝑡, which is the change rate of 𝑋⁽¹⁾ at time 𝑡. Then the slope of the tangent is divided by the mean value to establish the generation rate which eliminates the order of magnitudes.

Definition 4. Generation rate of 𝑋⁽⁰⁾ at time 𝑡 is defined as (11), where 𝑡∈[𝑎,𝑏]: 𝑠(𝑡)=ℎ(𝑡)𝑋⁽¹⁾. (11)

Definition 5. Let 𝑠₀(𝑘)=ℎ₀(𝑘)/𝑋₀⁽¹⁾ and 𝑠_𝑖(𝑘)=(ℎ_𝑖(𝑘)/𝑋_𝑖⁽¹⁾)(𝑖=1,2,…,𝑚) be generation rates of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾(𝑖=1,2,…,𝑚) at time 𝑘; then 𝑆₀=(𝑠₀(1),𝑠₀(2),…,𝑠₀(𝑛)) is the generation rate sequence of system characteristic sequence 𝑋₀ and 𝑆_𝑖=(𝑠_𝑖(1),𝑠_𝑖(2),…,𝑠_𝑖(𝑛))(𝑖=1,2,…,𝑚) are the generation rate sequences of system behavioral sequence 𝑋_𝑖(𝑖=1,2,…,𝑚), where ℎ₀(𝑘)=−𝑎₀𝑐₀𝑒^{−𝑎0(𝑘−𝑡10)},ℎ_𝑖(𝑘)=−𝑎_𝑖𝑐_𝑖𝑒^{−𝑎𝑖(𝑘−𝑡1𝑖)},𝑘=1,2,…,𝑛,𝑋₀⁽¹⁾=𝑐₀(𝑛−1)𝑎₀𝑒^{−𝑎0(1−𝑡10)}−𝑒^{−𝑎0(𝑛−𝑡10)}+𝑏₀𝑎₀,𝑋_𝑖⁽¹⁾=𝑐_𝑖(𝑛−1)𝑎_𝑖𝑒^{−𝑎𝑖(1−𝑡1𝑖)}−𝑒^{−𝑎𝑖(𝑛−𝑡1𝑖)}+𝑏_𝑖𝑎_𝑖,𝑖=1,2,…,𝑚. (12)

Definition 6. 𝑋₀⁽⁰⁾=(𝑥₀⁽⁰⁾(𝑡¹₀),𝑥₀⁽⁰⁾(𝑡²₀),…,𝑥₀⁽⁰⁾(𝑡^𝑞0₀)) is the nonnegative system characteristic sequence and 𝑋_𝑖⁽⁰⁾=(𝑥_𝑖⁽⁰⁾(𝑡¹_𝑖),𝑥_𝑖⁽⁰⁾(𝑡²_𝑖),…,𝑥_𝑖⁽⁰⁾(𝑡^𝑞𝑖_𝑖))(𝑖=1,2,…,𝑚) are nonnegative system behavioral sequence sequences. Equidistance equal-length sequences 𝑋^′₀=(𝑥^′₀(1),𝑥^′₀(2),…𝑥^′₀(𝑛)) and 𝑋^′_𝑖=(𝑥^′_𝑖(1),𝑥^′_𝑖(2),…,𝑥^′_𝑖(𝑛)) (𝑖=1,2,…,𝑚) are generated from 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ according to Section 2.1. Then, 𝜉_𝑖(𝑘)=𝑠₀(𝑘)𝑠₀(𝑘)+𝑠₀(𝑘)−𝑠_𝑖(𝑘,(),)𝑖=1,2,…,𝑚 (13) is named as the grey relational coefficient of AGRA between 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾at time 𝑘.

|𝑠₀(𝑘)−𝑠_𝑖(𝑘)| reflects the proximity of change trend at time 𝑘 between 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾, so the closer development tendency at time 𝑘 of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ is, the smaller |𝑠₀(𝑘)−𝑠_𝑖(𝑘)| is, meanwhile, the greater the relational coefficient of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ is, and vice versa. The constriction of function for the relational coefficient is consistent with the meaning of grey relational analysis.

Definition 7. Grey relational degree of AGRA between 𝑋_𝑖 and 𝑋_𝑗 is constructed as 𝜁_𝑖=1𝑛^𝑛_𝑘=1𝜉_𝑖(𝑘),(𝑖=1,2,…,𝑚). (14)

The grey relational coefficient of AGRA between sequences reflects the degree of consensus of sequences with change trend at one moment, and the grey relational degree of AGRA between sequences is the mean of the grey relational coefficient, so the grey relational degree of AGRA between sequences is measurement of consensus of sequences with development tendency for the whole time.

2.4. The Step of AGRA Model for NDUL Sequences

The steps to obtain the order of sequences by AGRA model for NDUL sequences are concluded as follows.(1)Make qualitative analysis on system characteristics sequence 𝑋₀ to confirm system behavioral sequences 𝑋_𝑖.(2)Get the accumulation generation sequences of 𝑋₀ and 𝑋_𝑖 by (1).(3)Obtain the fitting curve functions of generation sequences according to (4)–(8).(4)Construct the generation rate sequences using (9)–(11).(5)Calculate the grey relational coefficients of AGRA between 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ at time 𝑡 (𝑡=1,2,…𝑛) by (13).(6)Determine the grey relational degree between 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾(𝑖=1,2,…,𝑚) based on grey relational coefficients and (14). Confirm the order of behavioral sequences between characteristics sequence according to the grey relational degree of AGRA.

3. Properties of AGRA Model for NDUL Sequences

Theorem 8. AGRA model for NDUL sequences has the following properties.(1)0<𝜁_𝑖≤1. 𝜁_𝑖=1, iff 𝑠₀(𝑘)=𝑠_𝑖(𝑘) for ∀𝑘∈{1,2,…,𝑛}.(2)Uniqueness and Independence. 𝜁_𝑖 is unique for fixed two sequences and is not influenced by other sequences.(3)Proximity. The closer the development tendency of sequences is, the greater 𝜁_𝑖 is.(4)Transitivity. If 𝜁_𝑖≥𝜁_𝑗 and 𝜁_𝑗≥𝜁_𝑞, then 𝜁_𝑖≥𝜁_𝑞.

Proof. (1)𝜁_𝑖=0⇔|𝑠₀(𝑘)|=0, ∀𝑘∈{1,2,…,𝑛}⇔𝑥₀⁽¹⁾(𝑘)=𝑟, ∀𝑘∈{1,2,…,𝑛} where 𝑟=const; then 𝑥₀⁽⁰⁾(𝑘)=𝑥₀⁽¹⁾(𝑘)−𝑥₀⁽¹⁾(𝑘−1)=𝑟−𝑟=0,𝑘=2,3,…,𝑛. (15)
There is no sense to research correlation between these sequences. So 𝜁_𝑖≠0.
As |𝑠₀(𝑘)|>0, |𝑠₀(𝑘)−𝑠_𝑖(𝑘)|≥0, so 0<|𝑠₀(𝑘)|/(|𝑠₀(𝑘)|+|𝑠₀(𝑘)−𝑠_𝑖(𝑘)|)≤1; that is 0<𝜉_𝑖(𝑘)≤1, and 0<(1/𝑛)∑^𝑛_𝑘=1𝜉_𝑖(𝑘)≤1; therefore 0<𝜁_𝑖≤1. (16)
If 𝑠₀(𝑘)=𝑠_𝑖(𝑘), then 𝜉_𝑖(𝑘)=1; if 𝑠₀(𝑘)=𝑠_𝑖(𝑘), ∀𝑘∈{1,2,…,𝑛}, then 𝜁_𝑖=1.
(2) According to the definition of the grey relational coefficient and grey relational degree of AGRA, 𝜁_𝑖 is unique for fixed 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ and is not influenced by other sequences.
(3) The closer change trends at time 𝑘 of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ is, the smaller |𝑠₀(𝑘)−𝑠_𝑖(𝑘)| is. meanwhile, the greater the relational coefficient of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ is, and vice versa. The closer development tendency of whole 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ is, the greater 𝜁_𝑖 is.
(4) Transitivity can be known by Properties 2 and 3.

Theorem 9. Let 𝑡∈[𝑎,𝑏], 𝑋₀⁽⁰⁾=(𝑥₀⁽⁰⁾(𝑡¹₀),𝑥₀⁽⁰⁾(𝑡²₀),…,𝑥₀⁽⁰⁾(𝑡^𝑞0₀)) is the system characteristics sequence, 𝑋_𝑖⁽⁰⁾=(𝑥_𝑖⁽⁰⁾(𝑡¹_𝑖),𝑥_𝑖⁽⁰⁾(𝑡²_𝑖),…,𝑥_𝑖⁽⁰⁾(𝑡^𝑞𝑖_𝑖))(𝑖=1,2,…,𝑚) are the behavioral sequences, and equidistance equal-length sequences based on 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ are 𝑋^′₀=(𝑥^′₀(1),𝑥^′₀(2),…𝑥^′₀(𝑛)) and 𝑋^′_𝑖=(𝑥^′_𝑖(1),𝑥^′_𝑖(2),…,𝑥^′_𝑖(𝑛)), 𝑛=max{𝑞_𝑖}, (𝑖=0,1,2,…,𝑚). The grey relational degree of AGRA will remain the same after 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ are divided by the initial value or the mean value, or multiplied; in other words, the grey relational degree of AGRA will not change by dimensionless of sequences.

