422
N. Gorelikova, N. Patyk-Kara, I. Chizova, J.Plakht, A. Sivtsov
geological age. For the description of this relationship we used an equation of
nonlinear relation of the following form:
which characterizes the relation between absolute age (Y) and concentration
values of elements Xl> X 2 ' ••• , xN. Since it is impossible to restore a true form of a
function, it is possible to design its approximation in a class of the given form.
In this connection, the task of search modeling of this relation has been set, taking into account not only elemental concentrations but their ratio and products.
For calculating numerical ages, the absolute age scale constructed on the basis
of thermoluminescence data was applied. The design of the equation of nonlinear regression is very complicated, and handling it traditionally by the method
of the least-squares assures minimal error of determination on the average for a
sampling, in this case high deviations are possible for some points. Therefore,
the task of an designing equation has been attempted which gives minimal error
for every point of determination. The description of the desired model is the following:
N
N
Y = a oo + I.aoixi + I.(bijXiXj +cijxi / Xj +dijxj / Xi)
i=l
i,j=l
where x=(xl> X 2 ' ••• ' xN) is a description of a sample in a set of initial signs; Y is
the calculated age (million years).
In designing this equation the main task was to find coefficients satisfying set
conditions. The study of the desired coefficients a, b, c, d is conducted on the
condition of distinction minimization / Y k - V K / (distinction between calculated values Y K and initial values V k' characterizing the age of a sample k for every
sample).
Such a method of attack is reasonable if there is plausible dating of samples.
Otherwise, the elaborated mathematical rules using logic informative analysis
provide identification of the age group to which an analyzed sample is closest.
This can be considered as a task of linear programming:
minH
,
H>O,IYk -Vk I
sampling.
For the solution of this equation, a simplex method is used. As a result, coefficient values a, b, c, d were estimated, which describe a model of a chosen form,
and the value H characterises the accuracy of this description. For the model
built on the provided data H =0.064, that is an error in estimation of absolute
age does not exceed 64000 years. The values of elemental concentrations are considered as an argument xrv =1,2, ... ,10). To facilitate forming the resultant
equation, the following designations are used:
N. Gorelikova, N. Patyk-Kara, I. Chizova, J.Plakht, A. Sivtsov
geological age. For the description of this relationship we used an equation of
nonlinear relation of the following form:
which characterizes the relation between absolute age (Y) and concentration
values of elements Xl> X 2 ' ••• , xN. Since it is impossible to restore a true form of a
function, it is possible to design its approximation in a class of the given form.
In this connection, the task of search modeling of this relation has been set, taking into account not only elemental concentrations but their ratio and products.
For calculating numerical ages, the absolute age scale constructed on the basis
of thermoluminescence data was applied. The design of the equation of nonlinear regression is very complicated, and handling it traditionally by the method
of the least-squares assures minimal error of determination on the average for a
sampling, in this case high deviations are possible for some points. Therefore,
the task of an designing equation has been attempted which gives minimal error
for every point of determination. The description of the desired model is the following:
N
N
Y = a oo + I.aoixi + I.(bijXiXj +cijxi / Xj +dijxj / Xi)
i=l
i,j=l
where x=(xl> X 2 ' ••• ' xN) is a description of a sample in a set of initial signs; Y is
the calculated age (million years).
In designing this equation the main task was to find coefficients satisfying set
conditions. The study of the desired coefficients a, b, c, d is conducted on the
condition of distinction minimization / Y k - V K / (distinction between calculated values Y K and initial values V k' characterizing the age of a sample k for every
sample).
Such a method of attack is reasonable if there is plausible dating of samples.
Otherwise, the elaborated mathematical rules using logic informative analysis
provide identification of the age group to which an analyzed sample is closest.
This can be considered as a task of linear programming:
minH
,
H>O,IYk -Vk I
sampling.
For the solution of this equation, a simplex method is used. As a result, coefficient values a, b, c, d were estimated, which describe a model of a chosen form,
and the value H characterises the accuracy of this description. For the model
built on the provided data H =0.064, that is an error in estimation of absolute
age does not exceed 64000 years. The values of elemental concentrations are considered as an argument xrv =1,2, ... ,10). To facilitate forming the resultant
equation, the following designations are used: