Cross-correlation analysis of the productivity of sugar beets and factors, that it is formed

     Stocked functions which can be the regressions utillized for a construction, limited enough. For this purpose it costs to utillize functions, linear in relation to parameters.

     Will consider a function which is applied during the analysis of agricultural activity more frequent than other:

     лінійна — Y = а₀ + а_1х

     Параметр  а₀ лінійного рівняння регресії — це значення у при х = 0. Якщо нуль перебуває в рамках фактичної варіації ознаки х, то а₀ — одне із теоретичних значень у, якщо х у досліджуваній сукупності не приймає значень, близьких до 0, то параметр а не має реального змісту.

     Parameter а₁ named the coefficient of regression and shows, on how many units will change on the average у at a change х on unit.

     The parameters of equalization of regression calculate the method of leastsquares. The basic condition of this method consists in that a sum of squares of rejections of theoretical values of Y from empiric must be minimum:

     Σ (Y – y)² = min

     Parameters of equalization regressions which respond to this condition expect the systems of normal equalizations by a decision. This system, for example, for a linear function (at a calculation from frill data) has such kind:

     Σy = na₀ + а₁Σx

     Σxy = а₀Σx + а₁Σx².

     Will calculate данні necessary for a decision systems of equalizations, will erect calculations in a table. 
 

Таблица 27. A information and calculation information is for the calculation of cross-correlation connection between     a yield and charges of forages

 

     Utillizing the expected information we can untie the system of equalizations described higher: 
 

     

      30а₀+1172,8а₁=1099,4             30

      1172,8а₀+45975,8а₁=43086,6   1172,8

     а₀ + 39,09а₁=36,65             ІІ - І

     а₀  + 39,2а₁=36,74

     0,11а₁=0,09

     

     

     Consequently equalization of cross-correlation connection between a yield and charges of forages will have such kind:

     

     Economic maintenance of this equalization is such: the coefficient of regression of а1 shows that in our aggregate with an increase on unit of charges of forages of hopes is increased on the average on 0,8182 c. The parameter of a₀=4,67, as a free member of equalization, has only calculation values and not interpreted.

     Putting  in equalization regression of value х, will get the theoretical levels of yield. In our case the parameters of equalization are certain correctly, because .

       
 

 
 
 
 

     

      30а₀+2811а₁=1099,4                30

      2811а₀+263521а₂=103107,3    2811

     а₀ + 93,7а₁=36,65             ІІ - І

     а₀  + 93,75а₁=36,68

     0,05а₁=0,03

     

       

     Consequently equalization of cross-correlation connection between a yield and charges of forages will have such kind:

     

     Economic maintenance of this equalization is such: coefficient of regression а₁ shows that in our aggregate with the increase of output of calves on one head of hopes increased on the average on 0,6с. Parameter а₀=-19,57, as a free member of equalization, has only calculation values and not interpreted.

     Putting  in equalization regression of value of õ, will get the theoretical levels of yield. In our case the parameters of equalization are certain correctly, because .

       
 
 
 
 
 
 
 
 
 
 

     3.3. Measuring of intensity of correlation. Coefficient of simple correlation and his maintenance 

     Next to determination of character of connection and effects of influence of factors х on a result the estimation of closeness of connection has an important value, that estimation of co-ordination of variation of associate signs. If influence of factor sign

  _x on effective considerable, it will appear in the appropriate change of values _y with the change of values , that factor forms variation the influence . In default of connection variation does not depend on variation .

     For the estimation of closeness of connection of statistician utillizes the row of coefficients with such general properties:

     1) in default of any connection the value of coefficient approaches a zero; at functional connection - to unit;

     2) at presence of cross-correlation connection a coefficient is expressed a shot which after an absolute value the greater, than more dense connection.

     Among the measures of closeness of connection most widespread is a coefficient of correlation of Pirsona. This coefficient is reflected by character r. As a sphere of his use is limited to linear dependence, in the name appears a word «linear». Calculation of linear coefficient of correlation r based on the rejections of values of associate signs і from averages.

       The coefficient of correlation, estimating the closeness of connection, specifies also on his direction: at direct connection —a size is positive, at reverse - negative. The signs of coefficients of correlation and regression are identical, sizes them associate functionally.

