Implementing stochastic distribution within the utopia plane of

68 Int. J. Computer Applications in Technology, Vol. 47, No. 1, 2013

Copyright © 2013 Inderscience Enterprises Ltd.

Implementing stochastic distribution within the utopia plane of primary producers using a hybrid genetic algorithm

James Nicholas Furze* Faculty of Environment and Technology, Department of Engineering, Design and Mathematics, Department of Geography and Environmental Management, University of the West of England, Frenchay Campus, Bristol, BS16 1QY, UK E-mail: [email protected] E-mail: [email protected] *Corresponding author

Quanmin Zhu Faculty of Environment and Technology, Department of Engineering, Design and Mathematics, University of the West of England, Frenchay Campus, Bristol, BS16 1QY, UK E-mail: [email protected]

Feng Qiao Faculty of Information and Control Engineering, Shenyang JianZhu University, 9 Hunnan East Road, Hunnan New District, Shenyang, 110168, China E-mail: [email protected]

Jennifer Hill Faculty of Environment and Technology, Department of Geography and Environmental Management, University of the West of England, Frenchay Campus, Bristol, BS16 1QY, UK E-mail: [email protected]

Abstract: The two key variables in estimating the water-energy dynamic, which determines proportions of plant strategy components on a macro basis, are temperature and precipitation. Additionally, use of high-resolution elevation data facilitates formation of the fuzzy rule base for ordination of the strategical nodes. Application of adaptive neural fuzzy inference systems produces sets of rules, which may be minimised to increase the efficiency of modelling the distribution of plants and their characters. A modified objective genetic evolutionary algorithm was employed in this study to show the distribution of elements of strategies within a strength Pareto. Distribution of the elements showed an approximate Poisson curve in objective space that may be extrapolated to a real-numbered population via application of optimisation algorithms to reflect the stochastic organisation of the populations.

Keywords: water-energy dynamic; adaptive neural fuzzy inference systems; genetic algorithm; optimisation; stochastic organisation.

Reference to this paper should be made as follows: Furze, J.N., Zhu, Q., Qiao, F. and Hill, J. (2013) ‘Implementing stochastic distribution within the utopia plane of primary producers using a hybrid genetic algorithm’, Int. J. Computer Applications in Technology, Vol. 47, No. 1, pp.68–77.

Biographical notes: James Nicholas Furze graduated from the University of Wales, UK in Botany in 1998, completed his MSc in Applied Genetics at the University of Birmingham, UK in 1999 and is currently with the Faculty of Environment and Technology, University of the West

Implementing stochastic distribution within the utopia plane of primary producers 69

of England, Bristol, UK. His research interests include fuzzy logic, genetic algorithms, sustainability and national environmental policy formation.

Quanmin Zhu is a Professor of Control Systems at the Department of Engineering Design and Mathematics, University of the West of England, Bristol, UK. He obtained his MSc from Harbin Institute of Technology, China, in 1983 and PhD from the Faculty of Engineering, University of Warwick, UK, in 1989. His main research interest is in the area of non-linear system modelling, identification and control. He has published over 160 papers on these topics and provided consultancy to various industries. Currently, he is acting as a member of the editorial board of International Journal of Systems Science, member of the editorial board of Chinese Journal of Scientific Instrument, editor (and founder) of International Journal of Modelling, Identification and Control, and editor of International Journal of Computer Applications in Technology.

Feng Qiao received his BEng in Electrical Engineering and MSE in Systems Engineering from the Northeastern University, Shenyang, China, in 1982 and 1987, respectively, and his PhD in Intelligent Modelling and Control from the University of the West of England, UK, in 2005. From 1987 to 2001, he worked at the Automation Research Institute of Metallurgical Industry, China, as a Senior Engineer in Electrical and Computer Engineering. He is currently a Professor of Control Systems at the Shenyang JianZhu University, China. His research interests include fuzzy systems, neural networks, non-linear systems, stochastic systems, sliding mode control, wind energy conversion systems, structural vibration control and robotic manipulation. He is acting as an Associate Editor of International Journal of Modelling, Identification and Control.

