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Development of a mathematical model for forecasting pastural fertility to remote sensing data

Variants of developing a mathematical model for predicting pasture fertility based on the use of remote sensing data from the Earth and ground based measurements on test field plots are presented. The NDVI data, which is a time series with pronounced seasonal features in the form of one or two maxima, are the basis for obtaining the vegetation index forecast model (NDVI) from the currently available measurements. To predict the parameters of this maximum, we used the position on the time axis, width, amplitude, and its shape. Using various model regressions over the entire available time interval, it is possible to obtain models, for example, in the form of rational and polynomial functions. As model, in addition to the rational and polynomial ones presented above, we used the following functions with characteristic impulse behavior: Cauchy, arctg (x)’, Verhulst, Gompertz differential functions (derivatives) with left and right asymmetries, as well as the sum fs and the product fp of logistic functions Verhulst. Of all the options on the evaluation interval [56 and 180] days, the model in the form of the sum of the Verhulst logistic functions with 7 evaluated parameters turned out to be more effective. The results show that all parameters are statistically significant, and the corrected determination coefficient is large enough: 0.9545. However, to parameterize the model, observations are needed “on the decline” of the time series - 150-180 days from the beginning of the year. The use of mathematical models is necessary for managing pasture plots and the need for timely management decisions of various kinds - alternating corrals, mowing terms for grass stands, and fertilizing. To generate reliable information about the state of pastures, the use of remote monitoring is relevant.


Introduction
A modern, innovative approach to the development of animal husbandry should provide high predictability of the final result, information support in informed management decisions, reducing the average possible risks, and ultimately, increasing labor productivity in this important area for the country [1][2].
The availability of modern information-measuring systems for aerospace pasture monitoring, as well as relevant information technologies, require further improvement of predictive mathematical models of both pasture productivity and productive qualities of pasture animals [3].
Other things being equal, the indicated characteristics in the operational time scale are determined mainly by weather and climatic conditions.
The aim of our research was to develop a mathematical model for predicting pasture fertility based on remote sensing data [4][5].

Materials and methods
In our work we used the deviation of the observed factor characteristics from their changes averaged over the years of observation to build a model for predicting pasture fertility. Such deviations in a particular year obviously contain climatic features of a given year, which should be used for forecasting.
To determine an adequate model of temperature changes during the year, model regressions were considered in the form of trigonometric series of the 1st, 2nd and 3rd orders, as well as models of the moving average (smooth) with different periods of averaging.
Creating a model for predicting the biological productivity of the pasture and the nutritional value of the feed received from it from the temperature trajectory of the season, the amount of precipitation, as well as from the levels of the normalized vegetation index NDVI (Normalized difference vegetation index) recorded using satellites [6][7].

Results and Discussion
To solve the problem of developing a mathematical model for predicting pasture fertility, the vegetative index NDVI was used (figure 1). At the first stage, we tried to obtain a forecast model for the NDVI index based on the currently available measurements. NDVI data represent a time series with pronounced seasonal features in the form of one or two maxima. It is necessary to predict the parameters of this maximum -position on the time axis, width, amplitude and its shape.
Using various model regressions on the existing time interval, it is possible to obtain models, for example, in the form of rational and polynomial functions (figure 2). On the left side are the equations of the models, optimal on method least squares values of their parameters, and also in parentheses, 95% confidence intervals for the values of these parameters.
On the right side: At the same time, it can be seen that of the above models, only the third-order polynomial model is statistically significant (with a reliability of 95%). A similar result is obtained for other so-called custom models.
If we limit the time interval on the right, simulating the limited nature of observations for forecasting problems, then it becomes completely impossible to obtain any meaningful model. This statement corresponds to the well-known requirement for the volume of statistical data -for each estimated parameter of the model should be at least 10 independent observations. As one of the possible ways out of the situation, it is possible to propose a not quite methodically correct way. Let's try to generate "observations" dependent on existing ones, for example, using piecewise linear interpolation. In the following figure, such "observations" are indicated by blue stars (figure 3).  Figure 3. Generated observations using piecewise linear interpolation.
In addition to the above rational and polynomial functions, the following functions with characteristic impulse behavior were used as model ones: Cauchy, arctg (x)', Verhulst, Gompertz differential functions (derivatives) with left and right asymmetries, as well as the sum fs and the product fp of logistic functions Verhulst (figure 4).

Conclusion
Based on the foregoing:  The results show that all parameters are statistically significant, and the corrected determination coefficient is large enough: 0.9545. However, to parameterize the model, observations are needed "on the decline" of the time series -150-180 days from the beginning of the year.  The use of mathematical models is necessary for managing pasture plots and the need for timely management decisions of various kinds -alternating corrals, mowing terms for grass stands, and fertilizing. For the formation of reliable information about the state of pastures, the use of remote monitoring is relevant.