     Obviously, that at functional connection the actual sum of rejections is evened to maximum, and coefficient of correlation , at cross-correlation connection his absolute value will be the more than more dense connection.

     In practice utillize different modifications of formula of coefficient of correlation. For the estimation of closeness of connection between the amount of charges of forages and yield will take advantage of one of modifications from a formula: 

     

     Separately will expect the index of correlation: 

     

     Divergence between an index and coefficient of correlation is small.

     Will expect the coefficient of determination: 

     

     Consequently, we can assert that in our cross-correlation model of hopes on a cow depends on the expense of forages on 57,76%, and other 42,24% it is other невраховані factors.

     For help F – will check the criterion of Fishera for importance the coefficient of детермінації. At first will formulate a null-hypothesis:

     Н₀ – a coefficient of determination is unimportant.

     

     Search a critical point after a table:

     F_{0,95(p-1;n-p)}=F_0,95(1;28)=4,2

       To insert ripened calculation value with alternative:

     F_ф>Fт, that 38,52>4,2 – a tabular value less than is actual, that decline a null-hypothesis, consequently a coefficient of детермінації is substantial.

     Like taking advantage of information of tables of previous section and will expound them higher will estimate the closeness of connection material between the output of calves and yield:

     

     Separately will expect the index of correlation: 

     

     Will expect the coefficient of determination: 

     

     Consequently, we can assert that in our cross-correlation model of hopes  depends on the output of calves on 35,76%, and other 62,24% it is other невраховані factors.

     For help F – will check the criterion of Fishera for importance the coefficient of детермінації. At first will formulate a null-hypothesis:

     Н₀- a coefficient of детермінації is unimportant.

     

     Search a critical point after a table:

     F_{0,95(p-1;n-p)}=F_0,95(1;28)=4,2

     Stavlyaemo ripened calculation value with alternative:

     F_ф>Fт, that 15,55>4,2 – a tabular value less than is actual, that decline a null-hypothesis, consequently a coefficient of детермінації is substantial.  
 
 
 
 
 
 

3.4 Plural correlation 

      Plural correlation enables to estimate connection of effective sign with any factor at the fixed value other, plugged in a regressive model. In practice often utillize plural, multivariable equalizations of regression, when two influence on the size of effective sign, three, and more factors.

      At the theoretical ground of model and choice of factor signs it follows to take into account the crowd conditions of cross-correlation connection between signs. At presence of connection, what near to functional, estimations of parameters of multivariable equalization of regression will be unreliable. For the estimation of multicolinearity between signs it is enough to calculate the proper coefficients of correlation. If the coefficient of correlation of two factor signs is near to unit, one of them it is needed to eliminate. On this stage it is important not only to choose factors but also expose the structure of intercommunication between them.

      Difficult is a problem of ground of functional connection of type of multivariable type of equalization of regression. The analysis of парних connections is useless, because factors are associate, and to define connection between In and X at the fixed values of other factor signs very difficultly. Therefore in practice more frequent all utillize multivariable linear equalizations and equalizations which over are brought to the linear kind the proper transformations, namely to the kind:

     У_х=а₀+а₁х₁+а₂х₂+а_nx_n

     _where У_х - theoretical values of effective sign;

     а_1,а_2,а_n - parameters of equalization;

     х_1,х_2,x_n - factor signs.

     The parameter of equalization of а1 is named the coefficient of part of regression. He shows, as an effective sign changes in middle у with the change of factor sign х₁ на unit, on condition that other factor signs remain unchanging.

     For determination of parameters а_1,а_2, that at presence of 2 factor signs, it is needed to untie the system of normal equalizations: 

      

     For determination of crowd conditions and form of connection between the probed signs utillize the followings coefficients:

     Parni coefficients of correlation - utillize for measuring of closeness a copula between two probed signs without the account of their co-operating with other signs, plugged in a cross-correlation model:

           

     Partial coefficients of correlation - characterize the closeness of connection of effective sign with the first factor sign on condition that other factor signs of еліміновані:  

     

      The coefficient of plural correlation characterizes the crowd conditions of connection between the probed all models of factor and effective signs: 

      

 

     Coefficient   of determination:  shows  crowd conditions   connection  between effective and by the aggregate of factor signs, determine after a formula:  

     

     In same queue the plural coefficient of детермінації is laid out on partial coefficients which characterize on how many % variation of effective sign depends on each of factor: