Quantitative Analysis of Dynamic Behaviours of Rural Areas at Provincial Level Using Public Data of Gross Domestic Product

: A spatial approach that incorporates three economic components and one environmental factor has been developed to evaluate the dynamic behaviours of the rural areas at a provincial level. An artiﬁcial


Introduction
During the past two decades, China's rapid economic growth has been making China become one of the largest contributors to the global economy and natural environment issues [1].Combining with China's large population, rapid urbanisation, increasing household wealth demands and local economic redisbtribution, unbalanced pressure has deployed on regional development and caused low efficient environmental recycle.Issues of the local impact of various air pollutants on the regional economic development, such as carbon monoxide (CO), nitrogen oxides (N O x ), sulphur dioxide (SO 2 ), have been extensively discussed.
The global warming is one of the most serious environmental issues that human currently confronts.The mechanism of global warming has not yet been fully understood.However, it has been considered closely related to the emission of the anthropogenic greenhouse gases, such as carbon dioxide (CO 2 ) and methane (CH 4 ).These issues has attracted researchers' extensive interest, and in reality various countermeasures, such as technological development, abatement policies based on economic mechanisms and international negotiations, have been discussed and partially applied into practices [2,3].
As it has been widely accepted, CO 2 emission is connected with the human activities mainly through the consumption of fossil fuel type energy, for example the operation of a fossil fuel power station.To evaluate the dynamic behaviours of the regional areas coupled with energy consumption and CO 2 emissions, a few interdisciplinary studies have been conducted.Berling-wolff and Wu outlined the historical development of urban growth models, which indicated the different disciplines and diverse theories could be brought together to hybrid the newly developed models including fuzzy logic theory and neural network theory [4].Nejadkoorki et al. proposed a model for the road traffic CO 2 emissions of an urban area with three components [5].Weiss et al. presented a bottom-up model for the non-energy use of fossil fuels and its CO 2 emissions [6].Bala discussed rural energy supply projections and assessed the contributions to global warming.Dynamic system models have provided an overall energy balance analysis [7].Generally speaking, these study integrated interdisciplinary factors and in particular used the interaction between economic activities and surrounding environmental factors.
The artificial fish swarm algorithm (AFSA) [23] is one of the swarm algorithms, which is inspired by the social behaviours of the fish school in searching, swarming and following.A schooling fish can response quickly to the changes in the direction and speed of their neighbours, and the information of their behaviours can be passed to others and help them move from one configuration to another almost as one unit.By borrowing this intelligence of the social behaviours, the AFSA is parallel and independent to the initial values, and able to achieve a global optimum.In this paper, an AFSA method with the variable of population size (AFSAVP) is proposed, which is applied to the intelligent approach for a spatial analysis that incorporates three economic components and one environmental factor of carbon dioxide emission.

Variable Population Size Fish Swarm Algorithm
Inspired by the swarm intelligence of the fish schooling behaviours, the AFSA is an artificial intelligent algorithm that firstly simulates the behaviour of an individual artificial fish (AF) and then constructs an AF schooling.Each AF searches its own local optimal solution and passes information to its self-organized system, and finally achieves the global optimal solution.
The AFSAVP provides a one-variable AF population P .As stated in Equation ( 1), the AF school population P is made up of two sub-schools: the replaceable school P R and the non-replaceable school P N .Equation (2) gives P R at generation t, which is calculated by its own size plus the off-size ∆P R at generation t − 1.As stated in Equation ( 3), the off-size replaceable population is proportional to the size of P R , where λ is a reproduction rate set by users.
The AFSAVP work-flow is given in Figure 2, which is similar to the basic AFSA [23,24].The AFSAVP includes five steps of operations: (1) behaviour selection; (2) searching behaviour; (3) swarming behaviour; (4) following behaviour and (5) bulletin.A "max-generation" is the trial number of an AF school searching for food under a given initial condition, which is one of the widely used criteria for the AFSAVP simulation termination.
Behaviour Selection: in the AFSAVP, the behaviour selection step takes searching behaviour as the default or initial behaviour for each AF.According to the food density, the number of companion and the visual conditions, the AF school select their behaviours, which includes searching behaviour, following behaviour and swarming behaviour.
Searching Behaviour: for a certain AF individual k, S k = {s 1 , . . ., s M } is its finite state set, there is M states that an AF can perform in.Within the AF's visual field, if the current state of this AF is S i and the next state is S j , the AF moves from S i to S j randomly and check the state updating conditions as stated in Equations ( 4) and (5).As demonstrated in Figure 1, r ij = S j − S i is the distance between the i th and j th individual AF.= f (S) is the food density for this AF, where is the fitness function.δ is the iterate step, υ is the AF visual constant.
Figure 1.The state distance between the i th and j th individual.
Swarming Behaviour: suppose the number of this AF's neighbours is α, the central state is S c , the food density is c = f (S c ) and η is the crowd factor.Within its visual field (r ij < υ), if the c /α > η i and η ≥ 1, the AF implements the central state driven step; otherwise, when the c /α ≤ η i or η = 1, the AF will go on with the searching behaviour, as expressed in Equation (6).
Following Behaviour: when the AF's companions reach "max" state S max with the number α within the neighbourhood, the food density reaches max at the mean time.As stated in Equation (7), with the same conditions as Equation ( 6), the AF updates its state in highest food density area; otherwise, the AF will go on with the searching behaviour, as expressed in Equation (6).
Bulletin: the bulletin operation is a step to compare each AF's current state S i with the historical state data, the bulletin data will be replaced and updated only when the current state is better than the last one, as described by Equation (8).
Figure 2. The variable population size fish swarm algorithm workflow.

Quantitative Analysis Framework
A framework of this quantitative analysis of dynamic behaviours is given in Figure 3, which summarises five steps for the overall process.Step 1 is to introduce the data source, which includes statistical yearbooks, independent reports and other big data sources.Step 2 is data management, which is to capture, clean, store, manage, and process the data within an applicable time scale.Step 3 is the data analytics, in which the gross domestic product (GDP) is used to estimate CO 2 emission and the AFSAVP method is utilised to optimise the parameters of the multi-state dynamic behaviours model.
Step 4 is to visualise the outputs and present the results.In step 5, the result is employed for advising policy makers and data enabled applications.According to "Revised 1996 IPCC Guidelines: Reference Manual, Volume 3" [25], the estimation of the CO 2 emissions, which is related to the productive activity for the rural area of each province, requires three major data for each of fuel type: the heat conversion factors (HCF), the carbon emission factor (CEF) and the fraction of carbon oxidised (FOC) in each sector.As listed in Table 1, the three types of data are provided by different data sources [26,27] for the rural areas with local impacts, which includes three aspects: liquid (row A), solid (row B) and gaseous (row C).

Conversion Factor Emission Factor Fraction of Carbon Oxidised i
Fuel α i (T J/10 To set the local specific values of the FOC, Qu [27] suggests that the sector specific values of FOC for coal vary between 80%-95%, which are smaller than the IPCC default value of 98%.Equation ( 9) is the CO 2 emission estimation (E) approach provided by IPCC [25], which includes main fuel combustion as listed in Table 1.China typically converts all its energy statistics into the metric tons of standard coal equivalent (TCE), a unit that bears little relation to the heating value of coals actually being used in China [28], whose HCF, CEF and FOC are given in the row i = 0 in Table 1. where, • E is the CO 2 emission, 10 • α i is the heat conversion factor, T J/10 3 t; • β i is the carbon emission factor, which is considered as the carbon content per unit of energy for its close link between the carbon content and energy value of the fuel, tC/T J; • S i is the non-energy use, 10 3 t; • A i is the apparent consumption, t; • O i is the energy production, t; • I i is the import energy, t; • X i is the export energy, t; • B i is the international bunkers, t; • R i is the stock change, t; Subject to the requirements outlined above, to ensure the comparability of country inventories, the IPCC approach to the calculation of emission encourages the use of fuel statistics collected by an officially recognised national body.However, practically, the IPCC approach can hardly be utilised for the CO 2 emission estimation with two main reasons.Firstly, recent satellite data has shown that the drop in coal consumption is probably unrealistic and the coal consumption data should not be used [2,3].Secondly, the CO 2 emissions data by fuel combustion and each local province are not directly available.
To handle this problem, Chen et al. have proposed a GDP approach [12], which employs the GDP data to estimate the CO 2 emission.Empirical studies indicate that the GDP approach provides a reasonable CO 2 emission result that is consistent with other researchers' works [29,30].As given in Equation ( 11), the TCE/GDP energy statistics of provincial data are listed in Table 2. κ j is the CO 2 emission estimation of each province j, as expressed in Equation ( 12), j = 1,2,...n 2 , n 2 = 31, which is also shown in Figure 4.The energy consumption data constructed from the GDP g j and its related energy consumption e j and electricity consumption f j of the province j can be collected and are officially provided by the "Chinese Energy Statistical Yearbook" [31].Âj is the CO 2 estimation of the equivalent energy consumption of province j, Bj is the CO 2 estimation of the equivalent electricity consumption, as given in Equations ( 13) and (14).
Âj = e j g j (13) where, • Ê is the total TCE CO 2 emission estimation of the provincial rural areas, 10 3 t; • κ j is the CO 2 emission estimation of province j by GDP, as listed in Table 2, 10 3 t; • g j is the gross domestic product (GDP) of province j, 10 4 CN Y ; • e j is the TCE energy consumption per GDP unit of province j, TCE/10 4 CN Y ; • f j is the electricity consumption per GDP unit of province j, kW h/10 4 CN Y ; • ν 0 is the ratio of electricity to TCE, which is 0.01182 in this case (Electricity is converted to TCE by the equation 104 kW h = 1.229TCE, that is, ν 0 = 1.229/104 = 0.01182 [31]); • τ , γ 0 , α 0 and β 0 are the molecular weight ratio of CO 2 to C, the TCE fraction of carbon oxidised, the TCE heat conversion factor and the TCE carbon emission factor, as given in Table 1, j = 1,...,n 2 .

Multi-State Dynamic Behaviours Modelling
As demonstrated in Figure 5, the multi-state system (MSS) is expressed by a power function (y = x a ), in which a is the multi-state factor (MSF) that can provide different multi-state combinations by different a's values.Figure 5(a) and Figure 5(b) represent an increase (a > 0) and a decrease (a < 0) in economic activities, respectively.The power function MSS representation can provide a qualitative tool to perform a functional distance related interactive analysis and can be utilised as a fitness function for the AFSAVP optimisation process.In this paper, the statistics data of 31 provinces are adopted from China Statistical Yearbook 2009 [32].In order to estimate the rural-urban(R-U) spatial interactions, a functional region affecting index (Θ) with a "law-of-gravity" interpretation [12] can be characterised as Equation ( 15) by using the basic form of a Cobb-Douglas (C-D) production function [33], which can provide a qualitative tool to perform a distance-related interactive analysis and can be utilised as the fitness function for AFSAVP optimisation.
A case in Figure 6 shows the Sichuan province location and its functional distance measurement.Taking Sichuang province (province j = 23) as an example, O is the capital of China (Beijing), r 1 and r 2 are the shortest and longest radii (the distances r 1 and r 2 are provided by the GIS system China Map, http://www.51ditu.com/)from O to Sichuan, A and B are the closest and farthest locating points of Sichuan to Beijing. where, • Θ is the functional region affecting index; • x j is the number of enterprises in the province j, n 2 is the number of the provinces; • y j is the total profits of province j, 10 4 CN Y ; • z j is the non-agricultural employment of province j, 10 4 persons; • α j = α 1j /α 0j is the population effect coefficient of province j, α 1j is the local non-agricultural population, α 0j is the population of province j, 10 4 persons; • κ j is the CO 2 emission of province j, as listed in Table 2; • η j is the multi-state factor of province j, η j ∈ [-1,1], as defined in Equation ( 16), which is the normalised g j , g min and g max are the min and max values of g j .According to the historical data and statistics, η j is utilised to describe the MSS weighted behaviour of the uncertainty analysis; • β is the factor of the functional distance r j , β ∈ [0.5,3]; • is the floating point relative accuracy, which prevents the singularity in case of the r β j is approaching to 0 and Θ is approaching to ∞, numerically; • r j is the mean value of point-to-point distance from the point O to each province j, as defined in Equation ( 17), kilometre (km).
Figure 6.Functional distance measurement of Sichuan province.

Fitness Function Definition
In order to evaluate the dynamic behaviour related to the functional distance, the CO 2 emission and population in the provincial economic production, the Θ feature is utilised as the fitness function defined in Equation ( 18) by the form of Equation ( 15), which is a cumulative measurement for the functional region and CO 2 emission spatial determination of the agreement between the statistical data and the AFSAVP driven data pool.All the statistical data for the parameters of Equation ( 18) are listed in Table 3, and the data source is the "China Statistical Yearbook 2009" [32].

Empirical Results and Discussions
The empirical data analysis for the AFSAVP driven hybrid modelling of the dynamic behaviours are implemented by the SwarmF ish [35] toolbox and the parameters of the AFSAVP calculation are stated in Table 4, in which the termination condition is 100,000 max generations, the coefficients of variation δ j for all x j , y j and z j are set to 15%, the experiment number is 100, the non-replaceable P N and replaceable population P R are 50 and 10, iterate step δ is 0.5, visual υ is 10, crowd η is 0.618, try number is 5 for each AF.  the fitness plots for 100, 1000, 10,000 and 100,000 generations, whose "x-axis" is the evolution generations and "y-axis" is the fitness values.These figures demonstrate the short term (generation = 100), short-middle term (generation = 1000), middle term (generation = 10,000) and long term (generation = 100,000) dynamic behaviours, in which the solid line, dotted line and dash-dot line are the "max", "min" and "mean" values of fitness evolution process.Different from the fitness shape for engineering applications, the fluctuations of the non-monotonic fitness curves indicate the dynamic behaviours of the provincial areas.Figure 11 is the phase portrait of the "max" fitness at 100,000 generations, which demonstrate an isolated closed "limit cycle" phenomenon.This closed and isolated cycle indicates that the limiting nature of motions of the provincial dynamic system is in an overall stable state with limit active trajectories, depending on the motion patterns and evolutionary uncertainties.3, the negative percentage errors of ∆x j (%), ∆y j (%) and ∆z j (%) indicate that the scale of the non-agricultural activities in the rural areas are in a decreasing trend, which matches the practical situation, that is, the rural areas of provinces are designed as the agriculture oriented functional regions administratively [32].In recent years, the quantitative analysis is widely utilised to assist the functional region planning decisions, which tries to take good advantage of the available statistical data.This paper provides an evolutionary alternative of dynamic behaviour modelling for functional region with distance factor and CO 2 emission factor.In this paper, the AFSAVP is firstly introduced, then the GDP-based TCE approach for the CO 2 emission estimation is discussed.For the hybrid modelling of provincial regions, the Θ factor is proposed to describe the interaction of the production inputs and outputs with the source data uncertainties, which indicates a sort of distance-based relationship between a Θ combination of the X, Y and Z inputs and the possible maximum output under current productive capability, in which the experience of a policy-maker can be incorporated in a natural or artificial way.It is clear that the CO 2 emission is related to the complicated dynamic behaviours of the functional region.The GDP/TCE approach to the CO 2 emission estimation is a practical way to deal with the lack of direct fossil fuel data source.This paper's results are consistent with the actual behaviours of the functional regions, which suggests some interesting issues for further investigation on the real-time public sector policy making, in which the consumer price index (CPI) will be one of the dynamic decision criteria for the further reliability analysis of a multi-state Θ modelling or the analysis of the regional system maintenance and sustainability.

Figure 3 .
Figure 3. Framework of quantitative analysis of dynamic behaviours.

Figure 11 .
Figure 11.Fitness phase portrait of the dynamic behaviours, 100,000 generations.
[25]• τ = 44/12 is the molecular weight ratio of CO 2 to C (Conversion between C and CO 2 can be calculated using the relative atomic weights of the carbon and oxygen atoms C = 12, O = 16.The atomic weight of carbon is 12 and that of CO 2 is 12 + 16 + 16 = 44, i.e., one carbon and two oxygen atoms.To convert from C to CO 2 thus requires multiplication by 44/12, or 3.67[25].); • γ i is the fraction of carbon oxidised (i = 1,2,..., n 1 ), n 1 = 17;

Table 4 .
Parameters for the AFSAVP simulations.

Table 5 .
An estimation X, Ŷ and Ẑ optimised by the AFSAVP.