Jennifer Hill holds her MA in Geography from the University of Oxford, UK and her PhD in Biogeography from the University of Swansea, UK. She is an Associate Professor and Associate Head (Research and Scholarship) in the Department of Geography and Environmental Management at UWE, Bristol. She has authored close to 30 refereed journal papers, co-edited three books and published over 100 other items concerning natural resource management. She specialises in landscape ecology and biodiversity conservation. She is a Fellow of the Royal Geographical Society (with the Institute of British Geographers) and is a member of the Steering Committee of the International Geographical Union Commission on Biogeography and Biodiversity. She holds a number of editorial board positions and has refereed for 26 journals.

1 Introduction

Fuzzy logic and logic-based techniques are rooted in probability theory (Zadeh, 1965). The principal area of set theory with relation to development of evolutionary networks is stochaism. A stochastic process is generation of random variables, the key point being that evolution of variables is not uni-variate, but may potentially develop in many different shapes. Stochastic networks are well suited in the field of evolutionary algorithms and have extensive use in non-linear system modelling, computer technology and biological systems (Silvera et al., 2009). Just as any stochastic group, plants may be said to be functions of one or several deterministic arguments. To apply stochastic processes, the key point is that the variables determining the measured characteristics or vectors must share the same functional domain. In other words, they may be seen to show certain probability distributions such as Poisson, Gaussian or other continuous or discrete pattern, hence they often share complex statistical relationships (Zhang et al., 2012). Plants show chaotic patterns of evolution in terms of their individual growth processes and whole plant numbers (Cui et al., 2012; Furze et al., 2011; Su et al., 2009). Patterning of plant species may be determined by key factors of the water-energy dynamic (Hawkins et al., 2003; Kreft and Jetz, 2007; Sommer et al., 2010). Climatic variables such as rainfall and temperature often show discrete patterns across timescales, so are often made use of

within fixed time ranges. As such, they can be said to be discrete stochastic patterns, which facilitate macro-level modelling of plants (Grime and Pierce, 2012; New et al., 1999; Silvert, 2000).

Gaussian functions or normally deviated variables are those that may be plotted with a bell-shaped curve or parabola; they are deeply rooted in probabilistic and statistical theory and have been made use of by many authors for patterning of natural or artificially created variables (Herrera, 2005). In particular, Lotfi Zadeh made use of Gaussian distribution functions in the inception of fuzzy techniques. The effect of the Gaussian pattern is that it can normalise a distorted statistical view of a variable or individual vector. The derivative of Gaussian or normal deviation is the central sample of the variation, which is proportional to the standard deviation or greatest incline of the population range of variables under consideration. Gaussian functions arise by applying the exponential function of variables to a general quadratic function; the key premise is hence the natural logarithm of the system under considerations total. The latter makes Gaussian distribution a central function to minimise the error within Boolean statistics, as such it may be seen to be a central premise of higher (logic based) mathematics of great application in the analysis of complex systems through inferential/differential equations (Angelov and Feliv, 2004; Zadeh, 1973; Zhao, 2012). Furthermore, the normal deviation can be estimated most effectively by an iterative estimation of differential

70 J.N. Furze et al.

weighted least squares until the tail ends of the bell curve are produced (Guo, 2011). Gaussian distribution is related to other functional spreads of variation such as Poisson distribution.

Evolutionary computing methods include genetic algorithms (Broekhoven et al., 2007; Zadeh, 1973). These algorithms make use of representation of the components of variation as vectors (chromosomes) within strings (populations). The chromosomes recombine in an iterative process under specified conditions involving the same elements of genetic recombination as in natural systems. As in biological systems, operators are selection, crossover and mutation. Genetic algorithms were originally proposed by Lotfi Zadeh; however, they have proved to be exceptionally useful in more recent studies. In combination with fuzzy set theory, they are robust, stochastic evolutionary computational algorithms (Su et al., 2009). GAs are adaptive algorithms for finding the best (global) solution to optimisation problems. The stages of GA are as follows: start with a population of randomly generated ‘chromosomes’ (not actual chromosomes but sequences of defined length); initialisation, the collection of ‘chromosomes’ (sequences) evolve through a form of natural selection. Each iteration of selection is known as a generation, and the chromosomes are rated for their ‘adaption for solutions’ or potential to solve the problem. On the basis of the evaluation, a new population of sequences is formed using a process of selection. At this point, genetic processes analogous to crossover and mutation take place. After further selection, given the solution is found an output is given. Evaluation or fitness function must be devised for each problem to be solved. Given a particular sequence or chromosome solution, the evaluation function returns a single numeric value proportional to the adaptation of the solution represented by the chromosome or sequence (Cordón et al., 2004). Populations may go through continual cycles of the GA depending on the complexity of the population itself and the changing conditions in which it exists. In fact, running a GA may produce a series of the best solutions for each chromosome of any given population known as the Pareto front. Recent work on GAs makes use of the Multiple Objective Genetic Algorithm (MOGA) to produce a Strength Pareto Evolutionary Algorithm (SPEA), which has been advanced in further work to become the most popularly used form of GA in categorisation of complex systems with multiple elements (Wendt et al., 2010). Similar approaches to GA are made use of in recent modelling of plant systems (Cieslak et al., 2011; Cui et al., 2012). It is proposed that the elements of plant species such as life form (Raunkier, 1934), metabolism type (Silvera et al., 2009) and life-history strategies of plant species are effective candidates for the use of hybrid GA systems (Furze et al., 2012b). Plants are primary producers affecting every trophic level of life on earth.

A natural progression of GA and fuzzy systems is the estimation of population’s resultant distribution; this forms an additional method of approximation: Estimation of Distribution Algorithms (EDAs). These algorithms

have applications in computing, industry and natural systems. Copula theory is a concise, robust form of EDAs (Nelson, 2006). Copulas join multivariate distributions to one-dimensional marginal distribution functions, which make them ideal candidates for the analysis of patterns of genetic variation such as that which is produced by the Multi Objective Optimisation (MOO) or Pareto front of a MOGA process (Wang et al., 2012).

The objective of this study is to show proportions of plant strategies nested within fixed population sizes, which have been determined by a genetic (generational) algorithm fuzzy rule base. We also hope to estimate the Pareto front for plant strategies and make further exploration into the utopia hyperplane (Erfani and Utyuzhnikov, 2011) of objective space for plant strategies via the construction of code in the technical computing platform, Matlab (version 2010a©). As we aim to show a minimised (efficient) modelling framework, it is envisaged that techniques employed within this paper may increase the accuracy of both data sources of plant variation employed and climatic modelling data.

This paper is organised as follows. Section 2 presents the data used to form the fuzzy algorithm of plant strategies. Section 3 gives a brief ten-point summary of the methods used for the hybrid MOGA. Section 4 shows the Takagi-Sugeno-Kang (TSK) nodal structure, gives a rule base structure and algorithm for plant strategies. Section 5 shows quantification of ideal solutions of elements of plant strategies. Section 6 details the multi-objective genetic algorithm (MOGA) process and the Pareto frontier resulting from the MOGA is shown along with objective quantification. Section 7 provides a discussion and concluding remarks for further work.

2 Use of climatic and Digital Elevation Model (DEM) data to establish fuzzy inferential systems

The data presented in this section is an example of data that may be used to form fuzzy algorithms, on which we operated a genetic (generational) structure. We successfully predicted water-energy objective dispersion of plant species strategies (Furze et al., 2012a). As the process of quantification has been covered in two recent publications by the first author (Furze et al., 2012a, 2012b), it is not included in this paper, should further detail be required contact the first author. The area covered is that of Azerbaijan, southern Europe, which falls in between latitude 42.14 degrees north, 38.33 degrees south and longitude 50.53 degrees east, 44.78 degrees west. Coordinates of the location were obtained using United States Geological Survey (USGS) data and extracted using Matlab (version 2010a©).

Azerbaijan quarterly mean temperature (1961–1990) at 18.5 km resolution is shown in Figure 1, characteristic of the energy component of the water-energy dynamic used to form the algorithmic framework for modelling of strategies.


Environment E6, Azerbaijan quarterly mean precipitation (1961–1990) is shown in Figure 2. Data of Figures 1 and 2 was originally recorded by New et al. (1999) and has been used by the Inter Governmental Panel on Climate Change (IPCC) for the use of climate modelling. It was made use of in the current study after having been validated as a characteristic of the water-energy dynamic (Sommer et al., 2010). The legend of the figure was used to obtain a unit percentage and subsequently the figure was summarised as shown in Section 4.

The DEM data of GTOPO30, published through the United States Geological Survey (USGS), is shown graphically in Figure 3, being at 30 s (1 km resolution). Data was processed using Matlab (version 2010a). Code was constructed for mapping and image-processing sections of Matlab and is available on request.

Figures 1–3 were processed to follow the fuzzy algorithm summarised through the adaptive neural fuzzy inference system shown in Section 4. The following section describes fuzzy rule bases and proceeds to elaborate on the application of multi-objective genetic algorithm to plant strategies.

Figure 1 Azerbaijan energy (mean temperature 1961–1990) data, 18.5 km resolution (see online version for colours)

Figure 2 Azerbaijan water (mean precipitation 1961–1990) data, 18.5 km resolution (see online version for colours)

Figure 3 Area of Azerbaijan, Europe DEM data (see online version for colours)

3 Multi-objective genetic algorithm and the application process for strategies

The development of rule-based (eR) systems (FRBS) may use a summary of the information, in terms of the potential of a new data sample (such as accumulated spatial proximity information), to trigger the new rule base. Greater generality of the structural changes to the data can, therefore, be catered for (Angelov and Filev, 2004). In terms of plant strategy estimation for example, we may state the membership functions in terms of the seven environments (as demonstrated by Barreto (2008)) and obtain the data for the placement of species within each by considering optimisation limits (given that the initial number of species is known (Broekhoven et al., 2007)). Using this method, we may obtain stochastic matrices, effectively Kp or rK continuums. In combination with a generational algorithm approach, the rule base may be successfully used to generate the distribution present in natural systems (Elith et al., 2011). Specific detail of the processes involved is covered in subsequent sections; a ten-point summary for the combined methods is included here:

• Determine structure via identification of complete sets of background/environmental data.

• Derive minimal TSK IF–THEN Rule base for strategical nodes of network organisation.

• Identify elements of strategies (node structure).

• Use expert-based intuition to rank ideal solutions of each element, represent within chromosome population structure.

• Identify constraints (stopping conditions), key objectives and total number (maximum) individuals/generations for generation of ideal solution combination (MOO) to achieve the Z Utopia Hyperplane.

• Generate random population of chromosomes and operate random selection of chromosomal values via MOGA.

Long

itude


• Allow algorithm to run until sets of ideal solutions have spaced in objective (z) space – this determines the Pareto Frontier.

• Identify distribution of Ideal Pareto Frontier, fit linear, quadratic and polynomial functions as required to precisely describe relative distribution of chromosomes within maximum number of individuals. The linear fit of the Pareto frontier is defined as the Utopia Line (The Utopia objective space is represented by the error of each individual Pareto solution; this may be determined by binary conversion of each Pareto (P) generating set, equal to 0–1).

• Approximation of the distribution may determine real proportions of strategies within strategical nodes (stochastic distribution in this case).

• Apply solutions to ranges identified through TSK FRBS, additionally structure may be checked via pattern identification (Angelov and Feliv, 2004; Juang and Hseih, 2012; Salah and Abdalla, 2011) to determine the network density.

4 Structure of the adaptive neuro fuzzy inference system (ANFIS) for plant strategies

Ordination of plant strategies 1–7 (environments) was carried out in specific locations, which were chosen at random, though are representative of the strategies

according to the water-energy dynamic (Furze et al., 2012a; Grime and Pierce, 2012; Hawkins et al., 2003; Jetz et al., 2009). The ANFIS for E6 (Competitive-Stress Tolerant, C-S plants), found in Azerbaijan, Southern Europe, is shown in Figure 4.

In layer 1 of the above-mentioned figure, input variables were estimated and quantified from the data sources (Intergovernmental Panel on Climate Change for climate variables, United States Geological Survey for altitude). The original data was partitioned into 5 linear ranges across 100% for the purposes of the algorithmic statement, the ranges being 0–20%, Low (A11); 20–40%, Low-Medium (A12); 40–60%, Medium (A13); 60–80%, Medium-High (A14) and 80–100%, High (A15) in the case of temperature. Precipitation, ground-frost frequency and altitude were all quantified in the same way. Weights were added to the variable ranges to make an accurate inference of the plant strategy present in Azerbaijan (E6). Layer 2 transferred the variable ranges through 5 equally partitioned triangular functions between 0 and 1, in the fuzzification process. The fuzzy ranges were separated in layer 3 according to the rules applied (19 in the case of E6 shown here). Layer 4 returned fuzzy output depending on the combined distributed range as instructed; finally, layer 5 gives the crisp output, sourced from the Global Biodiversity Information Facility. The corresponding author may be consulted for the modelling framework from which the ANFIS was constructed; alternatively, details are given in Furze et al. (2012a).

Figure 4 ANFIS node structure for E6 (see online version for colours)

4.1 Rules and interpretation for strategy present

in E6

The ANFIS structure of Figure 4 operated under 19 rules with different weights as shown:

1 If (Temperature is Medium) then (Strategy is S-C) (0.25)

2 If (Temperature is Medium-High) then (Strategy is S-C) (0.25)





6 If (Precipitation is Low) then (Strategy is S-C) (1)

7 If (Precipitation is Low-Medium) then (Strategy is S-C) (1)

8 If (GFF is Medium) then (Strategy is S-C) (0.25)

9 If (GFF is Medium-High) then (Strategy is S-C) (0.25)

10 If (GFF is High) then (Strategy is S-C) (0.25)

11 If (GFF is Low) then (Strategy is S-C) (0.5)

12 If (GFF is Low-Medium) then (Strategy is S-C) (0.5)

13 If (GFF is Medium) then (Strategy is S-C) (0.5)

14 If (GFF is Low) then (Strategy is S-C) (0.25)

15 If (GFF is Low-Medium) then (Strategy is S-C) (0.25)

16 If (Altitude is Low) then (Strategy is S-C) (1)

17 If (Altitude is Low-Medium) then (Strategy is S-C) (1)

18 If (Altitude is Medium) then (Strategy is S-C) (1)

19 If (Altitude is Medium-High) then (Strategy is S-C) (1)

The above-mentioned rules were concisely expressed in the algorithm statement for E6 as follows:

IF 0.25A1(3) – A1(4), 0.5A1(4) IS ≤ A1(4), 0.25A1(4) – A1(5) AND A2(1) – A2(2) AND 0.25A3(3) – A3(5), 0.5A3(1) – A3(3), 0.25A3(1) – A3(2) AND A4(1) – A4(4) THEN B(8805) = E6 (1)

In equation (1), A1 was the representative of temperature (Figure 1), A2 was the representative of precipitation (Figure 2), A3 was the representative of ground frost frequency (data not shown), A4 was the representative of elevation (Figure 3) The network diagram of Figure 4 was constructed using the technical computing platform Matlab (version 2010a©). The ‘stand-alone’ code, which may be operated independently of Matlab, is available from the corresponding author on request.

Algorithms for plant strategies (Furze et al., 2012a) conform to the original definition of logic-based systems, the statement being concisely given by Zadeh (1965):

( ) /AUA y yµ= ∫ (2)

where the Laplace transform union membership of input(s) A is equal to the constituent singletons (y), in this case being variable data temperature, precipitation, ground frost frequency and altitude (Furze et al., 2012a; Hawkins et al., 2003). Furthermore, during a recent study (Furze et al., 2012b), it was shown that the essential elements of the water-energy dynamic may be efficiently reduced down to temperature and precipitation in the derivation of the fuzzy control algorithm for E2 (e.g., Guyana, South America). This was consistent with the present work as may be seen in Figure 5, which shows the surface view of the algorithm for competitive-stress-tolerant plants in E6 (C-S).

Figure 5 3-dimensional view of the surface of E6 algorithm (see online version for colours)

In Figure 5, it is shown that the Competitive-Stress-tolerant strategy of E6 is efficiently instructed by reduced variables of the water-energy dynamic. In addition to the earlier statement of equation (1), we may subsequently state the input vectors of Figure 5 to be:

1 2[ , ,... ]nx x x x= (3)

1 2[ , , ]ny y y y= … (4)

where x and y are the singletons temperature and precipitation, respectively, hence we state:

[ * ]y x R= (5)

where the relational index (R) is seen to be a function of input variables x and y (Omizegba and Monikang, 2009).

Defining the relational index confers that there is an objective space within which the combination of input


variables is able to propagate in variable conditions. Global optimisation techniques are often used to calculate the dispersal of elements within objective space following application of the modelling technique (ANFIS). In order that we may apply the genetic algorithm on the defined plant population, we identify and quantify ideal solutions of the elements of strategies (Zadeh, 1973) in the next section.

5 Plant strategy solutions and quantification In this part, elements of plant strategies are detailed, defined and given quantifiable values.

In Table 1, the elements of plant strategies are: PT = plant type, sm = shoot morphology, lf = leaf form, c = canopy, loep = length of established phase, lor = lifetime of roots, lp = leaf phenology, rop = reproductive organ

phenology, ff = flowering frequency, poaps = proportion of annual production for seeds, podup = perennating organs during unfavourable periods, rs = regenerative strategy, mpgr = mean potential growth rate, rrd = response to resource depletion, pumn = photosynthetic uptake of mineral nutrients, ac = acclimation capacity, sop = storage of photosynthates, lc = litter characteristic, psh = palatability to non-specific herbivores and nDNA = nuclear DNA amount. Ideal quantification is seen in brackets in the table. The elements identified in Table 1 represented a chromosomal population and were used to form a MOGA, in which the chromosomes cycled (via roulette wheel selection) through 1–5, resulting in a Pareto front. The process was carried out in order that estimation of the distribution of each plant strategy in the previously algorithmically defined environments could take place.

Table 1 Solutions and ranges for plant strategy chromosomes

Character/ chromosome Competitive (1, …, 5) Stress tolerating (1, …, 5) Ruderal (1, …, 5) PT Herbs, shrubs, trees (1, 2, 3) Lichens, bryophytes, herbs, shrubs and

trees (4, 5, 1, 2, 3) Herbs, bryophytes (1, 5)

sm Long with extensive above and below ground (3)

Long, short and intermediate (3, 1 and 2)

Short stem, limited lateral spread (1)

lf Robust, large often require high water (5)

Small (1), leatherly (1) or needle-like (2) low water requirement (1) (1, 1 or 2, 1)

Variable, often require high water (1, …, 5)

c Rapid upward growth of one layer (5, 1)

Multi layered (5) if mono (1) layered, slow (1) upward growth (1, 5, 1)

Variable (1, …, 5)

loep Long or relatively short (4 or 2) Long to very long (4 ≤ 5) Variable (1, …, 5) lor Relatively short (2, 3) Long (4) Very short (0, 1) lp Well defined peaks of leaf production

coincides with periods of maximum productivity (5 ≡ 5)

Short phase of production within period of high productivity (1, …, 4)

Evergreen, with various patterns of leaf generation (5 ≡ 1 ≤ 5)

rop Flowers produced after periods of maximum productivity (1, …, 5)

No relationship between productivity and flowering time (1, …, 5 ≠ 1, …, 5)

Flowers produced early in life-history (often before maximum growth) (5 = 1)

ff Established plants flower every year (5)

Flowering intermittently over a long life-history (4)

High frequency of flowering (>1 a year)

poaps Small (1) Small (1) Large (4) podup Buds and seeds (1, …, 5) None (0) Seeds (5) rs Vegetative (1), seasonal regeneration

in gaps (1), wind dispersal of small seeds (1), persistent seed bank (5) (1, 1, 1, 5)

Vegetative, wind dispersal of small seeds, persistent juvenile bank (1, 1, 5)

Seasonal regeneration in gaps (2), wind dispersal of small seeds (1), persistent juvenile bank (5)

mpgr High (5) Low (1) High (5) rrd Rapid morphogenetic responses in

form and distribution of leaves and roots (5, 5)

Slow, small morphogenetic responses (1, 1)

Rapid cessation of vegetative growth and reallocation of resources into flowering (5 ≡ 1)

pumn Strongly seasonal coinciding with long continuous period of vegetative growth (5 = 1)

Opportunistic, uncoupled from vegetative growth (1 ≠ 1)

Opportunistic, coinciding with vegetative growth (3 = 1)

ac Weak (0, …, 1) Strong (5) Weak (0, …, 1) sop Rapid incorporation in vegetative

structure and compartmentalised storage for growth in next season (1)

Storage in leaves, stems, or both (2, 3 or 5)

Seeds (4)

lc High volume (5), non-persistent (0) Sparse (2), persistent (5) Sparse (2), non-persistent (0) psh Various (0, …, 5) Low (1) High (5) nDNA Small (1) Small (1) and high (5) Small (1) to very small (0, … ≤1)


6 Process of multi-objective genetic algorithm coding for plant strategy

The process of multi-objective genetic algorithm involved the following main process steps:

1 Define each vector for plant strategies.

2 Randomly generate an initial population of 20 solutions (chromosomes).

3 Evaluate each solution according to how well it fits into the desired evironment (as defined in equation (1)).

4 Select chromosomes randomly (tournament selection), keep those with the highest fitness function to improve the population, discard those with too low (value may be previously calculated) fitness.

5 Create new chromosomes by crossing selected solutions to produce new individual chromosomes.

6 Mutate a previously determined proportion of the populations chromosomes.

7 Go back to step 3.

Code was constructed in Matlab, which determined the number of parameters varying to be 4 (temperature, precipitation, ground frost frequency and altitude). The variables were directed through two main objective types (water and energy) and expressed as a double vector population in Matlab. The algorithm was programmed to stop when the total number of individuals had reached that of C-S, E6. Code is available on request from the first author.

The number of iterations required was recorded and the distribution of ideal solutions for each of the 20 chromosomes across objective space was noted. Linear and quadratic lines were fitted to the resulting Pareto in order that an approximation of the distribution be made. A plot was carried out in order that the position of the linear

fit (representing the Utopia line) and the shape of the curve be visualised in order that the evolution of the 20 characters could be estimated through future transgressions.

Figure 6 is the combination of ‘good rules’ and ‘bad rules’ (Cordón et al., 2004) in the formation of a dispersive pattern of chromosomes (see Table 1). The lower part of the figure shows the residual error, which resulted from both the linear and the quadratic fits. These errors should be considered over the range 0–1 and represent the potential expansion of the Pareto front through objective Utopia space.

The Pareto shows approximate Poisson distribution (probability) and may be summarised using the following statement for complex dynamic dispersive vectors (Angelov and Feliv, 2004).

2e [ ] : {1, } {1, }.aij j jjx x i R j nµ −= − = = (6)

Equation (6) is often used to describe antecedent sets (Angelov and Feliv, 2004). In the equation, e is the exponential function, a = 4/R2 and R is a positive constant, which defines the spread of the antecedent and the zone of influence of the ith model (in this study, we may use the covariance or relational index R as discussed in Section 2.1 (equation (5)). The objective space that Figure 6 is plotted over is a representation of the two main driving variables – temperature and precipitation.

In the case of Figure 6, temperature and precipitation are n objective functions, which may be expressed as Z in the following:

Z ⊆ Rn (7)

where vectors of Z are seen to be within the correlation (Relational) matrix multiplied by the number of objectives. From the latter, it followed that there was a design variable (D) formed from the MOO, which may determine vectors within the objective space. The design variable is the image of the Z space (Erfani and Utyuzhnikov, 2011).

Figure 6 Plant strategy evolutionary strength pareto (see online version for colours)


The following expression is used as a generic form of the optimisation:

Min F = {F1(x), F2(x), …, Fn(x)}, subject to x ∈ D (8)

where F represents different optimisation solutions (or functions in the case of F1, F2, Fn, with x representing set values of individuals). The error of D in equation (8) was seen to be 0.02717. The optimisation may be expressed across a given total number of individuals following a Poisson distribution with use of quadratic or varying degrees of polynomial expressions with greater differentiation to express greater numbers of categories within the optimisation (Zhao, 2012). We may approximate a similar distribution with a lesser number of grouped characters (such as strategies, or photosynthetic metabolism type) or a greater number of characters such as life form owing to expanding/contracting dimensional relations given through fuzzy and interval type 2 fuzzy inference systems (Raunkier, 1934; Zadeh, 1965; Zhao, 2012). The beauty of the MOGA Strength Pareto is that one may substitute any value of objective 1, temperature and return a value for objective 2; these values being subject to the parameters entered (population size, number of characters to be dispersed). This method may, therefore, be used to identify accurate prediction of temperature or precipitation given a finite population number.

7 Discussion Genetic algorithms are a proved method of dispersing random variables in an objective space (Liao et al., 2011; Notario et al., 2012). This study has made elegant use of a fuzzy-genetic algorithm approach to characterise the dispersal of elements of plant strategies in objective space defined by the water-energy dynamic (Hawkins et al., 2003). The 20 elements of plant strategies were randomly generated using previously determined scales and selected in conformations. Chromosomes are measured against the objective functions of temperature and precipitation. Before evaluation under the fitness function, the variables are selected and recombined to generate the most efficient combination. Given that the number of generations required in the final expression is met, the algorithm stopped with the vectors spaced in the objective (Z) space. The dispersal of the elements displayed the Pareto front. The scale of the plot is arbitrary as it is a result of the combination of objectives. Therefore, one may impose any number scale on the axis of the Pareto. It is most efficient to imagine that the scale is 0–1, and the error of the Pareto distribution is, therefore, less than 1. Approximating the distribution (probability) means that the distribution can be further applied to a larger or smaller number of variables, giving a similar result.

The result of the strength Pareto displays the natural dispersal of unweighted vectors (elements of plant strategies) within the natural systems of primary producers. The Poisson-type dispersal may be summarised by the Linear (weighted least squares) expression. The Utopia Front is one that may be described with fundamental

principles of exponentiality and is hence of a factorial nature. Explorations of data within the error shown by the linear and quadratic expressions given are of the Utopia Hyperplane (Erfani and Utyuzhnikov, 2011) and may be extrapolated onto different scales.

Application of logic-based mathematics (fuzzy systems) in combination with genetic computational methods provides an ideal method by which one may form a statement of the stochastic distribution of elements of plant strategies. The neural-network-based approach determines the crisp definition of each strategy within variable locations (Azerbaijan – E6 is used in this paper). Subsequently, MOGA demonstrates that elements of the strategies disperse within each defined node of the network. It is essential to make use of both techniques, as one cannot quantify dispersal of elements of any strategy before having identified the parameters within which the strategy is present.

This work enhances our ability to make predictions of numbers of plant strategies formed in previously defined environments, effectively combining fuzzy rule bases and the process of genetic algorithms; as such the term stochastic distribution within the Utopia plane is highly appropriate. One possible direction for further work is the employment of fast computational algorithms, which can generate stochastic networks for a chosen number of classes within a fixed final population number (Marsaglia and Tsang, 2000). Additionally, exploration of the copular distribution may elucidate a possible covariance/relationship to investigate the growth of biological populations (Schölzel and Friedrichs, 2008; Wang et al., 2012), particularly within the water-energy dynamic. The potential of this work unveils the method by which one may explore proportions of metabolism type and secondary metabolites (e.g., expression of medicinal compounds) with multiple uses. This study is proposed to be of assistance in the formation of national policies towards ultimate conservation goals for the good of countries inhabitants and international conservation objectives. The inference on climatic data used to form the original fuzzy rule bases is also enhanced.

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Implementing stochastic distribution within the utopia plane of