Proof. The simulation functions of 𝑋₀⁽¹⁾ and 𝑋_𝑖⁽¹⁾ which are the accumulation generation sequences of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾, 𝑖=1,2,…,𝑚, are 𝑥₀⁽¹⁾(𝑡)=𝑥₀⁽⁰⁾𝑡¹₀−𝑏₀𝑎₀𝑒^{−𝑎0(𝑡−𝑡10)}+𝑏₀𝑎₀,𝑥_𝑖⁽¹⁾(𝑡)=𝑥_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏_𝑖𝑎_𝑖𝑒^{−𝑎𝑖(𝑡−𝑡1𝑖)}+𝑏_𝑖𝑎_𝑖. (17)
Then the generation rates of 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾(𝑖=1,2,…,𝑚) at time 𝑘 are 𝑠₀(𝑘)=−𝑎²₀𝑥₀⁽⁰⁾𝑡¹₀−𝑏₀/𝑎₀(𝑛−1)𝑒^{−𝑎0(𝑘−𝑡10)}×𝑥₀⁽⁰⁾𝑡¹₀−𝑏₀/𝑎₀𝑒^{−𝑎0(1−𝑡10)}−𝑒^{−𝑎0(𝑛−𝑡10)}+(𝑛−1)𝑏₀⁻¹,𝑠_𝑖(𝑘)=−𝑎²_𝑖𝑥_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏_𝑖/𝑎_𝑖(𝑛−1)𝑒^{−𝑎𝑖(𝑘−𝑡1𝑖)}×𝑥_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏_𝑖/𝑎_𝑖𝑒^{−𝑎𝑖(1−𝑡1𝑖)}−𝑒^{−𝑎𝑖(𝑛−𝑡1𝑖)}+(𝑛−1)𝑏_𝑖⁻¹,𝑘=1,2,…,𝑛,𝑖=1,2,…,𝑚. (18) Now let 𝑌₀⁽⁰⁾=𝜆𝑋₀⁽⁰⁾, 𝑌_𝑖⁽⁰⁾=𝛽𝑋_𝑖⁽⁰⁾(𝑖=1,2,…,𝑚), and 𝑌₀⁽¹⁾, 𝑌_𝑖⁽¹⁾(𝑖=1,2,…,𝑚) are the accumulation generation sequences of 𝑌₀⁽⁰⁾ and 𝑌_𝑖⁽⁰⁾. The simulation functions of 𝑌₀⁽¹⁾ and 𝑌_𝑖⁽¹⁾(𝑖=1,2,…,𝑚) are 𝑦₀⁽¹⁾(𝑡)=𝑦₀⁽⁰⁾𝑡¹₀−𝑏^′₀𝑎^′₀𝑒^{−𝑎0(𝑡−𝑡10)}+𝑏^′₀𝑎^′₀,𝑦_𝑖⁽¹⁾(𝑡)=𝑦_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏^′_𝑖𝑎^′_𝑖𝑒^{−𝑎𝑖(𝑡−𝑡1𝑖)}+𝑏^′_𝑖𝑎^′_𝑖,(𝑖=1,2,…,𝑚). (19) Then the generation rates after the dimensionless of dates are 𝑠^′₀(𝑘)=−𝑎₀^′2𝑦₀⁽⁰⁾𝑡¹₀−𝑏^′₀/𝑎^′₀(𝑛−1)𝑒^{−𝑎′0(𝑘−𝑡10)}×𝑦₀⁽⁰⁾𝑡¹₀−𝑏^′₀/𝑎^′₀𝑒^{−𝑎′0(1−𝑡10)}−𝑒^{−𝑎′0(𝑛−𝑡10)}+(𝑛−1)𝑏^′₀⁻¹,𝑠^′_𝑖(𝑘)=−𝑎_𝑖^′2𝑦_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏^′_𝑖/𝑎^′_𝑖(𝑛−1)𝑒^{−𝑎′𝑖(𝑘−𝑡1𝑖)}×𝑦_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏^′_𝑖/𝑎^′_𝑖𝑒^{−𝑎′𝑖(1−𝑡1𝑖)}−𝑒^{−𝑎′𝑖(𝑛−𝑡1𝑖)}+(𝑛−1)𝑏^′_𝑖⁻¹,𝑘=1,2,…,𝑛,𝑖=1,2,…,𝑚. (20) Meanwhile, 𝑎₀=𝑎^′₀, 𝑎_𝑖=𝑎^′_𝑖, 𝑏^′_𝑜=𝜆𝑏_𝑜, 𝑏^′_𝑖=𝛽𝑏_𝑖(𝑖=1,2,…,𝑚) can be got by Li [31].
So, 𝑠^′₀(𝑘)=𝑠₀(𝑘), 𝑠^′_𝑖(𝑘)=𝑠_𝑖(𝑘); then 𝜉^′_𝑖(𝑘)=𝑠^′₀(𝑘)𝑠^′₀(𝑘)+𝑠^′₀(𝑘)−𝑠^′_𝑖(𝑘=𝑠₀)(𝑘)𝑠₀(𝑘)+𝑠₀(𝑘)−𝑠_𝑖(𝑘)=𝜉_𝑖(𝑘). (21)
Therefore, the grey relational coefficients of generation rate between 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ will remain the same after 𝑋₀⁽⁰⁾ and 𝑋_𝑖⁽⁰⁾ are divided by the initial value or the mean value, or multiplied; in other words, the grey relational coefficients of AGRA will not be affected by dimensionless so do the grey relational degree of AGRA. Another conclusion can be got; the AGRA model for NDUL sequences is suitable for both positive sequences and negative sequences.

