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ISSN: 2090-0902
Journal of Physical Mathematics
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Factor Analysis of Influence of Parameters of Water Regime and Hydrological Changes on Pastures

Mazurkin PM*

Doctor of Engineering, Academician of Russian Academy of Natural History Russian, Russia

*Corresponding Author:
Mazurkin PM
Academy of Natural Sciences
Member of the European Academy of Natural Sciences
Volga State University of Technology, Russia
Tel: 7 836 245-53-44
E-mail: [email protected]

Received Date: November 14, 2016; Accepted Date: March 29, 2017; Published Date: March 31, 2017

Citation: Mazurkin PM (2017) Factor Analysis of Influence of Parameters of Water Regime and Hydrological Changes on Pastures. J Phys Math 8:220. doi: 10.4172/2090-0902.1000220

Copyright: © 2017 Mazurkin PM. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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Abstract

According to basic tabular data (DEFRA-commissioned project BD1310. Final report to the Department for Environment, Food and Rural Affairs) steady regularities between biochemical substances of the soil of pastures and hydrological parameters of their water mode are revealed shows the ranking methodology affecting and dependent factors and identification of deterministic models of the relationship between the 10 factors according to the general equation consisting of the sum of two biotechnical laws.

Keywords

Pastures; Options; Rating; Mutual Influence; Factor; Analysis; Patterns

Introduction

Analysis of binary relations between the 10 factors conducted on the data [1] (Appendix C). Soil parameters used in the hydrological modelling). Statistical modeling was performed by identification of General algebraic formula containing the sum of two biotechnical laws [2-5].

Factor analysis is understood as the identification of stable patterns of changes of values of each of the plurality of parameters of the studied systems, as well as mathematical relations between the factors. In comparison with the approximation in the methodology of identifying the truth of stable laws is accepted as an axiom. So there is no need of using empirical formula it is set in advance.

Our method of factor analysis allows not only to establish a posteriori causality, but also to give them a quantitative characteristic, provides an assessment of the level of influence of factors (influence parameters) on the results of functioning (dependent parameters). This makes factor analysis accurate method, and conclusions quantitatively valid and meaningful in the identification of regularities.

The source data

Us, it is assumed that the factors the researcher selected and the corresponding tabular model (Table 1) was compiled. Then factor analysis is the identification of the algebraic relationships between the selected factors.

Site name Topsoil
hydraulic
conductivity
(m day)
Subsoil
hydraulic
conductivity
(m day)
Topsoil
drainable
porosity
Subsoil
drainable
pororsity
Unsaturated
Hydraulic
conductivity
exponent
Rainfall
(mm)
Potential
transpiration
(mm)
SMD (mm)
(end July)
Drought
threshold
(cm)
Aeration
threshold
(cm)
x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
Belaugh 3 3 0.3 0.3 8 575 531 107 49.4 35.7
Blackthorn 0.22 0.01 0.06 0.03 3 669 511 90 48.5 23.5
Broaddale 0.7 0.35 0.14 0.09 4 1663 375 0 47.7 30.4
Cricklade 0.24 3.5 0.12 0.12 7 726 503 82 44.6 34.1
Dancing Gate _ _ _ _ _ 1045 444 39 46.4 35.9
East Cottingwith _ _ _ _ _ 643 486 85 - -
East Harnham 5.7 _ 0.11 _ 8 799 511 86 49.6 44.3
Moorlinch 0.6 0.6 0.16 0.16 8 865 523 85 46.8 27.3
Mottey Meadows 1 _ 0.13 _ 8 700 498 86 46.4 25.6
Nethercote 0.41 0.73 0.1 _ _ 726 503 82 49.1 28.9
Portholme 0.2 3.5 0.12 0.1 7 574 523 103 48.3 38.7
Southlake 0.08 1 0.12 0.14 7 865 523 85 48.7 42
Stonygillfoot 2.3 2.3 0.1 0.1 11 1068 404 33 47.4 23.3
Tadham 2.5 1.75 0.15 0.15 8 865 523 85 48.8 35.6
Tadham ESA 2.5 1.75 0.15 0.15 8 865 523 85 48.8 35.6
Upton Ham 0.9 0.7 0.11 0.11 5 775 514 78 48.2 35.6
Upwood 0.22 0.01 0.06 0.02 3 574 523 103 48.5 23.5
Westhay ESA 2.5 1.75 0.15 0.15 8 865 523 85 48.8 35.6
West Sedgemoor 1.5 1.5 0.27 0.27 6.4 865 523 85 49.3 44.7
Wet Moor 0.1 3.35 0.06 0.15 8 865 523 85 49.3 42.7

Table 1: Soil parameters used in the hydrological modelling.

This will show a specific example [1]. It is clear that they received some kind of grouping. Most often the grouping is performed by calculating the arithmetic mean value (Table 1). If at our disposal were the primary measurement data, it would be possible to identify a more accurate statistical model on 10 indicators with consideration of the wave components. On average factors, there is a coarsening of the desired biotechnical regularities. Therefore, the wave functions do not identify and the only deterministic model binary relations between factors. Rank relationship is not detected, for a one-dimensional relationship; the correlation coefficient is equal to 1.

General biotechnical regularity

Inductively, on the basis of tens of thousands of examples of identification of statistical selections from various areas of science, two generalized mathematical models [2-5] were revealed:

a) The generalized determined (trend) model for identification on values of factors and communications between them (it is shown in this article).

b) The general wave function of oscillatory indignation of the studied system in the form of an asymmetric wavelet signal (it is offered according to primary not grouped data).

All tendencies are modelled by binomial regularity of a look

y=a1 xa 2exp(-a3 xa 4)+a5 xa6 exp(-a7 xa8) (1)

where, y: estimated parameter (parameter is an indicator of the studied system);

x: the influencing parameter, in an example of ref. [1] 10 measured factors on 20 values.

A rating of the influencing and dependent factors

The full correlation matrix (without rank distributions) binary (between couples of mutually influencing factors) communications between 10 factors is given in Table 2.

The influencing factors (parameters x) Dependent factors (indicators y) Parameters   Sum Place
X Dependent factors (indicators y) r Ix
x1 x2 x3 x4 x5   X6 x7 x8 x9 x10    
Topsoil hydraulic conductivity (m day) x1 1 0.5382 0.6028 0.6337 0.6409 x1 0.3702 0.3698 0.4575 0.5232 0.4469 5.5832 4
Subsoil hydraulic conductivity (m day) x2 0.8652 1 0.4973 0.6772 0.8634 x2 0.3517 0.379 0.6024 0.6294 0.8915 6.7571 1
Topsoil drainable porosity x3 0.3666 0.2321 1 0.9079 0.4495 x3 0.0033 0.3565 0.1194 0.4246 0.3474 4.2073 9
Subsoil drainable porosity x4 0.591 0.5206 0.9293 1 0.7528 x4 0.3726 0.6964 0.6342 0.4903 0.7067 6.6939 2
Unsaturated Hydraulic conductivity exponent x5 0.6068 0.6989 0.4786 0.6994 1 x5 0.2785 0.6658 0.5374 0.1581 0.7606 5.8841 3
Rainfall (mm) x6 0.255 0.225 0.0739 0.2373 0.5864 x6 1 0.9021 0.9525 0.1825 0.4483 4.863 7
Potential transpiration (mm) x7 0.0176 0.149 0.2092 0.3188 0.4963 x7 0.8767 1 0.9514 0.5864 0.3902 4.9956 6
SMD (mm) (end July) x8 0.0167 0.2099 0.1194 0.2223 0.6612 x8 0.9363 0.9683 1 0.4164 0.9553 5.5058 5
Drought threshold (cm) x9 0.5277 0.1634 0.1783 0.3824 0.0596 x9 0.2545 0.5205 0.4263 1 0.6013 4.114 10
Aeration threshold (cm) x10 0.3202 0.4632 0.3848 0.6131 0.2246 x10 0.3716 0.3828 0.3417 0.5029 1 4.6049 8
Sum of coefficients of correlation Sr 4.5668 4.2003 4.4736 5.6921 5.7347 r 4.8154 6.2412 6.0228 4.9138 6.5482 53.2089 -
Indicator place I y 8 10 9 5 4 Place I y 7 2 3 6 1 - 0.5321

Table 2: Correlation matrix of the full factorial analysis and rating of factors.

Follows from the concept of a correlative variation of Ch. Darwin that in other conditions of dwelling other combinations of values of factors of the soil can be stronger (Darwin calls factors hereditary evasion). Therefore, weak factorial communications can be stronger on other objects of research. You need to compare pastures from different regions of the Earth. In the end there is a mathematical tool to compare the environmental systems of pastures between them.

The coefficient of functional connectivity (in the wider biotechnological meaning - correlative variation) 10 factors equal 53.2089/102=0.5321. This criterion is applied by comparison of different phytocenoses, in particular of pastures from different regions.

According to Table 2 among the influencing factors first place was won by a factor x2 (Subsoil hydraulic conductivity), the second - x4 (Subsoil drainable porosity) and the third - x5 (Unsaturated Hydraulic conductivity exponent). Among dependent factors (indicators) in ranked first factor (Aeration threshold), and the second - (Potential transpiration) and the third - (SMD end July).

Correlation matrix of the binary relations

We will exclude a rating and cages with units on a diagonal Table 3 from data of Table 2. Then we will receive a set of coefficients of correlation at binary communications. The maximum adequacy 0.9683 is observed at dependence x7=f (x8). The minimum coefficient of correlation 0.0033 is equal for the relation x6=f (x3). Above the known level of adequacy 0.7 there are 15 binary communications (Table 4) that makes 100 × 15/(102–10)=16.67%. Thus the share of the strongest bi nary relations is very far from a gold proportion of 61.80%.

  x1 x2 x3 x4 x5 x6 x7 x8 x9 x10
x1   0.5382 0.6028 0.6337 0.6409 0.3702 0.3698 0.4575 0.5232 0.4469
x2 0.8652   0.4973 0.6772 0.8634 0.3517 0.379 0.6024 0.6294 0.8915
x3 0.3666 0.2321   0.9079 0.4495 0.0033 0.3565 0.1194 0.4246 0.3474
x4 0.591 0.5206 0.9293   0.7528 0.3726 0.6964 0.6342 0.4903 0.7067
x5 0.6068 0.6989 0.4786 0.6994   0.2785 0.6658 0.5374 0.1581 0.7606
x6 0.255 0.225 0.0739 0.2373 0.5864   0.9021 0.9525 0.1825 0.4483
x7 0.0176 0.149 0.2092 0.3188 0.4963 0.8767   0.9514 0.5864 0.3902
x8 0.0167 0.2099 0.1194 0.2223 0.6612 0.9363 0.9683   0.4164 0.9553
x9 0.5277 0.1634 0.1783 0.3824 0.0596 0.2545 0.5205 0.4263   0.6013
x10 0.3202 0.4632 0.3848 0.6131 0.2246 0.3716 0.3828 0.3417 0.5029  

Table 3: Correlation matrix of the binary relations between factors.

  x1 x3 x4 x5 x6 x7 x8 x10
x2 0.8652     0.8634       0.8915
x3     0.9079          
x4   0.9293   0.7528       0.7067
x5               0.7606
x6           0.9021 0.9525  
x7         0.8767   0.9514  
x8         0.9363 0.9683   0.9553

Table 4: Correlation matrix of the binary relations at correlation r ≥ 0,7.

The grayed out the block from pair communications between three factors is allocated x6, x7 and x8. In Table 5 compact records of values of parameters of model are given (1). In total three clusters of regularities were formed.

Factors General law y=a1 xa2exp(-a3 xa4 )+a5 xa6 exp(-a7 xa8 ) Correl.
coef.
r
First component Second component
x y a1 a2 a3 a4 a5 a6 a7 a8  
x8 x7 374.7037 0 0.026208 1 6.51818е6 5.0856 16.64811 0.14794 0.9683
x8 x10 3.22802е7 8.18164 23.41995 0.17105 0 0 0 0 0.9553
x6 x8 40.23982 0 -0.00231 1 -4.55658е-8 3.29629 0 0 0.9525
x7 x8 -1.14232е7 0 1.53724 0.3491 0.004412 1.60871 0 0 0.9514
x8 x6 1652.688 0 0.028053 0.74615 0 0 0 0 0.9363
x4 x3 0.05782 0 -5.07272 0.92808 0 0 0 0 0.9293
x3 x4 0.033043 0 0 0 1.26462 1.28611 0 0 0.9079
x6 x7 227.6021 0 -0.00201 1 -5.09798е-7 3.12871 0 0 0.9021
x2 x10 23.75135 0 -0.01731 2.67978 17.15156 1.95212 0.051754 6.18305 0.8915
x7 x6 386020.2 0 0.007819 0.94425 -6774.02 0.83325 0.014992 0.89617 0.8767
x2 x1 0.49583 0 0.45055 1 39.61873 12.39564 5.60147 1.00944 0.8652
x2 x5 3.02871 0 0.37467 1 7.85863 1.18715 0.47018 1 0.8634
x5 x10 4.28454 2.3827 0.27272 1.1259 0 0 0 0 0.7606
x4 x5 2.79236 0 -3.31653 18.59787 1.75307е8 4.77904 23.86727 0.55696 0.7528
x4 x10 23.48359 0 8.2251 1 996.423 1.43515 5.22215 1 0.7067

Table 5: Parameters of regularities of mutual influence of indicators.

First cluster

It possesses good symmetry (Table 6). Here the strongest biotechnical regularities settle down. In fact three factors x6 (Rainfall), x7 (Potential transpiration) and x8 (SMD end July) become a kernel for all set of factors. Schedules are shown in Figure 1.

  x6 x7 x8
x6   0.9021 0.9525
x7 0.8767   0.9514
x8 0.9363 0.9683  

Table 6: First cluster of the strongest communications.

physical-mathematics-schedules-first-cluster

Figure 1: Schedules of the binary relations of the first cluster of regularities.

Because of repetitions of values only 12 points were formed of 20 names of the sites. Apparently from Figure 1, these 12 points form a big cluster of basic data of nine points located closely to each other. Thus, the clustering occurs not only on regularities, but also on values of the factors entering these biotechnical regularities. As a result when using all primary data there will be also wave indignations.

Second cluster

It was defined (Table 7) influence on an indicator x10 (Aeration threshold) four influencing variable x2 (Subsoil hydraulic conductivity), x4 (Subsoil drainable pororsity), x5 (Unsaturated Hydraulic conductivity exponent) and x8 (SMD end July).

  x10
x2 0.8915
x4 0.7067
x5 0.7606
x8 0.9553

Table 7: Second cluster of strong communications.

Figure 2 schedules of the binary relations are shown. The coefficient of correlation r is given in the right top corner of the schedule.

physical-mathematics-schedules-second-cluster

Figure 2: Schedules of the binary relations of the second cluster of regularities.

Third cluster

It (Table 8) is received by addition to the group of regularities of influence of x5 which is available in Table 4 (Unsaturated Hydraulic conductivity exponent).

  x1 x2 x3 x4 x5
x2 0.8652       0.8634
x3       0.9079  
x4     0.9293   0.7528
x5 0.6068 0.6989   0.6994  

Table 8: Third cluster of the binary relations.

This influence on three indicators x1 (Topsoil hydraulic conductivity), x2 (Subsoil hydraulic conductivity) and x4 (Subsoil drainable pororsity) happens to correlation coefficients less than 0,7.

Figure 3 schedules of eight binary relations are given. In addition to Table 5 influence happens on formulas:

Equation

When the same design of the general model (1) specific design patterns varies greatly, so get differing in complexity graphics.

physical-mathematics-schedules-third-cluster

Figure 3: Schedules of the binary relations of the third cluster of regularities.

Conclusion

Between biochemical substances of the soil of pastures and hydrological parameters of the water mode always there is a homeostasis. On the general tabular model [1], after identification of strong factorial communications, it is quite possible to define optimum or rational values at all 10 factors. And then on statistical models to predict the productivity of hayfields and pastures on the rational values of the factors. You can then proceed to parametric substantiation of measures for improvement of the water regime of riparian areas.

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