Doctor of Engineering, Academician of Russian Academy of Natural History Russian, Russia
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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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.
Pastures; Options; Rating; Mutual Influence; Factor; Analysis; Patterns
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) |
---|---|---|---|---|---|---|---|---|---|---|
x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | |
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=a_{1} x^{a 2}exp(-a_{3} x^{a 4})+a_{5} x^{a6} exp(-a_{7} x^{a8}) (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} | I_{x} | ||||||||||
x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | X_{6} | x_{7} | x_{8} | x_{9} | x_{10} | ||||
Topsoil hydraulic conductivity (m day) x_{1} | 1 | 0.5382 | 0.6028 | 0.6337 | 0.6409 | x_{1} | 0.3702 | 0.3698 | 0.4575 | 0.5232 | 0.4469 | 5.5832 | 4 |
Subsoil hydraulic conductivity (m day) x_{2} | 0.8652 | 1 | 0.4973 | 0.6772 | 0.8634 | x_{2} | 0.3517 | 0.379 | 0.6024 | 0.6294 | 0.8915 | 6.7571 | 1 |
Topsoil drainable porosity x_{3} | 0.3666 | 0.2321 | 1 | 0.9079 | 0.4495 | x_{3} | 0.0033 | 0.3565 | 0.1194 | 0.4246 | 0.3474 | 4.2073 | 9 |
Subsoil drainable porosity x_{4} | 0.591 | 0.5206 | 0.9293 | 1 | 0.7528 | x_{4} | 0.3726 | 0.6964 | 0.6342 | 0.4903 | 0.7067 | 6.6939 | 2 |
Unsaturated Hydraulic conductivity exponent x_{5} | 0.6068 | 0.6989 | 0.4786 | 0.6994 | 1 | x_{5} | 0.2785 | 0.6658 | 0.5374 | 0.1581 | 0.7606 | 5.8841 | 3 |
Rainfall (mm) x_{6} | 0.255 | 0.225 | 0.0739 | 0.2373 | 0.5864 | x_{6} | 1 | 0.9021 | 0.9525 | 0.1825 | 0.4483 | 4.863 | 7 |
Potential transpiration (mm) x_{7} | 0.0176 | 0.149 | 0.2092 | 0.3188 | 0.4963 | x_{7} | 0.8767 | 1 | 0.9514 | 0.5864 | 0.3902 | 4.9956 | 6 |
SMD (mm) (end July) x_{8} | 0.0167 | 0.2099 | 0.1194 | 0.2223 | 0.6612 | x_{8} | 0.9363 | 0.9683 | 1 | 0.4164 | 0.9553 | 5.5058 | 5 |
Drought threshold (cm) x_{9} | 0.5277 | 0.1634 | 0.1783 | 0.3824 | 0.0596 | x_{9} | 0.2545 | 0.5205 | 0.4263 | 1 | 0.6013 | 4.114 | 10 |
Aeration threshold (cm) x_{10} | 0.3202 | 0.4632 | 0.3848 | 0.6131 | 0.2246 | x_{10} | 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 53.2089/10^{2}=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 x_{2} (Subsoil hydraulic conductivity), the second - x_{4} (Subsoil drainable porosity) and the third - x_{5} (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 x_{7}=f (x_{8}). The minimum coefficient of correlation 0.0033 is equal for the relation x_{6}=f (x_{3}). 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%.
x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | |
---|---|---|---|---|---|---|---|---|---|---|
x_{1} | 0.5382 | 0.6028 | 0.6337 | 0.6409 | 0.3702 | 0.3698 | 0.4575 | 0.5232 | 0.4469 | |
x_{2} | 0.8652 | 0.4973 | 0.6772 | 0.8634 | 0.3517 | 0.379 | 0.6024 | 0.6294 | 0.8915 | |
x_{3} | 0.3666 | 0.2321 | 0.9079 | 0.4495 | 0.0033 | 0.3565 | 0.1194 | 0.4246 | 0.3474 | |
x_{4} | 0.591 | 0.5206 | 0.9293 | 0.7528 | 0.3726 | 0.6964 | 0.6342 | 0.4903 | 0.7067 | |
x_{5} | 0.6068 | 0.6989 | 0.4786 | 0.6994 | 0.2785 | 0.6658 | 0.5374 | 0.1581 | 0.7606 | |
x_{6} | 0.255 | 0.225 | 0.0739 | 0.2373 | 0.5864 | 0.9021 | 0.9525 | 0.1825 | 0.4483 | |
x_{7} | 0.0176 | 0.149 | 0.2092 | 0.3188 | 0.4963 | 0.8767 | 0.9514 | 0.5864 | 0.3902 | |
x_{8} | 0.0167 | 0.2099 | 0.1194 | 0.2223 | 0.6612 | 0.9363 | 0.9683 | 0.4164 | 0.9553 | |
x_{9} | 0.5277 | 0.1634 | 0.1783 | 0.3824 | 0.0596 | 0.2545 | 0.5205 | 0.4263 | 0.6013 | |
x_{10} | 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.
x_{1} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{10} | |
---|---|---|---|---|---|---|---|---|
x_{2} | 0.8652 | 0.8634 | 0.8915 | |||||
x_{3} | 0.9079 | |||||||
x_{4} | 0.9293 | 0.7528 | 0.7067 | |||||
x_{5} | 0.7606 | |||||||
x_{6} | 0.9021 | 0.9525 | ||||||
x_{7} | 0.8767 | 0.9514 | ||||||
x_{8} | 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 x_{6}, x_{7} and x_{8}. 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=a_{1} x^{a2}exp(-a_{3} x^{a4} )+a_{5} x^{a6} exp(-a_{7} x^{a8} ) | Correl. coef. r |
||||||||
---|---|---|---|---|---|---|---|---|---|---|
First component | Second component | |||||||||
x | y | a_{1} | a_{2} | a_{3} | a_{4} | a_{5} | a_{6} | a_{7} | a_{8} | |
x_{8} | x_{7} | 374.7037 | 0 | 0.026208 | 1 | 6.51818Ðµ6 | 5.0856 | 16.64811 | 0.14794 | 0.9683 |
x_{8} | x_{10} | 3.22802Ðµ7 | 8.18164 | 23.41995 | 0.17105 | 0 | 0 | 0 | 0 | 0.9553 |
x_{6} | x_{8} | 40.23982 | 0 | -0.00231 | 1 | -4.55658Ðµ-8 | 3.29629 | 0 | 0 | 0.9525 |
x_{7} | x_{8} | -1.14232Ðµ7 | 0 | 1.53724 | 0.3491 | 0.004412 | 1.60871 | 0 | 0 | 0.9514 |
x_{8} | x_{6} | 1652.688 | 0 | 0.028053 | 0.74615 | 0 | 0 | 0 | 0 | 0.9363 |
x_{4} | x_{3} | 0.05782 | 0 | -5.07272 | 0.92808 | 0 | 0 | 0 | 0 | 0.9293 |
x_{3} | x_{4} | 0.033043 | 0 | 0 | 0 | 1.26462 | 1.28611 | 0 | 0 | 0.9079 |
x_{6} | x_{7} | 227.6021 | 0 | -0.00201 | 1 | -5.09798Ðµ-7 | 3.12871 | 0 | 0 | 0.9021 |
x_{2} | x_{10} | 23.75135 | 0 | -0.01731 | 2.67978 | 17.15156 | 1.95212 | 0.051754 | 6.18305 | 0.8915 |
x_{7} | x_{6} | 386020.2 | 0 | 0.007819 | 0.94425 | -6774.02 | 0.83325 | 0.014992 | 0.89617 | 0.8767 |
x_{2} | x_{1} | 0.49583 | 0 | 0.45055 | 1 | 39.61873 | 12.39564 | 5.60147 | 1.00944 | 0.8652 |
x_{2} | x_{5} | 3.02871 | 0 | 0.37467 | 1 | 7.85863 | 1.18715 | 0.47018 | 1 | 0.8634 |
x_{5} | x_{10} | 4.28454 | 2.3827 | 0.27272 | 1.1259 | 0 | 0 | 0 | 0 | 0.7606 |
x_{4} | x_{5} | 2.79236 | 0 | -3.31653 | 18.59787 | 1.75307Ðµ8 | 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 x_{6} (Rainfall), x_{7} (Potential transpiration) and x_{8} (SMD end July) become a kernel for all set of factors. Schedules are shown in Figure 1.
x_{6} | x_{7} | x_{8} | |
---|---|---|---|
x_{6} | 0.9021 | 0.9525 | |
x_{7} | 0.8767 | 0.9514 | |
x_{8} | 0.9363 | 0.9683 |
Table 6: First cluster of the strongest communications.
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 x_{10} (Aeration threshold) four influencing variable x_{2} (Subsoil hydraulic conductivity), x_{4} (Subsoil drainable pororsity), x_{5} (Unsaturated Hydraulic conductivity exponent) and x_{8} (SMD end July).
x_{10} | |
---|---|
x_{2} | 0.8915 |
x_{4} | 0.7067 |
x_{5} | 0.7606 |
x_{8} | 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.
Third cluster
It (Table 8) is received by addition to the group of regularities of influence of x_{5} which is available in Table 4 (Unsaturated Hydraulic conductivity exponent).
x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | |
---|---|---|---|---|---|
x_{2} | 0.8652 | 0.8634 | |||
x_{3} | 0.9079 | ||||
x_{4} | 0.9293 | 0.7528 | |||
x_{5} | 0.6068 | 0.6989 | 0.6994 |
Table 8: Third cluster of the binary relations.
This influence on three indicators x_{1} (Topsoil hydraulic conductivity), x_{2} (Subsoil hydraulic conductivity) and x_{4} (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:
When the same design of the general model (1) specific design patterns varies greatly, so get differing in complexity graphics.
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.