Theorem 10. Let behavioral sequences be 𝑋_𝑖⁽⁰⁾=(𝑥_𝑖⁽⁰⁾(𝑡₁),𝑥_𝑖⁽⁰⁾(𝑡₂),…𝑥_𝑖⁽⁰⁾(𝑡_𝑛)), and 𝑥_𝑖⁽⁰⁾(𝑡_𝑘)=𝜃𝑥₀⁽⁰⁾(𝑡_𝑘), 𝜃=const, 𝑘=1,2,…𝑛; then 𝜁_𝑖=1.

Proof. The generation rates of 𝑋₀⁽¹⁾ and 𝑋_𝑖⁽¹⁾ at time 𝑘 are 𝑠₀(𝑘)=−𝑎²₀𝑥₀⁽⁰⁾𝑡¹₀−𝑏₀/𝑎₀(𝑛−1)𝑒^{−𝑎0(𝑘−𝑡10)}×𝑥₀⁽⁰⁾𝑡¹₀−𝑏₀/𝑎₀𝑒^{−𝑎0(1−𝑡10)}−𝑒^{−𝑎0(𝑛−𝑡10)}+(𝑛−1)𝑏₀⁻¹,𝑠_𝑖(𝑘)=−𝑎²_𝑖𝑥_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏_𝑖/𝑎_𝑖(𝑛−1)𝑒^{−𝑎𝑖(𝑘−𝑡1𝑖)}×𝑥_𝑖⁽⁰⁾𝑡¹_𝑖−𝑏_𝑖/𝑎_𝑖𝑒^{−𝑎𝑖(1−𝑡1𝑖)}−𝑒^{−𝑎𝑖(𝑛−𝑡1𝑖)}+(𝑛−1)𝑏_𝑖⁻¹,𝑘=1,2,…,𝑛,𝑖=1,2,…,𝑚. (22)
As 𝑥_𝑖⁽⁰⁾(𝑡_𝑘)=𝜃𝑥₀⁽⁰⁾(𝑡_𝑘), 𝑡¹₀=𝑡¹_𝑖, so 𝑎₀=𝑎_𝑖, 𝑏_𝑖=𝜃𝑏_𝑜(𝑖=1,2,…,𝑚), then 𝑠_𝑖(𝑘)=𝑠₀(𝑘); therefore 𝜉_𝑖(𝑘)=𝑠₀(𝑘)𝑠₀(𝑘)+𝑠₀(𝑘)−𝑠_𝑖(𝑘=𝑠₀)(𝑘)𝑠₀(𝑘)=1. (23) So 𝜁_𝑖=1𝑛^𝑛_𝑘=1𝜉_𝑖(𝑘)=1. (24)

Theorem 11. The AGRA model for NDUL sequences has noise reduction effect and is stable for noise.

