The article presents the application results of the combinatorial-genetic algorithm COMBI-GA for modeling from experimental data of amylolytic microorganism quantity in a soil plot contaminated by copper. The constructed nonlinear mathematical models describe dependence of microorganisms concentration in soil from basic vital environmental factors and the copper concentration in the soil.
One of the main components of most ecosystems is soil,
in which microorganisms play an important role in the
evolution and formation of fertility. Anthropogenic pollution
of the biosphere affects all living components of
biogeocenoses, including soil microorganisms [
Modern agroecosystems are subject to the considerable
technogenic influence resulting frequently in pollution of
arable soils. Pollutants influence negatively on soil
mikrobiotics, which causes the necessity to carry out
longterm observations after its current status. In parallel with
monitoring, there is the task of determination of critical
deviations and prediction of the mikrobiotum state
dependently on pollutants concentration in soil. Solving this
task is possible only by formalization of monitoring data in
the form of mathematical models [
Investigation of the effect of specific anthropogenic factors, such as heavy metals, on microbial community functioning is very important. Negative influence of heavy metals on microbial kenosis, soil and biological activity is well-known. In this connection, the organization and carrying out of regular control of soil state for the purpose of critical situation detecting and forecasting are actual.
Thus the dynamics of change of microorganism quantity in soil is studied under influence of different ecological factors. Modeling from observation data is the necessary condition of the ecological monitoring as it allows operative estimating current ecological situations and forecasting their development.
In this research we find models of dependence of microorganisms functioning from supporting weather conditions, this means kind of models, where inputs are hydrometeorological variables and outputs ecological indexes. Such models can be used both for restoring the omitted data and for forecasting the development of the controlled ecological processes on the basis of weather terms prediction. For construction of such models we use the Group Method of Data Handling (GMDH) as the most effective method for the analysis, modeling and forecasting of complex processes from experimental data under conditions of incompleteness of a priori information and short samples given.
To analyse of soil microbiotics of dark gray podzol soil
(Kyiv region) polluted with heavy metals, the automated
system of simulation ASTRID [
In [
Section II of this paper describes briefly task of modelling. Section III considers hybrid combinatorial-genetic GMDH algorithm COMBI-GA and their features. Section IV presents modelling results.
Experiments regarding functioning amylolytic
microorganisms under copper contamination were carried out
on small plots in deep-gray podzolic soil (Kyiv region). The
The traditional chart [
Model contamination of soil was carried out by the annual one time bringing in soil solutions of Cu2+ salts at the beginning of a vegetation season. The amount of the applied metal (computed as content of their ions) corresponds to contamination doses of 2 maximum permissible concentrations (MPC). The soil pieces for the analyses was taken during vegetation periods from 1993 to 1996 from the arable layer depth (0-20 cm) approximately in 2nd, 30th and 90th day after bringing of the metal salt. It was hence received three measuring points during four years or 12 points together.
The amount of amylolytic microorganisms in the control and polluted soils was determined by the method of sowing of soil suspension on a nourishing medium consisting of a starch-ammonia agar.
Based on observations data of below listed variables, the
linear mathematical models were built in [
As input (independent) variables (factors) for construction of models were used: concentration of mobile forms of Cu2+, decade average values of temperature, humidity of soil and air, and number of microorganisms in soil of control unpolluted plot. Quantity data of amylolytic microorganisms were output (dependent) variables in plots with model pollution of soil by copper salt. Based on the obtained data, the models of microorganisms number in soil were built.
For construction of model of changing quantity of amylolytic microorganisms such list of input variables was formed: x1 – quantity of microorganisms in the control plot (millions in 1 g of dry soil); x2 – concentration of copper (mg/cg soil); x3 – number of days from the date of pollution; x4 – current decade average temperature of air (oC); x5 – previous decade average air temperature (oC); x6 – current decade average humidity of air (%); x7 – previous decade average humidity of air (%)
A definition of the inductive modelling problem in this task may be done as follows. Let us given: a data set of n observations after 7 inputs x1, x2, …, x7 and one output y variables. The GMDH task is to find a model y=f(x1, x2,…, x7, θ) with minimum value of a given model quality criterion C(f), where θ is unknown vector of model parameters. The optimal model is defined as f*=argminΦC(f), where Φ is a set of models of various complexity, f∈Φ.
The genetic algorithm [
Since GA is based on the principles of biological evolution and genetics, biological terms are used actively (and sometimes incorrectly) to describe them. Here are some of these terms. Individual is the potential solution to the problem; population is a set of individuals; offspring is usually improved copy potential solution (father); fitness is usually a quality characteristic of the solution. Chromosome is encoded data structure of an individual in the form of an array of fixed lengths. In the simplest case it’s a binary string of fixed length. The gene is an element of this array.
Formally, GA can be represented in such a way:
GA = {P0 , M , L, F , G, s} , where P0= (a10 ,..., aM0 ) is an initial population; ai0 is an individual of this population treated as a candidate for the solution of the optimization problem presented in the form of a chromosome; M is the population size (integer number); L is the length of each chromosome of the population (integer number); F is a fitness function of an individual; G is a set of genetic operators; s is the algorithm stopping rule.
As input data for any GA initial population Р0, a finite set of chromosomes is used each of which represents a potential solution of the problem. Then the first population of offspring Р1 is formed from the parent chromosomes Р0 using some genetic operators, similarly the next population Р2 is formed from the population Р1 and so on. The process continues until the specified stopping rule of the algorithm will be satisfied.
An important feature of the GA work is that with each step the mean FF value of the current population improved and converges to the solution of the optimization problem.
The effectiveness of GA's work depends on the method of encoding genes, the composition of the initial population used by genetic operators, GA parameters, such as population size, number of chromosomes selected during selection and for crossover, probability of using genetic operators. The most important in GA are genetic operators especially the selection of which stores a certain amount of chromosomes with the best values of FF for each iteration of GA, and the operators of the creation of new offspring-chromosomes such as crossover and mutation. The crossover operator creates offspring by exchanging genetic material between the parent chromosomes, and the mutation operators by changing one chromosome in accordance with certain rules.
Formally, the hybrid of COMBI [
COMBI-GA = 〈Z, y, f, X, D, CR, Р0 , H , M , G, k, F 〉 , where Z[n×r] is the measurement matrix of input variables of an object, r and n are numbers of inputs and measurements respectively; y[n×1] is vector of measurements of an output variable; f [m×1] is vector of a given m base functions of input variables; X[n×m] is the measurement matrix of base set of arguments; D is a given rule of dividing matrix X[n×m] and vector y[n×1] to testing A and checking B parts; CR is an external selection criterion (as fitness function) based on dividing the sample (X, y); Р0 is a set of model structures of GA initial population consisting of binary chromosomes (encoded structure of partial models); Н is size of initial population of models, Н<m; M is size of any next population, М>H; G is set of genetic operators;k is stopping rule of GA; F is number of best partial models (freedom of choice) monitored during all iterations of the algorithm, 1< F ≤ H.
This algorithm consists of the following steps:
Step 1. Calculating the matrix of the base set of arguments X[n×m] using the input matrix Z and the vector of base functions f and dividing it and the output vector of measurements y[n×1] according to the rule D in testing XА[nA×m] and checking XВ[nB×m] submatrices (nA + nB = n). Obviously, in the case of linear polynomial, matrices X and Z are identical (m = r).
Step 2. Random generating the initial population Р0 of the genetic algorithm.
Step 3. Calculating the coefficients of each partial model by LSM or another method using the training matrix of base arguments XА and output vector yА.
Step 4. Calculating the value of an external criterion CR (as the GA fitness function) for each partial model using the checking matrix XВ and output yВ.
Step 5. Forming the current population of partial models (chromosomes) of the size H with better criterion values to form the next offspring. In addition, selection the best F partial models that are potential solutions of the task of model building.
Step 6. Forming new population of M individuals applying genetic operators of crossover and mutation to individuals of the current population.
Step 7. Checking a given GA stopping rule. If it is satisfied, then go to step 8, otherwise go to step 3.
Step 8. Choosing F best models from the current population of the size Н.
Step 9. The end.
Based on experimental data, models of quantity of amylolytic microorganisms were built in the control as well as in the copper polluted soils. In all cases we use the COMBI-GA algorithm with the following division of all data sample (12 points of observation during 4 years for the vegetation period 1993-1996): 6 points (2 years) as training set A, 3 points (1 year) as checking set B, and 3 points (1 year) as validation set C.
Results in the class of linear models. In case of linear
modelling, the quantity of amylolytic in a control soil Ycontr=
x1 is described by the model obtained in [
Ycontr = 0,2136x4 – 0,7149x5 + 0,5412x7 .
Models for the quantity of microorganisms in polluted soil were built taking into account the quantity of amylolytic microorganisms at the observation of the control soil x1.
The linear model for quantity of amylolytic
microorganisms in the copper polluted soil [
Y = 0,743x1+1,6516x2 – 0,9182x5 – 0,2845x7
Proper graphs of experimental and model data are given on Fig. 2. The characteristics of accuracy of these models are presented in the table below. This accuracy level is insufficient for quality monitoring needs. That is why we decide to build more complex nonlinear (polynomial) models. The results of this stage of modeling are presented below.
Results in the class of nonlinear models. In the case of nonlinear modelling, the quantity of amylolytics in a control soil is described by the model:
x1 = – 0.19297x5 – 2.0976x7 + 0.130x4x5 + 0.111x5x7 Fig. 3 shows graphs of measured and modeled data .
The model of dynamics of changing the amylolytic microorganisms in soil polluted by copper:
Y = 1,1759x5 + 0,2159x7 + 0,1638x1x5 – 0,0261x2x7 – 0,1252x4x5
The characteristics of all obtained models quality are presented in the table calculated according to next formulas: MSE = 1 1∑2 ( xi − x )2 , ARB =|| y B − X B θˆ A ||2 .
12 i =1
The designation “Valid. err.” means error on the independent validation set C calculated like ARB.
Measured Predicted
Valid. err.
Proper graphs of experimental and model data are given on Fig. 4. These graphs shows that in most points the data measured and predicted by the model coincide, that is the models adequately represents the change of microorganisms quantity. Three last three validation points on the graphs testify good results of models verification in the forecasting mode. Some distinctions can be accounted for by spatial heterogeneity of soil and other terms what could cause irregular variability of quantity. Fig. 3 Graphs of change of quantity of amylolytic microorganisms on the control plot (nonlinear model)
As it is evident from the results, the functioning of amylolytic bacteria in soil is substantially influenced by the temperature and humidity of air.
In nonlinear case we obtain much more accurate results that can further help to effectively solve different ecological tasks based on microbial monitoring.
The carried out research manifests the possibility of formalization of the given ecological observations by construction of mathematical models. For the modeling of quantity of microorganisms in soil, application of inductive approach on the basis of GMDH is effective.
The obtained nonlinear models in a high degree coincide with experimental data that enables to use them in the system of the experiments for estimation of degree of soil contamination, renewal of intermediate or omitted data and operative forecasting the dynamics of microorganisms under various ecological conditions. Equally, these models will be helpful also for the data restoration with the purpose of obtaining the uniform series of ecological observations.
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