Proof. As initial sequences in AGRA model for NDUL sequences are accumulation generated before taken into the model, and accumulation generation has noise reduction effect, so the model reduces the impact of noise. Stability and validity for noise interference in this model will be demonstrated as follows.
Let 𝑥=𝑥_𝑜+𝜀 be the sequence which has noise, 𝑥_𝑜 the sequence without noise, and 𝜀 the noise; then 𝑠(𝑡)=ℎ(𝑡)𝑋⁽¹⁾=𝑑𝑥₀+𝜀/𝑑𝑡𝑋₀⁽¹⁾=+𝜀𝑑𝑥₀/𝑑𝑡+𝑑𝜀/𝑑𝑡𝑋₀⁽¹⁾.+𝜀 (25) As 𝜀 can be ignored when compared with 𝑥_𝑜 and 𝑋₀⁽¹⁾, and the derivative of 𝜀 is infinitesimal, so 𝑠(𝑡)=𝑑𝑥₀/𝑑𝑡+𝑑𝜀/𝑑𝑡𝑋₀⁽¹⁾≈+𝜀𝑑𝑥₀/𝑑𝑡𝑋₀⁽¹⁾=𝑠₀(𝑡). (26) Therefore, the AGRA model for NDUL sequences is stable for noise.

4. Case Study

Traffic jam is becoming one of the most difficult problems in which cities are facing as the urban space is expanding and the population is accumulating. Traffic accidents, air pollution, noise pollution, resource shortages, and other issues which are brought by traffic jam are hindering the global economy and even threatening human survival; that is why traffic jam is one of the most focused on problems for academic organizations and governments. It is significant to sort factors which affect the amount of commercial passenger cars for traffic management and congestion controlling. The road mileage, turnover passengers on highway, and the number of civil aircrafts are selected as affecting factors according to the existing literatures and expert advices. The original data of affecting factors are in Table 1.

Table 1 

The original data of factors which affect the number of commercial passenger cars.

Due to the limitation of data source, the numbers of commercial passenger cars and civil aircrafts in 2006 and 2007 are lacking. Common GRA models fail to handle this situation. In order to make full use of existing data, the AGRA model for NDUL sequences is required to rank factors which affect the number of commercial passenger cars. The parameters during the mathematical calculation are provided in Table 2. The grey generation rate degrees of incidence and the ranking list are in Table 3.

Table 2 

The parameters during the mathematical calculation.

Table 3 

The grey relational degree of AGRA and the ranking of factors which affect the number of commercial passenger cars.

As shown in Table 3, the number of civil aircrafts is more affected by requirement of society than social capacity. The driving action of requirement of society is greater than the restriction of social capacity in front of the increasing of cars.

5. Conclutions and Future Studies

AGRA model for NDUL sequences is established in this paper. The original data is accumulated firstly and the curve is fitted by optimized GM (1,1) model for nonequidistance sequences; then the fitting equation is obtained. Generation rate is established as the ratio of the tangent slope and the mean of the fitting equation. The similarity of dynamic change trend of the original time series is characterized by the proximity of generation sequences. The AGRA model for NDUL sequences is available for both equal interval sequences and nonequidistance sequences. Meanwhile, the new model can handle sequences which have relationship before transformation and sequences which have relationship after accumulation. Accumulation generation reflects development tendency of data and expands the scope of application of grey relational analysis. Meanwhile, properties of AGRA model for NDUL sequences are researched. It is shown that AGRA model for NDUL sequences is not limited to positive sequences. In the last section, factors which affect the number of passenger cars in China are ordered using the model proposed in this paper. It is illustrated that the proposed model is effective and practical.

Because the economic system, ecological system, and agricultural system could be treated as generalized energy system, the accumulation of energy follows the exponential rule which is used to establish AGRA model. Meanwhile, data missing is common in practical problem so that NDUL sequences are general. In conclusion, AGRA model for NDUL sequences can be applied but not limited to economic system, ecological system, and agricultural system problems with NDUL sequences.

There are some potential studies that can be done in the future according to the ideas presented in this paper, including conducting a new AGRA model based on other kinds of GRA models; in this way, more applications of other kinds of AGRA models can be determined. The AGRA model for NDUL sequences can also be combined with clustering methods and decision making methods in the presence of NDUL sequences.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to express their gratitude to the editor and anonymous reviewers for their helpful comments which improved the quality of the paper. The paper is sponsored by National Natural Science Foundation of China (71071077, 71371098); Fundamental Research Funds for the Central Universities (NC2012001, NZ2010006); Funding of Jiangsu Base of Major Projects on Philosophy and Social Sciences (2012JDXM005); Funding of Jiangsu Innovation Program for Graduate Education (CXZZ12_0174); and Funding for Outstanding Doctoral Dissertation in NUAA (BCXJ12-11).